Subgroups — C₂×C₂₃×#3295 #19208739

Order 5888.

2642 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3295	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#481	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#465	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#450	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×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	C₂³×D₄×C₂₃	—	2	1	no	—
14	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
17	C₂³×C₈×C₂₃	C₂²	1	2	yes	complemented
18	C₂₃×#554	—	2	4	no	—
19	C₂₃×#554	C₂²	1	8	yes	complemented
20	D₄²×C₂₃	—	2	4	no	—
21	D₄²×C₂₃	C₂²	1	8	yes	—
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#125	—	2	1	no	—
25	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₂×C₂₃×#121	—	2	1	no	—
28	C₂×C₂₃×#121	C₂²	1	2	yes	—
29	C₂×C₂₃×#120	—	2	2	no	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄×D₄×C₂₃	—	2	1	no	—
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
35	C₂₃×#486	C₂²	1	8	yes	complemented
36	C₂₃×#481	C₂²	1	16	yes	complemented
37	C₂₃×#476	C₂²	1	8	yes	complemented
38	C₂₃×#465	C₂²	1	16	yes	complemented
39	C₂₃×#460	C₂²	1	8	yes	complemented
40	C₂₃×#450	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
42	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
43	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
44	C₂×C₂₃×#106	C₂²	1	2	yes	complemented
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#102	C₂²	1	2	yes	—
49	C₂×C₂₃×#102	C₂²	1	16	yes	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	C₂²×Q₈×C₂₃	—	2	1	no	—
57	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₂₃	—	2	8	no	—
59	C₂²×D₄×C₂₃	—	2	2	no	—
60	C₂²×D₄×C₂₃	—	2	8	no	—
61	C₂²×D₄×C₂₃	—	2	1	no	—
62	C₂²×D₄×C₂₃	—	2	2	no	—
63	C₂²×D₄×C₂₃	—	2	4	no	—
64	C₂²×D₄×C₂₃	—	2	1	no	—
65	C₂²×D₄×C₂₃	—	4	4	no	—
66	C₂²×D₄×C₂₃	—	4	8	no	—
67	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
68	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
69	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
70	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	—	2	2	no	—
72	C₂³×C₄×C₂₃	—	2	1	no	—
73	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
74	C₂×SD₁₆×C₂₃	—	2	4	no	—
75	C₂×SD₁₆×C₂₃	—	4	4	no	—
76	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
78	C₂²×C₈×C₂₃	—	4	4	no	—
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
81	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₂₃	C₂³	1	8	yes	complemented
83	C₂₃×C₄⋊Q₈	—	2	2	no	—
84	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
85	C₂₃×#127	—	2	2	no	—
86	C₂₃×#127	C₂³	1	4	yes	—
87	C₂₃×#125	—	2	8	no	—
88	C₂₃×C₂²⋊Q₈	—	2	4	no	—
89	C₂₃×C₂²⋊Q₈	—	2	16	no	—
90	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	complemented
91	C₂₃×#121	—	2	4	no	—
92	C₂₃×#121	—	2	8	no	—
93	C₂₃×#121	C₂³	1	8	yes	—
94	C₂₃×#120	—	2	16	no	—
95	C₄×Q₈×C₂₃	—	2	2	no	—
96	C₄×Q₈×C₂₃	C₂³	1	4	yes	complemented
97	C₄×D₄×C₂₃	—	2	8	no	—
98	C₄×D₄×C₂₃	—	2	2	no	—
99	C₄×D₄×C₂₃	D₄	1	4	yes	—
100	C₄×D₄×C₂₃	C₂³	1	4	yes	—
101	C₄×D₄×C₂₃	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
112	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
116	C₂×C₄²×C₂₃	—	2	1	no	—
117	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
118	C₂₃×#106	C₂³	1	4	yes	complemented
119	C₂₃×#106	C₂³	1	8	yes	complemented
120	C₂₃×C₄⋊C₈	C₂³	1	4	yes	complemented
121	C₂₃×#103	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	8	yes	complemented
127	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	complemented
128	C₄×C₈×C₂₃	C₂³	1	4	yes	complemented
