Subgroups — C₂×C₂₃×#3294 #19208738

Order 5888.

2858 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3294	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#480	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#466	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#447	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂₃×#102	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₂₃	—	2	1	no	—
14	C₂³×D₄×C₂₃	—	2	1	no	—
15	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
16	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₂₃	C₂²	1	2	yes	complemented
18	C₂³×C₈×C₂₃	C₂²	1	2	yes	complemented
19	C₂₃×#547	—	2	4	no	—
20	C₂₃×#547	C₂²	1	8	yes	complemented
21	D₄²×C₂₃	—	2	4	no	—
22	D₄²×C₂₃	C₂²	1	8	yes	complemented
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#123	—	2	2	no	—
26	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
27	C₂×C₂₃×#121	—	2	1	no	—
28	C₂×C₂₃×#121	C₂²	1	2	yes	—
29	C₂×C₂₃×#121	C₂²	1	2	yes	complemented
30	C₂×C₂₃×#120	—	2	2	no	—
31	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
35	C₂₃×#486	C₂²	1	8	yes	complemented
36	C₂₃×#480	C₂²	1	16	yes	complemented
37	C₂₃×#474	C₂²	1	8	yes	complemented
38	C₂₃×#466	C₂²	1	16	yes	complemented
39	C₂₃×#459	C₂²	1	8	yes	complemented
40	C₂₃×#447	C₂²	1	16	yes	complemented
41	C₄×D₈×C₂₃	C₂²	1	8	yes	complemented
42	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
43	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
44	C₂×C₂₃×#107	C₂²	1	2	yes	complemented
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×#102	C₂²	1	2	yes	—
48	C₂×C₂₃×#102	C₂²	1	16	yes	complemented
49	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
50	C₂×C₂₃×#102	C₂²	1	2	yes	complemented
51	C₂×C₂₃×#102	C₂²	1	2	yes	complemented
52	C₂×C₂₃×#102	C₂²	1	2	yes	complemented
53	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
54	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
55	C₂⁵×C₂₃	—	2	2	no	—
56	C₂×#41×C₂₃	—	2	8	no	—
57	C₂×#41×C₂₃	—	4	8	no	—
58	C₂²×Q₈×C₂₃	—	2	1	no	—
59	C₂²×D₄×C₂₃	—	2	8	no	—
60	C₂²×D₄×C₂₃	—	2	2	no	—
61	C₂²×D₄×C₂₃	—	2	2	no	—
62	C₂²×D₄×C₂₃	—	2	8	no	—
63	C₂²×D₄×C₂₃	—	2	1	no	—
64	C₂²×D₄×C₂₃	—	2	2	no	—
65	C₂²×D₄×C₂₃	—	2	4	no	—
66	C₂²×D₄×C₂₃	—	2	1	no	—
67	C₂²×D₄×C₂₃	—	4	4	no	—
68	C₂²×D₄×C₂₃	—	4	8	no	—
69	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
70	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
72	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
73	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₂₃	—	2	2	no	—
75	C₂³×C₄×C₂₃	—	2	1	no	—
76	C₂³×C₄×C₂₃	—	2	1	no	—
77	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
78	C₂×D₈×C₂₃	—	2	4	no	—
79	C₂×D₈×C₂₃	—	4	4	no	—
80	C₂×D₈×C₂₃	C₂³	1	4	yes	—
81	C₂×D₈×C₂₃	C₂³	1	8	yes	complemented
82	C₂²×C₈×C₂₃	—	4	4	no	—
83	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
