Subgroups — C₂₃×#60612 #19208238

Order 5888.

2050 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3449	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3137	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3136	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3135	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3131	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2896	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#416	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	2	yes	—
11	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#544	—	2	2	no	—
14	C₂₃×#544	C₂²	1	8	yes	complemented
15	C₂₃×#543	—	2	2	no	—
16	C₂₃×#537	—	2	2	no	—
17	C₂₃×#536	C₂²	1	4	yes	—
18	C₂₃×#535	C₂²	1	4	yes	—
19	C₂₃×#534	—	2	2	no	—
20	C₂₃×#533	—	2	4	no	—
21	C₂₃×#533	C₂²	1	4	yes	complemented
22	C₂₃×#532	C₂²	1	4	yes	—
23	C₂₃×#525	—	2	4	no	—
24	C₂×C₂₃×#117	C₂²	1	2	yes	—
25	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
26	C₂₃×#491	C₂²	1	8	yes	complemented
27	C₂₃×#490	C₂²	1	8	yes	complemented
28	C₂₃×#489	C₂²	1	16	yes	complemented
29	C₂₃×#488	C₂²	1	16	yes	complemented
30	C₂₃×#487	C₂²	1	8	yes	complemented
31	C₂₃×#486	C₂²	1	8	yes	complemented
32	C₂₃×#419	C₂²	1	4	yes	complemented
33	C₂₃×#418	C₂²	1	4	yes	complemented
34	C₂₃×#417	C₂²	1	8	yes	—
35	C₂₃×#416	C₂²	1	8	yes	—
36	C₂₃×#416	C₂²	1	4	yes	complemented
37	C₂₃×#416	C₂²	1	8	yes	complemented
38	C₂₃×#416	C₂²	1	4	yes	complemented
39	C₂×C₂₃×#103	C₂²	1	4	yes	complemented
40	C₂×C₂₃×#102	C₂²	1	4	yes	complemented
41	C₂₃×#405	C₂²	1	8	yes	complemented
42	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
43	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
44	C₂×#41×C₂₃	—	2	2	no	—
45	C₂×#41×C₂₃	—	2	2	no	—
46	C₂×#41×C₂₃	—	2	8	no	—
47	C₂×#41×C₂₃	—	2	8	no	—
48	C₂×#41×C₂₃	C₂³	1	4	yes	—
49	C₂×#41×C₂₃	C₂³	1	4	yes	—
50	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
51	C₂²×D₄×C₂₃	—	2	2	no	—
52	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₂₃	—	2	2	no	—
54	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₂₃	—	2	4	no	—
56	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
57	C₂²×C₈×C₂₃	—	2	4	no	—
58	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
61	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂₃×C₄⋊Q₈	—	2	8	no	—
63	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
64	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
65	C₂₃×#127	C₂³	1	4	yes	—
66	C₂₃×#125	—	2	8	no	—
67	C₂₃×#125	C₂³	1	8	yes	—
68	C₂₃×#124	C₂³	1	8	yes	—
69	C₂₃×#123	—	2	8	no	—
70	C₂₃×C₂²⋊Q₈	—	2	8	no	—
71	C₂₃×C₂²⋊Q₈	—	2	8	no	—
72	C₂₃×C₂²⋊Q₈	—	2	16	no	—
73	C₂₃×#121	—	2	8	no	—
74	C₂₃×#121	—	2	8	no	—
75	C₂₃×#120	—	2	8	no	—
76	C₄×Q₈×C₂₃	—	2	8	no	—
77	C₄×D₄×C₂₃	—	2	8	no	—
78	C₂₃×#117	—	2	2	no	—
79	C₂₃×#117	—	2	2	no	—
80	C₂₃×#117	D₄	1	4	yes	—
81	C₂₃×#117	C₂³	1	4	yes	—
82	C₂₃×#117	C₂³	1	4	yes	—
83	C₂₃×#117	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
88	C₂×C₄²×C₂₃	—	2	2	no	—
89	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
90	C₂₃×#103	C₂³	1	8	yes	—
91	C₂₃×#103	C₂³	1	8	yes	—
92	C₂₃×#103	C₂³	1	16	yes	—
