Subgroups — C₂₃×#60524 #19208150

Order 5888.

2116 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3489	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3099	C₂	1	3	yes	maximal, complemented
4	C₂₃×#3098	C₂	1	3	yes	maximal, complemented
5	C₂₃×#3097	C₂	1	3	yes	maximal, complemented
6	C₂₃×#3097	C₂	1	3	yes	maximal, complemented
7	C₂₃×#3095	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3093	C₂	1	3	yes	maximal, complemented
9	C₂₃×#3091	C₂	1	3	yes	maximal, complemented
10	C₂×C₂₃×#485	C₂	1	3	yes	maximal, complemented
11	C₂₃×#2913	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#2912	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#2898	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#2898	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#2880	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
17	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
20	D₄×Q₈×C₂₃	—	2	1	no	—
21	C₂₃×#548	—	2	1	no	—
22	C₂₃×#547	—	2	3	no	—
23	C₂₃×#546	—	2	3	no	—
24	C₂₃×#537	C₂²	1	3	yes	—
25	C₂₃×#536	C₂²	1	3	yes	—
26	C₂₃×#525	C₂²	1	3	yes	—
27	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	3	yes	—
28	C₂₃×#520	C₂²	1	1	yes	characteristic
29	C₂₃×#520	C₂²	1	1	yes	characteristic
30	C₂₃×#520	C₂²	1	1	yes	characteristic
31	C₂₃×#520	C₂²	1	3	yes	—
32	C₂₃×#520	C₂²	1	3	yes	—
33	C₂₃×#519	—	2	1	no	—
34	C₂₃×#518	—	2	3	no	—
35	C₄×#41×C₂₃	—	2	3	no	—
36	C₄×#41×C₂₃	—	2	1	no	—
37	C₂×C₂₃×#117	C₂²	1	3	yes	—
38	C₂×C₂₃×#117	C₂²	1	3	yes	—
39	C₂₃×#485	C₂²	1	3	yes	—
40	C₂₃×#485	C₂²	1	6	yes	—
41	C₂₃×#485	C₂²	1	3	yes	—
42	C₂₃×#485	C₂²	1	6	yes	complemented
43	C₂₃×#485	C₂²	1	6	yes	complemented
44	C₂₃×#485	C₂²	1	6	yes	complemented
45	C₂₃×#484	C₂²	1	3	yes	complemented
46	C₂₃×#484	C₂²	1	6	yes	complemented
47	C₂₃×#483	C₂²	1	3	yes	complemented
48	C₂₃×#483	C₂²	1	6	yes	complemented
49	C₂₃×#482	C₂²	1	1	yes	characteristic
50	C₂₃×#482	C₂²	1	3	yes	—
51	C₂₃×#482	C₂²	1	6	yes	complemented
52	C₂₃×#481	C₂²	1	3	yes	complemented
53	C₂₃×#480	C₂²	1	3	yes	complemented
54	C₂₃×#428	C₂²	1	3	yes	complemented
55	C₂₃×#428	C₂²	1	3	yes	complemented
56	C₂₃×#428	C₂²	1	3	yes	complemented
57	C₂₃×#428	C₂²	1	3	yes	complemented
58	C₂₃×#427	C₂²	1	3	yes	complemented
59	C₂₃×#427	C₂²	1	3	yes	complemented
60	C₂×C₂₃×#107	C₂²	1	3	yes	complemented
61	C₂×C₂₃×#106	C₂²	1	3	yes	complemented
62	C₂₃×#419	C₂²	1	3	yes	complemented
63	C₂₃×#419	C₂²	1	3	yes	complemented
64	C₂₃×#417	C₂²	1	3	yes	complemented
65	C₂₃×#417	C₂²	1	3	yes	complemented
66	C₂₃×#416	C₂²	1	3	yes	complemented
67	C₂₃×#416	C₂²	1	3	yes	complemented
68	C₂×C₂₃×#103	C₂²	1	3	yes	complemented
69	C₂×C₂₃×#103	C₂²	1	3	yes	complemented
70	C₂₃×#408	C₂²	1	3	yes	—
71	C₂₃×#408	C₂²	1	3	yes	—
72	C₂₃×#407	C₂²	1	3	yes	—
73	C₂×C₂₃×C₂²⋊C₈	C₂²	1	3	yes	—
74	C₂₃×2₋¹⁺⁴	—	2	4	no	—
75	C₂₃×2₋¹⁺⁴	—	2	4	no	—
76	C₂×#41×C₂₃	—	2	4	no	—
77	C₂×#41×C₂₃	—	2	4	no	—
78	C₂×#41×C₂₃	—	2	3	no	—
79	C₂×#41×C₂₃	D₄	1	3	yes	—
80	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
81	C₂×#41×C₂₃	C₂³	1	3	yes	—
82	C₂×#41×C₂₃	C₂³	1	3	yes	—
83	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₂₃	—	2	1	no	—
