Subgroups — C₂₃×#60622 #19208248

Order 5888.

2178 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3498	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3141	C₂	1	3	yes	maximal, complemented
4	C₂₃×#3138	C₂	1	3	yes	maximal, complemented
5	C₂₃×#3136	C₂	1	12	yes	maximal, complemented
6	C₂₃×#3131	C₂	1	3	yes	maximal, complemented
7	C₂×C₂₃×#486	C₂	1	3	yes	maximal, complemented
8	C₂₃×#2897	C₂	1	4	yes	maximal, complemented
9	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	2	yes	—
11	C₂×C₂₃×#131	C₂²	1	2	yes	—
12	Q₈²×C₂₃	—	2	1	no	—
13	C₂₃×#557	—	2	1	no	—
14	C₂₃×#554	—	2	3	no	—
15	C₂₃×#550	—	2	1	no	—
16	C₂₃×#550	—	2	1	no	—
17	D₄×Q₈×C₂₃	—	2	3	no	—
18	D₄²×C₂₃	—	2	3	no	—
19	D₄²×C₂₃	—	2	3	no	—
20	C₂₃×#544	C₂²	1	6	yes	complemented
21	C₂₃×#535	C₂²	1	6	yes	—
22	C₂₃×#533	C₂²	1	3	yes	complemented
23	C₂₃×#532	C₂²	1	3	yes	—
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
25	C₂×C₂₃×#127	C₂²	1	3	yes	—
26	C₂₃×#520	C₂²	1	2	yes	—
27	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#490	C₂²	1	12	yes	complemented
29	C₂₃×#489	C₂²	1	12	yes	complemented
30	C₂₃×#488	C₂²	1	12	yes	complemented
31	C₂₃×#487	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#487	C₂²	1	3	yes	complemented
33	C₂₃×#486	C₂²	1	3	yes	complemented
34	C₂₃×#486	C₂²	1	12	yes	complemented
35	C₂₃×#486	C₂²	1	6	yes	complemented
36	C₂₃×#486	C₂²	1	3	yes	complemented
37	C₂₃×#418	C₂²	1	12	yes	complemented
38	C₂₃×#417	C₂²	1	12	yes	complemented
39	C₂₃×#416	C₂²	1	12	yes	complemented
40	C₂×C₂₃×#102	C₂²	1	12	yes	complemented
41	C₂₃×#405	C₂²	1	6	yes	complemented
42	C₄×M₁₆×C₂₃	C₂²	1	3	yes	complemented
43	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	complemented
44	C₂₃×H(5,𝔽₂)	—	2	16	no	—
45	C₂×#41×C₂₃	—	2	2	no	—
46	C₂×#41×C₂₃	C₂³	1	6	yes	—
47	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₂₃	—	2	3	no	—
49	C₂²×D₄×C₂₃	—	2	6	no	—
50	C₂²×D₄×C₂₃	—	2	3	no	—
51	C₂²×D₄×C₂₃	—	2	3	no	—
52	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
53	C₂₃×#131	—	2	8	no	—
54	C₂×M₁₆×C₂₃	C₂³	1	6	yes	—
55	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
56	C₂₃×C₄⋊Q₈	—	2	6	no	—
57	C₂₃×C₄⋊Q₈	—	2	3	no	—
58	C₂₃×C₄⋊Q₈	—	2	3	no	—
59	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
61	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
63	C₂₃×#127	—	2	3	no	—
64	C₂₃×#127	—	2	3	no	—
65	C₂₃×#127	C₂³	1	3	yes	—
66	C₂₃×#127	C₂³	1	6	yes	—
67	C₂₃×#125	—	2	6	no	—
68	C₂₃×#125	C₂³	1	6	yes	—
69	C₂₃×#124	C₂³	1	6	yes	—
70	C₂₃×C₂²⋊Q₈	—	2	3	no	—
71	C₂₃×C₂²⋊Q₈	—	2	3	no	—
72	C₂₃×C₂²⋊Q₈	—	2	12	no	—
73	C₂₃×#121	—	2	3	no	—
74	C₂₃×#121	—	2	3	no	—
