Subgroups — C₂₃×#3275 #15670894

Order 2944.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#548	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#469	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#464	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#463	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#452	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#449	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#448	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#137	—	2	4	no	—
21	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
22	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#124	—	2	1	no	—
34	C₂₃×#124	C₂²	1	1	yes	characteristic
35	C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#123	—	2	1	no	—
37	C₂₃×C₂²⋊Q₈	—	2	1	no	—
38	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
39	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
40	C₂₃×#121	—	2	1	no	—
41	C₂₃×#121	—	2	1	no	—
42	C₂₃×#121	C₂²	1	1	yes	characteristic
43	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
44	C₂₃×#120	—	2	1	no	—
45	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₂₃	—	2	1	no	—
47	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
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×#103	C₂²	1	1	yes	characteristic, complemented
54	C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
55	C₂₃×#102	C₂²	1	1	yes	characteristic
56	C₂₃×#102	C₂²	1	1	yes	characteristic
57	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
58	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
59	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
60	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
61	C₂⁴×C₂₃	—	2	1	no	—
62	#41×C₂₃	—	2	2	no	—
63	#41×C₂₃	—	2	2	no	—
64	#41×C₂₃	—	4	1	no	—
65	#41×C₂₃	—	4	1	no	—
66	C₂×Q₈×C₂₃	—	2	1	no	—
67	C₂×Q₈×C₂₃	—	2	2	no	—
68	C₂×Q₈×C₂₃	—	4	1	no	—
69	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₂₃	—	2	2	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	—	4	1	no	—
79	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
80	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	—	2	1	no	—
84	C₂²×C₄×C₂₃	—	2	1	no	—
85	C₂²×C₄×C₂₃	—	2	1	no	—
86	C₂²×C₄×C₂₃	—	2	1	no	—
87	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
89	Q₁₆×C₂₃	—	2	2	no	—
90	Q₁₆×C₂₃	—	2	2	no	—
91	SD₁₆×C₂₃	—	2	2	no	—
92	SD₁₆×C₂₃	—	2	2	no	—
93	SD₁₆×C₂₃	—	4	1	no	—
94	SD₁₆×C₂₃	—	4	1	no	—
95	SD₁₆×C₂₃	D₄	1	4	yes	complemented
96	D₈×C₂₃	—	2	2	no	—
97	D₈×C₂₃	—	2	2	no	—
98	M₁₆×C₂₃	—	2	2	no	—
99	M₁₆×C₂₃	—	2	2	no	—
100	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₄⋊C₄×C₂₃	—	2	1	no	—
107	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
108	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂²⋊C₄×C₂₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
120	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
121	C₄²×C₂₃	—	2	1	no	—
122	C₄²×C₂₃	C₂³	1	1	yes	characteristic
123	C₂³×C₂₃	—	2	1	no	—
124	C₂³×C₂₃	—	2	1	no	—
125	C₂³×C₂₃	—	2	1	no	—
126	C₂³×C₂₃	—	4	2	no	—
127	C₂³×C₂₃	—	4	2	no	—
128	C₂³×C₂₃	—	8	1	no	—
129	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
131	Q₈×C₂₃	—	2	1	no	—
132	Q₈×C₂₃	—	2	1	no	—
133	Q₈×C₂₃	—	2	1	no	—
134	Q₈×C₂₃	—	2	1	no	—
135	Q₈×C₂₃	—	2	1	no	—
136	Q₈×C₂₃	—	4	2	no	—
137	Q₈×C₂₃	C₂×D₄	1	2	yes	—
138	D₄×C₂₃	—	2	1	no	—
139	D₄×C₂₃	—	2	1	no	—
140	D₄×C₂₃	—	2	1	no	—
141	D₄×C₂₃	—	2	1	no	—
142	D₄×C₂₃	—	2	1	no	—