85	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
86	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
87	C₂²×D₄×C₂₃	D₄	1	3	yes	—
88	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
89	C₂₃×#131	—	2	4	no	—
90	C₂₃×#131	—	2	4	no	—
91	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
92	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
93	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
94	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
95	C₂₃×C₄⋊Q₈	—	2	3	no	—
96	C₂₃×#124	—	2	3	no	—
97	C₂₃×#123	—	2	6	no	—
98	C₂₃×#123	—	2	3	no	—
99	C₂₃×C₂²⋊Q₈	—	2	3	no	—
100	C₂₃×C₂²⋊Q₈	—	2	3	no	—
101	C₂₃×C₂²⋊Q₈	—	2	3	no	—
102	C₂₃×C₂²⋊Q₈	C₂³	1	6	yes	—
103	C₂₃×C₂²⋊Q₈	C₂³	1	3	yes	—
104	C₂₃×C₂²⋊Q₈	C₂³	1	3	yes	—
105	C₂₃×#121	—	2	6	no	—
106	C₂₃×#121	—	2	3	no	—
107	C₂₃×#121	C₂³	1	3	yes	—
108	C₂₃×#121	C₂³	1	1	yes	characteristic
109	C₂₃×#120	—	2	3	no	—
110	C₄×Q₈×C₂₃	—	2	1	no	—
111	C₄×Q₈×C₂₃	—	2	3	no	—
112	C₄×Q₈×C₂₃	—	2	3	no	—
113	C₄×Q₈×C₂₃	—	2	1	no	—
114	C₄×Q₈×C₂₃	—	2	1	no	—
115	C₄×D₄×C₂₃	—	2	3	no	—
116	C₄×D₄×C₂₃	—	2	6	no	—
117	C₄×D₄×C₂₃	—	2	6	no	—
118	C₄×D₄×C₂₃	—	2	3	no	—
119	C₄×D₄×C₂₃	—	2	3	no	—
120	C₄×D₄×C₂₃	—	2	3	no	—
121	C₄×D₄×C₂₃	—	2	3	no	—
122	C₄×D₄×C₂₃	—	2	3	no	—
123	C₄×D₄×C₂₃	—	2	3	no	—
124	C₂₃×#117	—	2	3	no	—
125	C₂₃×#117	—	2	3	no	—
126	C₂₃×#117	—	2	3	no	—
127	C₂₃×#117	—	2	3	no	—
128	C₂₃×#117	C₂³	1	3	yes	—
129	C₂₃×#117	C₂³	1	3	yes	—
130	C₂₃×#117	C₂³	1	3	yes	—
131	C₂₃×#117	C₂³	1	1	yes	characteristic
132	C₂₃×#117	C₂³	1	3	yes	—
133	C₂₃×#117	C₂³	1	6	yes	—
134	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
135	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
136	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
137	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
138	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
139	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
140	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
141	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
142	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
143	C₂×C₄²×C₂₃	—	2	3	no	—
144	C₂×C₄²×C₂₃	—	2	3	no	—
145	C₂₃×#107	C₂³	1	3	yes	—
146	C₂₃×#107	C₂³	1	6	yes	—
147	C₂₃×#107	C₂³	1	1	yes	characteristic
148	C₂₃×#107	C₂³	1	3	yes	—
149	C₂₃×#107	C₂³	1	3	yes	—
150	C₂₃×#106	C₂³	1	3	yes	—
151	C₂₃×#106	C₂³	1	6	yes	—
152	C₂₃×#106	C₂³	1	1	yes	characteristic
153	C₂₃×#106	C₂³	1	3	yes	—
154	C₂₃×#106	C₂³	1	3	yes	—
155	C₂₃×#103	C₂³	1	3	yes	—
156	C₂₃×#103	C₂³	1	3	yes	—
157	C₂₃×#103	C₂³	1	6	yes	—
158	C₂₃×#103	C₂³	1	6	yes	—
159	C₂₃×#103	C₂³	1	3	yes	—
160	C₂₃×#103	C₂³	1	3	yes	—
161	C₂₃×#102	C₂³	1	1	yes	characteristic
162	C₂₃×#102	C₂³	1	1	yes	characteristic
163	C₂₃×#102	C₂³	1	3	yes	—
164	C₂₃×#102	C₂³	1	3	yes	—
165	C₂₃×C₂²⋊C₈	C₂³	1	1	yes	characteristic
166	C₂₃×C₂²⋊C₈	C₂³	1	3	yes	—
167	C₂₃×C₂²⋊C₈	C₂³	1	3	yes	—
168	C₂₃×C₂²⋊C₈	C₂³	1	3	yes	—
169	C₂₃×C₂²⋊C₈	C₂³	1	6	yes	—
170	C₂⁴×C₂₃	C₂×D₄	1	3	yes	—
171	#41×C₂₃	—	2	4	no	—
172	#41×C₂₃	—	2	12	no	—
173	#41×C₂₃	—	2	6	no	—
174	#41×C₂₃	—	2	6	no	—
175	#41×C₂₃	—	2	6	no	—