129	C₂⁴×C₂₃	—	2	1	no	—
130	C₂⁴×C₂₃	—	2	2	no	—
131	C₂⁴×C₂₃	—	2	8	no	—
132	C₂⁴×C₂₃	—	4	8	no	—
133	C₂⁴×C₂₃	—	4	8	no	—
134	C₂⁴×C₂₃	—	8	4	no	—
135	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	4	no	—
138	C₂×Q₈×C₂₃	—	2	2	no	—
139	C₂×Q₈×C₂₃	—	2	2	no	—
140	C₂×Q₈×C₂₃	—	8	2	no	—
141	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
142	C₂×D₄×C₂₃	—	2	16	no	—
143	C₂×D₄×C₂₃	—	2	8	no	—
144	C₂×D₄×C₂₃	—	2	1	no	—
145	C₂×D₄×C₂₃	—	2	8	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	4	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	—	4	8	no	—
153	C₂×D₄×C₂₃	—	4	16	no	—
154	C₂×D₄×C₂₃	—	4	8	no	—
155	C₂×D₄×C₂₃	—	4	8	no	—
156	C₂×D₄×C₂₃	—	4	32	no	—
157	C₂×D₄×C₂₃	—	4	4	no	—
158	C₂×D₄×C₂₃	—	8	2	no	—
159	C₂×D₄×C₂₃	SD₁₆	1	8	yes	complemented
160	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
161	C₂×D₄×C₂₃	#41	1	4	yes	—
162	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
163	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₂₃	—	2	4	no	—
165	C₂²×C₄×C₂₃	—	2	4	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	2	no	—
170	C₂²×C₄×C₂₃	—	2	8	no	—
171	C₂²×C₄×C₂₃	—	2	4	no	—
172	C₂²×C₄×C₂₃	—	4	4	no	—
173	C₂²×C₄×C₂₃	—	8	4	no	—
174	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
175	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
180	SD₁₆×C₂₃	—	2	8	no	—
181	SD₁₆×C₂₃	—	4	8	no	—
182	C₂×C₈×C₂₃	—	2	2	no	—
183	C₂×C₈×C₂₃	—	2	2	no	—
184	C₂×C₈×C₂₃	—	2	4	no	—
185	C₂×C₈×C₂₃	—	4	8	no	—
186	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
187	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
188	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	8	no	—
192	C₄⋊C₄×C₂₃	—	2	4	no	—
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	—	2	8	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
198	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
199	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
200	C₂²⋊C₄×C₂₃	—	2	8	no	—
201	C₂²⋊C₄×C₂₃	—	2	4	no	—
202	C₂²⋊C₄×C₂₃	—	2	8	no	—
203	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
204	C₄²×C₂₃	—	2	4	no	—
205	C₄²×C₂₃	C₂×D₄	1	2	yes	—
206	C₄²×C₂₃	C₂⁴	1	2	yes	—
207	C₂×#3295	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂³×C₂₃	—	2	4	no	—
209	C₂³×C₂₃	—	2	4	no	—
210	C₂³×C₂₃	—	2	4	no	—
211	C₂³×C₂₃	—	2	2	no	—
212	C₂³×C₂₃	—	2	4	no	—
213	C₂³×C₂₃	—	4	16	no	—
214	C₂³×C₂₃	—	4	2	no	—
215	C₂³×C₂₃	—	4	2	no	—
216	C₂³×C₂₃	—	4	4	no	—
217	C₂³×C₂₃	—	4	16	no	—
218	C₂³×C₂₃	—	4	4	no	—
219	C₂³×C₂₃	—	4	4	no	—
220	C₂³×C₂₃	—	8	8	no	—
221	C₂³×C₂₃	—	8	32	no	—
222	C₂³×C₂₃	C₂×SD₁₆	1	8	yes	—
223	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
224	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₂₃	—	2	2	no	—
226	Q₈×C₂₃	—	2	2	no	—
227	Q₈×C₂₃	—	4	2	no	—
228	Q₈×C₂₃	—	8	4	no	—
229	D₄×C₂₃	—	2	4	no	—
230	D₄×C₂₃	—	2	4	no	—
231	D₄×C₂₃	—	2	2	no	—
232	D₄×C₂₃	—	2	4	no	—
233	D₄×C₂₃	—	4	16	no	—
234	D₄×C₂₃	—	4	16	no	—
235	D₄×C₂₃	—	4	4	no	—
236	D₄×C₂₃	—	4	8	no	—
237	D₄×C₂₃	—	8	4	no	—
238	D₄×C₂₃	C₂×SD₁₆	1	16	yes	complemented
239	D₄×C₂₃	C₂×#41	1	4	yes	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	8	no	—
243	C₂×C₄×C₂₃	—	2	4	no	—
244	C₂×C₄×C₂₃	—	2	4	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	4	no	—
249	C₂×C₄×C₂₃	—	4	2	no	—
250	C₂×C₄×C₂₃	—	4	2	no	—
251	C₂×C₄×C₂₃	—	4	2	no	—
252	C₂×C₄×C₂₃	—	4	4	no	—
253	C₂×C₄×C₂₃	—	4	8	no	—
254	C₂×C₄×C₂₃	—	8	8	no	—
255	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
256	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
258	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
