Subgroups — C₂₃×#60882 #19208508

Order 5888.

1650 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3512	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3242	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3241	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3233	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3232	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3142	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#3099	C₂	1	2	yes	maximal, complemented
12	C₂₃×#3098	C₂	1	2	yes	maximal, complemented
13	C₂₃×#3084	C₂	1	2	yes	maximal, complemented
14	C₂₃×#3083	C₂	1	2	yes	maximal, complemented
15	C₂₃×#2922	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#557	—	2	2	no	—
17	C₂₃×#554	—	2	4	no	—
18	C₂₃×#554	—	2	2	no	—
19	C₂₃×#553	—	2	4	no	—
20	C₂₃×#553	—	2	2	no	—
21	C₂₃×#552	—	2	2	no	—
22	C₂₃×#544	C₂²	1	4	yes	—
23	C₂₃×#544	C₂²	1	2	yes	—
24	C₂₃×#541	C₂²	1	2	yes	—
25	C₂₃×#540	C₂²	1	2	yes	—
26	C₂₃×#537	C₂²	1	2	yes	—
27	C₂₃×#533	C₂²	1	1	yes	characteristic
28	C₂₃×#533	C₂²	1	2	yes	—
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₂₃×#529	C₂²	1	4	yes	—
31	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#125	C₂²	1	2	yes	—
33	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
34	C₂₃×#520	C₂²	1	2	yes	—
35	C₂₃×#520	C₂²	1	2	yes	—
36	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#501	C₂²	1	4	yes	—
38	C₂₃×#499	C₂²	1	2	yes	—
39	C₂₃×#499	C₂²	1	2	yes	—
40	C₂₃×#499	C₂²	1	4	yes	—
41	C₂₃×#499	C₂²	1	4	yes	—
42	C₂₃×#491	C₂²	1	2	yes	—
43	C₂₃×#490	C₂²	1	2	yes	—
44	C₂₃×#488	C₂²	1	2	yes	—
45	C₂₃×#488	C₂²	1	4	yes	—
46	C₂₃×#488	C₂²	1	2	yes	—
47	C₂₃×#487	C₂²	1	2	yes	—
48	C₂₃×#486	C₂²	1	2	yes	—
49	C₂₃×#485	C₂²	1	4	yes	—
50	C₂₃×#485	C₂²	1	4	yes	—
51	C₂₃×#484	C₂²	1	4	yes	—
52	C₂₃×#483	C₂²	1	4	yes	—
53	C₂₃×#482	C₂²	1	4	yes	—
54	C₂₃×#482	C₂²	1	4	yes	—
55	C₂₃×#481	C₂²	1	4	yes	—
56	C₂₃×#480	C₂²	1	4	yes	—
57	C₂₃×#479	C₂²	1	4	yes	—
58	C₂₃×#479	C₂²	1	4	yes	—
59	C₂₃×#478	C₂²	1	4	yes	—
60	C₂₃×#478	C₂²	1	4	yes	—
61	C₂₃×#477	C₂²	1	4	yes	—
62	C₂₃×#476	C₂²	1	4	yes	—
63	C₂₃×#475	C₂²	1	4	yes	—
64	C₂₃×#474	C₂²	1	4	yes	—
65	C₂₃×#433	C₂²	1	4	yes	—
66	C₂₃×#431	C₂²	1	4	yes	—
67	C₂₃×#428	C₂²	1	2	yes	—
68	C₂₃×#428	C₂²	1	2	yes	—
69	C₂×C₂₃×#107	C₂²	1	2	yes	—
70	C₂×C₂₃×#106	C₂²	1	2	yes	—
71	C₂₃×#424	C₂²	1	1	yes	characteristic
72	C₂₃×#424	C₂²	1	1	yes	characteristic
73	C₂₃×#417	C₂²	1	2	yes	—
74	C₂₃×#417	C₂²	1	2	yes	—
75	C₂₃×#416	C₂²	1	2	yes	—
76	C₂₃×#416	C₂²	1	2	yes	—
77	C₂₃×#408	C₂²	1	1	yes	characteristic
78	C₂₃×#408	C₂²	1	1	yes	characteristic
79	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
80	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
81	C₂×#41×C₂₃	C₂³	1	2	yes	—
82	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊Q₈	—	2	2	no	—