93	C₂₃×#102	C₂³	1	8	yes	—
94	C₂₃×#102	C₂³	1	8	yes	—
95	C₂₃×#102	C₂³	1	16	yes	—
96	C₂₃×#97	C₂³	1	8	yes	—
97	C₄×C₈×C₂₃	C₂³	1	8	yes	—
98	C₂⁴×C₂₃	—	2	2	no	—
99	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
100	#41×C₂₃	—	2	8	no	—
101	#41×C₂₃	—	2	8	no	—
102	#41×C₂₃	—	2	8	no	—
103	#41×C₂₃	—	2	8	no	—
104	#41×C₂₃	—	4	8	no	—
105	#41×C₂₃	—	4	8	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	2	8	no	—
108	C₂×Q₈×C₂₃	—	2	4	no	—
109	C₂×Q₈×C₂₃	—	2	4	no	—
110	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
111	C₂×D₄×C₂₃	—	2	4	no	—
112	C₂×D₄×C₂₃	—	2	4	no	—
113	C₂×D₄×C₂₃	—	2	2	no	—
114	C₂×D₄×C₂₃	—	2	8	no	—
115	C₂×D₄×C₂₃	—	2	4	no	—
116	C₂×D₄×C₂₃	—	2	4	no	—
117	C₂×D₄×C₂₃	—	4	8	no	—
118	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	2	4	no	—
121	C₂²×C₄×C₂₃	—	2	4	no	—
122	C₂²×C₄×C₂₃	—	2	4	no	—
123	C₂²×C₄×C₂₃	—	4	8	no	—
124	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₂₃	#41	1	4	yes	—
128	C₂²×C₄×C₂₃	#41	1	4	yes	—
129	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
130	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
131	M₁₆×C₂₃	—	2	8	no	—
132	C₂×C₈×C₂₃	—	2	8	no	—
133	C₂×C₈×C₂₃	—	2	2	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	2	4	no	—
136	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
137	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
138	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
139	C₄⋊C₄×C₂₃	—	2	4	no	—
140	C₄⋊C₄×C₂₃	—	2	8	no	—
141	C₄⋊C₄×C₂₃	—	2	8	no	—
142	C₄⋊C₄×C₂₃	—	2	16	no	—
143	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
144	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
145	C₂²⋊C₄×C₂₃	—	2	4	no	—
146	C₂²⋊C₄×C₂₃	—	2	4	no	—
147	C₂²⋊C₄×C₂₃	—	2	16	no	—
148	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
149	C₄²×C₂₃	—	2	4	no	—
150	C₄²×C₂₃	—	2	4	no	—
151	C₄²×C₂₃	C₂×D₄	1	4	yes	—
152	C₄²×C₂₃	C₂⁴	1	4	yes	—
153	#60612	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₂₃	—	2	4	no	—
155	C₂³×C₂₃	—	4	2	no	—
156	C₂³×C₂₃	—	4	4	no	—
157	C₂³×C₂₃	—	8	4	no	—
158	C₂³×C₂₃	#124	1	4	yes	—
159	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
160	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
161	Q₈×C₂₃	—	2	4	no	—
162	Q₈×C₂₃	—	2	4	no	—
163	Q₈×C₂₃	—	4	4	no	—
164	Q₈×C₂₃	—	8	8	no	—
165	D₄×C₂₃	—	2	4	no	—
166	D₄×C₂₃	—	2	4	no	—
167	D₄×C₂₃	—	4	8	no	—
168	D₄×C₂₃	—	4	8	no	—
169	D₄×C₂₃	—	4	4	no	—
170	D₄×C₂₃	—	8	8	no	—
171	C₂×C₄×C₂₃	—	2	8	no	—
172	C₂×C₄×C₂₃	—	2	4	no	—
173	C₂×C₄×C₂₃	—	4	4	no	—
174	C₂×C₄×C₂₃	—	4	4	no	—
175	C₂×C₄×C₂₃	—	4	8	no	—
176	C₂×C₄×C₂₃	—	4	8	no	—
177	C₂×C₄×C₂₃	—	8	4	no	—
178	C₂×C₄×C₂₃	#124	1	8	yes	—
179	C₂×C₄×C₂₃	#124	1	4	yes	—
180	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
181	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
182	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
184	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
185	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
186	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
187	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
188	C₈×C₂₃	—	2	2	no	—