85	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₂₃	C₂³	1	8	yes	complemented
87	C₂₃×C₄⋊Q₈	—	2	2	no	—
88	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
89	C₂₃×#127	—	2	2	no	—
90	C₂₃×#127	C₂³	1	4	yes	—
91	C₂₃×#123	—	2	16	no	—
92	C₂₃×C₂²⋊Q₈	—	2	8	no	—
93	C₂₃×#121	—	2	4	no	—
94	C₂₃×#121	—	2	4	no	—
95	C₂₃×#121	—	2	8	no	—
96	C₂₃×#121	C₂³	1	8	yes	—
97	C₂₃×#121	C₂³	1	8	yes	complemented
98	C₂₃×#120	—	2	16	no	—
99	C₄×D₄×C₂₃	—	2	2	no	—
100	C₄×D₄×C₂₃	—	2	2	no	—
101	C₄×D₄×C₂₃	D₄	1	4	yes	—
102	C₄×D₄×C₂₃	C₂³	1	4	yes	—
103	C₄×D₄×C₂₃	C₂³	1	4	yes	—
104	C₄×D₄×C₂₃	C₂³	1	4	yes	complemented
105	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
113	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
118	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
119	C₂₃×#107	C₂³	1	4	yes	complemented
120	C₂₃×#107	C₂³	1	8	yes	complemented
121	C₂₃×C₄⋊C₈	C₂³	1	4	yes	complemented
122	C₂₃×#102	C₂³	1	4	yes	—
123	C₂₃×#102	C₂³	1	8	yes	—
124	C₂₃×#102	C₂³	1	8	yes	complemented
125	C₂₃×#102	C₂³	1	8	yes	complemented
126	C₂₃×#102	C₂³	1	4	yes	complemented
127	C₂₃×#102	C₂³	1	8	yes	complemented
128	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	complemented
129	C₄×C₈×C₂₃	C₂³	1	4	yes	complemented
130	C₂⁴×C₂₃	—	2	1	no	—
131	C₂⁴×C₂₃	—	2	1	no	—
132	C₂⁴×C₂₃	—	2	2	no	—
133	C₂⁴×C₂₃	—	2	8	no	—
134	C₂⁴×C₂₃	—	4	8	no	—
135	C₂⁴×C₂₃	—	4	8	no	—
136	C₂⁴×C₂₃	—	8	4	no	—
137	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
138	#41×C₂₃	—	4	32	no	—
139	C₂×Q₈×C₂₃	—	2	4	no	—
140	C₂×Q₈×C₂₃	—	4	4	no	—
141	C₂×D₄×C₂₃	—	2	16	no	—
142	C₂×D₄×C₂₃	—	2	8	no	—
143	C₂×D₄×C₂₃	—	2	8	no	—
144	C₂×D₄×C₂₃	—	2	1	no	—
145	C₂×D₄×C₂₃	—	2	1	no	—
146	C₂×D₄×C₂₃	—	2	8	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	2	4	no	—
149	C₂×D₄×C₂₃	—	2	4	no	—
150	C₂×D₄×C₂₃	—	2	2	no	—
151	C₂×D₄×C₂₃	—	2	4	no	—
152	C₂×D₄×C₂₃	—	2	2	no	—
153	C₂×D₄×C₂₃	—	2	2	no	—
154	C₂×D₄×C₂₃	—	2	4	no	—
155	C₂×D₄×C₂₃	—	4	8	no	—
156	C₂×D₄×C₂₃	—	4	16	no	—
157	C₂×D₄×C₂₃	—	4	8	no	—
158	C₂×D₄×C₂₃	—	4	8	no	—
159	C₂×D₄×C₂₃	—	4	8	no	—
160	C₂×D₄×C₂₃	—	4	32	no	—
161	C₂×D₄×C₂₃	—	4	4	no	—
162	C₂×D₄×C₂₃	—	8	2	no	—
163	C₂×D₄×C₂₃	D₈	1	8	yes	complemented
164	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
165	C₂×D₄×C₂₃	#41	1	4	yes	complemented
166	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
167	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
168	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₂₃	—	2	4	no	—
170	C₂²×C₄×C₂₃	—	2	4	no	—
171	C₂²×C₄×C₂₃	—	2	8	no	—
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	1	no	—
174	C₂²×C₄×C₂₃	—	2	2	no	—
175	C₂²×C₄×C₂₃	—	2	4	no	—
176	C₂²×C₄×C₂₃	—	2	4	no	—
177	C₂²×C₄×C₂₃	—	4	8	no	—
178	C₂²×C₄×C₂₃	—	4	4	no	—
179	C₂²×C₄×C₂₃	—	4	4	no	—
180	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
186	D₈×C₂₃	—	2	8	no	—