176	#41×C₂₃	—	2	6	no	—
177	#41×C₂₃	—	4	12	no	—
178	#41×C₂₃	#41	1	4	yes	—
179	#41×C₂₃	#41	1	4	yes	—
180	C₂×Q₈×C₂₃	—	2	4	no	—
181	C₂×Q₈×C₂₃	—	2	3	no	—
182	C₂×Q₈×C₂₃	—	2	6	no	—
183	C₂×Q₈×C₂₃	—	2	6	no	—
184	C₂×Q₈×C₂₃	—	4	4	no	—
185	C₂×Q₈×C₂₃	C₂×D₄	1	3	yes	—
186	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
188	C₂×D₄×C₂₃	—	2	12	no	—
189	C₂×D₄×C₂₃	—	2	1	no	—
190	C₂×D₄×C₂₃	—	2	3	no	—
191	C₂×D₄×C₂₃	—	2	3	no	—
192	C₂×D₄×C₂₃	C₂×D₄	1	6	yes	—
193	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
194	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
195	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₂₃	—	2	6	no	—
197	C₂²×C₄×C₂₃	—	2	6	no	—
198	C₂²×C₄×C₂₃	—	2	3	no	—
199	C₂²×C₄×C₂₃	—	2	3	no	—
200	C₂²×C₄×C₂₃	—	2	3	no	—
201	C₂²×C₄×C₂₃	—	2	3	no	—
202	C₂²×C₄×C₂₃	—	2	3	no	—
203	C₂²×C₄×C₂₃	—	2	3	no	—
204	C₂²×C₄×C₂₃	—	2	3	no	—
205	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
206	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
207	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
208	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
209	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
211	M₁₆×C₂₃	—	2	6	no	—
212	M₁₆×C₂₃	—	2	6	no	—
213	C₂×C₈×C₂₃	—	2	6	no	—
214	C₂×C₈×C₂₃	—	2	6	no	—
215	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
216	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
217	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
218	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
219	C₄⋊C₄×C₂₃	—	2	3	no	—
220	C₄⋊C₄×C₂₃	—	2	6	no	—
221	C₄⋊C₄×C₂₃	—	2	3	no	—
222	C₄⋊C₄×C₂₃	—	2	3	no	—
223	C₄⋊C₄×C₂₃	—	2	3	no	—
224	C₄⋊C₄×C₂₃	—	2	3	no	—
225	C₄⋊C₄×C₂₃	—	2	3	no	—
226	C₄⋊C₄×C₂₃	—	2	3	no	—
227	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
228	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
229	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
230	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
231	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
232	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
233	C₂²⋊C₄×C₂₃	—	2	3	no	—
234	C₂²⋊C₄×C₂₃	—	2	1	no	—
235	C₂²⋊C₄×C₂₃	—	2	3	no	—
236	C₂²⋊C₄×C₂₃	—	2	6	no	—
237	C₂²⋊C₄×C₂₃	—	2	3	no	—
238	C₂²⋊C₄×C₂₃	—	2	3	no	—
239	C₂²⋊C₄×C₂₃	—	2	6	no	—
240	C₂²⋊C₄×C₂₃	—	2	3	no	—
241	C₂²⋊C₄×C₂₃	—	2	3	no	—
242	C₂²⋊C₄×C₂₃	—	2	1	no	—
243	C₂²⋊C₄×C₂₃	—	2	3	no	—
244	C₄²×C₂₃	—	2	3	no	—
245	C₄²×C₂₃	—	2	1	no	—
246	C₄²×C₂₃	—	2	6	no	—
247	C₄²×C₂₃	—	2	3	no	—
248	C₄²×C₂₃	—	2	3	no	—
249	C₄²×C₂₃	—	2	3	no	—
250	#60524	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂³×C₂₃	—	2	6	no	—
252	C₂³×C₂₃	—	2	1	no	—
253	C₂³×C₂₃	—	4	3	no	—
254	C₂³×C₂₃	—	4	3	no	—
255	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
256	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
257	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
258	Q₈×C₂₃	—	2	6	no	—
259	Q₈×C₂₃	—	2	3	no	—
260	Q₈×C₂₃	—	2	3	no	—
261	Q₈×C₂₃	—	4	6	no	—
262	Q₈×C₂₃	#123	1	4	yes	—
263	D₄×C₂₃	—	2	12	no	—
264	D₄×C₂₃	—	2	6	no	—
265	D₄×C₂₃	—	2	1	no	—
266	D₄×C₂₃	—	2	1	no	—
267	D₄×C₂₃	—	4	2	no	—
268	D₄×C₂₃	—	4	6	no	—
