Subgroups — C₂₃×#18542 #19166168

Order 5888.

918 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#556	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3113	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3110	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1652	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1650	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1556	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1491	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1490	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#556	—	2	2	no	—
11	C₂₃×#556	—	2	2	no	—
12	C₂₃×#556	—	2	2	no	—
13	C₂₃×#556	C₂²	1	4	yes	—
14	C₂₃×#533	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#126	—	2	2	no	—
17	C₂×C₂₃×#124	—	2	1	no	—
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#117	—	2	1	no	—
23	C₂₃×#501	—	2	2	no	—
24	C₂₃×#500	C₂²	1	2	yes	complemented
25	C₂₃×#498	C₂²	1	2	yes	complemented
26	C₂₃×#484	—	2	1	no	—
27	C₂₃×#483	—	2	1	no	—
28	C₂₃×#475	C₂²	1	2	yes	complemented
29	C₂₃×#474	C₂²	1	2	yes	—
30	C₂₃×#462	C₂²	1	2	yes	complemented
31	C₂₃×#461	C₂²	1	2	yes	—
32	C₂₃×#451	—	2	1	no	—
33	C₂₃×#450	—	2	1	no	—
34	C₂₃×#441	—	2	2	no	—
35	C₂₃×#440	—	2	2	no	—
36	C₂₃×#432	C₂²	1	1	yes	characteristic
37	C₂₃×#432	C₂²	1	1	yes	characteristic
38	C₂₃×#428	—	2	2	no	—
39	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#332	C₂²	1	4	yes	complemented
41	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂²×Q₈×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	—	2	1	no	—
47	C₂×Q₁₆×C₂₃	—	2	2	no	—
48	C₂×SD₁₆×C₂₃	—	2	2	no	—
49	C₂×M₁₆×C₂₃	—	2	1	no	—
50	C₂₃×C₄⋊Q₈	—	2	1	no	—
51	C₂₃×C₄⋊Q₈	—	2	1	no	—
52	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
53	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₂₃×#126	—	2	8	no	—
57	C₂₃×#126	—	2	8	no	—
58	C₂₃×#125	—	2	1	no	—
59	C₂₃×#124	—	2	4	no	—
60	C₂₃×#124	—	2	4	no	—
61	C₂₃×C₂²⋊Q₈	—	2	4	no	—
62	C₂₃×C₂²⋊Q₈	—	2	4	no	—
63	C₂₃×C₂²⋊Q₈	—	2	1	no	—
64	C₂₃×C₂²⋊Q₈	—	2	1	no	—
65	C₄×Q₈×C₂₃	—	2	4	no	—
66	C₄×Q₈×C₂₃	—	2	4	no	—
67	C₄×Q₈×C₂₃	—	2	1	no	—
68	C₄×Q₈×C₂₃	—	2	1	no	—
69	C₄×Q₈×C₂₃	—	2	1	no	—
70	C₄×Q₈×C₂₃	—	2	1	no	—
71	C₄×Q₈×C₂₃	D₄	1	2	yes	—
72	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
73	C₄×D₄×C₂₃	—	2	1	no	—
74	C₄×D₄×C₂₃	—	2	1	no	—
75	C₄×D₄×C₂₃	D₄	1	2	yes	—
76	C₄×D₄×C₂₃	C₂³	1	2	yes	—
77	C₂₃×#117	—	2	4	no	—
78	C₂₃×#117	—	2	4	no	—
79	C₂₃×#117	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
84	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄²×C₂₃	—	2	2	no	—
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
92	C₂₃×#107	—	2	4	no	—
93	C₂₃×#107	—	2	2	no	—
94	C₂₃×#106	—	2	4	no	—
95	C₂₃×#106	—	2	2	no	—
96	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
97	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
98	C₂₃×#103	—	2	2	no	—
99	C₂₃×#103	—	2	2	no	—
100	C₂₃×#103	—	2	2	no	—
101	C₂₃×#102	—	2	2	no	—
102	C₂₃×C₂²⋊C₈	—	2	1	no	—
103	C₂₃×C₂²⋊C₈	—	2	1	no	—
104	C₂₃×#97	—	2	1	no	—
105	C₂₃×#97	—	2	1	no	—
106	C₂₃×#97	—	2	1	no	—
107	C₄×C₈×C₂₃	C₂³	1	2	yes	—
108	C₂⁴×C₂₃	—	2	1	no	—
109	C₂×Q₈×C₂₃	—	2	1	no	—
110	C₂×Q₈×C₂₃	—	2	2	no	—
111	C₂×Q₈×C₂₃	—	2	2	no	—
112	C₂×Q₈×C₂₃	—	2	1	no	—
113	C₂×Q₈×C₂₃	—	2	1	no	—
114	C₂×Q₈×C₂₃	—	4	2	no	—