75	C₂₃×#121	—	2	6	no	—
76	C₂₃×#121	—	2	6	no	—
77	C₂₃×#120	—	2	12	no	—
78	C₄×Q₈×C₂₃	—	2	2	no	—
79	C₄×Q₈×C₂₃	—	2	3	no	—
80	C₄×Q₈×C₂₃	—	2	3	no	—
81	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
83	C₄×D₄×C₂₃	—	2	6	no	—
84	C₄×D₄×C₂₃	—	2	3	no	—
85	C₄×D₄×C₂₃	—	2	3	no	—
86	C₄×D₄×C₂₃	D₄	1	3	yes	—
87	C₄×D₄×C₂₃	D₄	1	3	yes	—
88	C₂₃×#117	C₂³	1	6	yes	—
89	C₂₃×#117	C₂³	1	3	yes	—
90	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
92	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
93	C₂₃×#103	C₂³	1	4	yes	—
94	C₂₃×#103	C₂³	1	12	yes	—
95	C₂₃×#102	C₂³	1	12	yes	—
96	C₂₃×#102	C₂³	1	24	yes	—
97	C₂₃×#102	C₂³	1	12	yes	—
98	C₂₃×#97	C₂³	1	6	yes	—
99	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₂₃	C₂³	1	3	yes	—
101	C₄×C₈×C₂₃	C₂³	1	6	yes	—
102	C₂⁴×C₂₃	—	2	6	no	—
103	#41×C₂₃	—	2	24	no	—
104	#41×C₂₃	—	4	8	no	—
105	#41×C₂₃	#41	1	8	yes	—
106	C₂×Q₈×C₂₃	—	2	1	no	—
107	C₂×Q₈×C₂₃	—	2	3	no	—
108	C₂×Q₈×C₂₃	—	2	6	no	—
109	C₂×Q₈×C₂₃	—	8	6	no	—
110	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
112	C₂×D₄×C₂₃	—	2	6	no	—
113	C₂×D₄×C₂₃	—	2	3	no	—
114	C₂×D₄×C₂₃	—	2	3	no	—
115	C₂×D₄×C₂₃	—	2	6	no	—
116	C₂×D₄×C₂₃	—	2	6	no	—
117	C₂×D₄×C₂₃	—	2	6	no	—
118	C₂×D₄×C₂₃	—	2	24	no	—
119	C₂×D₄×C₂₃	—	4	24	no	—
120	C₂×D₄×C₂₃	—	8	6	no	—
121	C₂×D₄×C₂₃	—	8	6	no	—
122	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
123	C₂×D₄×C₂₃	C₂⁴	1	6	yes	—
124	C₂²×C₄×C₂₃	—	2	6	no	—
125	C₂²×C₄×C₂₃	—	2	6	no	—
126	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
127	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
128	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
129	M₁₆×C₂₃	—	2	12	no	—
130	C₂×C₈×C₂₃	—	2	12	no	—
131	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
132	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
133	C₄⋊C₄×C₂₃	—	2	3	no	—
134	C₄⋊C₄×C₂₃	—	2	6	no	—
135	C₄⋊C₄×C₂₃	—	2	6	no	—
136	C₄⋊C₄×C₂₃	—	2	1	no	—
137	C₄⋊C₄×C₂₃	—	2	1	no	—
138	C₄⋊C₄×C₂₃	—	2	1	no	—
139	C₄⋊C₄×C₂₃	—	2	6	no	—
140	C₄⋊C₄×C₂₃	—	2	6	no	—
141	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
142	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
143	C₄⋊C₄×C₂₃	C₂⁴	1	6	yes	—
144	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
145	C₂²⋊C₄×C₂₃	—	2	6	no	—
146	C₂²⋊C₄×C₂₃	—	2	6	no	—
147	C₂²⋊C₄×C₂₃	—	2	6	no	—
148	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
149	C₄²×C₂₃	—	2	6	no	—
150	C₄²×C₂₃	C₂×D₄	1	3	yes	—
151	C₄²×C₂₃	C₂×D₄	1	3	yes	—
152	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
153	C₄²×C₂₃	C₂⁴	1	3	yes	—
154	#60622	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₂₃	—	2	6	no	—