269	D₄×C₂₃	—	8	3	no	—
270	D₄×C₂₃	#123	1	12	yes	—
271	C₂×C₄×C₂₃	—	2	6	no	—
272	C₂×C₄×C₂₃	—	2	6	no	—
273	C₂×C₄×C₂₃	—	2	6	no	—
274	C₂×C₄×C₂₃	—	2	1	no	—
275	C₂×C₄×C₂₃	—	2	1	no	—
276	C₂×C₄×C₂₃	—	2	3	no	—
277	C₂×C₄×C₂₃	—	2	3	no	—
278	C₂×C₄×C₂₃	—	2	3	no	—
279	C₂×C₄×C₂₃	—	2	1	no	—
280	C₂×C₄×C₂₃	—	2	3	no	—
281	C₂×C₄×C₂₃	—	2	1	no	—
282	C₂×C₄×C₂₃	—	2	1	no	—
283	C₂×C₄×C₂₃	—	2	3	no	—
284	C₂×C₄×C₂₃	—	2	3	no	—
285	C₂×C₄×C₂₃	—	4	6	no	—
286	C₂×C₄×C₂₃	—	4	6	no	—
287	C₂×C₄×C₂₃	—	8	3	no	—
288	C₂×C₄×C₂₃	—	8	3	no	—
289	C₂×C₄×C₂₃	—	8	3	no	—
290	C₂×C₄×C₂₃	—	8	3	no	—
291	C₂×C₄×C₂₃	—	8	3	no	—
292	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
294	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
295	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
296	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
297	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
298	C₈×C₂₃	—	2	1	no	—
299	C₈×C₂₃	—	2	1	no	—
300	C₈×C₂₃	—	2	3	no	—
301	C₈×C₂₃	—	2	3	no	—
302	#3489	C₂×C₂₃	1	1	yes	characteristic
303	#3428	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#3099	C₂×C₂₃	1	3	yes	complemented
305	#3098	C₂×C₂₃	1	3	yes	complemented
306	#3097	C₂×C₂₃	1	3	yes	complemented
307	#3097	C₂×C₂₃	1	3	yes	complemented
308	#3095	C₂×C₂₃	1	3	yes	complemented
309	#3093	C₂×C₂₃	1	3	yes	complemented
310	#3091	C₂×C₂₃	1	3	yes	complemented
311	C₂×#485	C₂×C₂₃	1	3	yes	complemented
312	#2913	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#2912	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#2898	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#2898	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#2880	C₂×C₂₃	1	1	yes	characteristic, complemented
317	C₂²×C₂₃	—	2	3	no	—
318	C₂²×C₂₃	—	2	3	no	—
319	C₂²×C₂₃	—	2	3	no	—
320	C₂²×C₂₃	—	2	1	no	—
321	C₂²×C₂₃	—	2	3	no	—
322	C₂²×C₂₃	—	2	1	no	—
323	C₂²×C₂₃	—	2	1	no	—
324	C₂²×C₂₃	—	2	3	no	—
325	C₂²×C₂₃	—	4	1	no	—
326	C₂²×C₂₃	—	4	1	no	—
327	C₂²×C₂₃	—	4	1	no	—
328	C₂²×C₂₃	—	8	6	no	—
329	C₂²×C₂₃	C₂×#123	1	6	yes	—
330	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
331	C₄×C₂₃	—	2	1	no	—
332	C₄×C₂₃	—	2	3	no	—
333	C₄×C₂₃	—	4	3	no	—
334	C₄×C₂₃	—	4	1	no	—
335	C₄×C₂₃	—	4	3	no	—
336	C₄×C₂₃	—	4	3	no	—
337	C₄×C₂₃	—	4	1	no	—
338	C₄×C₂₃	—	4	1	no	—
339	C₄×C₂₃	—	4	3	no	—
340	C₄×C₂₃	C₂×#123	1	6	yes	—
341	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
342	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
343	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
344	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
347	D₄×Q₈	—	2	1	no	—
348	#548	—	2	1	no	—
349	#547	—	2	3	no	—
350	#546	—	2	3	no	—
351	#537	C₂²×C₂₃	1	3	yes	—
352	#536	C₂²×C₂₃	1	3	yes	—
353	#525	C₂²×C₂₃	1	3	yes	—
354	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	3	yes	—
355	#520	C₂²×C₂₃	1	3	yes	—
356	#520	C₂²×C₂₃	1	3	yes	—
357	#520	C₂²×C₂₃	1	1	yes	characteristic
358	#520	C₂²×C₂₃	1	1	yes	characteristic
359	#520	C₂²×C₂₃	1	1	yes	characteristic
360	#519	—	2	1	no	—
361	#518	—	2	3	no	—
362	C₄×#41	—	2	3	no	—
363	C₄×#41	—	2	1	no	—
364	C₂×#117	C₂²×C₂₃	1	3	yes	—