261	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
262	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
263	C₈×C₂₃	—	2	2	no	—
264	C₈×C₂₃	—	2	2	no	—
265	C₈×C₂₃	—	2	4	no	—
266	C₂×#554	C₂×C₂₃	1	1	yes	characteristic, complemented
267	C₂×D₄²	C₂×C₂₃	1	1	yes	characteristic
268	#3295	C₂×C₂₃	1	16	yes	direct product factor
269	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
270	C₂×#481	C₂×C₂₃	1	2	yes	complemented
271	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
272	C₂×#465	C₂×C₂₃	1	2	yes	complemented
273	C₂×#460	C₂×C₂₃	1	1	yes	characteristic, complemented
274	C₂×#450	C₂×C₂₃	1	2	yes	complemented
275	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
276	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
277	C₂²×#102	C₂×C₂₃	1	2	yes	complemented
278	C₂²×C₂₃	—	2	4	no	—
279	C₂²×C₂₃	—	2	16	no	—
280	C₂²×C₂₃	—	2	4	no	—
281	C₂²×C₂₃	—	2	4	no	—
282	C₂²×C₂₃	—	4	8	no	—
283	C₂²×C₂₃	—	4	2	no	—
284	C₂²×C₂₃	—	4	2	no	—
285	C₂²×C₂₃	—	4	4	no	—
286	C₂²×C₂₃	—	4	4	no	—
287	C₂²×C₂₃	—	4	4	no	—
288	C₂²×C₂₃	—	8	32	no	—
289	C₂²×C₂₃	—	8	8	no	—
290	C₂²×C₂₃	#546	1	2	yes	—
291	C₂²×C₂₃	C₂²×SD₁₆	1	8	yes	—
292	C₂²×C₂₃	C₂²×SD₁₆	1	2	yes	—
293	C₂²×C₂₃	#576	1	2	yes	—
294	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
295	C₄×C₂₃	—	2	4	no	—
296	C₄×C₂₃	—	2	2	no	—
297	C₄×C₂₃	—	4	2	no	—
298	C₄×C₂₃	—	4	2	no	—
299	C₄×C₂₃	—	4	2	no	—
300	C₄×C₂₃	—	4	4	no	—
301	C₄×C₂₃	#546	1	2	yes	—
302	C₄×C₂₃	C₂²×SD₁₆	1	4	yes	—
303	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
304	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
305	C₂³×D₄	—	2	1	no	—
306	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
307	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
308	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
309	C₂³×C₈	C₂²×C₂₃	1	2	yes	complemented
310	#554	—	2	4	no	—
311	#554	C₂²×C₂₃	1	8	yes	complemented
312	D₄²	—	2	4	no	—
313	D₄²	C₂²×C₂₃	1	8	yes	—
314	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
315	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×#125	—	2	1	no	—
317	C₂×C₂²⋊Q₈	—	2	2	no	—
318	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
319	C₂×#121	—	2	1	no	—
320	C₂×#121	C₂²×C₂₃	1	2	yes	—
321	C₂×#120	—	2	2	no	—
322	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
323	C₂×C₄×D₄	—	2	1	no	—
324	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
326	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
327	#486	C₂²×C₂₃	1	8	yes	complemented
328	#481	C₂²×C₂₃	1	16	yes	complemented
329	#476	C₂²×C₂₃	1	8	yes	complemented
330	#465	C₂²×C₂₃	1	16	yes	complemented
331	#460	C₂²×C₂₃	1	8	yes	complemented
332	#450	C₂²×C₂₃	1	16	yes	complemented
333	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
334	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
335	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
336	C₂×#106	C₂²×C₂₃	1	2	yes	complemented
337	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
338	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
339	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×#102	C₂²×C₂₃	1	2	yes	—
341	C₂×#102	C₂²×C₂₃	1	16	yes	complemented
342	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
343	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
344	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
345	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
346	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
347	C₂×C₂₃	—	2	8	no	—
348	C₂×C₂₃	—	4	4	no	—
349	C₂×C₂₃	—	8	4	no	—
350	C₂×C₂₃	#3295	1	4	yes	direct product factor
