Subgroups — C₂₃×#3264 #15670883

Order 2944.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#546	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#465	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#463	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#449	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#447	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#447	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#442	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₄×C₂₃	—	2	1	no	—
18	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
21	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
22	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
25	C₂×D₈×C₂₃	—	2	4	no	—
26	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
32	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
33	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
35	C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#123	—	2	1	no	—
37	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#121	—	2	1	no	—
39	C₂₃×#121	—	2	1	no	—
40	C₂₃×#121	C₂²	1	1	yes	characteristic
41	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
42	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
43	C₂₃×#120	—	2	1	no	—
44	C₂₃×#120	—	2	1	no	—
45	C₂₃×#120	—	2	1	no	—
46	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
47	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
48	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
50	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
51	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
52	C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
53	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
54	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
55	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
56	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
57	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
58	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
59	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
60	C₂₃×#97	C₂²	1	1	yes	characteristic, complemented
61	C₂⁴×C₂₃	—	2	1	no	—
62	C₂⁴×C₂₃	—	2	1	no	—
63	C₂⁴×C₂₃	—	2	1	no	—
64	#41×C₂₃	—	2	2	no	—
65	#41×C₂₃	—	4	1	no	—
66	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₂₃	—	2	2	no	—
68	C₂×D₄×C₂₃	—	2	2	no	—
69	C₂×D₄×C₂₃	—	2	2	no	—
70	C₂×D₄×C₂₃	—	2	1	no	—
71	C₂×D₄×C₂₃	—	2	1	no	—
72	C₂×D₄×C₂₃	—	2	1	no	—
73	C₂×D₄×C₂₃	—	2	1	no	—
74	C₂×D₄×C₂₃	—	2	1	no	—
75	C₂×D₄×C₂₃	—	2	1	no	—
76	C₂×D₄×C₂₃	—	2	1	no	—
77	C₂×D₄×C₂₃	—	2	1	no	—
78	C₂×D₄×C₂₃	—	2	1	no	—
79	C₂×D₄×C₂₃	—	4	1	no	—
80	C₂×D₄×C₂₃	—	4	1	no	—
81	C₂×D₄×C₂₃	—	4	1	no	—
82	C₂×D₄×C₂₃	—	4	4	no	—
83	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
84	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₂₃	—	2	1	no	—
90	C₂²×C₄×C₂₃	—	2	1	no	—
91	C₂²×C₄×C₂₃	—	2	1	no	—
92	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
94	SD₁₆×C₂₃	—	2	2	no	—
95	SD₁₆×C₂₃	—	2	2	no	—
96	SD₁₆×C₂₃	—	4	1	no	—
97	D₈×C₂₃	—	2	2	no	—
98	D₈×C₂₃	—	2	2	no	—
99	D₈×C₂₃	—	2	2	no	—
100	D₈×C₂₃	—	2	2	no	—
101	D₈×C₂₃	—	4	1	no	—
102	D₈×C₂₃	D₄	1	4	yes	complemented
103	M₁₆×C₂₃	—	2	2	no	—
104	C₂×C₈×C₂₃	—	2	2	no	—
105	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂²⋊C₄×C₂₃	—	2	1	no	—
116	C₂²⋊C₄×C₂₃	—	2	1	no	—