365	C₂×#117	C₂²×C₂₃	1	3	yes	—
366	#485	C₂²×C₂₃	1	3	yes	—
367	#485	C₂²×C₂₃	1	6	yes	—
368	#485	C₂²×C₂₃	1	3	yes	—
369	#485	C₂²×C₂₃	1	6	yes	complemented
370	#485	C₂²×C₂₃	1	6	yes	complemented
371	#485	C₂²×C₂₃	1	6	yes	complemented
372	#484	C₂²×C₂₃	1	3	yes	complemented
373	#484	C₂²×C₂₃	1	6	yes	complemented
374	#483	C₂²×C₂₃	1	3	yes	complemented
375	#483	C₂²×C₂₃	1	6	yes	complemented
376	#482	C₂²×C₂₃	1	1	yes	characteristic
377	#482	C₂²×C₂₃	1	3	yes	—
378	#482	C₂²×C₂₃	1	6	yes	complemented
379	#481	C₂²×C₂₃	1	3	yes	complemented
380	#480	C₂²×C₂₃	1	3	yes	complemented
381	#428	C₂²×C₂₃	1	3	yes	complemented
382	#428	C₂²×C₂₃	1	3	yes	complemented
383	#428	C₂²×C₂₃	1	3	yes	complemented
384	#428	C₂²×C₂₃	1	3	yes	complemented
385	#427	C₂²×C₂₃	1	3	yes	complemented
386	#427	C₂²×C₂₃	1	3	yes	complemented
387	C₂×#107	C₂²×C₂₃	1	3	yes	complemented
388	C₂×#106	C₂²×C₂₃	1	3	yes	complemented
389	#419	C₂²×C₂₃	1	3	yes	complemented
390	#419	C₂²×C₂₃	1	3	yes	complemented
391	#417	C₂²×C₂₃	1	3	yes	complemented
392	#417	C₂²×C₂₃	1	3	yes	complemented
393	#416	C₂²×C₂₃	1	3	yes	complemented
394	#416	C₂²×C₂₃	1	3	yes	complemented
395	C₂×#103	C₂²×C₂₃	1	3	yes	complemented
396	C₂×#103	C₂²×C₂₃	1	3	yes	complemented
397	#408	C₂²×C₂₃	1	3	yes	—
398	#408	C₂²×C₂₃	1	3	yes	—
399	#407	C₂²×C₂₃	1	3	yes	—
400	C₂×C₂²⋊C₈	C₂²×C₂₃	1	3	yes	—
401	C₂×C₂₃	—	2	6	no	—
402	C₂×C₂₃	—	4	3	no	—
403	C₂×C₂₃	—	4	1	no	—
404	C₂×C₂₃	—	8	1	no	—
405	C₂×C₂₃	C₂²×#123	1	1	yes	characteristic
406	C₂×C₂₃	#3584	1	1	yes	characteristic
407	C₂×C₂₃	#3586	1	1	yes	characteristic
408	2₋¹⁺⁴	—	2	4	no	—
409	2₋¹⁺⁴	—	2	4	no	—
410	C₂×#41	—	2	4	no	—
411	C₂×#41	—	2	4	no	—
412	C₂×#41	—	2	3	no	—
413	C₂×#41	D₄×C₂₃	1	3	yes	—
414	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
415	C₂×#41	C₂³×C₂₃	1	3	yes	—
416	C₂×#41	C₂³×C₂₃	1	3	yes	—
417	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
418	C₂²×Q₈	—	2	1	no	—
419	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
420	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
421	C₂²×D₄	D₄×C₂₃	1	3	yes	—
422	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
423	#131	—	2	4	no	—
424	#131	—	2	4	no	—
425	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
426	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
427	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
428	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
429	C₄⋊Q₈	—	2	3	no	—
430	#124	—	2	3	no	—
431	#123	—	2	6	no	—
432	#123	—	2	3	no	—
433	C₂²⋊Q₈	—	2	3	no	—
434	C₂²⋊Q₈	—	2	3	no	—
435	C₂²⋊Q₈	—	2	3	no	—
436	C₂²⋊Q₈	C₂³×C₂₃	1	3	yes	—
437	C₂²⋊Q₈	C₂³×C₂₃	1	6	yes	—
438	C₂²⋊Q₈	C₂³×C₂₃	1	3	yes	—
439	#121	—	2	6	no	—
440	#121	—	2	3	no	—
441	#121	C₂³×C₂₃	1	1	yes	characteristic
442	#121	C₂³×C₂₃	1	3	yes	—
443	#120	—	2	3	no	—
444	C₄×Q₈	—	2	3	no	—
445	C₄×Q₈	—	2	3	no	—
446	C₄×Q₈	—	2	1	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	—	2	1	no	—
449	C₄×D₄	—	2	6	no	—
450	C₄×D₄	—	2	6	no	—
451	C₄×D₄	—	2	3	no	—
452	C₄×D₄	—	2	3	no	—
453	C₄×D₄	—	2	3	no	—
454	C₄×D₄	—	2	3	no	—
455	C₄×D₄	—	2	3	no	—
456	C₄×D₄	—	2	3	no	—
457	C₄×D₄	—	2	3	no	—
458	#117	—	2	3	no	—
