Subgroups — C₂×C₂₃×#3302 #19208746

Order 5888.

2146 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3302	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#485	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#467	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#452	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	C₂×C₂₃×#416	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₂₃	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#556	—	2	4	no	—
18	C₂₃×#556	C₂²	1	8	yes	complemented
19	C₂₃×#547	—	2	4	no	—
20	C₂₃×#547	C₂²	1	8	yes	—
21	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#126	—	2	2	no	—
24	C₂×C₂₃×#124	—	2	1	no	—
25	C₂×C₂₃×#123	—	2	2	no	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₂×C₂₃×#121	—	2	1	no	—
29	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#117	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
35	C₂₃×#487	C₂²	1	8	yes	complemented
36	C₂₃×#485	C₂²	1	16	yes	complemented
37	C₂₃×#474	C₂²	1	8	yes	complemented
38	C₂₃×#467	C₂²	1	16	yes	complemented
39	C₂₃×#462	C₂²	1	8	yes	complemented
40	C₂₃×#452	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×#107	C₂²	1	2	yes	—
44	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂₃×#416	C₂²	1	16	yes	complemented
47	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#103	C₂²	1	2	yes	—
49	C₂×C₂₃×#103	C₂²	1	2	yes	—
50	C₂×C₂₃×#103	C₂²	1	2	yes	—
51	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
52	C₂×C₂₃×#102	C₂²	1	2	yes	—
53	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×#41×C₂₃	—	2	4	no	—
56	C₂×#41×C₂₃	—	2	8	no	—
57	C₂×#41×C₂₃	—	4	4	no	—
58	C₂×#41×C₂₃	C₂³	1	8	yes	—
59	C₂²×Q₈×C₂₃	—	2	1	no	—
60	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
62	C₂²×D₄×C₂₃	—	2	2	no	—
63	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	—	2	2	no	—
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
68	C₂×Q₁₆×C₂₃	—	4	4	no	—
69	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
70	C₂×SD₁₆×C₂₃	—	2	4	no	—
71	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
72	C₂²×C₈×C₂₃	—	4	4	no	—
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
77	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	—	2	2	no	—
79	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
80	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
81	C₂₃×#126	—	2	16	no	—
82	C₂₃×#124	—	2	8	no	—
83	C₂₃×#123	—	2	16	no	—
84	C₂₃×C₂²⋊Q₈	—	2	4	no	—
85	C₂₃×C₂²⋊Q₈	—	2	4	no	—
86	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
87	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
88	C₂₃×#121	—	2	8	no	—
89	C₄×Q₈×C₂₃	—	2	8	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
92	C₄×D₄×C₂₃	—	2	2	no	—
93	C₄×D₄×C₂₃	D₄	1	4	yes	—
94	C₄×D₄×C₂₃	C₂³	1	4	yes	—
95	C₄×D₄×C₂₃	C₂³	1	4	yes	—
96	C₂₃×#117	—	2	4	no	—
97	C₂₃×#117	C₂³	1	8	yes	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
105	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
114	C₂×C₄²×C₂₃	—	2	2	no	—
115	C₂×C₄²×C₂₃	—	2	1	no	—
116	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
117	C₂₃×#107	C₂³	1	8	yes	—
118	C₂₃×#106	C₂³	1	4	yes	—
119	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
120	C₂₃×#103	C₂³	1	8	yes	—
121	C₂₃×#103	C₂³	1	8	yes	—
122	C₂₃×#103	C₂³	1	4	yes	—
123	C₂₃×#103	C₂³	1	8	yes	—
124	C₂₃×#102	C₂³	1	4	yes	—
125	C₂₃×#102	C₂³	1	8	yes	—
126	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