156	C₂³×C₂₃	—	4	12	no	—
157	C₂³×C₂₃	—	8	6	no	—
158	C₂³×C₂₃	—	8	6	no	—
159	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
160	Q₈×C₂₃	—	2	4	no	—
161	Q₈×C₂₃	—	4	6	no	—
162	Q₈×C₂₃	—	4	2	no	—
163	Q₈×C₂₃	—	8	6	no	—
164	Q₈×C₂₃	#124	1	4	yes	—
165	D₄×C₂₃	—	2	12	no	—
166	D₄×C₂₃	—	4	12	no	—
167	D₄×C₂₃	—	4	12	no	—
168	D₄×C₂₃	—	4	6	no	—
169	D₄×C₂₃	—	4	6	no	—
170	D₄×C₂₃	—	8	6	no	—
171	D₄×C₂₃	—	8	6	no	—
172	D₄×C₂₃	#124	1	12	yes	—
173	C₂×C₄×C₂₃	—	2	6	no	—
174	C₂×C₄×C₂₃	—	2	6	no	—
175	C₂×C₄×C₂₃	—	2	2	no	—
176	C₂×C₄×C₂₃	—	2	6	no	—
177	C₂×C₄×C₂₃	—	2	2	no	—
178	C₂×C₄×C₂₃	—	4	12	no	—
179	C₂×C₄×C₂₃	—	8	6	no	—
180	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
181	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
182	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
183	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₂₃	C₂×#41	1	12	yes	—
186	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
187	C₈×C₂₃	—	2	2	no	—
188	C₈×C₂₃	—	2	6	no	—
189	#3503	C₂×C₂₃	1	1	yes	characteristic, complemented
190	#3498	C₂×C₂₃	1	1	yes	characteristic
191	#3141	C₂×C₂₃	1	3	yes	complemented
192	#3138	C₂×C₂₃	1	3	yes	complemented
193	#3136	C₂×C₂₃	1	12	yes	complemented
194	#3131	C₂×C₂₃	1	3	yes	complemented
195	C₂×#486	C₂×C₂₃	1	3	yes	complemented
196	#2897	C₂×C₂₃	1	4	yes	complemented
197	C₄×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
198	C₂²×C₂₃	—	2	3	no	—
199	C₂²×C₂₃	—	2	3	no	—
200	C₂²×C₂₃	—	4	6	no	—
201	C₂²×C₂₃	—	4	6	no	—
202	C₂²×C₂₃	—	4	6	no	—
203	C₂²×C₂₃	—	16	12	no	—
204	C₂²×C₂₃	C₂×#124	1	6	yes	—
205	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
206	C₄×C₂₃	—	2	3	no	—
207	C₄×C₂₃	—	2	3	no	—
208	C₄×C₂₃	—	4	2	no	—
209	C₄×C₂₃	—	4	6	no	—
210	C₄×C₂₃	—	4	2	no	—
211	C₄×C₂₃	C₂×#124	1	6	yes	—
212	C₄×C₂₃	#576	1	2	yes	—
213	C₄×C₂₃	#577	1	2	yes	—
214	C₄×C₂₃	C₂²×#41	1	2	yes	—
215	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	2	yes	—
216	C₂×#131	C₂²×C₂₃	1	2	yes	—
217	Q₈²	—	2	1	no	—
218	#557	—	2	1	no	—
219	#554	—	2	3	no	—
220	#550	—	2	1	no	—
221	#550	—	2	1	no	—
222	D₄×Q₈	—	2	3	no	—
223	D₄²	—	2	3	no	—
224	D₄²	—	2	3	no	—
225	#544	C₂²×C₂₃	1	6	yes	complemented
226	#535	C₂²×C₂₃	1	6	yes	—
227	#533	C₂²×C₂₃	1	3	yes	complemented
228	#532	C₂²×C₂₃	1	3	yes	—
229	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	complemented
230	C₂×#127	C₂²×C₂₃	1	3	yes	—
231	#520	C₂²×C₂₃	1	2	yes	—
232	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
233	#490	C₂²×C₂₃	1	12	yes	complemented
234	#489	C₂²×C₂₃	1	12	yes	complemented
235	#488	C₂²×C₂₃	1	12	yes	complemented
236	#487	C₂²×C₂₃	1	1	yes	characteristic, complemented