459	#117	—	2	3	no	—
460	#117	—	2	3	no	—
461	#117	—	2	3	no	—
462	#117	C₂³×C₂₃	1	3	yes	—
463	#117	C₂³×C₂₃	1	1	yes	characteristic
464	#117	C₂³×C₂₃	1	3	yes	—
465	#117	C₂³×C₂₃	1	3	yes	—
466	#117	C₂³×C₂₃	1	6	yes	—
467	#117	C₂³×C₂₃	1	3	yes	—
468	C₂×C₄⋊C₄	—	2	3	no	—
469	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
470	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
471	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
472	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
473	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
474	C₂×C₂²⋊C₄	—	2	3	no	—
475	C₂×C₂²⋊C₄	—	2	3	no	—
476	C₂×C₂²⋊C₄	—	2	3	no	—
477	C₂×C₄²	—	2	3	no	—
478	C₂×C₄²	—	2	3	no	—
479	#107	C₂³×C₂₃	1	3	yes	—
480	#107	C₂³×C₂₃	1	6	yes	—
481	#107	C₂³×C₂₃	1	1	yes	characteristic
482	#107	C₂³×C₂₃	1	3	yes	—
483	#107	C₂³×C₂₃	1	3	yes	—
484	#106	C₂³×C₂₃	1	3	yes	—
485	#106	C₂³×C₂₃	1	6	yes	—
486	#106	C₂³×C₂₃	1	1	yes	characteristic
487	#106	C₂³×C₂₃	1	3	yes	—
488	#106	C₂³×C₂₃	1	3	yes	—
489	#103	C₂³×C₂₃	1	3	yes	—
490	#103	C₂³×C₂₃	1	3	yes	—
491	#103	C₂³×C₂₃	1	6	yes	—
492	#103	C₂³×C₂₃	1	6	yes	—
493	#103	C₂³×C₂₃	1	3	yes	—
494	#103	C₂³×C₂₃	1	3	yes	—
495	#102	C₂³×C₂₃	1	1	yes	characteristic
496	#102	C₂³×C₂₃	1	1	yes	characteristic
497	#102	C₂³×C₂₃	1	3	yes	—
498	#102	C₂³×C₂₃	1	3	yes	—
499	C₂²⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
500	C₂²⋊C₈	C₂³×C₂₃	1	3	yes	—
501	C₂²⋊C₈	C₂³×C₂₃	1	3	yes	—
502	C₂²⋊C₈	C₂³×C₂₃	1	3	yes	—
503	C₂²⋊C₈	C₂³×C₂₃	1	6	yes	—
504	C₂₃	#60524	1	1	yes	characteristic, direct product factor, Sylow
505	C₂⁴	C₂×D₄×C₂₃	1	3	yes	—
506	#41	—	2	4	no	—
507	#41	—	2	12	no	—
508	#41	—	2	6	no	—
509	#41	—	2	6	no	—
510	#41	—	2	6	no	—
511	#41	—	2	6	no	—
512	#41	—	4	12	no	—
513	#41	#41×C₂₃	1	4	yes	—
514	#41	#41×C₂₃	1	4	yes	—
515	C₂×Q₈	—	2	4	no	—
516	C₂×Q₈	—	2	3	no	—
517	C₂×Q₈	—	2	6	no	—
518	C₂×Q₈	—	2	6	no	—
519	C₂×Q₈	—	4	4	no	—
520	C₂×Q₈	C₂×D₄×C₂₃	1	3	yes	—
521	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
522	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
523	C₂×D₄	—	2	12	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	3	no	—
526	C₂×D₄	—	2	3	no	—
527	C₂×D₄	C₂×D₄×C₂₃	1	6	yes	—
528	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
529	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
530	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
531	C₂²×C₄	—	2	6	no	—
532	C₂²×C₄	—	2	6	no	—
533	C₂²×C₄	—	2	3	no	—
534	C₂²×C₄	—	2	3	no	—
535	C₂²×C₄	—	2	3	no	—
536	C₂²×C₄	—	2	3	no	—
537	C₂²×C₄	—	2	3	no	—
538	C₂²×C₄	—	2	3	no	—
539	C₂²×C₄	—	2	3	no	—
540	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
541	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
542	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
543	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
544	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
545	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
546	M₁₆	—	2	6	no	—
547	M₁₆	—	2	6	no	—
548	C₂×C₈	—	2	6	no	—
549	C₂×C₈	—	2	6	no	—
550	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