127	C₄×C₈×C₂₃	C₂³	1	4	yes	—
128	C₂⁴×C₂₃	—	2	1	no	—
129	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
130	#41×C₂₃	—	2	16	no	—
131	#41×C₂₃	—	4	8	no	—
132	#41×C₂₃	—	4	16	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	4	no	—
135	C₂×Q₈×C₂₃	—	2	2	no	—
136	C₂×Q₈×C₂₃	—	2	2	no	—
137	C₂×Q₈×C₂₃	—	2	4	no	—
138	C₂×Q₈×C₂₃	—	2	2	no	—
139	C₂×Q₈×C₂₃	—	2	4	no	—
140	C₂×Q₈×C₂₃	—	4	4	no	—
141	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
142	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
143	C₂×D₄×C₂₃	—	2	8	no	—
144	C₂×D₄×C₂₃	—	2	4	no	—
145	C₂×D₄×C₂₃	—	2	1	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	4	8	no	—
148	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
149	C₂×D₄×C₂₃	#41	1	4	yes	—
150	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₂₃	—	2	4	no	—
152	C₂²×C₄×C₂₃	—	2	4	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	2	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	2	no	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	2	8	no	—
160	C₂²×C₄×C₂₃	—	4	8	no	—
161	C₂²×C₄×C₂₃	—	4	4	no	—
162	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₂₃	—	4	8	no	—
169	SD₁₆×C₂₃	—	2	8	no	—
170	C₂×C₈×C₂₃	—	2	2	no	—
171	C₂×C₈×C₂₃	—	2	2	no	—
172	C₂×C₈×C₂₃	—	2	4	no	—
173	C₂×C₈×C₂₃	—	4	8	no	—
174	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
175	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
176	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
177	C₄⋊C₄×C₂₃	—	2	4	no	—
178	C₄⋊C₄×C₂₃	—	2	8	no	—
179	C₄⋊C₄×C₂₃	—	2	8	no	—
180	C₄⋊C₄×C₂₃	—	2	4	no	—
181	C₄⋊C₄×C₂₃	—	2	4	no	—
182	C₄⋊C₄×C₂₃	—	2	4	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	2	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
187	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
188	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
189	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
190	C₂²⋊C₄×C₂₃	—	2	8	no	—
191	C₂²⋊C₄×C₂₃	—	2	8	no	—
192	C₂²⋊C₄×C₂₃	—	2	8	no	—
193	C₂²⋊C₄×C₂₃	—	2	4	no	—
194	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
195	C₄²×C₂₃	—	2	8	no	—
196	C₄²×C₂₃	—	2	4	no	—
197	C₄²×C₂₃	C₂×D₄	1	2	yes	—
198	C₄²×C₂₃	C₂⁴	1	2	yes	—
199	C₂×#3302	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₂₃	—	2	4	no	—
201	C₂³×C₂₃	—	2	4	no	—
202	C₂³×C₂₃	—	2	4	no	—
203	C₂³×C₂₃	—	2	4	no	—
204	C₂³×C₂₃	—	2	2	no	—
205	C₂³×C₂₃	—	4	2	no	—
206	C₂³×C₂₃	—	4	2	no	—
207	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
208	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₂₃	—	2	2	no	—
210	Q₈×C₂₃	—	2	4	no	—
211	Q₈×C₂₃	—	2	2	no	—
212	Q₈×C₂₃	—	2	4	no	—
213	Q₈×C₂₃	—	4	2	no	—
214	Q₈×C₂₃	—	4	4	no	—
215	Q₈×C₂₃	—	4	8	no	—
216	D₄×C₂₃	—	2	4	no	—
217	D₄×C₂₃	—	2	8	no	—
218	D₄×C₂₃	—	2	2	no	—
219	D₄×C₂₃	—	4	16	no	—
220	D₄×C₂₃	C₂×#41	1	4	yes	—
221	C₂×C₄×C₂₃	—	2	4	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	8	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	4	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	4	no	—
231	C₂×C₄×C₂₃	—	4	16	no	—
232	C₂×C₄×C₂₃	—	4	4	no	—
233	C₂×C₄×C₂₃	—	4	2	no	—
234	C₂×C₄×C₂₃	—	4	2	no	—
235	C₂×C₄×C₂₃	—	4	4	no	—
236	C₂×C₄×C₂₃	—	4	2	no	—
237	C₂×C₄×C₂₃	—	4	8	no	—
238	C₂×C₄×C₂₃	#135	1	4	yes	—
239	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
240	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
244	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