117	C₂²⋊C₄×C₂₃	—	2	1	no	—
118	C₂²⋊C₄×C₂₃	—	2	1	no	—
119	C₂²⋊C₄×C₂₃	—	2	1	no	—
120	C₂²⋊C₄×C₂₃	—	2	1	no	—
121	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
122	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
123	C₄²×C₂₃	C₂³	1	1	yes	characteristic
124	C₂³×C₂₃	—	2	1	no	—
125	C₂³×C₂₃	—	2	1	no	—
126	C₂³×C₂₃	—	2	1	no	—
127	C₂³×C₂₃	—	2	1	no	—
128	C₂³×C₂₃	—	2	1	no	—
129	C₂³×C₂₃	—	4	2	no	—
130	C₂³×C₂₃	—	4	2	no	—
131	C₂³×C₂₃	—	4	2	no	—
132	C₂³×C₂₃	—	4	2	no	—
133	C₂³×C₂₃	—	4	2	no	—
134	C₂³×C₂₃	—	4	2	no	—
135	C₂³×C₂₃	—	8	1	no	—
136	C₂³×C₂₃	—	8	1	no	—
137	C₂³×C₂₃	—	8	1	no	—
138	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
140	Q₈×C₂₃	—	2	1	no	—
141	Q₈×C₂₃	—	2	1	no	—
142	D₄×C₂₃	—	2	2	no	—
143	D₄×C₂₃	—	2	1	no	—
144	D₄×C₂₃	—	2	1	no	—
145	D₄×C₂₃	—	2	1	no	—
146	D₄×C₂₃	—	2	1	no	—
147	D₄×C₂₃	—	2	1	no	—
148	D₄×C₂₃	—	2	1	no	—
149	D₄×C₂₃	—	2	1	no	—
150	D₄×C₂₃	—	2	1	no	—
151	D₄×C₂₃	—	4	2	no	—
152	D₄×C₂₃	—	4	2	no	—
153	D₄×C₂₃	—	4	2	no	—
154	D₄×C₂₃	—	4	2	no	—
155	D₄×C₂₃	—	4	2	no	—
156	D₄×C₂₃	—	4	2	no	—
157	D₄×C₂₃	—	4	2	no	—
158	D₄×C₂₃	—	8	1	no	—
159	D₄×C₂₃	—	8	1	no	—
160	D₄×C₂₃	C₂×D₄	1	2	yes	—
161	D₄×C₂₃	C₂×D₄	1	2	yes	complemented
162	C₂×C₄×C₂₃	—	2	1	no	—
163	C₂×C₄×C₂₃	—	2	1	no	—
164	C₂×C₄×C₂₃	—	2	1	no	—
165	C₂×C₄×C₂₃	—	2	1	no	—
166	C₂×C₄×C₂₃	—	2	1	no	—
167	C₂×C₄×C₂₃	—	2	1	no	—
168	C₂×C₄×C₂₃	—	2	1	no	—
169	C₂×C₄×C₂₃	—	4	2	no	—
170	C₂×C₄×C₂₃	—	4	2	no	—
171	C₂×C₄×C₂₃	—	8	1	no	—
172	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
177	C₈×C₂₃	—	2	1	no	—
178	C₈×C₂₃	—	2	1	no	—
179	C₈×C₂₃	—	2	1	no	—
180	C₈×C₂₃	C₂×D₄	1	2	yes	complemented
181	#3264	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂²×C₂₃	—	2	1	no	—
183	C₂²×C₂₃	—	2	1	no	—
184	C₂²×C₂₃	—	2	1	no	—
185	C₂²×C₂₃	—	2	2	no	—
186	C₂²×C₂₃	—	2	2	no	—
187	C₂²×C₂₃	—	2	1	no	—
188	C₂²×C₂₃	—	2	1	no	—
189	C₂²×C₂₃	—	4	1	no	—
190	C₂²×C₂₃	—	4	1	no	—
191	C₂²×C₂₃	—	4	1	no	—
192	C₂²×C₂₃	—	4	1	no	—
193	C₂²×C₂₃	—	4	1	no	—
194	C₂²×C₂₃	—	4	1	no	—
195	C₂²×C₂₃	—	4	1	no	—
196	C₂²×C₂₃	—	4	1	no	—
197	C₂²×C₂₃	—	4	1	no	—
198	C₂²×C₂₃	—	4	1	no	—
199	C₂²×C₂₃	—	4	1	no	—
200	C₂²×C₂₃	—	4	1	no	—
201	C₂²×C₂₃	—	4	1	no	—
202	C₂²×C₂₃	—	8	4	no	—
203	C₂²×C₂₃	—	8	4	no	—
204	C₂²×C₂₃	—	8	4	no	—
205	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
206	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
207	C₄×C₂₃	—	2	1	no	—
208	C₄×C₂₃	—	2	1	no	—
209	C₄×C₂₃	—	2	1	no	—
210	C₄×C₂₃	—	2	1	no	—
211	C₄×C₂₃	—	4	1	no	—
212	C₄×C₂₃	—	4	1	no	—
213	C₄×C₂₃	—	4	1	no	—
214	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
215	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂²×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#546	C₂×C₂₃	1	1	yes	characteristic, complemented
219	D₄²	C₂×C₂₃	1	1	yes	characteristic, complemented
220	#496	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#469	C₂×C₂₃	1	1	yes	characteristic, complemented
222	#465	C₂×C₂₃	1	1	yes	characteristic, complemented
223	#463	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#459	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#450	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#449	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#447	C₂×C₂₃	1	1	yes	characteristic, complemented
228	#447	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#442	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#435	C₂×C₂₃	1	1	yes	characteristic, complemented