115	C₂×Q₈×C₂₃	—	4	4	no	—
116	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
117	C₂×D₄×C₂₃	—	2	1	no	—
118	C₂×D₄×C₂₃	—	4	2	no	—
119	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₂₃	—	2	1	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	2	no	—
124	C₂²×C₄×C₂₃	—	2	2	no	—
125	C₂²×C₄×C₂₃	—	2	2	no	—
126	C₂²×C₄×C₂₃	—	4	4	no	—
127	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
130	Q₁₆×C₂₃	—	4	4	no	—
131	SD₁₆×C₂₃	—	4	4	no	—
132	M₁₆×C₂₃	—	2	4	no	—
133	C₂×C₈×C₂₃	—	2	2	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	2	2	no	—
136	C₄⋊C₄×C₂₃	—	2	8	no	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	—	2	2	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	2	1	no	—
141	C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₄⋊C₄×C₂₃	—	2	1	no	—
146	C₄⋊C₄×C₂₃	—	2	1	no	—
147	C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₄⋊C₄×C₂₃	—	4	1	no	—
149	C₄⋊C₄×C₂₃	—	4	1	no	—
150	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
151	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
152	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
153	C₂²⋊C₄×C₂₃	—	2	8	no	—
154	C₂²⋊C₄×C₂₃	—	2	8	no	—
155	C₂²⋊C₄×C₂₃	—	2	2	no	—
156	C₂²⋊C₄×C₂₃	—	2	2	no	—
157	C₂²⋊C₄×C₂₃	—	4	1	no	—
158	C₄²×C₂₃	—	2	8	no	—
159	C₄²×C₂₃	—	2	2	no	—
160	C₄²×C₂₃	—	2	2	no	—
161	C₄²×C₂₃	—	4	1	no	—
162	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
166	#18542	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₂₃	—	2	2	no	—
168	C₂³×C₂₃	—	4	2	no	—
169	C₂³×C₂₃	—	4	2	no	—
170	C₂³×C₂₃	—	4	2	no	—
171	C₂³×C₂₃	#120	1	1	yes	characteristic
172	Q₈×C₂₃	—	2	2	no	—
173	Q₈×C₂₃	—	2	2	no	—
174	Q₈×C₂₃	—	4	2	no	—
175	Q₈×C₂₃	—	4	4	no	—
176	Q₈×C₂₃	—	4	4	no	—
177	Q₈×C₂₃	—	4	2	no	—
178	Q₈×C₂₃	—	4	2	no	—
179	D₄×C₂₃	—	2	2	no	—
180	D₄×C₂₃	—	2	2	no	—
181	D₄×C₂₃	—	4	2	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	2	no	—
184	C₂×C₄×C₂₃	—	2	2	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	4	4	no	—
190	C₂×C₄×C₂₃	—	4	2	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	4	4	no	—
193	C₂×C₄×C₂₃	—	4	2	no	—
194	C₂×C₄×C₂₃	—	4	4	no	—
195	C₂×C₄×C₂₃	—	4	1	no	—
196	C₂×C₄×C₂₃	—	4	1	no	—
197	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
198	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
199	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
200	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
203	C₈×C₂₃	—	2	4	no	—
204	C₈×C₂₃	—	4	2	no	—
205	C₈×C₂₃	—	4	2	no	—
206	C₂×#556	C₂×C₂₃	1	1	yes	characteristic
207	#3165	C₂×C₂₃	1	1	yes	characteristic, complemented
208	#3113	C₂×C₂₃	1	1	yes	characteristic, complemented
209	#3110	C₂×C₂₃	1	1	yes	characteristic
210	#1652	C₂×C₂₃	1	4	yes	complemented
211	#1650	C₂×C₂₃	1	4	yes	complemented
212	#1556	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#1491	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#1490	C₂×C₂₃	1	1	yes	characteristic
215	C₂²×C₂₃	—	2	1	no	—
216	C₂²×C₂₃	—	2	1	no	—
217	C₂²×C₂₃	—	4	2	no	—
218	C₂²×C₂₃	—	4	2	no	—
219	C₂²×C₂₃	—	4	2	no	—
220	C₂²×C₂₃	—	8	4	no	—
221	C₂²×C₂₃	#456	1	2	yes	—
222	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
223	C₄×C₂₃	—	2	1	no	—
224	C₄×C₂₃	—	2	1	no	—
225	C₄×C₂₃	—	2	1	no	—
226	C₄×C₂₃	—	2	1	no	—
227	C₄×C₂₃	—	4	2	no	—
228	C₄×C₂₃	—	4	2	no	—