351	C₂×C₂₃	C₂×#546	1	1	yes	characteristic
352	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
353	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
354	C₂⁵	—	2	2	no	—
355	C₂²×Q₈	—	2	1	no	—
356	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
357	C₂²×D₄	—	2	8	no	—
358	C₂²×D₄	—	2	2	no	—
359	C₂²×D₄	—	2	8	no	—
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	—	2	1	no	—
362	C₂²×D₄	—	2	4	no	—
363	C₂²×D₄	—	2	1	no	—
364	C₂²×D₄	—	4	4	no	—
365	C₂²×D₄	—	4	8	no	—
366	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
367	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
368	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
369	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
370	C₂³×C₄	—	2	2	no	—
371	C₂³×C₄	—	2	1	no	—
372	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
373	C₂×SD₁₆	—	2	4	no	—
374	C₂×SD₁₆	—	4	4	no	—
375	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
376	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
377	C₂²×C₈	—	4	4	no	—
378	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
380	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₂₃	1	8	yes	complemented
382	C₄⋊Q₈	—	2	2	no	—
383	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
384	#127	—	2	2	no	—
385	#127	C₂³×C₂₃	1	4	yes	—
386	#125	—	2	8	no	—
387	C₂²⋊Q₈	—	2	4	no	—
388	C₂²⋊Q₈	—	2	16	no	—
389	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	complemented
390	#121	—	2	4	no	—
391	#121	—	2	8	no	—
392	#121	C₂³×C₂₃	1	8	yes	—
393	#120	—	2	16	no	—
394	C₄×Q₈	—	2	2	no	—
395	C₄×Q₈	C₂³×C₂₃	1	4	yes	complemented
396	C₄×D₄	—	2	8	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	D₄×C₂₃	1	4	yes	—
399	C₄×D₄	C₂³×C₂₃	1	4	yes	—
400	C₄×D₄	C₂³×C₂₃	1	4	yes	—
401	C₂×C₄⋊C₄	—	2	2	no	—
402	C₂×C₄⋊C₄	—	2	4	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	—	2	2	no	—
406	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	complemented
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
415	C₂×C₄²	—	2	1	no	—
416	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
417	#106	C₂³×C₂₃	1	4	yes	complemented
418	#106	C₂³×C₂₃	1	8	yes	complemented
419	C₄⋊C₈	C₂³×C₂₃	1	4	yes	complemented
420	#103	C₂³×C₂₃	1	4	yes	complemented
421	#102	C₂³×C₂₃	1	4	yes	—
422	#102	C₂³×C₂₃	1	8	yes	—
423	#102	C₂³×C₂₃	1	8	yes	complemented
424	#102	C₂³×C₂₃	1	8	yes	complemented
425	#102	C₂³×C₂₃	1	8	yes	complemented
426	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	complemented
427	C₄×C₈	C₂³×C₂₃	1	4	yes	complemented
428	C₂₃	C₂×#3295	1	1	yes	characteristic, direct product factor, Sylow
429	C₂⁴	—	2	1	no	—
430	C₂⁴	—	2	2	no	—
431	C₂⁴	—	2	8	no	—
432	C₂⁴	—	4	8	no	—
433	C₂⁴	—	4	8	no	—
434	C₂⁴	—	8	4	no	—
435	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	2	2	no	—
440	C₂×Q₈	—	8	2	no	—
441	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
442	C₂×D₄	—	2	16	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	8	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	4	8	no	—
453	C₂×D₄	—	4	16	no	—
454	C₂×D₄	—	4	8	no	—
455	C₂×D₄	—	4	8	no	—
456	C₂×D₄	—	4	32	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	8	2	no	—
459	C₂×D₄	SD₁₆×C₂₃	1	8	yes	complemented
460	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
461	C₂×D₄	#41×C₂₃	1	4	yes	—
462	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
463	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	8	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	—	8	4	no	—
474	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
480	SD₁₆	—	2	8	no	—
481	SD₁₆	—	4	8	no	—