231	C₂×C₂₃	—	2	2	no	—
232	C₂×C₂₃	—	4	1	no	—
233	C₂×C₂₃	—	4	2	no	—
234	C₂×C₂₃	—	4	2	no	—
235	C₂×C₂₃	—	8	1	no	—
236	C₂×C₂₃	—	8	1	no	—
237	C₂×C₂₃	—	8	1	no	—
238	C₂×C₂₃	D₄²	1	1	yes	characteristic
239	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂×C₂₃	#576	1	1	yes	characteristic
241	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic, complemented
242	C₂²×D₄	—	2	1	no	—
243	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
244	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
245	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
246	AGL(1,ℤ/8)	—	2	4	no	—
247	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
248	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
249	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
250	C₂×D₈	—	2	4	no	—
251	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
254	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
255	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
256	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
257	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
258	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
259	#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
260	#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
261	#123	—	2	1	no	—
262	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
263	#121	—	2	1	no	—
264	#121	—	2	1	no	—
265	#121	C₂²×C₂₃	1	1	yes	characteristic
266	#121	C₂²×C₂₃	1	1	yes	characteristic, complemented
267	#121	C₂²×C₂₃	1	1	yes	characteristic, complemented
268	#120	—	2	1	no	—
269	#120	—	2	1	no	—
270	#120	—	2	1	no	—
271	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
272	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
273	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
274	C₂×C₂²⋊C₄	—	2	1	no	—
275	#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
276	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
277	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
278	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
279	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
280	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
281	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
282	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
283	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
284	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
285	#97	C₂²×C₂₃	1	1	yes	characteristic, complemented
286	C₂₃	#3264	1	1	yes	characteristic, direct product factor, Sylow
287	C₂⁴	—	2	1	no	—
288	C₂⁴	—	2	1	no	—
289	C₂⁴	—	2	1	no	—
290	#41	—	2	2	no	—
291	#41	—	4	1	no	—
292	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	1	no	—
297	C₂×D₄	—	2	1	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	—	2	1	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	4	1	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	—	4	4	no	—
309	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
310	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
311	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
312	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
313	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