187	D₈×C₂₃	—	4	8	no	—
188	C₂×C₈×C₂₃	—	2	2	no	—
189	C₂×C₈×C₂₃	—	2	2	no	—
190	C₂×C₈×C₂₃	—	2	4	no	—
191	C₂×C₈×C₂₃	—	4	8	no	—
192	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
193	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
194	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	2	no	—
197	C₄⋊C₄×C₂₃	—	2	8	no	—
198	C₄⋊C₄×C₂₃	—	2	4	no	—
199	C₄⋊C₄×C₂₃	—	2	1	no	—
200	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
201	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
203	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	8	no	—
207	C₂²⋊C₄×C₂₃	—	2	8	no	—
208	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
209	C₄²×C₂₃	C₂×D₄	1	2	yes	—
210	C₄²×C₂₃	C₂⁴	1	2	yes	—
211	C₂×#3294	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₂₃	—	2	4	no	—
213	C₂³×C₂₃	—	2	4	no	—
214	C₂³×C₂₃	—	2	4	no	—
215	C₂³×C₂₃	—	2	2	no	—
216	C₂³×C₂₃	—	2	2	no	—
217	C₂³×C₂₃	—	2	4	no	—
218	C₂³×C₂₃	—	4	16	no	—
219	C₂³×C₂₃	—	4	2	no	—
220	C₂³×C₂₃	—	4	2	no	—
221	C₂³×C₂₃	—	4	4	no	—
222	C₂³×C₂₃	—	4	2	no	—
223	C₂³×C₂₃	—	4	2	no	—
224	C₂³×C₂₃	—	4	16	no	—
225	C₂³×C₂₃	—	4	4	no	—
226	C₂³×C₂₃	—	4	4	no	—
227	C₂³×C₂₃	—	4	2	no	—
228	C₂³×C₂₃	—	8	8	no	—
229	C₂³×C₂₃	—	8	32	no	—
230	C₂³×C₂₃	C₂×D₈	1	8	yes	—
231	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
232	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
233	Q₈×C₂₃	—	4	8	no	—
234	D₄×C₂₃	—	2	4	no	—
235	D₄×C₂₃	—	2	2	no	—
236	D₄×C₂₃	—	2	4	no	—
237	D₄×C₂₃	—	2	2	no	—
238	D₄×C₂₃	—	2	2	no	—
239	D₄×C₂₃	—	2	4	no	—
240	D₄×C₂₃	—	4	16	no	—
241	D₄×C₂₃	—	4	16	no	—
242	D₄×C₂₃	—	4	16	no	—
243	D₄×C₂₃	—	4	2	no	—
244	D₄×C₂₃	—	4	4	no	—
245	D₄×C₂₃	—	4	8	no	—
246	D₄×C₂₃	—	8	4	no	—
247	D₄×C₂₃	C₂×D₈	1	16	yes	complemented
248	D₄×C₂₃	C₂×#41	1	4	yes	complemented
249	C₂×C₄×C₂₃	—	2	4	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	8	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	2	4	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	4	no	—
257	C₂×C₄×C₂₃	—	4	2	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	4	no	—
260	C₂×C₄×C₂₃	—	4	16	no	—
261	C₂×C₄×C₂₃	—	4	8	no	—
262	C₂×C₄×C₂₃	—	4	8	no	—
263	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
264	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
266	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
269	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
270	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
271	C₈×C₂₃	—	2	2	no	—
272	C₈×C₂₃	—	2	2	no	—
273	C₈×C₂₃	—	2	4	no	—
274	C₂×#547	C₂×C₂₃	1	1	yes	characteristic, complemented
275	C₂×D₄²	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#3294	C₂×C₂₃	1	16	yes	direct product factor
277	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₂×#480	C₂×C₂₃	1	2	yes	complemented
279	C₂×#474	C₂×C₂₃	1	1	yes	characteristic, complemented
280	C₂×#466	C₂×C₂₃	1	2	yes	complemented
