Subgroups — C₂₃×#22666 #19170292

Order 5888.

978 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#428	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2352	C₂	1	2	yes	maximal
4	C₂₃×#1949	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1854	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1853	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1809	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#117	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#436	C₄	1	8	yes	complemented
15	C₂₃×#428	—	2	8	no	—
16	C₂×C₂₃×#107	C₂²	1	2	yes	—
17	C₂×C₂₃×#107	C₂²	1	2	yes	—
18	C₂×C₂₃×#106	C₂²	1	2	yes	—
19	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#388	—	2	2	no	—
27	C₂₃×#387	—	2	2	no	—
28	C₂₃×#386	—	2	2	no	—
29	C₂₃×#382	C₂²	1	2	yes	—
30	C₂₃×#380	C₂²	1	2	yes	—
31	C₂₃×#380	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
33	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
34	C₂₃×#340	C₂²	1	2	yes	—
35	C₂₃×#340	C₂²	1	2	yes	—
36	C₂₃×#340	C₂²	1	2	yes	—
37	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₂₃	—	2	2	no	—
39	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×M₁₆×C₂₃	—	2	4	no	—
41	C₂×M₁₆×C₂₃	—	2	4	no	—
42	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
47	C₄×D₄×C₂₃	Q₈	1	4	yes	—
48	C₂₃×#117	—	2	4	no	—
49	C₂₃×#117	—	4	2	no	—
50	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
53	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
62	C₂×C₄²×C₂₃	—	2	2	no	—
63	C₂×C₄²×C₂₃	—	2	2	no	—
64	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₂₃×#107	—	2	4	no	—
66	C₂₃×#107	—	2	4	no	—
67	C₂₃×#106	—	2	2	no	—
68	C₂₃×#106	—	2	4	no	—
69	C₂₃×#106	—	4	1	no	—
70	C₂₃×#106	—	4	1	no	—
71	C₂₃×#106	D₄	1	4	yes	complemented
72	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
73	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
74	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
75	C₂₃×#95	—	2	2	no	—
76	C₂₃×#95	—	2	2	no	—
77	C₂₃×#95	—	2	2	no	—
78	C₂₃×#95	—	2	2	no	—
79	C₂₃×#95	—	2	2	no	—
80	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×D₄×C₂₃	—	4	2	no	—
82	C₂×D₄×C₂₃	C₄⋊C₄	1	4	yes	—
83	C₂²×C₄×C₂₃	—	2	2	no	—
84	C₂²×C₄×C₂₃	—	2	2	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	—	2	2	no	—
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	—	2	2	no	—
89	C₂²×C₄×C₂₃	—	2	2	no	—
90	C₂²×C₄×C₂₃	—	2	2	no	—
91	C₂²×C₄×C₂₃	—	4	4	no	—
92	C₂²×C₄×C₂₃	—	4	4	no	—
93	C₂²×C₄×C₂₃	—	8	2	no	—
94	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
99	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	M₁₆×C₂₃	—	2	16	no	—
101	C₂×C₈×C₂₃	—	2	2	no	—
102	C₂×C₈×C₂₃	—	2	2	no	—
103	C₂×C₈×C₂₃	—	2	1	no	—
104	C₂×C₈×C₂₃	—	2	1	no	—
105	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
106	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₂₃	#41	1	2	yes	—
110	C₄⋊C₄×C₂₃	—	2	2	no	—
111	C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₄⋊C₄×C₂₃	—	2	2	no	—
113	C₄⋊C₄×C₂₃	—	2	2	no	—
114	C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₄⋊C₄×C₂₃	—	4	4	no	—
118	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
119	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
120	C₂²⋊C₄×C₂₃	—	4	4	no	—
121	C₂²⋊C₄×C₂₃	—	4	4	no	—
122	C₂²⋊C₄×C₂₃	—	8	2	no	—
123	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
124	C₄²×C₂₃	—	4	4	no	—
125	C₄²×C₂₃	—	8	2	no	—
126	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
127	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
128	#22666	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₂₃	—	2	2	no	—
130	C₂³×C₂₃	—	2	2	no	—
131	C₂³×C₂₃	—	2	2	no	—
132	C₂³×C₂₃	—	2	2	no	—
133	C₂³×C₂₃	—	2	2	no	—
134	C₂³×C₂₃	—	2	2	no	—
135	C₂³×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
136	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
137	D₄×C₂₃	—	4	4	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	2	2	no	—
141	C₂×C₄×C₂₃	—	2	2	no	—
142	C₂×C₄×C₂₃	—	2	4	no	—
143	C₂×C₄×C₂₃	—	2	4	no	—
144	C₂×C₄×C₂₃	—	2	4	no	—
145	C₂×C₄×C₂₃	—	2	4	no	—
146	C₂×C₄×C₂₃	—	2	4	no	—
147	C₂×C₄×C₂₃	—	4	2	no	—
148	C₂×C₄×C₂₃	—	4	2	no	—
149	C₂×C₄×C₂₃	—	4	2	no	—
150	C₂×C₄×C₂₃	—	4	2	no	—
151	C₂×C₄×C₂₃	—	4	2	no	—
152	C₂×C₄×C₂₃	—	4	2	no	—
153	C₂×C₄×C₂₃	—	4	2	no	—