88	C₂₃×C₄⋊Q₈	—	2	4	no	—
89	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
91	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
92	C₂₃×#127	C₂³	1	1	yes	characteristic
93	C₂₃×#126	—	2	4	no	—
94	C₂₃×#126	—	2	4	no	—
95	C₂₃×#125	—	2	4	no	—
96	C₂₃×#125	—	2	2	no	—
97	C₂₃×#125	—	2	4	no	—
98	C₂₃×#125	—	2	2	no	—
99	C₂₃×#125	—	2	4	no	—
100	C₂₃×#125	—	2	2	no	—
101	C₂₃×#125	C₂³	1	4	yes	—
102	C₂₃×#125	C₂³	1	1	yes	characteristic
103	C₂₃×#125	C₂³	1	2	yes	—
104	C₂₃×#125	C₂³	1	1	yes	characteristic
105	C₂₃×#125	C₂³	1	2	yes	—
106	C₂₃×#125	C₂³	1	2	yes	—
107	C₂₃×#125	C₂³	1	2	yes	—
108	C₂₃×#124	C₂³	1	2	yes	—
109	C₂₃×#123	—	2	4	no	—
110	C₂₃×#123	—	2	4	no	—
111	C₂₃×C₂²⋊Q₈	—	2	2	no	—
112	C₂₃×C₂²⋊Q₈	—	2	2	no	—
113	C₂₃×C₂²⋊Q₈	—	2	4	no	—
114	C₂₃×C₂²⋊Q₈	—	2	4	no	—
115	C₂₃×C₂²⋊Q₈	—	2	4	no	—
116	C₂₃×C₂²⋊Q₈	—	2	4	no	—
117	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
118	C₂₃×#121	—	2	2	no	—
119	C₂₃×#121	—	2	4	no	—
120	C₂₃×#121	—	2	2	no	—
121	C₂₃×#121	—	2	4	no	—
122	C₂₃×#121	C₂³	1	4	yes	—
123	C₄×Q₈×C₂₃	—	2	2	no	—
124	C₄×Q₈×C₂₃	—	2	4	no	—
125	C₄×Q₈×C₂₃	—	2	2	no	—
126	C₄×Q₈×C₂₃	D₄	1	2	yes	—
127	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
128	C₄×D₄×C₂₃	—	2	4	no	—
129	C₄×D₄×C₂₃	—	2	2	no	—
130	C₄×D₄×C₂₃	—	2	4	no	—
131	C₄×D₄×C₂₃	—	2	4	no	—
132	C₄×D₄×C₂₃	—	2	2	no	—
133	C₄×D₄×C₂₃	D₄	1	4	yes	—
134	C₄×D₄×C₂₃	D₄	1	2	yes	—
135	C₄×D₄×C₂₃	C₂³	1	4	yes	—
136	C₂₃×#117	—	2	2	no	—
137	C₂₃×#117	—	2	4	no	—
138	C₂₃×#117	C₂³	1	2	yes	—
139	C₂₃×#117	C₂³	1	4	yes	—
140	C₂₃×#117	C₂³	1	1	yes	characteristic
141	C₂₃×#117	C₂³	1	2	yes	—
142	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
145	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
147	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
148	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
149	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
151	C₂₃×#107	C₂³	1	4	yes	—
152	C₂₃×#107	C₂³	1	4	yes	—
153	C₂₃×#107	C₂³	1	4	yes	—
154	C₂₃×#107	C₂³	1	4	yes	—
155	C₂₃×#106	C₂³	1	4	yes	—
156	C₂₃×#106	C₂³	1	4	yes	—
157	C₂₃×#106	C₂³	1	4	yes	—
158	C₂₃×#106	C₂³	1	4	yes	—
159	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
160	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
161	C₂₃×#103	C₂³	1	2	yes	—
162	C₂₃×#103	C₂³	1	2	yes	—
163	C₂₃×#103	C₂³	1	2	yes	—
164	C₂₃×#103	C₂³	1	2	yes	—
165	C₂₃×#103	C₂³	1	4	yes	—
166	C₂₃×#103	C₂³	1	4	yes	—
167	C₂₃×#102	C₂³	1	2	yes	—
168	C₂₃×#102	C₂³	1	2	yes	—
169	C₂₃×#102	C₂³	1	2	yes	—
170	C₂₃×#102	C₂³	1	2	yes	—
171	C₂₃×#102	C₂³	1	4	yes	—
172	C₂₃×#102	C₂³	1	4	yes	—
173	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
174	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
175	C₂₃×#97	C₂³	1	2	yes	—
176	C₄×C₈×C₂₃	C₂³	1	2	yes	—