245	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
246	C₈×C₂₃	—	2	2	no	—
247	C₈×C₂₃	—	2	2	no	—
248	C₈×C₂₃	—	2	4	no	—
249	C₂×#556	C₂×C₂₃	1	1	yes	characteristic, complemented
250	C₂×#547	C₂×C₂₃	1	1	yes	characteristic
251	#3302	C₂×C₂₃	1	16	yes	direct product factor
252	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
253	C₂×#485	C₂×C₂₃	1	2	yes	complemented
254	C₂×#474	C₂×C₂₃	1	1	yes	characteristic, complemented
255	C₂×#467	C₂×C₂₃	1	2	yes	complemented
256	C₂×#462	C₂×C₂₃	1	1	yes	characteristic, complemented
257	C₂×#452	C₂×C₂₃	1	2	yes	complemented
258	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
260	C₂×#416	C₂×C₂₃	1	2	yes	complemented
261	C₂²×C₂₃	—	2	4	no	—
262	C₂²×C₂₃	—	2	4	no	—
263	C₂²×C₂₃	—	2	4	no	—
264	C₂²×C₂₃	—	2	4	no	—
265	C₂²×C₂₃	—	4	8	no	—
266	C₂²×C₂₃	—	4	8	no	—
267	C₂²×C₂₃	—	4	2	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	#546	1	2	yes	—
270	C₂²×C₂₃	C₂×#135	1	2	yes	—
271	C₂²×C₂₃	#578	1	2	yes	—
272	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
273	C₄×C₂₃	—	2	4	no	—
274	C₄×C₂₃	—	2	2	no	—
275	C₄×C₂₃	—	4	4	no	—
276	C₄×C₂₃	—	4	2	no	—
277	C₄×C₂₃	—	4	2	no	—
278	C₄×C₂₃	—	4	4	no	—
279	C₄×C₂₃	—	4	2	no	—
280	C₄×C₂₃	#546	1	2	yes	—
281	C₄×C₂₃	C₂×#135	1	4	yes	—
282	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
283	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
284	C₂²×#41	C₂²×C₂₃	1	2	yes	—
285	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
286	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
287	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
288	#556	—	2	4	no	—
289	#556	C₂²×C₂₃	1	8	yes	complemented
290	#547	—	2	4	no	—
291	#547	C₂²×C₂₃	1	8	yes	—
292	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
293	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
294	C₂×#126	—	2	2	no	—
295	C₂×#124	—	2	1	no	—
296	C₂×#123	—	2	2	no	—
297	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
298	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
299	C₂×#121	—	2	1	no	—
300	C₂×C₄×Q₈	—	2	1	no	—
301	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×#117	C₂²×C₂₃	1	2	yes	—
305	C₂²×C₄⋊C₄	—	2	1	no	—
306	#487	C₂²×C₂₃	1	8	yes	complemented
307	#485	C₂²×C₂₃	1	16	yes	complemented
308	#474	C₂²×C₂₃	1	8	yes	complemented
309	#467	C₂²×C₂₃	1	16	yes	complemented
310	#462	C₂²×C₂₃	1	8	yes	complemented
311	#452	C₂²×C₂₃	1	16	yes	complemented
312	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
313	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
314	C₂×#107	C₂²×C₂₃	1	2	yes	—
315	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
317	#416	C₂²×C₂₃	1	16	yes	complemented
318	C₂×#103	C₂²×C₂₃	1	2	yes	—
319	C₂×#103	C₂²×C₂₃	1	2	yes	—
320	C₂×#103	C₂²×C₂₃	1	2	yes	—
321	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×#102	C₂²×C₂₃	1	2	yes	—
324	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
325	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×C₂₃	—	4	4	no	—
327	C₂×C₂₃	—	4	4	no	—
328	C₂×C₂₃	#3302	1	4	yes	direct product factor
329	C₂×C₂₃	C₂×#546	1	1	yes	characteristic
330	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
331	C₂×C₂₃	C₂×#578	1	1	yes	characteristic
332	C₂×#41	—	2	4	no	—
333	C₂×#41	—	2	8	no	—
334	C₂×#41	—	4	4	no	—
335	C₂×#41	C₂³×C₂₃	1	8	yes	—
336	C₂²×Q₈	—	2	1	no	—
337	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