281	C₂×#459	C₂×C₂₃	1	1	yes	characteristic, complemented
282	C₂×#447	C₂×C₂₃	1	2	yes	complemented
283	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
284	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
285	C₂²×#102	C₂×C₂₃	1	2	yes	complemented
286	C₂²×C₂₃	—	2	4	no	—
287	C₂²×C₂₃	—	2	16	no	—
288	C₂²×C₂₃	—	2	4	no	—
289	C₂²×C₂₃	—	2	4	no	—
290	C₂²×C₂₃	—	4	8	no	—
291	C₂²×C₂₃	—	4	2	no	—
292	C₂²×C₂₃	—	4	2	no	—
293	C₂²×C₂₃	—	4	4	no	—
294	C₂²×C₂₃	—	4	2	no	—
295	C₂²×C₂₃	—	4	2	no	—
296	C₂²×C₂₃	—	4	4	no	—
297	C₂²×C₂₃	—	4	4	no	—
298	C₂²×C₂₃	—	4	2	no	—
299	C₂²×C₂₃	—	8	32	no	—
300	C₂²×C₂₃	—	8	4	no	—
301	C₂²×C₂₃	—	8	8	no	—
302	C₂²×C₂₃	#546	1	2	yes	—
303	C₂²×C₂₃	C₂²×D₈	1	8	yes	—
304	C₂²×C₂₃	C₂²×D₈	1	2	yes	—
305	C₂²×C₂₃	#577	1	2	yes	—
306	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
307	C₄×C₂₃	—	2	4	no	—
308	C₄×C₂₃	—	2	2	no	—
309	C₄×C₂₃	—	4	2	no	—
310	C₄×C₂₃	—	4	2	no	—
311	C₄×C₂₃	—	4	4	no	—
312	C₄×C₂₃	#546	1	2	yes	—
313	C₄×C₂₃	C₂²×D₈	1	4	yes	—
314	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
315	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
316	C₂²×#41	—	2	1	no	—
317	C₂³×D₄	—	2	1	no	—
318	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
319	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
320	C₂²×D₈	C₂²×C₂₃	1	2	yes	complemented
321	C₂³×C₈	C₂²×C₂₃	1	2	yes	complemented
322	#547	—	2	4	no	—
323	#547	C₂²×C₂₃	1	8	yes	complemented
324	D₄²	—	2	4	no	—
325	D₄²	C₂²×C₂₃	1	8	yes	complemented
326	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
327	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×#123	—	2	2	no	—
329	C₂×C₂²⋊Q₈	—	2	1	no	—
330	C₂×#121	—	2	1	no	—
331	C₂×#121	C₂²×C₂₃	1	2	yes	—
332	C₂×#121	C₂²×C₂₃	1	2	yes	complemented
333	C₂×#120	—	2	2	no	—
334	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
337	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
338	#486	C₂²×C₂₃	1	8	yes	complemented
339	#480	C₂²×C₂₃	1	16	yes	complemented
340	#474	C₂²×C₂₃	1	8	yes	complemented
341	#466	C₂²×C₂₃	1	16	yes	complemented
342	#459	C₂²×C₂₃	1	8	yes	complemented
343	#447	C₂²×C₂₃	1	16	yes	complemented
344	C₄×D₈	C₂²×C₂₃	1	8	yes	complemented
345	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
346	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic, complemented
347	C₂×#107	C₂²×C₂₃	1	2	yes	complemented
348	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
349	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#102	C₂²×C₂₃	1	2	yes	—
351	C₂×#102	C₂²×C₂₃	1	16	yes	complemented
352	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
353	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
354	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
355	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
356	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