177	C₄×C₈×C₂₃	C₂³	1	2	yes	—
178	C₄×C₈×C₂₃	C₂³	1	2	yes	—
179	#41×C₂₃	—	2	2	no	—
180	#41×C₂₃	—	2	2	no	—
181	#41×C₂₃	—	2	4	no	—
182	#41×C₂₃	—	2	4	no	—
183	C₂×Q₈×C₂₃	—	2	4	no	—
184	C₂×Q₈×C₂₃	—	2	2	no	—
185	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
187	C₂×D₄×C₂₃	—	2	2	no	—
188	C₂×D₄×C₂₃	—	2	4	no	—
189	C₂×D₄×C₂₃	—	2	2	no	—
190	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
191	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₂₃	—	2	2	no	—
194	C₂²×C₄×C₂₃	—	2	4	no	—
195	C₂²×C₄×C₂₃	—	2	2	no	—
196	C₂²×C₄×C₂₃	—	2	2	no	—
197	C₂²×C₄×C₂₃	—	2	4	no	—
198	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
199	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
200	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
202	M₁₆×C₂₃	—	2	2	no	—
203	M₁₆×C₂₃	—	2	2	no	—
204	C₂×C₈×C₂₃	—	2	2	no	—
205	C₂×C₈×C₂₃	—	2	2	no	—
206	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
207	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
208	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
209	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
210	C₄⋊C₄×C₂₃	—	2	4	no	—
211	C₄⋊C₄×C₂₃	—	2	2	no	—
212	C₄⋊C₄×C₂₃	—	2	2	no	—
213	C₄⋊C₄×C₂₃	—	2	4	no	—
214	C₄⋊C₄×C₂₃	—	2	4	no	—
215	C₄⋊C₄×C₂₃	—	2	4	no	—
216	C₄⋊C₄×C₂₃	—	2	4	no	—
217	C₄⋊C₄×C₂₃	—	2	4	no	—
218	C₄⋊C₄×C₂₃	—	2	4	no	—
219	C₄⋊C₄×C₂₃	—	2	4	no	—
220	C₄⋊C₄×C₂₃	—	2	2	no	—
221	C₄⋊C₄×C₂₃	—	2	4	no	—
222	C₄⋊C₄×C₂₃	—	2	2	no	—
223	C₄⋊C₄×C₂₃	—	2	2	no	—
224	C₄⋊C₄×C₂₃	—	2	4	no	—
225	C₄⋊C₄×C₂₃	—	2	2	no	—
226	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
227	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
228	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
229	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
230	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
231	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
232	C₂²⋊C₄×C₂₃	—	2	4	no	—
233	C₂²⋊C₄×C₂₃	—	2	2	no	—
234	C₂²⋊C₄×C₂₃	—	2	4	no	—
235	C₂²⋊C₄×C₂₃	—	2	2	no	—
236	C₂²⋊C₄×C₂₃	—	2	4	no	—
237	C₂²⋊C₄×C₂₃	—	2	4	no	—
238	C₂²⋊C₄×C₂₃	—	2	2	no	—
239	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
240	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
241	C₄²×C₂₃	—	2	4	no	—
242	C₄²×C₂₃	—	2	4	no	—
243	C₄²×C₂₃	—	2	2	no	—
244	C₄²×C₂₃	C₂×D₄	1	4	yes	—
245	C₄²×C₂₃	C₂×D₄	1	2	yes	—
246	C₄²×C₂₃	C₂⁴	1	2	yes	—
247	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
248	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
249	#60882	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂³×C₂₃	—	2	2	no	—
251	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
252	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
253	Q₈×C₂₃	—	2	2	no	—
254	Q₈×C₂₃	—	2	2	no	—
255	Q₈×C₂₃	—	2	2	no	—
256	Q₈×C₂₃	—	8	4	no	—
257	Q₈×C₂₃	—	8	2	no	—