237	#487	C₂²×C₂₃	1	3	yes	complemented
238	#486	C₂²×C₂₃	1	3	yes	complemented
239	#486	C₂²×C₂₃	1	12	yes	complemented
240	#486	C₂²×C₂₃	1	6	yes	complemented
241	#486	C₂²×C₂₃	1	3	yes	complemented
242	#418	C₂²×C₂₃	1	12	yes	complemented
243	#417	C₂²×C₂₃	1	12	yes	complemented
244	#416	C₂²×C₂₃	1	12	yes	complemented
245	C₂×#102	C₂²×C₂₃	1	12	yes	complemented
246	#405	C₂²×C₂₃	1	6	yes	complemented
247	C₄×M₁₆	C₂²×C₂₃	1	3	yes	complemented
248	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	complemented
249	C₂×C₂₃	—	2	6	no	—
250	C₂×C₂₃	—	8	6	no	—
251	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
252	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
253	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
254	H(5,𝔽₂)	—	2	16	no	—
255	C₂×#41	—	2	2	no	—
256	C₂×#41	C₂³×C₂₃	1	6	yes	—
257	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
258	C₂²×Q₈	—	2	3	no	—
259	C₂²×D₄	—	2	6	no	—
260	C₂²×D₄	—	2	3	no	—
261	C₂²×D₄	—	2	3	no	—
262	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
263	#131	—	2	8	no	—
264	C₂×M₁₆	C₂³×C₂₃	1	6	yes	—
265	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
266	C₄⋊Q₈	—	2	6	no	—
267	C₄⋊Q₈	—	2	3	no	—
268	C₄⋊Q₈	—	2	3	no	—
269	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
270	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
271	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
272	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
273	#127	—	2	3	no	—
274	#127	—	2	3	no	—
275	#127	C₂³×C₂₃	1	3	yes	—
276	#127	C₂³×C₂₃	1	6	yes	—
277	#125	—	2	6	no	—
278	#125	C₂³×C₂₃	1	6	yes	—
279	#124	C₂³×C₂₃	1	6	yes	—
280	C₂²⋊Q₈	—	2	3	no	—
281	C₂²⋊Q₈	—	2	3	no	—
282	C₂²⋊Q₈	—	2	12	no	—
283	#121	—	2	3	no	—
284	#121	—	2	3	no	—
285	#121	—	2	6	no	—
286	#121	—	2	6	no	—
287	#120	—	2	12	no	—
288	C₄×Q₈	—	2	2	no	—
289	C₄×Q₈	—	2	3	no	—
290	C₄×Q₈	—	2	3	no	—
291	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
292	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
293	C₄×D₄	—	2	6	no	—
294	C₄×D₄	—	2	3	no	—
295	C₄×D₄	—	2	3	no	—
296	C₄×D₄	D₄×C₂₃	1	3	yes	—
297	C₄×D₄	D₄×C₂₃	1	3	yes	—
298	#117	C₂³×C₂₃	1	6	yes	—
299	#117	C₂³×C₂₃	1	3	yes	—
300	C₂×C₄⋊C₄	—	2	3	no	—
301	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
302	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
303	#103	C₂³×C₂₃	1	4	yes	—
304	#103	C₂³×C₂₃	1	12	yes	—
305	#102	C₂³×C₂₃	1	12	yes	—
306	#102	C₂³×C₂₃	1	24	yes	—
307	#102	C₂³×C₂₃	1	12	yes	—
308	#97	C₂³×C₂₃	1	6	yes	—
309	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
310	C₄×C₈	C₂³×C₂₃	1	3	yes	—
311	C₄×C₈	C₂³×C₂₃	1	6	yes	—