357	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
358	C₂×C₂₃	—	2	8	no	—
359	C₂×C₂₃	—	4	4	no	—
360	C₂×C₂₃	—	8	2	no	—
361	C₂×C₂₃	—	8	4	no	—
362	C₂×C₂₃	#3294	1	4	yes	direct product factor
363	C₂×C₂₃	C₂×#546	1	1	yes	characteristic
364	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
365	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
366	C₂⁵	—	2	2	no	—
367	C₂×#41	—	2	8	no	—
368	C₂×#41	—	4	8	no	—
369	C₂²×Q₈	—	2	1	no	—
370	C₂²×D₄	—	2	8	no	—
371	C₂²×D₄	—	2	2	no	—
372	C₂²×D₄	—	2	2	no	—
373	C₂²×D₄	—	2	8	no	—
374	C₂²×D₄	—	2	2	no	—
375	C₂²×D₄	—	2	1	no	—
376	C₂²×D₄	—	2	4	no	—
377	C₂²×D₄	—	2	1	no	—
378	C₂²×D₄	—	4	4	no	—
379	C₂²×D₄	—	4	8	no	—
380	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
381	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
382	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
383	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
384	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
385	C₂³×C₄	—	2	2	no	—
386	C₂³×C₄	—	2	1	no	—
387	C₂³×C₄	—	2	1	no	—
388	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
389	C₂×D₈	—	2	4	no	—
390	C₂×D₈	—	4	4	no	—
391	C₂×D₈	C₂³×C₂₃	1	4	yes	—
392	C₂×D₈	C₂³×C₂₃	1	8	yes	complemented
393	C₂²×C₈	—	4	4	no	—
394	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
395	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
396	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₂₃	1	8	yes	complemented
398	C₄⋊Q₈	—	2	2	no	—
399	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
400	#127	—	2	2	no	—
401	#127	C₂³×C₂₃	1	4	yes	—
402	#123	—	2	16	no	—
403	C₂²⋊Q₈	—	2	8	no	—
404	#121	—	2	4	no	—
405	#121	—	2	4	no	—
406	#121	—	2	8	no	—
407	#121	C₂³×C₂₃	1	8	yes	—
408	#121	C₂³×C₂₃	1	8	yes	complemented
409	#120	—	2	16	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	D₄×C₂₃	1	4	yes	—
413	C₄×D₄	C₂³×C₂₃	1	4	yes	—
414	C₄×D₄	C₂³×C₂₃	1	4	yes	—
415	C₄×D₄	C₂³×C₂₃	1	4	yes	complemented
416	C₂×C₄⋊C₄	—	2	2	no	—
417	C₂×C₄⋊C₄	—	2	4	no	—
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	complemented
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	2	no	—
427	C₂×C₂²⋊C₄	—	2	2	no	—
428	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
429	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
430	#107	C₂³×C₂₃	1	4	yes	complemented
431	#107	C₂³×C₂₃	1	8	yes	complemented
432	C₄⋊C₈	C₂³×C₂₃	1	4	yes	complemented
433	#102	C₂³×C₂₃	1	4	yes	—
434	#102	C₂³×C₂₃	1	8	yes	—
435	#102	C₂³×C₂₃	1	8	yes	complemented
436	#102	C₂³×C₂₃	1	8	yes	complemented
437	#102	C₂³×C₂₃	1	4	yes	complemented
438	#102	C₂³×C₂₃	1	8	yes	complemented
439	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	complemented
440	C₄×C₈	C₂³×C₂₃	1	4	yes	complemented
441	C₂₃	C₂×#3294	1	1	yes	characteristic, direct product factor, Sylow
442	C₂⁴	—	2	1	no	—
443	C₂⁴	—	2	1	no	—
444	C₂⁴	—	2	2	no	—