229	C₄×C₂₃	—	4	4	no	—
230	C₄×C₂₃	—	4	2	no	—
231	C₄×C₂₃	—	8	1	no	—
232	C₄×C₂₃	—	8	1	no	—
233	C₄×C₂₃	#575	1	2	yes	—
234	C₄×C₂₃	#575	1	2	yes	—
235	#556	—	2	2	no	—
236	#556	—	2	2	no	—
237	#556	—	2	2	no	—
238	#556	C₂²×C₂₃	1	4	yes	—
239	#533	C₂²×C₂₃	1	1	yes	characteristic
240	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
241	C₂×#126	—	2	2	no	—
242	C₂×#124	—	2	1	no	—
243	C₂×C₂²⋊Q₈	—	2	1	no	—
244	C₂×C₄×Q₈	—	2	1	no	—
245	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
246	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
247	C₂×#117	—	2	1	no	—
248	#501	—	2	2	no	—
249	#500	C₂²×C₂₃	1	2	yes	complemented
250	#498	C₂²×C₂₃	1	2	yes	complemented
251	#484	—	2	1	no	—
252	#483	—	2	1	no	—
253	#475	C₂²×C₂₃	1	2	yes	complemented
254	#474	C₂²×C₂₃	1	2	yes	—
255	#462	C₂²×C₂₃	1	2	yes	complemented
256	#461	C₂²×C₂₃	1	2	yes	—
257	#451	—	2	1	no	—
258	#450	—	2	1	no	—
259	#441	—	2	2	no	—
260	#440	—	2	2	no	—
261	#432	C₂²×C₂₃	1	1	yes	characteristic
262	#432	C₂²×C₂₃	1	1	yes	characteristic
263	#428	—	2	2	no	—
264	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
265	#332	C₂²×C₂₃	1	4	yes	complemented
266	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
267	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
268	C₂×C₂₃	—	2	2	no	—
269	C₂×C₂₃	—	8	2	no	—
270	C₂×C₂₃	#3005	1	1	yes	characteristic
271	C₂×C₂₃	#3005	1	1	yes	characteristic
272	C₂×C₂₃	C₂×#456	1	1	yes	characteristic
273	C₂²×Q₈	—	2	1	no	—
274	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
275	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
276	C₂³×C₄	—	2	1	no	—
277	C₂×Q₁₆	—	2	2	no	—
278	C₂×SD₁₆	—	2	2	no	—
279	C₂×M₁₆	—	2	1	no	—
280	C₄⋊Q₈	—	2	1	no	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
283	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
284	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
285	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
286	#126	—	2	8	no	—
287	#126	—	2	8	no	—
288	#125	—	2	1	no	—
289	#124	—	2	4	no	—
290	#124	—	2	4	no	—
291	C₂²⋊Q₈	—	2	4	no	—
292	C₂²⋊Q₈	—	2	4	no	—
293	C₂²⋊Q₈	—	2	1	no	—
294	C₂²⋊Q₈	—	2	1	no	—
295	C₄×Q₈	—	2	4	no	—
296	C₄×Q₈	—	2	4	no	—
297	C₄×Q₈	—	2	1	no	—
298	C₄×Q₈	—	2	1	no	—
299	C₄×Q₈	—	2	1	no	—
300	C₄×Q₈	—	2	1	no	—
301	C₄×Q₈	D₄×C₂₃	1	2	yes	—
302	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
303	C₄×D₄	—	2	1	no	—
304	C₄×D₄	—	2	1	no	—
305	C₄×D₄	D₄×C₂₃	1	2	yes	—
306	C₄×D₄	C₂³×C₂₃	1	2	yes	—
307	#117	—	2	4	no	—
308	#117	—	2	4	no	—
309	#117	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
314	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
315	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
322	#107	—	2	4	no	—
323	#107	—	2	2	no	—
324	#106	—	2	4	no	—
325	#106	—	2	2	no	—
326	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
327	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
328	#103	—	2	2	no	—
329	#103	—	2	2	no	—
330	#103	—	2	2	no	—
331	#102	—	2	2	no	—
332	C₂²⋊C₈	—	2	1	no	—
333	C₂²⋊C₈	—	2	1	no	—
334	#97	—	2	1	no	—
335	#97	—	2	1	no	—
336	#97	—	2	1	no	—
337	C₄×C₈	C₂³×C₂₃	1	2	yes	—
338	C₂₃	#18542	1	1	yes	characteristic, direct product factor, Sylow
339	C₂⁴	—	2	1	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	—	2	1	no	—
345	C₂×Q₈	—	4	2	no	—
346	C₂×Q₈	—	4	4	no	—