482	C₂×C₈	—	2	2	no	—
483	C₂×C₈	—	2	2	no	—
484	C₂×C₈	—	2	4	no	—
485	C₂×C₈	—	4	8	no	—
486	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
487	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
488	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	8	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	8	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
497	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
498	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
499	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
504	C₄²	—	2	4	no	—
505	C₄²	C₂×D₄×C₂₃	1	2	yes	—
506	C₄²	C₂⁴×C₂₃	1	2	yes	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	4	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	4	16	no	—
513	C₂³	—	4	2	no	—
514	C₂³	—	4	2	no	—
515	C₂³	—	4	16	no	—
516	C₂³	—	4	4	no	—
517	C₂³	—	4	4	no	—
518	C₂³	—	4	4	no	—
519	C₂³	—	8	8	no	—
520	C₂³	—	8	32	no	—
521	C₂³	C₂×SD₁₆×C₂₃	1	8	yes	—
522	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
523	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
524	Q₈	—	2	2	no	—
525	Q₈	—	2	2	no	—
526	Q₈	—	4	2	no	—
527	Q₈	—	8	4	no	—
528	D₄	—	2	4	no	—
529	D₄	—	2	2	no	—
530	D₄	—	2	4	no	—
531	D₄	—	2	4	no	—
532	D₄	—	4	16	no	—
533	D₄	—	4	16	no	—
534	D₄	—	4	8	no	—
535	D₄	—	4	4	no	—
536	D₄	—	8	4	no	—
537	D₄	C₂×SD₁₆×C₂₃	1	16	yes	complemented
538	D₄	C₂×#41×C₂₃	1	4	yes	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	8	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	8	no	—
553	C₂×C₄	—	8	8	no	—
554	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
555	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
557	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
558	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
559	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
560	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
561	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
562	C₈	—	2	2	no	—
563	C₈	—	2	2	no	—
564	C₈	—	2	4	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	2	16	no	—
567	C₂²	—	2	4	no	—
568	C₂²	—	2	4	no	—
569	C₂²	—	4	8	no	—
570	C₂²	—	4	2	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	4	no	—
575	C₂²	—	8	32	no	—
576	C₂²	—	8	8	no	—
577	C₂²	C₂₃×#546	1	2	yes	—
578	C₂²	C₂²×SD₁₆×C₂₃	1	8	yes	—
579	C₂²	C₂²×SD₁₆×C₂₃	1	2	yes	—
580	C₂²	C₂₃×#576	1	2	yes	—
581	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
582	C₄	—	2	4	no	—
583	C₄	—	2	2	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	2	no	—
586	C₄	—	4	2	no	—
587	C₄	—	4	4	no	—
588	C₄	C₂₃×#546	1	2	yes	—
589	C₄	C₂²×SD₁₆×C₂₃	1	4	yes	—
590	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
591	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
592	C₂	—	2	8	no	—
593	C₂	—	4	4	no	—
594	C₂	—	8	4	no	—
595	C₂	C₂₃×#3295	1	4	yes	direct product factor
596	C₂	C₂×C₂₃×#546	1	1	yes	characteristic
597	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
598	C₂	C₂×C₂₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×D₄²×C₂₃	C₂
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×#554	C₂	semidirect
3. C₂₃×#3295	C₂	direct (N × H)
4. C₂×C₂₃×#486	C₂	semidirect
5. C₂×C₂₃×#481	C₂	semidirect
6. C₂×C₂₃×#476	C₂	semidirect
7. C₂×C₂₃×#465	C₂	semidirect
8. C₂×C₂₃×#460	C₂	semidirect
9. C₂×C₂₃×#450	C₂	semidirect
10. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
11. C₂×C₈×D₄×C₂₃	C₂	semidirect
12. C₂²×C₂₃×#102	C₂	semidirect
16. C₂²×SD₁₆×C₂₃	C₂²	semidirect
17. C₂³×C₈×C₂₃	C₂²	semidirect
19. C₂₃×#554	C₂²	semidirect
22. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
26. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
30. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
34. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
35. C₂₃×#486	C₂²	semidirect
36. C₂₃×#481	C₂²	semidirect
37. C₂₃×#476	C₂²	semidirect
38. C₂₃×#465	C₂²	semidirect
39. C₂₃×#460	C₂²	semidirect
40. C₂₃×#450	C₂²	semidirect
41. C₄×SD₁₆×C₂₃	C₂²	semidirect
42. C₈×D₄×C₂₃	C₂²	semidirect
43. C₂×C₂₃×#106	C₂²	semidirect
44. C₂×C₂₃×#106	C₂²	semidirect
45. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
46. C₂×C₂₃×#103	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
77. C₂×SD₁₆×C₂₃	C₂³	semidirect
82. C₂²×C₈×C₂₃	C₂³	semidirect
84. C₂₃×C₄⋊Q₈	C₂³	semidirect
90. C₂₃×C₂²⋊Q₈	C₂³	semidirect
96. C₄×Q₈×C₂₃	C₂³	semidirect
111. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
118. C₂₃×#106	C₂³	semidirect
119. C₂₃×#106	C₂³	semidirect
120. C₂₃×C₄⋊C₈	C₂³	semidirect
121. C₂₃×#103	C₂³	semidirect
124. C₂₃×#102	C₂³	semidirect
125. C₂₃×#102	C₂³	semidirect
126. C₂₃×#102	C₂³	semidirect
127. C₂₃×C₂²⋊C₈	C₂³	semidirect
128. C₄×C₈×C₂₃	C₂³	semidirect
159. C₂×D₄×C₂₃	SD₁₆	semidirect
207. C₂×#3295	C₂₃	direct (N × H)
238. D₄×C₂₃	C₂×SD₁₆	semidirect
266. C₂×#554	C₂×C₂₃	semidirect
268. #3295	C₂×C₂₃	direct (N × H)
269. C₂×#486	C₂×C₂₃	semidirect
270. C₂×#481	C₂×C₂₃	semidirect
271. C₂×#476	C₂×C₂₃	semidirect
272. C₂×#465	C₂×C₂₃	semidirect
273. C₂×#460	C₂×C₂₃	semidirect
274. C₂×#450	C₂×C₂₃	semidirect
275. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
276. C₂×C₈×D₄	C₂×C₂₃	semidirect
277. C₂²×#102	C₂×C₂₃	semidirect
308. C₂²×SD₁₆	C₂²×C₂₃	semidirect
309. C₂³×C₈	C₂²×C₂₃	semidirect
311. #554	C₂²×C₂₃	semidirect
314. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
318. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
322. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
326. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
327. #486	C₂²×C₂₃	semidirect
328. #481	C₂²×C₂₃	semidirect
329. #476	C₂²×C₂₃	semidirect
330. #465	C₂²×C₂₃	semidirect
331. #460	C₂²×C₂₃	semidirect
332. #450	C₂²×C₂₃	semidirect
333. C₄×SD₁₆	C₂²×C₂₃	semidirect
334. C₈×D₄	C₂²×C₂₃	semidirect
335. C₂×#106	C₂²×C₂₃	semidirect
336. C₂×#106	C₂²×C₂₃	semidirect
337. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
338. C₂×#103	C₂²×C₂₃	semidirect
341. C₂×#102	C₂²×C₂₃	semidirect
342. C₂×#102	C₂²×C₂₃	semidirect
343. C₂×#102	C₂²×C₂₃	semidirect
344. C₂×#102	C₂²×C₂₃	semidirect
345. C₂×C₂²⋊C₈	C₂²×C₂₃	semidirect
346. C₂×C₄×C₈	C₂²×C₂₃	semidirect
350. C₂×C₂₃	#3295	direct (N × H)
376. C₂×SD₁₆	C₂³×C₂₃	semidirect
381. C₂²×C₈	C₂³×C₂₃	semidirect
383. C₄⋊Q₈	C₂³×C₂₃	semidirect
389. C₂²⋊Q₈	C₂³×C₂₃	semidirect
395. C₄×Q₈	C₂³×C₂₃	semidirect
410. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
417. #106	C₂³×C₂₃	semidirect
418. #106	C₂³×C₂₃	semidirect
419. C₄⋊C₈	C₂³×C₂₃	semidirect
420. #103	C₂³×C₂₃	semidirect
423. #102	C₂³×C₂₃	semidirect
424. #102	C₂³×C₂₃	semidirect
425. #102	C₂³×C₂₃	semidirect
426. C₂²⋊C₈	C₂³×C₂₃	semidirect
427. C₄×C₈	C₂³×C₂₃	semidirect
428. C₂₃	C₂×#3295	direct (N × H)
459. C₂×D₄	SD₁₆×C₂₃	semidirect
537. D₄	C₂×SD₁₆×C₂₃	semidirect
595. C₂	C₂₃×#3295	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×#554	2
C₂×D₄²×C₂₃	2
C₂₃×#3295	2
C₂×C₂₃×#486	2
C₂×C₂₃×#481	2
C₂×C₂₃×#476	2
C₂×C₂₃×#465	2
C₂×C₂₃×#460	2
C₂×C₂₃×#450	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×C₈×D₄×C₂₃	2
C₂²×C₂₃×#102	2
C₂×#3295	23

Subgroups — C2×C23×#3295 #19208739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3295 #19208739