445	C₂⁴	—	2	8	no	—
446	C₂⁴	—	4	8	no	—
447	C₂⁴	—	4	8	no	—
448	C₂⁴	—	8	4	no	—
449	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
450	#41	—	4	32	no	—
451	C₂×Q₈	—	2	4	no	—
452	C₂×Q₈	—	4	4	no	—
453	C₂×D₄	—	2	16	no	—
454	C₂×D₄	—	2	8	no	—
455	C₂×D₄	—	2	8	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	8	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	4	8	no	—
468	C₂×D₄	—	4	16	no	—
469	C₂×D₄	—	4	8	no	—
470	C₂×D₄	—	4	8	no	—
471	C₂×D₄	—	4	8	no	—
472	C₂×D₄	—	4	32	no	—
473	C₂×D₄	—	4	4	no	—
474	C₂×D₄	—	8	2	no	—
475	C₂×D₄	D₈×C₂₃	1	8	yes	complemented
476	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
477	C₂×D₄	#41×C₂₃	1	4	yes	complemented
478	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
479	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
480	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	8	no	—
487	C₂²×C₄	—	2	4	no	—
488	C₂²×C₄	—	2	4	no	—
489	C₂²×C₄	—	4	8	no	—
490	C₂²×C₄	—	4	4	no	—
491	C₂²×C₄	—	4	4	no	—
492	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
494	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
495	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
497	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
498	D₈	—	2	8	no	—
499	D₈	—	4	8	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	4	no	—
503	C₂×C₈	—	4	8	no	—
504	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
505	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
506	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
513	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
514	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
515	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	8	no	—
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
521	C₄²	C₂×D₄×C₂₃	1	2	yes	—
522	C₄²	C₂⁴×C₂₃	1	2	yes	—
523	C₂³	—	2	4	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	4	16	no	—
530	C₂³	—	4	2	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	4	2	no	—
533	C₂³	—	4	16	no	—
534	C₂³	—	4	4	no	—
535	C₂³	—	4	4	no	—
536	C₂³	—	4	2	no	—
537	C₂³	—	4	4	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	8	8	no	—
540	C₂³	—	8	32	no	—
541	C₂³	C₂×D₈×C₂₃	1	8	yes	—
542	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
543	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
544	Q₈	—	4	8	no	—
545	D₄	—	2	2	no	—
546	D₄	—	2	4	no	—
547	D₄	—	2	2	no	—
548	D₄	—	2	2	no	—
549	D₄	—	2	4	no	—
550	D₄	—	2	4	no	—
551	D₄	—	4	16	no	—
552	D₄	—	4	16	no	—
553	D₄	—	4	16	no	—
554	D₄	—	4	8	no	—
555	D₄	—	4	4	no	—
556	D₄	—	4	2	no	—
557	D₄	—	8	4	no	—
558	D₄	C₂×D₈×C₂₃	1	16	yes	complemented
559	D₄	C₂×#41×C₂₃	1	4	yes	complemented
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	8	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	16	no	—
572	C₂×C₄	—	4	8	no	—
573	C₂×C₄	—	4	8	no	—