189	C₈×C₂₃	—	2	2	no	—
190	C₈×C₂₃	—	2	4	no	—
191	#3458	C₂×C₂₃	1	1	yes	characteristic, complemented
192	#3449	C₂×C₂₃	1	1	yes	characteristic
193	#3137	C₂×C₂₃	1	8	yes	complemented
194	#3136	C₂×C₂₃	1	8	yes	complemented
195	#3135	C₂×C₂₃	1	4	yes	complemented
196	#3131	C₂×C₂₃	1	4	yes	complemented
197	#2896	C₂×C₂₃	1	2	yes	complemented
198	C₂×#416	C₂×C₂₃	1	2	yes	complemented
199	C₂×#405	C₂×C₂₃	1	1	yes	characteristic, complemented
200	C₂²×C₂₃	—	2	2	no	—
201	C₂²×C₂₃	—	2	4	no	—
202	C₂²×C₂₃	—	4	4	no	—
203	C₂²×C₂₃	—	4	8	no	—
204	C₂²×C₂₃	—	4	4	no	—
205	C₂²×C₂₃	—	8	8	no	—
206	C₂²×C₂₃	C₂×#124	1	4	yes	—
207	C₂²×C₂₃	C₂×#124	1	2	yes	—
208	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
209	C₄×C₂₃	—	2	8	no	—
210	C₄×C₂₃	—	4	8	no	—
211	C₄×C₂₃	—	4	4	no	—
212	C₄×C₂₃	C₂×#124	1	2	yes	—
213	C₄×C₂₃	C₂×#124	1	4	yes	—
214	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
215	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
216	C₂²×#41	C₂²×C₂₃	1	2	yes	—
217	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
218	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
219	#544	—	2	2	no	—
220	#544	C₂²×C₂₃	1	8	yes	complemented
221	#543	—	2	2	no	—
222	#537	—	2	2	no	—
223	#536	C₂²×C₂₃	1	4	yes	—
224	#535	C₂²×C₂₃	1	4	yes	—
225	#534	—	2	2	no	—
226	#533	—	2	4	no	—
227	#533	C₂²×C₂₃	1	4	yes	complemented
228	#532	C₂²×C₂₃	1	4	yes	—
229	#525	—	2	4	no	—
230	C₂×#117	C₂²×C₂₃	1	2	yes	—
231	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
232	#491	C₂²×C₂₃	1	8	yes	complemented
233	#490	C₂²×C₂₃	1	8	yes	complemented
234	#489	C₂²×C₂₃	1	16	yes	complemented
235	#488	C₂²×C₂₃	1	16	yes	complemented
236	#487	C₂²×C₂₃	1	8	yes	complemented
237	#486	C₂²×C₂₃	1	8	yes	complemented
238	#419	C₂²×C₂₃	1	4	yes	complemented
239	#418	C₂²×C₂₃	1	4	yes	complemented
240	#417	C₂²×C₂₃	1	8	yes	—
241	#416	C₂²×C₂₃	1	8	yes	—
242	#416	C₂²×C₂₃	1	4	yes	complemented
243	#416	C₂²×C₂₃	1	8	yes	complemented
244	#416	C₂²×C₂₃	1	4	yes	complemented
245	C₂×#103	C₂²×C₂₃	1	4	yes	complemented
246	C₂×#102	C₂²×C₂₃	1	4	yes	complemented
247	#405	C₂²×C₂₃	1	8	yes	complemented
248	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
249	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
250	C₂×C₂₃	—	2	4	no	—
251	C₂×C₂₃	—	2	2	no	—
252	C₂×C₂₃	—	8	4	no	—
253	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
254	C₂×C₂₃	#3584	1	2	yes	—
255	C₂×#41	—	2	2	no	—
256	C₂×#41	—	2	2	no	—
257	C₂×#41	—	2	8	no	—
258	C₂×#41	—	2	8	no	—
259	C₂×#41	C₂³×C₂₃	1	4	yes	—
260	C₂×#41	C₂³×C₂₃	1	4	yes	—
261	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
264	C₂³×C₄	—	2	2	no	—
265	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
266	C₂×M₁₆	—	2	4	no	—
267	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
268	C₂²×C₈	—	2	4	no	—
269	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
272	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
273	C₄⋊Q₈	—	2	8	no	—
274	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
275	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
276	#127	C₂³×C₂₃	1	4	yes	—
277	#125	—	2	8	no	—
278	#125	C₂³×C₂₃	1	8	yes	—