314	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
319	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
320	SD₁₆	—	2	2	no	—
321	SD₁₆	—	2	2	no	—
322	SD₁₆	—	4	1	no	—
323	D₈	—	2	2	no	—
324	D₈	—	2	2	no	—
325	D₈	—	2	2	no	—
326	D₈	—	2	2	no	—
327	D₈	—	4	1	no	—
328	D₈	D₄×C₂₃	1	4	yes	complemented
329	M₁₆	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
332	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
333	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
334	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
337	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
338	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	2	1	no	—
341	C₂²⋊C₄	—	2	1	no	—
342	C₂²⋊C₄	—	2	1	no	—
343	C₂²⋊C₄	—	2	1	no	—
344	C₂²⋊C₄	—	2	1	no	—
345	C₂²⋊C₄	—	2	1	no	—
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
348	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
349	C₄²	C₂³×C₂₃	1	1	yes	characteristic
350	C₂³	—	2	1	no	—
351	C₂³	—	2	1	no	—
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	2	1	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	4	2	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	8	1	no	—
362	C₂³	—	8	1	no	—
363	C₂³	—	8	1	no	—
364	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
366	Q₈	—	2	1	no	—
367	Q₈	—	2	1	no	—
368	D₄	—	2	2	no	—
369	D₄	—	2	1	no	—
370	D₄	—	2	1	no	—
371	D₄	—	2	1	no	—
372	D₄	—	2	1	no	—
373	D₄	—	2	1	no	—
374	D₄	—	2	1	no	—
375	D₄	—	2	1	no	—
376	D₄	—	2	1	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	4	2	no	—
380	D₄	—	4	2	no	—
381	D₄	—	4	2	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	2	no	—
384	D₄	—	8	1	no	—
385	D₄	—	8	1	no	—
386	D₄	C₂×D₄×C₂₃	1	2	yes	—
387	D₄	C₂×D₄×C₂₃	1	2	yes	complemented
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	8	1	no	—
398	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
400	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
401	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
402	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
403	C₈	—	2	1	no	—
404	C₈	—	2	1	no	—
405	C₈	—	2	1	no	—
406	C₈	C₂×D₄×C₂₃	1	2	yes	complemented
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	2	1	no	—
411	C₂²	—	2	1	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	2	2	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	4	1	no	—
416	C₂²	—	4	1	no	—
417	C₂²	—	4	1	no	—
418	C₂²	—	4	1	no	—
419	C₂²	—	4	1	no	—
420	C₂²	—	4	1	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	1	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	4	1	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	8	4	no	—
428	C₂²	—	8	4	no	—
429	C₂²	—	8	4	no	—
430	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	complemented
431	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
432	C₄	—	2	1	no	—
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	2	1	no	—
436	C₄	—	4	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	1	no	—
439	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
440	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
441	C₂	—	2	2	no	—
442	C₂	—	4	1	no	—
443	C₂	—	4	2	no	—
444	C₂	—	4	2	no	—
445	C₂	—	8	1	no	—
446	C₂	—	8	1	no	—
447	C₂	—	8	1	no	—
448	C₂	D₄²×C₂₃	1	1	yes	characteristic
449	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