312	C₂₃	#60622	1	1	yes	characteristic, direct product factor, Sylow
313	C₂⁴	—	2	6	no	—
314	#41	—	2	24	no	—
315	#41	—	4	8	no	—
316	#41	#41×C₂₃	1	8	yes	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	3	no	—
319	C₂×Q₈	—	2	6	no	—
320	C₂×Q₈	—	8	6	no	—
321	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
322	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
323	C₂×D₄	—	2	6	no	—
324	C₂×D₄	—	2	3	no	—
325	C₂×D₄	—	2	3	no	—
326	C₂×D₄	—	2	6	no	—
327	C₂×D₄	—	2	6	no	—
328	C₂×D₄	—	2	6	no	—
329	C₂×D₄	—	2	24	no	—
330	C₂×D₄	—	4	24	no	—
331	C₂×D₄	—	8	6	no	—
332	C₂×D₄	—	8	6	no	—
333	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
334	C₂×D₄	C₂⁴×C₂₃	1	6	yes	—
335	C₂²×C₄	—	2	6	no	—
336	C₂²×C₄	—	2	6	no	—
337	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
338	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
339	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
340	M₁₆	—	2	12	no	—
341	C₂×C₈	—	2	12	no	—
342	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
343	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
344	C₄⋊C₄	—	2	3	no	—
345	C₄⋊C₄	—	2	6	no	—
346	C₄⋊C₄	—	2	6	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	6	no	—
351	C₄⋊C₄	—	2	6	no	—
352	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
353	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
354	C₄⋊C₄	C₂⁴×C₂₃	1	6	yes	—
355	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
356	C₂²⋊C₄	—	2	6	no	—
357	C₂²⋊C₄	—	2	6	no	—
358	C₂²⋊C₄	—	2	6	no	—
359	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
360	C₄²	—	2	6	no	—
361	C₄²	C₂×D₄×C₂₃	1	3	yes	—
362	C₄²	C₂×D₄×C₂₃	1	3	yes	—
363	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
364	C₄²	C₂⁴×C₂₃	1	3	yes	—
365	C₂³	—	2	6	no	—
366	C₂³	—	4	12	no	—
367	C₂³	—	8	6	no	—
368	C₂³	—	8	6	no	—
369	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
370	Q₈	—	2	4	no	—
371	Q₈	—	4	6	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	8	6	no	—
374	Q₈	C₂₃×#124	1	4	yes	—
375	D₄	—	2	12	no	—
376	D₄	—	4	12	no	—
377	D₄	—	4	6	no	—
378	D₄	—	4	6	no	—
379	D₄	—	4	12	no	—
380	D₄	—	8	6	no	—
381	D₄	—	8	6	no	—
382	D₄	C₂₃×#124	1	12	yes	—
383	C₂×C₄	—	2	6	no	—
384	C₂×C₄	—	2	6	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	6	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	12	no	—
389	C₂×C₄	—	8	6	no	—
390	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
391	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
392	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
395	C₂×C₄	C₂×#41×C₂₃	1	12	yes	—
396	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	6	no	—