154	C₂×C₄×C₂₃	—	4	2	no	—
155	C₂×C₄×C₂₃	—	4	2	no	—
156	C₂×C₄×C₂₃	—	4	2	no	—
157	C₂×C₄×C₂₃	—	4	2	no	—
158	C₂×C₄×C₂₃	—	8	8	no	—
159	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
162	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
168	C₈×C₂₃	—	2	4	no	—
169	C₈×C₂₃	—	2	2	no	—
170	C₈×C₂₃	C₄×D₄	1	4	yes	—
171	C₂×#436	C₂×C₂₃	1	1	yes	characteristic
172	C₂×#428	C₂×C₂₃	1	2	yes	complemented
173	#2352	C₂×C₂₃	1	2	yes	—
174	#1949	C₂×C₂₃	1	2	yes	complemented
175	#1854	C₂×C₂₃	1	2	yes	complemented
176	#1853	C₂×C₂₃	1	2	yes	complemented
177	#1846	C₂×C₂₃	1	1	yes	characteristic, complemented
178	#1809	C₂×C₂₃	1	2	yes	complemented
179	C₄×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
180	C₂²×C₂₃	—	2	4	no	—
181	C₂²×C₂₃	—	2	4	no	—
182	C₂²×C₂₃	—	2	4	no	—
183	C₂²×C₂₃	—	4	2	no	—
184	C₂²×C₂₃	—	4	2	no	—
185	C₂²×C₂₃	—	4	2	no	—
186	C₂²×C₂₃	—	4	2	no	—
187	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×C₂₃	D₄²	1	1	yes	characteristic
189	C₂²×C₂₃	#547	1	1	yes	characteristic
190	C₂²×C₂₃	#575	1	2	yes	—
191	C₂²×C₂₃	#577	1	2	yes	—
192	C₄×C₂₃	—	2	4	no	—
193	C₄×C₂₃	—	2	2	no	—
194	C₄×C₂₃	—	2	2	no	—
195	C₄×C₂₃	—	4	4	no	—
196	C₄×C₂₃	—	8	2	no	—
197	C₄×C₂₃	—	8	2	no	—
198	C₄×C₂₃	—	8	2	no	—
199	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₄×C₂₃	#520	1	1	yes	characteristic
201	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
202	C₄×C₂₃	#554	1	1	yes	characteristic
203	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
204	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
205	C₂×#117	C₂²×C₂₃	1	2	yes	—
206	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
207	#436	C₄×C₂₃	1	8	yes	complemented
208	#428	—	2	8	no	—
209	C₂×#107	C₂²×C₂₃	1	2	yes	—
210	C₂×#107	C₂²×C₂₃	1	2	yes	—
211	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
212	C₂×#106	C₂²×C₂₃	1	2	yes	—
213	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
214	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
215	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
217	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
218	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
219	#388	—	2	2	no	—
220	#387	—	2	2	no	—
221	#386	—	2	2	no	—
222	#382	C₂²×C₂₃	1	2	yes	—
223	#380	C₂²×C₂₃	1	2	yes	—
224	#380	C₂²×C₂₃	1	2	yes	—
225	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
226	C₄×C₂²⋊C₄	—	2	2	no	—
227	#340	C₂²×C₂₃	1	2	yes	—
228	#340	C₂²×C₂₃	1	2	yes	—
229	#340	C₂²×C₂₃	1	2	yes	—
230	C₂×C₂₃	—	4	4	no	—
231	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
232	C₂×C₂₃	#2911	1	1	yes	characteristic
233	C₂×C₂₃	#2912	1	1	yes	characteristic
234	C₂×C₂₃	#3275	1	2	yes	—
235	C₂×C₂₃	#3351	1	2	yes	—
236	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
237	C₂³×C₄	—	2	2	no	—
238	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
239	C₂×M₁₆	—	2	4	no	—
240	C₂×M₁₆	—	2	4	no	—
241	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
242	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
243	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
244	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
245	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
246	C₄×D₄	Q₈×C₂₃	1	4	yes	—
247	#117	—	2	4	no	—
248	#117	—	4	2	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	4	2	no	—
252	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
254	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
257	C₂×C₂²⋊C₄	—	2	2	no	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	—	2	2	no	—
263	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
264	#107	—	2	4	no	—
265	#107	—	2	4	no	—
266	#106	—	2	2	no	—
267	#106	—	2	4	no	—
268	#106	—	4	1	no	—
269	#106	—	4	1	no	—
270	#106	D₄×C₂₃	1	4	yes	complemented
271	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
272	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
273	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
274	#95	—	2	2	no	—
275	#95	—	2	2	no	—
276	#95	—	2	2	no	—
277	#95	—	2	2	no	—
278	#95	—	2	2	no	—
279	C₂₃	#22666	1	1	yes	characteristic, direct product factor, Sylow
280	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	C₄⋊C₄×C₂₃	1	4	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	4	4	no	—