551	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
552	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
553	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
554	C₄⋊C₄	—	2	3	no	—
555	C₄⋊C₄	—	2	6	no	—
556	C₄⋊C₄	—	2	3	no	—
557	C₄⋊C₄	—	2	3	no	—
558	C₄⋊C₄	—	2	3	no	—
559	C₄⋊C₄	—	2	3	no	—
560	C₄⋊C₄	—	2	3	no	—
561	C₄⋊C₄	—	2	3	no	—
562	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
563	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
564	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
565	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
566	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
567	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
568	C₂²⋊C₄	—	2	3	no	—
569	C₂²⋊C₄	—	2	1	no	—
570	C₂²⋊C₄	—	2	3	no	—
571	C₂²⋊C₄	—	2	6	no	—
572	C₂²⋊C₄	—	2	3	no	—
573	C₂²⋊C₄	—	2	3	no	—
574	C₂²⋊C₄	—	2	6	no	—
575	C₂²⋊C₄	—	2	3	no	—
576	C₂²⋊C₄	—	2	3	no	—
577	C₂²⋊C₄	—	2	1	no	—
578	C₂²⋊C₄	—	2	3	no	—
579	C₄²	—	2	3	no	—
580	C₄²	—	2	1	no	—
581	C₄²	—	2	6	no	—
582	C₄²	—	2	3	no	—
583	C₄²	—	2	3	no	—
584	C₄²	—	2	3	no	—
585	C₂³	—	2	6	no	—
586	C₂³	—	2	1	no	—
587	C₂³	—	4	3	no	—
588	C₂³	—	4	3	no	—
589	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
590	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
591	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
592	Q₈	—	2	6	no	—
593	Q₈	—	2	3	no	—
594	Q₈	—	2	3	no	—
595	Q₈	—	4	6	no	—
596	Q₈	C₂₃×#123	1	4	yes	—
597	D₄	—	2	12	no	—
598	D₄	—	2	1	no	—
599	D₄	—	2	1	no	—
600	D₄	—	2	6	no	—
601	D₄	—	4	2	no	—
602	D₄	—	4	6	no	—
603	D₄	—	8	3	no	—
604	D₄	C₂₃×#123	1	12	yes	—
605	C₂×C₄	—	2	6	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	3	no	—
609	C₂×C₄	—	2	3	no	—
610	C₂×C₄	—	2	6	no	—
611	C₂×C₄	—	2	6	no	—
612	C₂×C₄	—	2	3	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	3	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	3	no	—
618	C₂×C₄	—	2	3	no	—
619	C₂×C₄	—	4	6	no	—
620	C₂×C₄	—	4	6	no	—
621	C₂×C₄	—	8	3	no	—
622	C₂×C₄	—	8	3	no	—
623	C₂×C₄	—	8	3	no	—
624	C₂×C₄	—	8	3	no	—
625	C₂×C₄	—	8	3	no	—
626	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
627	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
628	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
629	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
630	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
631	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
632	C₈	—	2	1	no	—
633	C₈	—	2	1	no	—
634	C₈	—	2	3	no	—
635	C₈	—	2	3	no	—
636	C₂²	—	2	3	no	—
637	C₂²	—	2	3	no	—
638	C₂²	—	2	1	no	—
639	C₂²	—	2	3	no	—
640	C₂²	—	2	3	no	—
641	C₂²	—	2	1	no	—
642	C₂²	—	2	3	no	—
643	C₂²	—	2	1	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	1	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	8	6	no	—
648	C₂²	C₂×C₂₃×#123	1	6	yes	—
649	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
650	C₄	—	2	1	no	—
651	C₄	—	2	3	no	—
652	C₄	—	4	3	no	—
653	C₄	—	4	3	no	—
654	C₄	—	4	1	no	—
655	C₄	—	4	3	no	—
656	C₄	—	4	3	no	—
657	C₄	—	4	1	no	—
658	C₄	—	4	1	no	—
659	C₄	C₂×C₂₃×#123	1	6	yes	—
660	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