338	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
339	C₂²×D₄	—	2	2	no	—
340	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
341	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
342	C₂³×C₄	—	2	1	no	—
343	C₂³×C₄	—	2	2	no	—
344	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
345	C₂×Q₁₆	—	4	4	no	—
346	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
347	C₂×SD₁₆	—	2	4	no	—
348	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
349	C₂²×C₈	—	4	4	no	—
350	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
352	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
354	C₄⋊Q₈	—	2	2	no	—
355	C₄⋊Q₈	—	2	2	no	—
356	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
357	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
358	#126	—	2	16	no	—
359	#124	—	2	8	no	—
360	#123	—	2	16	no	—
361	C₂²⋊Q₈	—	2	4	no	—
362	C₂²⋊Q₈	—	2	4	no	—
363	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
364	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
365	#121	—	2	8	no	—
366	C₄×Q₈	—	2	8	no	—
367	C₄×Q₈	—	2	2	no	—
368	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	D₄×C₂₃	1	4	yes	—
371	C₄×D₄	C₂³×C₂₃	1	4	yes	—
372	C₄×D₄	C₂³×C₂₃	1	4	yes	—
373	#117	—	2	4	no	—
374	#117	C₂³×C₂₃	1	8	yes	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	8	no	—
382	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
394	#107	C₂³×C₂₃	1	8	yes	—
395	#106	C₂³×C₂₃	1	4	yes	—
396	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
397	#103	C₂³×C₂₃	1	8	yes	—
398	#103	C₂³×C₂₃	1	8	yes	—
399	#103	C₂³×C₂₃	1	4	yes	—
400	#103	C₂³×C₂₃	1	8	yes	—
401	#102	C₂³×C₂₃	1	4	yes	—
402	#102	C₂³×C₂₃	1	8	yes	—
403	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
404	C₄×C₈	C₂³×C₂₃	1	4	yes	—
405	C₂₃	C₂×#3302	1	1	yes	characteristic, direct product factor, Sylow
406	C₂⁴	—	2	1	no	—
407	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
408	#41	—	2	16	no	—
409	#41	—	4	8	no	—
410	#41	—	4	16	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	4	4	no	—
419	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
420	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
427	C₂×D₄	#41×C₂₃	1	4	yes	—
428	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	8	no	—
438	C₂²×C₄	—	4	8	no	—
439	C₂²×C₄	—	4	4	no	—
440	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
443	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
446	Q₁₆	—	4	8	no	—
447	SD₁₆	—	2	8	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	4	no	—
451	C₂×C₈	—	4	8	no	—
452	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
453	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
454	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
465	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
466	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
467	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
473	C₄²	—	2	8	no	—
474	C₄²	—	2	4	no	—
475	C₄²	C₂×D₄×C₂₃	1	2	yes	—
476	C₄²	C₂⁴×C₂₃	1	2	yes	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	4	2	no	—
484	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
485	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
486	Q₈	—	2	2	no	—
487	Q₈	—	2	4	no	—
488	Q₈	—	2	2	no	—
489	Q₈	—	2	4	no	—
490	Q₈	—	4	8	no	—
491	Q₈	—	4	4	no	—
492	Q₈	—	4	2	no	—
493	D₄	—	2	8	no	—
494	D₄	—	2	2	no	—
495	D₄	—	2	4	no	—
496	D₄	—	4	16	no	—