258	D₄×C₂₃	—	2	4	no	—
259	D₄×C₂₃	—	2	2	no	—
260	D₄×C₂₃	—	2	2	no	—
261	D₄×C₂₃	—	2	2	no	—
262	D₄×C₂₃	—	4	4	no	—
263	D₄×C₂₃	—	8	2	no	—
264	D₄×C₂₃	—	8	4	no	—
265	C₂×C₄×C₂₃	—	2	2	no	—
266	C₂×C₄×C₂₃	—	2	2	no	—
267	C₂×C₄×C₂₃	—	2	2	no	—
268	C₂×C₄×C₂₃	—	2	2	no	—
269	C₂×C₄×C₂₃	—	2	4	no	—
270	C₂×C₄×C₂₃	—	2	4	no	—
271	C₂×C₄×C₂₃	—	2	2	no	—
272	C₂×C₄×C₂₃	—	4	4	no	—
273	C₂×C₄×C₂₃	—	4	2	no	—
274	C₂×C₄×C₂₃	—	4	4	no	—
275	C₂×C₄×C₂₃	—	8	4	no	—
276	C₂×C₄×C₂₃	—	8	2	no	—
277	C₂×C₄×C₂₃	—	8	2	no	—
278	C₂×C₄×C₂₃	—	8	4	no	—
279	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
281	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
285	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
286	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
287	C₈×C₂₃	—	2	2	no	—
288	C₈×C₂₃	—	2	2	no	—
289	C₈×C₂₃	—	2	2	no	—
290	C₈×C₂₃	—	2	2	no	—
291	#3512	C₂×C₂₃	1	1	yes	characteristic
292	#3504	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#3242	C₂×C₂₃	1	4	yes	complemented
294	#3241	C₂×C₂₃	1	4	yes	complemented
295	#3233	C₂×C₂₃	1	4	yes	complemented
296	#3232	C₂×C₂₃	1	4	yes	complemented
297	#3166	C₂×C₂₃	1	1	yes	characteristic, complemented
298	C₂×#499	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#3142	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#3131	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#3099	C₂×C₂₃	1	2	yes	complemented
302	#3098	C₂×C₂₃	1	2	yes	complemented
303	#3084	C₂×C₂₃	1	2	yes	complemented
304	#3083	C₂×C₂₃	1	2	yes	complemented
305	#2922	C₂×C₂₃	1	1	yes	characteristic, complemented
306	C₂²×C₂₃	—	2	2	no	—
307	C₂²×C₂₃	—	2	4	no	—
308	C₂²×C₂₃	—	2	2	no	—
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	—	4	2	no	—
311	C₂²×C₂₃	—	4	2	no	—
312	C₂²×C₂₃	#536	1	2	yes	—
313	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
314	C₄×C₂₃	—	2	2	no	—
315	C₄×C₂₃	—	2	4	no	—
316	C₄×C₂₃	—	2	2	no	—
317	C₄×C₂₃	—	2	2	no	—
318	C₄×C₂₃	—	4	4	no	—
319	C₄×C₂₃	—	4	4	no	—
320	C₄×C₂₃	—	4	2	no	—
321	C₄×C₂₃	—	4	2	no	—
322	C₄×C₂₃	—	4	2	no	—
323	C₄×C₂₃	#536	1	2	yes	—
324	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
325	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
326	#557	—	2	2	no	—
327	#554	—	2	4	no	—
328	#554	—	2	2	no	—
329	#553	—	2	4	no	—
330	#553	—	2	2	no	—
331	#552	—	2	2	no	—
332	#544	C₂²×C₂₃	1	2	yes	—
333	#544	C₂²×C₂₃	1	4	yes	—
334	#541	C₂²×C₂₃	1	2	yes	—
335	#540	C₂²×C₂₃	1	2	yes	—
336	#537	C₂²×C₂₃	1	2	yes	—
337	#533	C₂²×C₂₃	1	1	yes	characteristic
338	#533	C₂²×C₂₃	1	2	yes	—
339	#532	C₂²×C₂₃	1	1	yes	characteristic
340	#529	C₂²×C₂₃	1	4	yes	—
341	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×#125	C₂²×C₂₃	1	2	yes	—
344	#520	C₂²×C₂₃	1	2	yes	—
345	#520	C₂²×C₂₃	1	2	yes	—
346	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