279	#124	C₂³×C₂₃	1	8	yes	—
280	#123	—	2	8	no	—
281	C₂²⋊Q₈	—	2	8	no	—
282	C₂²⋊Q₈	—	2	8	no	—
283	C₂²⋊Q₈	—	2	16	no	—
284	#121	—	2	8	no	—
285	#121	—	2	8	no	—
286	#120	—	2	8	no	—
287	C₄×Q₈	—	2	8	no	—
288	C₄×D₄	—	2	8	no	—
289	#117	—	2	2	no	—
290	#117	—	2	2	no	—
291	#117	D₄×C₂₃	1	4	yes	—
292	#117	C₂³×C₂₃	1	4	yes	—
293	#117	C₂³×C₂₃	1	4	yes	—
294	#117	C₂³×C₂₃	1	4	yes	—
295	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
297	C₂×C₂²⋊C₄	—	2	2	no	—
298	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
299	C₂×C₄²	—	2	2	no	—
300	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
301	#103	C₂³×C₂₃	1	8	yes	—
302	#103	C₂³×C₂₃	1	8	yes	—
303	#103	C₂³×C₂₃	1	16	yes	—
304	#102	C₂³×C₂₃	1	8	yes	—
305	#102	C₂³×C₂₃	1	8	yes	—
306	#102	C₂³×C₂₃	1	16	yes	—
307	#97	C₂³×C₂₃	1	8	yes	—
308	C₄×C₈	C₂³×C₂₃	1	8	yes	—
309	C₂₃	#60612	1	1	yes	characteristic, direct product factor, Sylow
310	C₂⁴	—	2	2	no	—
311	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
312	#41	—	2	8	no	—
313	#41	—	2	8	no	—
314	#41	—	2	8	no	—
315	#41	—	2	8	no	—
316	#41	—	4	8	no	—
317	#41	—	4	8	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	8	no	—
320	C₂×Q₈	—	2	4	no	—
321	C₂×Q₈	—	2	4	no	—
322	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
323	C₂×D₄	—	2	4	no	—
324	C₂×D₄	—	2	4	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	8	no	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	—	4	8	no	—
330	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	4	8	no	—
336	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
337	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
338	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
339	C₂²×C₄	#41×C₂₃	1	4	yes	—
340	C₂²×C₄	#41×C₂₃	1	4	yes	—
341	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
342	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
343	M₁₆	—	2	8	no	—
344	C₂×C₈	—	2	8	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	4	no	—
348	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
349	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
350	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	8	no	—
353	C₄⋊C₄	—	2	8	no	—
354	C₄⋊C₄	—	2	16	no	—
355	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
356	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₂²⋊C₄	—	2	16	no	—
360	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
361	C₄²	—	2	4	no	—
362	C₄²	—	2	4	no	—
363	C₄²	C₂×D₄×C₂₃	1	4	yes	—
364	C₄²	C₂⁴×C₂₃	1	4	yes	—
365	C₂³	—	2	4	no	—
366	C₂³	—	4	2	no	—
367	C₂³	—	4	4	no	—
368	C₂³	—	8	4	no	—
369	C₂³	C₂₃×#124	1	4	yes	—
370	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
371	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
372	Q₈	—	2	4	no	—
373	Q₈	—	2	4	no	—
374	Q₈	—	4	4	no	—
375	Q₈	—	8	8	no	—
376	D₄	—	2	4	no	—
377	D₄	—	2	4	no	—
378	D₄	—	4	8	no	—
379	D₄	—	4	8	no	—
380	D₄	—	4	4	no	—
381	D₄	—	8	8	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	4	4	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	—	4	8	no	—
388	C₂×C₄	—	8	4	no	—
389	C₂×C₄	C₂₃×#124	1	8	yes	—
390	C₂×C₄	C₂₃×#124	1	4	yes	—