497	D₄	C₂×#41×C₂₃	1	4	yes	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	8	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	4	16	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	8	no	—
515	C₂×C₄	C₂₃×#135	1	4	yes	—
516	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
517	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
519	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
520	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
521	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
522	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
523	C₈	—	2	2	no	—
524	C₈	—	2	2	no	—
525	C₈	—	2	4	no	—
526	C₂²	—	2	4	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	2	4	no	—
530	C₂²	—	4	8	no	—
531	C₂²	—	4	8	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	C₂₃×#546	1	2	yes	—
535	C₂²	C₂×C₂₃×#135	1	2	yes	—
536	C₂²	C₂₃×#578	1	2	yes	—
537	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
538	C₄	—	2	4	no	—
539	C₄	—	2	2	no	—
540	C₄	—	4	4	no	—
541	C₄	—	4	2	no	—
542	C₄	—	4	2	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	2	no	—
545	C₄	C₂₃×#546	1	2	yes	—
546	C₄	C₂×C₂₃×#135	1	4	yes	—
547	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
548	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
549	C₂	—	4	4	no	—
550	C₂	—	4	4	no	—
551	C₂	C₂₃×#3302	1	4	yes	direct product factor
552	C₂	C₂×C₂₃×#546	1	1	yes	characteristic
553	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
554	C₂	C₂×C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#547	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×#556	C₂	semidirect
3. C₂₃×#3302	C₂	direct (N × H)
4. C₂×C₂₃×#487	C₂	semidirect
5. C₂×C₂₃×#485	C₂	semidirect
6. C₂×C₂₃×#474	C₂	semidirect
7. C₂×C₂₃×#467	C₂	semidirect
8. C₂×C₂₃×#462	C₂	semidirect
9. C₂×C₂₃×#452	C₂	semidirect
10. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
11. C₂×C₈×D₄×C₂₃	C₂	semidirect
12. C₂×C₂₃×#416	C₂	semidirect
18. C₂₃×#556	C₂²	semidirect
35. C₂₃×#487	C₂²	semidirect
36. C₂₃×#485	C₂²	semidirect
37. C₂₃×#474	C₂²	semidirect
38. C₂₃×#467	C₂²	semidirect
39. C₂₃×#462	C₂²	semidirect
40. C₂₃×#452	C₂²	semidirect
41. C₄×SD₁₆×C₂₃	C₂²	semidirect
42. C₈×D₄×C₂₃	C₂²	semidirect
46. C₂₃×#416	C₂²	semidirect
199. C₂×#3302	C₂₃	direct (N × H)
249. C₂×#556	C₂×C₂₃	semidirect
251. #3302	C₂×C₂₃	direct (N × H)
252. C₂×#487	C₂×C₂₃	semidirect
253. C₂×#485	C₂×C₂₃	semidirect
254. C₂×#474	C₂×C₂₃	semidirect
255. C₂×#467	C₂×C₂₃	semidirect
256. C₂×#462	C₂×C₂₃	semidirect
257. C₂×#452	C₂×C₂₃	semidirect
258. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
259. C₂×C₈×D₄	C₂×C₂₃	semidirect
260. C₂×#416	C₂×C₂₃	semidirect
289. #556	C₂²×C₂₃	semidirect
306. #487	C₂²×C₂₃	semidirect
307. #485	C₂²×C₂₃	semidirect
308. #474	C₂²×C₂₃	semidirect
309. #467	C₂²×C₂₃	semidirect
310. #462	C₂²×C₂₃	semidirect
311. #452	C₂²×C₂₃	semidirect
312. C₄×SD₁₆	C₂²×C₂₃	semidirect
313. C₈×D₄	C₂²×C₂₃	semidirect
317. #416	C₂²×C₂₃	semidirect
328. C₂×C₂₃	#3302	direct (N × H)
405. C₂₃	C₂×#3302	direct (N × H)
551. C₂	C₂₃×#3302	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×#556	2
C₂×C₂₃×#547	2
C₂₃×#3302	2
C₂×C₂₃×#487	2
C₂×C₂₃×#485	2
C₂×C₂₃×#474	2
C₂×C₂₃×#467	2
C₂×C₂₃×#462	2
C₂×C₂₃×#452	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×C₈×D₄×C₂₃	2
C₂×C₂₃×#416	2
C₂×#3302	23

Subgroups — C2×C23×#3302 #19208746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3302 #19208746