450	C₂	C₂₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₂²×D₈×C₂₃	C₂	semidirect
3. C₂₃×#546	C₂	semidirect
4. D₄²×C₂₃	C₂	semidirect
5. C₂₃×#496	C₂	semidirect
6. C₂₃×#469	C₂	semidirect
7. C₂₃×#465	C₂	semidirect
8. C₂₃×#463	C₂	semidirect
9. C₂₃×#459	C₂	semidirect
10. C₂₃×#450	C₂	semidirect
11. C₂₃×#449	C₂	semidirect
12. C₂₃×#447	C₂	semidirect
13. C₂₃×#447	C₂	semidirect
14. C₂₃×#442	C₂	semidirect
15. C₂₃×#435	C₂	semidirect
16. C₂×#41×C₂₃	C₂²	semidirect
19. C₂²×D₄×C₂₃	C₂²	semidirect
20. C₂²×D₄×C₂₃	C₂²	semidirect
22. C₂×SD₁₆×C₂₃	C₂²	semidirect
23. C₂×SD₁₆×C₂₃	C₂²	semidirect
24. C₂×SD₁₆×C₂₃	C₂²	semidirect
28. C₂×D₈×C₂₃	C₂²	semidirect
29. C₂×D₈×C₂₃	C₂²	semidirect
30. C₂×D₈×C₂₃	C₂²	semidirect
31. C₂×D₈×C₂₃	C₂²	semidirect
32. C₂×M₁₆×C₂₃	C₂²	semidirect
33. C₂²×C₈×C₂₃	C₂²	semidirect
34. C₂₃×#127	C₂²	semidirect
35. C₂₃×#124	C₂²	semidirect
37. C₂₃×C₂²⋊Q₈	C₂²	semidirect
41. C₂₃×#121	C₂²	semidirect
42. C₂₃×#121	C₂²	semidirect
47. C₄×D₄×C₂₃	C₂²	semidirect
48. C₄×D₄×C₂₃	C₂²	semidirect
50. C₂₃×#106	C₂²	semidirect
51. C₂₃×C₄⋊C₈	C₂²	semidirect
52. C₂₃×#103	C₂²	semidirect
53. C₂₃×#102	C₂²	semidirect
54. C₂₃×#102	C₂²	semidirect
55. C₂₃×#102	C₂²	semidirect
56. C₂₃×#102	C₂²	semidirect
57. C₂₃×#102	C₂²	semidirect
58. C₂₃×C₂²⋊C₈	C₂²	semidirect
59. C₂₃×C₂²⋊C₈	C₂²	semidirect
60. C₂₃×#97	C₂²	semidirect
102. D₈×C₂₃	D₄	semidirect
161. D₄×C₂₃	C₂×D₄	semidirect
180. C₈×C₂₃	C₂×D₄	semidirect
181. #3264	C₂₃	direct (N × H)
205. C₂²×C₂₃	AGL(1,ℤ/8)	semidirect
216. C₂×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
217. C₂²×D₈	C₂×C₂₃	semidirect
218. #546	C₂×C₂₃	semidirect
219. D₄²	C₂×C₂₃	semidirect
220. #496	C₂×C₂₃	semidirect
221. #469	C₂×C₂₃	semidirect
222. #465	C₂×C₂₃	semidirect
223. #463	C₂×C₂₃	semidirect
224. #459	C₂×C₂₃	semidirect
225. #450	C₂×C₂₃	semidirect
226. #449	C₂×C₂₃	semidirect
227. #447	C₂×C₂₃	semidirect
228. #447	C₂×C₂₃	semidirect
229. #442	C₂×C₂₃	semidirect
230. #435	C₂×C₂₃	semidirect
241. C₂×#41	C₂²×C₂₃	semidirect
244. C₂²×D₄	C₂²×C₂₃	semidirect
245. C₂²×D₄	C₂²×C₂₃	semidirect
247. C₂×SD₁₆	C₂²×C₂₃	semidirect
248. C₂×SD₁₆	C₂²×C₂₃	semidirect
249. C₂×SD₁₆	C₂²×C₂₃	semidirect
253. C₂×D₈	C₂²×C₂₃	semidirect
254. C₂×D₈	C₂²×C₂₃	semidirect
255. C₂×D₈	C₂²×C₂₃	semidirect
256. C₂×D₈	C₂²×C₂₃	semidirect
257. C₂×M₁₆	C₂²×C₂₃	semidirect
258. C₂²×C₈	C₂²×C₂₃	semidirect
259. #127	C₂²×C₂₃	semidirect
260. #124	C₂²×C₂₃	semidirect
262. C₂²⋊Q₈	C₂²×C₂₃	semidirect
266. #121	C₂²×C₂₃	semidirect
267. #121	C₂²×C₂₃	semidirect
272. C₄×D₄	C₂²×C₂₃	semidirect
273. C₄×D₄	C₂²×C₂₃	semidirect
275. #106	C₂²×C₂₃	semidirect
276. C₄⋊C₈	C₂²×C₂₃	semidirect
277. #103	C₂²×C₂₃	semidirect
278. #102	C₂²×C₂₃	semidirect
279. #102	C₂²×C₂₃	semidirect
280. #102	C₂²×C₂₃	semidirect
281. #102	C₂²×C₂₃	semidirect
282. #102	C₂²×C₂₃	semidirect
283. C₂²⋊C₈	C₂²×C₂₃	semidirect
284. C₂²⋊C₈	C₂²×C₂₃	semidirect
285. #97	C₂²×C₂₃	semidirect
286. C₂₃	#3264	direct (N × H)
328. D₈	D₄×C₂₃	semidirect
387. D₄	C₂×D₄×C₂₃	semidirect
406. C₈	C₂×D₄×C₂₃	semidirect
430. C₂²	C₂₃×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×AGL(1,ℤ/8)	2
C₂²×D₈×C₂₃	2
C₂₃×#546	2
D₄²×C₂₃	2
C₂₃×#496	2
C₂₃×#469	2
C₂₃×#465	2
C₂₃×#463	2
C₂₃×#459	2
C₂₃×#450	2
C₂₃×#449	2
C₂₃×#447	2
C₂₃×#442	2
C₂₃×#435	2
#3264	23

Subgroups — C23×#3264 #15670883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3264 #15670883