661	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
662	C₂	—	2	6	no	—
663	C₂	—	4	1	no	—
664	C₂	—	4	3	no	—
665	C₂	—	8	1	no	—
666	C₂	C₂²×C₂₃×#123	1	1	yes	characteristic
667	C₂	C₂₃×#3584	1	1	yes	characteristic
668	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3489	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3428	C₂	semidirect
3. C₂₃×#3099	C₂	semidirect
4. C₂₃×#3098	C₂	semidirect
5. C₂₃×#3097	C₂	semidirect
6. C₂₃×#3097	C₂	semidirect
7. C₂₃×#3095	C₂	semidirect
8. C₂₃×#3093	C₂	semidirect
9. C₂₃×#3091	C₂	semidirect
10. C₂×C₂₃×#485	C₂	semidirect
11. C₂₃×#2913	C₂	semidirect
12. C₂₃×#2912	C₂	semidirect
13. C₂₃×#2898	C₂	semidirect
14. C₂₃×#2898	C₂	semidirect
15. C₂₃×#2880	C₂	semidirect
42. C₂₃×#485	C₂²	semidirect
43. C₂₃×#485	C₂²	semidirect
44. C₂₃×#485	C₂²	semidirect
45. C₂₃×#484	C₂²	semidirect
46. C₂₃×#484	C₂²	semidirect
47. C₂₃×#483	C₂²	semidirect
48. C₂₃×#483	C₂²	semidirect
51. C₂₃×#482	C₂²	semidirect
52. C₂₃×#481	C₂²	semidirect
53. C₂₃×#480	C₂²	semidirect
54. C₂₃×#428	C₂²	semidirect
55. C₂₃×#428	C₂²	semidirect
56. C₂₃×#428	C₂²	semidirect
57. C₂₃×#428	C₂²	semidirect
58. C₂₃×#427	C₂²	semidirect
59. C₂₃×#427	C₂²	semidirect
60. C₂×C₂₃×#107	C₂²	semidirect
61. C₂×C₂₃×#106	C₂²	semidirect
62. C₂₃×#419	C₂²	semidirect
63. C₂₃×#419	C₂²	semidirect
64. C₂₃×#417	C₂²	semidirect
65. C₂₃×#417	C₂²	semidirect
66. C₂₃×#416	C₂²	semidirect
67. C₂₃×#416	C₂²	semidirect
68. C₂×C₂₃×#103	C₂²	semidirect
69. C₂×C₂₃×#103	C₂²	semidirect
250. #60524	C₂₃	direct (N × H)
303. #3428	C₂×C₂₃	semidirect
304. #3099	C₂×C₂₃	semidirect
305. #3098	C₂×C₂₃	semidirect
306. #3097	C₂×C₂₃	semidirect
307. #3097	C₂×C₂₃	semidirect
308. #3095	C₂×C₂₃	semidirect
309. #3093	C₂×C₂₃	semidirect
310. #3091	C₂×C₂₃	semidirect
311. C₂×#485	C₂×C₂₃	semidirect
312. #2913	C₂×C₂₃	semidirect
313. #2912	C₂×C₂₃	semidirect
314. #2898	C₂×C₂₃	semidirect
315. #2898	C₂×C₂₃	semidirect
316. #2880	C₂×C₂₃	semidirect
369. #485	C₂²×C₂₃	semidirect
370. #485	C₂²×C₂₃	semidirect
371. #485	C₂²×C₂₃	semidirect
372. #484	C₂²×C₂₃	semidirect
373. #484	C₂²×C₂₃	semidirect
374. #483	C₂²×C₂₃	semidirect
375. #483	C₂²×C₂₃	semidirect
378. #482	C₂²×C₂₃	semidirect
379. #481	C₂²×C₂₃	semidirect
380. #480	C₂²×C₂₃	semidirect
381. #428	C₂²×C₂₃	semidirect
382. #428	C₂²×C₂₃	semidirect
383. #428	C₂²×C₂₃	semidirect
384. #428	C₂²×C₂₃	semidirect
385. #427	C₂²×C₂₃	semidirect
386. #427	C₂²×C₂₃	semidirect
387. C₂×#107	C₂²×C₂₃	semidirect
388. C₂×#106	C₂²×C₂₃	semidirect
389. #419	C₂²×C₂₃	semidirect
390. #419	C₂²×C₂₃	semidirect
391. #417	C₂²×C₂₃	semidirect
392. #417	C₂²×C₂₃	semidirect
393. #416	C₂²×C₂₃	semidirect
394. #416	C₂²×C₂₃	semidirect
395. C₂×#103	C₂²×C₂₃	semidirect
396. C₂×#103	C₂²×C₂₃	semidirect
504. C₂₃	#60524	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×#3489	2
C₂₃×#3428	2
C₂₃×#3099	2
C₂₃×#3098	2
C₂₃×#3097	2
C₂₃×#3095	2
C₂₃×#3093	2
C₂₃×#3091	2
C₂×C₂₃×#485	2
C₂₃×#2913	2
C₂₃×#2912	2
C₂₃×#2898	2
C₂₃×#2880	2
#60524	23

Subgroups — C23×#60524 #19208150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60524 #19208150