574	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
575	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
576	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
577	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
579	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
580	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
581	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
582	C₈	—	2	2	no	—
583	C₈	—	2	2	no	—
584	C₈	—	2	4	no	—
585	C₂²	—	2	4	no	—
586	C₂²	—	2	16	no	—
587	C₂²	—	2	4	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	4	8	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	8	32	no	—
599	C₂²	—	8	8	no	—
600	C₂²	—	8	4	no	—
601	C₂²	C₂₃×#546	1	2	yes	—
602	C₂²	C₂²×D₈×C₂₃	1	8	yes	—
603	C₂²	C₂²×D₈×C₂₃	1	2	yes	—
604	C₂²	C₂₃×#577	1	2	yes	—
605	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
606	C₄	—	2	4	no	—
607	C₄	—	2	2	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	2	no	—
610	C₄	—	4	4	no	—
611	C₄	C₂₃×#546	1	2	yes	—
612	C₄	C₂²×D₈×C₂₃	1	4	yes	—
613	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
614	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
615	C₂	—	2	8	no	—
616	C₂	—	4	4	no	—
617	C₂	—	8	4	no	—
618	C₂	—	8	2	no	—
619	C₂	C₂₃×#3294	1	4	yes	direct product factor
620	C₂	C₂×C₂₃×#546	1	1	yes	characteristic
621	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic
622	C₂	C₂×C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₂₃×#547	C₂	semidirect
2. C₂×D₄²×C₂₃	C₂	semidirect
3. C₂₃×#3294	C₂	direct (N × H)
4. C₂×C₂₃×#486	C₂	semidirect
5. C₂×C₂₃×#480	C₂	semidirect
6. C₂×C₂₃×#474	C₂	semidirect
7. C₂×C₂₃×#466	C₂	semidirect
8. C₂×C₂₃×#459	C₂	semidirect
9. C₂×C₂₃×#447	C₂	semidirect
10. C₂×C₄×D₈×C₂₃	C₂	semidirect
11. C₂×C₈×D₄×C₂₃	C₂	semidirect
12. C₂²×C₂₃×#102	C₂	semidirect
17. C₂²×D₈×C₂₃	C₂²	semidirect
18. C₂³×C₈×C₂₃	C₂²	semidirect
20. C₂₃×#547	C₂²	semidirect
22. D₄²×C₂₃	C₂²	semidirect
23. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
29. C₂×C₂₃×#121	C₂²	semidirect
33. C₂×C₄×D₄×C₂₃	C₂²	semidirect
34. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
35. C₂₃×#486	C₂²	semidirect
36. C₂₃×#480	C₂²	semidirect
37. C₂₃×#474	C₂²	semidirect
38. C₂₃×#466	C₂²	semidirect
39. C₂₃×#459	C₂²	semidirect
40. C₂₃×#447	C₂²	semidirect
41. C₄×D₈×C₂₃	C₂²	semidirect
42. C₈×D₄×C₂₃	C₂²	semidirect
43. C₂×C₂₃×#107	C₂²	semidirect
44. C₂×C₂₃×#107	C₂²	semidirect
45. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
48. C₂×C₂₃×#102	C₂²	semidirect
49. C₂×C₂₃×#102	C₂²	semidirect
50. C₂×C₂₃×#102	C₂²	semidirect
51. C₂×C₂₃×#102	C₂²	semidirect
52. C₂×C₂₃×#102	C₂²	semidirect
53. C₂×C₂₃×C₂²⋊C₈	C₂²	semidirect
54. C₂×C₄×C₈×C₂₃	C₂²	semidirect
81. C₂×D₈×C₂₃	C₂³	semidirect
86. C₂²×C₈×C₂₃	C₂³	semidirect
88. C₂₃×C₄⋊Q₈	C₂³	semidirect
97. C₂₃×#121	C₂³	semidirect
104. C₄×D₄×C₂₃	C₂³	semidirect
112. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
119. C₂₃×#107	C₂³	semidirect
120. C₂₃×#107	C₂³	semidirect
121. C₂₃×C₄⋊C₈	C₂³	semidirect
124. C₂₃×#102	C₂³	semidirect
125. C₂₃×#102	C₂³	semidirect
126. C₂₃×#102	C₂³	semidirect
127. C₂₃×#102	C₂³	semidirect
128. C₂₃×C₂²⋊C₈	C₂³	semidirect
129. C₄×C₈×C₂₃	C₂³	semidirect
163. C₂×D₄×C₂₃	D₈	semidirect
165. C₂×D₄×C₂₃	#41	semidirect
211. C₂×#3294	C₂₃	direct (N × H)
247. D₄×C₂₃	C₂×D₈	semidirect
248. D₄×C₂₃	C₂×#41	semidirect
274. C₂×#547	C₂×C₂₃	semidirect
275. C₂×D₄²	C₂×C₂₃	semidirect
276. #3294	C₂×C₂₃	direct (N × H)
277. C₂×#486	C₂×C₂₃	semidirect
278. C₂×#480	C₂×C₂₃	semidirect
279. C₂×#474	C₂×C₂₃	semidirect
280. C₂×#466	C₂×C₂₃	semidirect
281. C₂×#459	C₂×C₂₃	semidirect
282. C₂×#447	C₂×C₂₃	semidirect
283. C₂×C₄×D₈	C₂×C₂₃	semidirect
284. C₂×C₈×D₄	C₂×C₂₃	semidirect
285. C₂²×#102	C₂×C₂₃	semidirect
320. C₂²×D₈	C₂²×C₂₃	semidirect
321. C₂³×C₈	C₂²×C₂₃	semidirect
323. #547	C₂²×C₂₃	semidirect
325. D₄²	C₂²×C₂₃	semidirect
326. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
332. C₂×#121	C₂²×C₂₃	semidirect
336. C₂×C₄×D₄	C₂²×C₂₃	semidirect
337. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
338. #486	C₂²×C₂₃	semidirect
339. #480	C₂²×C₂₃	semidirect
340. #474	C₂²×C₂₃	semidirect
341. #466	C₂²×C₂₃	semidirect
342. #459	C₂²×C₂₃	semidirect
343. #447	C₂²×C₂₃	semidirect
344. C₄×D₈	C₂²×C₂₃	semidirect
345. C₈×D₄	C₂²×C₂₃	semidirect
346. C₂×#107	C₂²×C₂₃	semidirect
347. C₂×#107	C₂²×C₂₃	semidirect
348. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
351. C₂×#102	C₂²×C₂₃	semidirect
352. C₂×#102	C₂²×C₂₃	semidirect
353. C₂×#102	C₂²×C₂₃	semidirect
354. C₂×#102	C₂²×C₂₃	semidirect
355. C₂×#102	C₂²×C₂₃	semidirect
356. C₂×C₂²⋊C₈	C₂²×C₂₃	semidirect
357. C₂×C₄×C₈	C₂²×C₂₃	semidirect
362. C₂×C₂₃	#3294	direct (N × H)
392. C₂×D₈	C₂³×C₂₃	semidirect
397. C₂²×C₈	C₂³×C₂₃	semidirect
399. C₄⋊Q₈	C₂³×C₂₃	semidirect
408. #121	C₂³×C₂₃	semidirect
415. C₄×D₄	C₂³×C₂₃	semidirect
423. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
430. #107	C₂³×C₂₃	semidirect
431. #107	C₂³×C₂₃	semidirect
432. C₄⋊C₈	C₂³×C₂₃	semidirect
435. #102	C₂³×C₂₃	semidirect
436. #102	C₂³×C₂₃	semidirect
437. #102	C₂³×C₂₃	semidirect
438. #102	C₂³×C₂₃	semidirect
439. C₂²⋊C₈	C₂³×C₂₃	semidirect
440. C₄×C₈	C₂³×C₂₃	semidirect
441. C₂₃	C₂×#3294	direct (N × H)
475. C₂×D₄	D₈×C₂₃	semidirect
477. C₂×D₄	#41×C₂₃	semidirect
558. D₄	C₂×D₈×C₂₃	semidirect
559. D₄	C₂×#41×C₂₃	semidirect
619. C₂	C₂₃×#3294	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#547	2
C₂×D₄²×C₂₃	2
C₂₃×#3294	2
C₂×C₂₃×#486	2
C₂×C₂₃×#480	2
C₂×C₂₃×#474	2
C₂×C₂₃×#466	2
C₂×C₂₃×#459	2
C₂×C₂₃×#447	2
C₂×C₄×D₈×C₂₃	2
C₂×C₈×D₄×C₂₃	2
C₂²×C₂₃×#102	2
C₂×#3294	23

Subgroups — C2×C23×#3294 #19208738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3294 #19208738