292	C₂²×C₄	—	4	4	no	—
293	C₂²×C₄	—	8	2	no	—
294	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
295	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
298	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
299	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
300	M₁₆	—	2	16	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
306	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
307	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
308	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
309	C₂×C₈	#41×C₂₃	1	2	yes	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	4	4	no	—
318	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
319	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
320	C₂²⋊C₄	—	4	4	no	—
321	C₂²⋊C₄	—	4	4	no	—
322	C₂²⋊C₄	—	8	2	no	—
323	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
324	C₄²	—	4	4	no	—
325	C₄²	—	8	2	no	—
326	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
327	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
328	C₂³	—	2	2	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	2	2	no	—
332	C₂³	—	2	2	no	—
333	C₂³	—	2	2	no	—
334	C₂³	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
335	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
336	D₄	—	4	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	8	no	—
358	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
359	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
360	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
361	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
362	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
363	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
364	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
365	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
366	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
367	C₈	—	2	4	no	—
368	C₈	—	2	2	no	—
369	C₈	C₄×D₄×C₂₃	1	4	yes	—
370	C₂²	—	2	4	no	—
371	C₂²	—	2	4	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	2	no	—
377	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
378	C₂²	D₄²×C₂₃	1	1	yes	characteristic
379	C₂²	C₂₃×#547	1	1	yes	characteristic
380	C₂²	C₂₃×#575	1	2	yes	—
381	C₂²	C₂₃×#577	1	2	yes	—
382	C₄	—	2	2	no	—
383	C₄	—	2	2	no	—
384	C₄	—	2	4	no	—
385	C₄	—	4	4	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	8	2	no	—
389	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
390	C₄	C₂₃×#520	1	1	yes	characteristic
391	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
392	C₄	C₂₃×#554	1	1	yes	characteristic
393	C₂	—	4	4	no	—
394	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
395	C₂	C₂₃×#2911	1	1	yes	characteristic
396	C₂	C₂₃×#2912	1	1	yes	characteristic
397	C₂	C₂₃×#3275	1	2	yes	—
398	C₂	C₂₃×#3351	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
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
2. C₂×C₂₃×#428	C₂	semidirect
4. C₂₃×#1949	C₂	semidirect
5. C₂₃×#1854	C₂	semidirect
6. C₂₃×#1853	C₂	semidirect
7. C₂₃×#1846	C₂	semidirect
8. C₂₃×#1809	C₂	semidirect
9. C₄×C₂₃×#106	C₂	semidirect
14. C₂₃×#436	C₄	semidirect
42. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
71. C₂₃×#106	D₄	semidirect
72. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
73. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
74. C₄×C₈×C₂₃	C₂×C₄	semidirect
128. #22666	C₂₃	direct (N × H)
172. C₂×#428	C₂×C₂₃	semidirect
174. #1949	C₂×C₂₃	semidirect
175. #1854	C₂×C₂₃	semidirect
176. #1853	C₂×C₂₃	semidirect
177. #1846	C₂×C₂₃	semidirect
178. #1809	C₂×C₂₃	semidirect
179. C₄×#106	C₂×C₂₃	semidirect
207. #436	C₄×C₂₃	semidirect
241. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
270. #106	D₄×C₂₃	semidirect
271. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
272. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
273. C₄×C₈	C₂×C₄×C₂₃	semidirect
279. C₂₃	#22666	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×#436	2
C₂×C₂₃×#428	2
C₂₃×#2352	2
C₂₃×#1949	2
C₂₃×#1854	2
C₂₃×#1853	2
C₂₃×#1846	2
C₂₃×#1809	2
C₄×C₂₃×#106	2
#22666	23

Subgroups — C23×#22666 #19170292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22666 #19170292