347	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	4	2	no	—
350	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	4	4	no	—
358	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
361	Q₁₆	—	4	4	no	—
362	SD₁₆	—	4	4	no	—
363	M₁₆	—	2	4	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₄⋊C₄	—	2	8	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	4	1	no	—
380	C₄⋊C₄	—	4	1	no	—
381	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
382	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
383	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
384	C₂²⋊C₄	—	2	8	no	—
385	C₂²⋊C₄	—	2	8	no	—
386	C₂²⋊C₄	—	2	2	no	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	4	1	no	—
389	C₄²	—	2	8	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	4	1	no	—
393	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
394	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
397	C₂³	—	2	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	4	2	no	—
401	C₂³	C₂₃×#120	1	1	yes	characteristic
402	Q₈	—	2	2	no	—
403	Q₈	—	2	2	no	—
404	Q₈	—	4	2	no	—
405	Q₈	—	4	4	no	—
406	Q₈	—	4	4	no	—
407	Q₈	—	4	2	no	—
408	Q₈	—	4	2	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	2	no	—
411	D₄	—	4	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	4	4	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
428	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
429	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
430	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
431	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
433	C₈	—	2	4	no	—
434	C₈	—	4	2	no	—
435	C₈	—	4	2	no	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	8	4	no	—
442	C₂²	C₂₃×#456	1	2	yes	—
443	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
444	C₄	—	2	1	no	—
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	4	2	no	—
449	C₄	—	4	2	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	2	no	—
452	C₄	—	8	1	no	—
453	C₄	—	8	1	no	—
454	C₄	C₂₃×#575	1	2	yes	—
455	C₄	C₂₃×#575	1	2	yes	—
456	C₂	—	2	2	no	—
457	C₂	—	8	2	no	—
458	C₂	C₂₃×#3005	1	1	yes	characteristic
459	C₂	C₂₃×#3005	1	1	yes	characteristic
460	C₂	C₂×C₂₃×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3165	C₂	semidirect
3. C₂₃×#3113	C₂	semidirect
5. C₂₃×#1652	C₂	semidirect
6. C₂₃×#1650	C₂	semidirect
7. C₂₃×#1556	C₂	semidirect
8. C₂₃×#1491	C₂	semidirect
24. C₂₃×#500	C₂²	semidirect
25. C₂₃×#498	C₂²	semidirect
28. C₂₃×#475	C₂²	semidirect
30. C₂₃×#462	C₂²	semidirect
40. C₂₃×#332	C₂²	semidirect
41. C₂₃×Q₈⋊C₈	C₂²	semidirect
166. #18542	C₂₃	direct (N × H)
207. #3165	C₂×C₂₃	semidirect
208. #3113	C₂×C₂₃	semidirect
210. #1652	C₂×C₂₃	semidirect
211. #1650	C₂×C₂₃	semidirect
212. #1556	C₂×C₂₃	semidirect
213. #1491	C₂×C₂₃	semidirect
249. #500	C₂²×C₂₃	semidirect
250. #498	C₂²×C₂₃	semidirect
253. #475	C₂²×C₂₃	semidirect
255. #462	C₂²×C₂₃	semidirect
265. #332	C₂²×C₂₃	semidirect
266. Q₈⋊C₈	C₂²×C₂₃	semidirect
338. C₂₃	#18542	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×#556	2
C₂₃×#3165	2
C₂₃×#3113	2
C₂₃×#3110	2
C₂₃×#1652	2
C₂₃×#1650	2
C₂₃×#1556	2
C₂₃×#1491	2
C₂₃×#1490	2
#18542	23

Subgroups — C23×#18542 #19166168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18542 #19166168