143	D₄×C₂₃	—	4	1	no	—
144	D₄×C₂₃	—	4	2	no	—
145	D₄×C₂₃	—	4	2	no	—
146	D₄×C₂₃	—	4	2	no	—
147	D₄×C₂₃	—	4	2	no	—
148	D₄×C₂₃	—	8	1	no	—
149	D₄×C₂₃	C₂×D₄	1	2	yes	—
150	C₂×C₄×C₂₃	—	2	1	no	—
151	C₂×C₄×C₂₃	—	2	1	no	—
152	C₂×C₄×C₂₃	—	2	1	no	—
153	C₂×C₄×C₂₃	—	2	1	no	—
154	C₂×C₄×C₂₃	—	2	1	no	—
155	C₂×C₄×C₂₃	—	2	1	no	—
156	C₂×C₄×C₂₃	—	2	1	no	—
157	C₂×C₄×C₂₃	—	2	1	no	—
158	C₂×C₄×C₂₃	—	2	1	no	—
159	C₂×C₄×C₂₃	—	4	2	no	—
160	C₂×C₄×C₂₃	—	4	2	no	—
161	C₂×C₄×C₂₃	—	4	2	no	—
162	C₂×C₄×C₂₃	—	8	1	no	—
163	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
168	C₈×C₂₃	—	2	1	no	—
169	C₈×C₂₃	—	2	1	no	—
170	C₈×C₂₃	—	2	1	no	—
171	C₈×C₂₃	C₂×D₄	1	2	yes	—
172	#3275	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂²×C₂₃	—	2	1	no	—
174	C₂²×C₂₃	—	2	1	no	—
175	C₂²×C₂₃	—	2	2	no	—
176	C₂²×C₂₃	—	2	1	no	—
177	C₂²×C₂₃	—	2	1	no	—
178	C₂²×C₂₃	—	2	1	no	—
179	C₂²×C₂₃	—	2	1	no	—
180	C₂²×C₂₃	—	4	1	no	—
181	C₂²×C₂₃	—	4	1	no	—
182	C₂²×C₂₃	—	4	1	no	—
183	C₂²×C₂₃	—	4	1	no	—
184	C₂²×C₂₃	—	4	1	no	—
185	C₂²×C₂₃	—	4	1	no	—
186	C₂²×C₂₃	—	4	1	no	—
187	C₂²×C₂₃	—	4	1	no	—
188	C₂²×C₂₃	—	8	4	no	—
189	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	C₄×C₂₃	—	2	1	no	—
191	C₄×C₂₃	—	2	1	no	—
192	C₄×C₂₃	—	2	2	no	—
193	C₄×C₂₃	—	2	1	no	—
194	C₄×C₂₃	—	2	1	no	—
195	C₄×C₂₃	—	4	1	no	—
196	C₄×C₂₃	—	4	1	no	—
197	C₄×C₂₃	—	4	1	no	—
198	C₄×C₂₃	—	4	1	no	—
199	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
200	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×#137	C₂×C₂₃	1	1	yes	characteristic, complemented
202	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic, complemented
203	#548	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#546	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#495	C₂×C₂₃	1	1	yes	characteristic, complemented
206	#470	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#469	C₂×C₂₃	1	1	yes	characteristic, complemented
208	#464	C₂×C₂₃	1	1	yes	characteristic, complemented
209	#463	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#452	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#450	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#449	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#448	C₂×C₂₃	1	1	yes	characteristic, complemented
214	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
215	#436	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂×C₂₃	—	4	1	no	—
217	C₂×C₂₃	—	4	1	no	—
218	C₂×C₂₃	—	4	2	no	—
219	C₂×C₂₃	—	8	1	no	—
220	C₂×C₂₃	—	8	1	no	—
221	C₂×C₂₃	D₄²	1	1	yes	characteristic
222	C₂×C₂₃	#575	1	1	yes	characteristic
223	C₂×C₂₃	#577	1	1	yes	characteristic
224	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
225	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
226	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
227	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
228	#137	—	2	4	no	—
229	AGL(1,ℤ/8)	—	2	4	no	—
230	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
231	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
232	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
233	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
234	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
235	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
236	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
237	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
238	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