391	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
392	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
394	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
395	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
396	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
397	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
398	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
399	C₈	—	2	2	no	—
400	C₈	—	2	2	no	—
401	C₈	—	2	4	no	—
402	C₂²	—	2	2	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	4	4	no	—
405	C₂²	—	4	4	no	—
406	C₂²	—	4	8	no	—
407	C₂²	—	8	8	no	—
408	C₂²	C₂×C₂₃×#124	1	4	yes	—
409	C₂²	C₂×C₂₃×#124	1	2	yes	—
410	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
411	C₄	—	2	8	no	—
412	C₄	—	4	4	no	—
413	C₄	—	4	8	no	—
414	C₄	C₂×C₂₃×#124	1	2	yes	—
415	C₄	C₂×C₂₃×#124	1	4	yes	—
416	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
417	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
418	C₂	—	2	4	no	—
419	C₂	—	2	2	no	—
420	C₂	—	8	4	no	—
421	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
422	C₂	C₂₃×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂³×D₄
prime radical	C₂₃×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3458	C₂	semidirect
3. C₂₃×#3137	C₂	semidirect
4. C₂₃×#3136	C₂	semidirect
5. C₂₃×#3135	C₂	semidirect
6. C₂₃×#3131	C₂	semidirect
7. C₂₃×#2896	C₂	semidirect
8. C₂×C₂₃×#416	C₂	semidirect
9. C₂×C₂₃×#405	C₂	semidirect
14. C₂₃×#544	C₂²	semidirect
21. C₂₃×#533	C₂²	semidirect
26. C₂₃×#491	C₂²	semidirect
27. C₂₃×#490	C₂²	semidirect
28. C₂₃×#489	C₂²	semidirect
29. C₂₃×#488	C₂²	semidirect
30. C₂₃×#487	C₂²	semidirect
31. C₂₃×#486	C₂²	semidirect
32. C₂₃×#419	C₂²	semidirect
33. C₂₃×#418	C₂²	semidirect
36. C₂₃×#416	C₂²	semidirect
37. C₂₃×#416	C₂²	semidirect
38. C₂₃×#416	C₂²	semidirect
39. C₂×C₂₃×#103	C₂²	semidirect
40. C₂×C₂₃×#102	C₂²	semidirect
41. C₂₃×#405	C₂²	semidirect
42. C₂×C₂₃×#97	C₂²	semidirect
43. C₂×C₄×C₈×C₂₃	C₂²	semidirect
153. #60612	C₂₃	direct (N × H)
191. #3458	C₂×C₂₃	semidirect
193. #3137	C₂×C₂₃	semidirect
194. #3136	C₂×C₂₃	semidirect
195. #3135	C₂×C₂₃	semidirect
196. #3131	C₂×C₂₃	semidirect
197. #2896	C₂×C₂₃	semidirect
198. C₂×#416	C₂×C₂₃	semidirect
199. C₂×#405	C₂×C₂₃	semidirect
220. #544	C₂²×C₂₃	semidirect
227. #533	C₂²×C₂₃	semidirect
232. #491	C₂²×C₂₃	semidirect
233. #490	C₂²×C₂₃	semidirect
234. #489	C₂²×C₂₃	semidirect
235. #488	C₂²×C₂₃	semidirect
236. #487	C₂²×C₂₃	semidirect
237. #486	C₂²×C₂₃	semidirect
238. #419	C₂²×C₂₃	semidirect
239. #418	C₂²×C₂₃	semidirect
242. #416	C₂²×C₂₃	semidirect
243. #416	C₂²×C₂₃	semidirect
244. #416	C₂²×C₂₃	semidirect
245. C₂×#103	C₂²×C₂₃	semidirect
246. C₂×#102	C₂²×C₂₃	semidirect
247. #405	C₂²×C₂₃	semidirect
248. C₂×#97	C₂²×C₂₃	semidirect
249. C₂×C₄×C₈	C₂²×C₂₃	semidirect
309. C₂₃	#60612	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×#3458	2
C₂₃×#3449	2
C₂₃×#3137	2
C₂₃×#3136	2
C₂₃×#3135	2
C₂₃×#3131	2
C₂₃×#2896	2
C₂×C₂₃×#416	2
C₂×C₂₃×#405	2
#60612	23

Subgroups — C23×#60612 #19208238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60612 #19208238