399	C₂²	—	2	3	no	—
400	C₂²	—	2	3	no	—
401	C₂²	—	4	6	no	—
402	C₂²	—	4	6	no	—
403	C₂²	—	4	6	no	—
404	C₂²	—	16	12	no	—
405	C₂²	C₂×C₂₃×#124	1	6	yes	—
406	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
407	C₄	—	2	3	no	—
408	C₄	—	2	3	no	—
409	C₄	—	4	2	no	—
410	C₄	—	4	6	no	—
411	C₄	—	4	2	no	—
412	C₄	C₂×C₂₃×#124	1	6	yes	—
413	C₄	C₂₃×#576	1	2	yes	—
414	C₄	C₂₃×#577	1	2	yes	—
415	C₄	C₂²×#41×C₂₃	1	2	yes	—
416	C₂	—	2	6	no	—
417	C₂	—	8	6	no	—
418	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
419	C₂	C₂×C₂₃×#576	1	1	yes	characteristic
420	C₂	C₂×C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3503	C₂	semidirect
3. C₂₃×#3141	C₂	semidirect
4. C₂₃×#3138	C₂	semidirect
5. C₂₃×#3136	C₂	semidirect
6. C₂₃×#3131	C₂	semidirect
7. C₂×C₂₃×#486	C₂	semidirect
8. C₂₃×#2897	C₂	semidirect
9. C₄×C₂₃×#131	C₂	semidirect
20. C₂₃×#544	C₂²	semidirect
22. C₂₃×#533	C₂²	semidirect
24. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
28. C₂₃×#490	C₂²	semidirect
29. C₂₃×#489	C₂²	semidirect
30. C₂₃×#488	C₂²	semidirect
31. C₂₃×#487	C₂²	semidirect
32. C₂₃×#487	C₂²	semidirect
33. C₂₃×#486	C₂²	semidirect
34. C₂₃×#486	C₂²	semidirect
35. C₂₃×#486	C₂²	semidirect
36. C₂₃×#486	C₂²	semidirect
37. C₂₃×#418	C₂²	semidirect
38. C₂₃×#417	C₂²	semidirect
39. C₂₃×#416	C₂²	semidirect
40. C₂×C₂₃×#102	C₂²	semidirect
41. C₂₃×#405	C₂²	semidirect
42. C₄×M₁₆×C₂₃	C₂²	semidirect
43. C₂×C₄×C₈×C₂₃	C₂²	semidirect
154. #60622	C₂₃	direct (N × H)
189. #3503	C₂×C₂₃	semidirect
191. #3141	C₂×C₂₃	semidirect
192. #3138	C₂×C₂₃	semidirect
193. #3136	C₂×C₂₃	semidirect
194. #3131	C₂×C₂₃	semidirect
195. C₂×#486	C₂×C₂₃	semidirect
196. #2897	C₂×C₂₃	semidirect
197. C₄×#131	C₂×C₂₃	semidirect
225. #544	C₂²×C₂₃	semidirect
227. #533	C₂²×C₂₃	semidirect
229. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
233. #490	C₂²×C₂₃	semidirect
234. #489	C₂²×C₂₃	semidirect
235. #488	C₂²×C₂₃	semidirect
236. #487	C₂²×C₂₃	semidirect
237. #487	C₂²×C₂₃	semidirect
238. #486	C₂²×C₂₃	semidirect
239. #486	C₂²×C₂₃	semidirect
240. #486	C₂²×C₂₃	semidirect
241. #486	C₂²×C₂₃	semidirect
242. #418	C₂²×C₂₃	semidirect
243. #417	C₂²×C₂₃	semidirect
244. #416	C₂²×C₂₃	semidirect
245. C₂×#102	C₂²×C₂₃	semidirect
246. #405	C₂²×C₂₃	semidirect
247. C₄×M₁₆	C₂²×C₂₃	semidirect
248. C₂×C₄×C₈	C₂²×C₂₃	semidirect
312. C₂₃	#60622	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3503	2
C₂₃×#3498	2
C₂₃×#3141	2
C₂₃×#3138	2
C₂₃×#3136	2
C₂₃×#3131	2
C₂×C₂₃×#486	2
C₂₃×#2897	2
C₄×C₂₃×#131	2
#60622	23

Subgroups — C23×#60622 #19208248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60622 #19208248