239	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
240	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
241	#124	—	2	1	no	—
242	#124	C₂²×C₂₃	1	1	yes	characteristic
243	#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
244	#123	—	2	1	no	—
245	C₂²⋊Q₈	—	2	1	no	—
246	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
247	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
248	#121	—	2	1	no	—
249	#121	—	2	1	no	—
250	#121	C₂²×C₂₃	1	1	yes	characteristic
251	#121	C₂²×C₂₃	1	1	yes	characteristic, complemented
252	#120	—	2	1	no	—
253	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
254	C₄×D₄	—	2	1	no	—
255	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
256	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×C₂²⋊C₄	—	2	1	no	—
258	#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
259	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
260	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
261	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
262	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
263	#102	C₂²×C₂₃	1	1	yes	characteristic
264	#102	C₂²×C₂₃	1	1	yes	characteristic
265	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
266	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
267	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
268	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
269	C₂₃	#3275	1	1	yes	characteristic, direct product factor, Sylow
270	C₂⁴	—	2	1	no	—
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	4	1	no	—
274	#41	—	4	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
279	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
280	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
289	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
290	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
291	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
297	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
298	Q₁₆	—	2	2	no	—
299	Q₁₆	—	2	2	no	—
300	SD₁₆	—	2	2	no	—
301	SD₁₆	—	2	2	no	—
302	SD₁₆	—	4	1	no	—
303	SD₁₆	—	4	1	no	—
304	SD₁₆	D₄×C₂₃	1	4	yes	complemented
305	D₈	—	2	2	no	—
306	D₈	—	2	2	no	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	2	2	no	—
309	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
310	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
311	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
312	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
317	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
318	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
319	C₂²⋊C₄	—	2	1	no	—
320	C₂²⋊C₄	—	2	1	no	—
321	C₂²⋊C₄	—	2	1	no	—
322	C₂²⋊C₄	—	2	1	no	—
323	C₂²⋊C₄	—	2	1	no	—
324	C₂²⋊C₄	—	2	1	no	—
325	C₂²⋊C₄	—	2	1	no	—
326	C₂²⋊C₄	—	2	1	no	—
327	C₂²⋊C₄	—	2	1	no	—
328	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
329	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
330	C₄²	—	2	1	no	—
331	C₄²	C₂³×C₂₃	1	1	yes	characteristic
332	C₂³	—	2	1	no	—
333	C₂³	—	2	1	no	—
334	C₂³	—	2	1	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	8	1	no	—
338	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
339	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
340	Q₈	—	2	1	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	2	1	no	—
344	Q₈	—	2	1	no	—
345	Q₈	—	4	2	no	—
346	Q₈	C₂×D₄×C₂₃	1	2	yes	—
347	D₄	—	2	1	no	—
348	D₄	—	2	1	no	—
349	D₄	—	2	1	no	—
350	D₄	—	2	1	no	—
351	D₄	—	2	1	no	—
352	D₄	—	4	1	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	2	no	—
356	D₄	—	4	2	no	—
357	D₄	—	8	1	no	—
358	D₄	C₂×D₄×C₂₃	1	2	yes	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	8	1	no	—
372	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
373	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
375	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
376	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
377	C₈	—	2	1	no	—
378	C₈	—	2	1	no	—
379	C₈	—	2	1	no	—
380	C₈	C₂×D₄×C₂₃	1	2	yes	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	8	4	no	—
397	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	2	2	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	1	no	—
407	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
408	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
409	C₂	—	4	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	4	2	no	—
412	C₂	—	8	1	no	—
413	C₂	—	8	1	no	—
414	C₂	D₄²×C₂₃	1	1	yes	characteristic
415	C₂	C₂₃×#575	1	1	yes	characteristic
416	C₂	C₂₃×#577	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×#137	C₂	semidirect
2. C₂×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂₃×#548	C₂	semidirect
4. C₂₃×#546	C₂	semidirect
5. C₂₃×#495	C₂	semidirect
6. C₂₃×#470	C₂	semidirect
7. C₂₃×#469	C₂	semidirect
8. C₂₃×#464	C₂	semidirect
9. C₂₃×#463	C₂	semidirect
10. C₂₃×#452	C₂	semidirect
11. C₂₃×#450	C₂	semidirect
12. C₂₃×#449	C₂	semidirect
13. C₂₃×#448	C₂	semidirect
14. C₄×SD₁₆×C₂₃	C₂	semidirect
15. C₂₃×#436	C₂	semidirect
18. C₂²×Q₈×C₂₃	C₂²	semidirect
22. C₂×Q₁₆×C₂₃	C₂²	semidirect
23. C₂×Q₁₆×C₂₃	C₂²	semidirect
28. C₂×SD₁₆×C₂₃	C₂²	semidirect
32. C₂×M₁₆×C₂₃	C₂²	semidirect
35. C₂₃×#124	C₂²	semidirect
43. C₂₃×#121	C₂²	semidirect
45. C₄×Q₈×C₂₃	C₂²	semidirect
50. C₂₃×#106	C₂²	semidirect
51. C₂₃×C₄⋊C₈	C₂²	semidirect
52. C₂₃×#103	C₂²	semidirect
53. C₂₃×#103	C₂²	semidirect
54. C₂₃×#103	C₂²	semidirect
57. C₂₃×#102	C₂²	semidirect
59. C₂₃×C₂²⋊C₈	C₂²	semidirect
60. C₄×C₈×C₂₃	C₂²	semidirect
95. SD₁₆×C₂₃	D₄	semidirect
172. #3275	C₂₃	direct (N × H)
201. C₂×#137	C₂×C₂₃	semidirect
202. C₂×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
203. #548	C₂×C₂₃	semidirect
204. #546	C₂×C₂₃	semidirect
205. #495	C₂×C₂₃	semidirect
206. #470	C₂×C₂₃	semidirect
207. #469	C₂×C₂₃	semidirect
208. #464	C₂×C₂₃	semidirect
209. #463	C₂×C₂₃	semidirect
210. #452	C₂×C₂₃	semidirect
211. #450	C₂×C₂₃	semidirect
212. #449	C₂×C₂₃	semidirect
213. #448	C₂×C₂₃	semidirect
214. C₄×SD₁₆	C₂×C₂₃	semidirect
215. #436	C₂×C₂₃	semidirect
226. C₂²×Q₈	C₂²×C₂₃	semidirect
230. C₂×Q₁₆	C₂²×C₂₃	semidirect
231. C₂×Q₁₆	C₂²×C₂₃	semidirect
236. C₂×SD₁₆	C₂²×C₂₃	semidirect
240. C₂×M₁₆	C₂²×C₂₃	semidirect
243. #124	C₂²×C₂₃	semidirect
251. #121	C₂²×C₂₃	semidirect
253. C₄×Q₈	C₂²×C₂₃	semidirect
258. #106	C₂²×C₂₃	semidirect
259. C₄⋊C₈	C₂²×C₂₃	semidirect
260. #103	C₂²×C₂₃	semidirect
261. #103	C₂²×C₂₃	semidirect
262. #103	C₂²×C₂₃	semidirect
265. #102	C₂²×C₂₃	semidirect
267. C₂²⋊C₈	C₂²×C₂₃	semidirect
268. C₄×C₈	C₂²×C₂₃	semidirect
269. C₂₃	#3275	direct (N × H)
304. SD₁₆	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#137	2
C₂×C₂₃×AGL(1,ℤ/8)	2
C₂₃×#548	2
C₂₃×#546	2
C₂₃×#495	2
C₂₃×#470	2
C₂₃×#469	2
C₂₃×#464	2
C₂₃×#463	2
C₂₃×#452	2
C₂₃×#450	2
C₂₃×#449	2
C₂₃×#448	2
C₄×SD₁₆×C₂₃	2
C₂₃×#436	2
#3275	23

Subgroups — C23×#3275 #15670894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3275 #15670894