347	#501	C₂²×C₂₃	1	4	yes	—
348	#499	C₂²×C₂₃	1	4	yes	—
349	#499	C₂²×C₂₃	1	4	yes	—
350	#499	C₂²×C₂₃	1	2	yes	—
351	#499	C₂²×C₂₃	1	2	yes	—
352	#491	C₂²×C₂₃	1	2	yes	—
353	#490	C₂²×C₂₃	1	2	yes	—
354	#488	C₂²×C₂₃	1	2	yes	—
355	#488	C₂²×C₂₃	1	2	yes	—
356	#488	C₂²×C₂₃	1	4	yes	—
357	#487	C₂²×C₂₃	1	2	yes	—
358	#486	C₂²×C₂₃	1	2	yes	—
359	#485	C₂²×C₂₃	1	4	yes	—
360	#485	C₂²×C₂₃	1	4	yes	—
361	#484	C₂²×C₂₃	1	4	yes	—
362	#483	C₂²×C₂₃	1	4	yes	—
363	#482	C₂²×C₂₃	1	4	yes	—
364	#482	C₂²×C₂₃	1	4	yes	—
365	#481	C₂²×C₂₃	1	4	yes	—
366	#480	C₂²×C₂₃	1	4	yes	—
367	#479	C₂²×C₂₃	1	4	yes	—
368	#479	C₂²×C₂₃	1	4	yes	—
369	#478	C₂²×C₂₃	1	4	yes	—
370	#478	C₂²×C₂₃	1	4	yes	—
371	#477	C₂²×C₂₃	1	4	yes	—
372	#476	C₂²×C₂₃	1	4	yes	—
373	#475	C₂²×C₂₃	1	4	yes	—
374	#474	C₂²×C₂₃	1	4	yes	—
375	#433	C₂²×C₂₃	1	4	yes	—
376	#431	C₂²×C₂₃	1	4	yes	—
377	#428	C₂²×C₂₃	1	2	yes	—
378	#428	C₂²×C₂₃	1	2	yes	—
379	C₂×#107	C₂²×C₂₃	1	2	yes	—
380	C₂×#106	C₂²×C₂₃	1	2	yes	—
381	#424	C₂²×C₂₃	1	1	yes	characteristic
382	#424	C₂²×C₂₃	1	1	yes	characteristic
383	#417	C₂²×C₂₃	1	2	yes	—
384	#417	C₂²×C₂₃	1	2	yes	—
385	#416	C₂²×C₂₃	1	2	yes	—
386	#416	C₂²×C₂₃	1	2	yes	—
387	#408	C₂²×C₂₃	1	1	yes	characteristic
388	#408	C₂²×C₂₃	1	1	yes	characteristic
389	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×C₂₃	—	2	2	no	—
392	C₂×C₂₃	—	4	2	no	—
393	C₂×C₂₃	—	8	2	no	—
394	C₂×C₂₃	C₂×#536	1	1	yes	characteristic
395	C₂×C₂₃	#3584	1	2	yes	—
396	C₂×#41	C₂³×C₂₃	1	2	yes	—
397	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
399	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
400	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₄⋊Q₈	—	2	2	no	—
403	C₄⋊Q₈	—	2	4	no	—
404	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
406	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
407	#127	C₂³×C₂₃	1	1	yes	characteristic
408	#126	—	2	4	no	—
409	#126	—	2	4	no	—
410	#125	—	2	4	no	—
411	#125	—	2	2	no	—
412	#125	—	2	4	no	—
413	#125	—	2	2	no	—
414	#125	—	2	4	no	—
415	#125	—	2	2	no	—
416	#125	C₂³×C₂₃	1	2	yes	—
417	#125	C₂³×C₂₃	1	1	yes	characteristic
418	#125	C₂³×C₂₃	1	2	yes	—
419	#125	C₂³×C₂₃	1	2	yes	—
420	#125	C₂³×C₂₃	1	4	yes	—
421	#125	C₂³×C₂₃	1	2	yes	—
422	#125	C₂³×C₂₃	1	1	yes	characteristic
423	#124	C₂³×C₂₃	1	2	yes	—
424	#123	—	2	4	no	—
425	#123	—	2	4	no	—
426	C₂²⋊Q₈	—	2	2	no	—
427	C₂²⋊Q₈	—	2	2	no	—
428	C₂²⋊Q₈	—	2	4	no	—
429	C₂²⋊Q₈	—	2	4	no	—
430	C₂²⋊Q₈	—	2	4	no	—
431	C₂²⋊Q₈	—	2	4	no	—
432	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
433	#121	—	2	2	no	—
434	#121	—	2	4	no	—
435	#121	—	2	2	no	—
436	#121	—	2	4	no	—
437	#121	C₂³×C₂₃	1	4	yes	—
438	C₄×Q₈	—	2	2	no	—
439	C₄×Q₈	—	2	4	no	—
440	C₄×Q₈	—	2	2	no	—
441	C₄×Q₈	D₄×C₂₃	1	2	yes	—
442	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
443	C₄×D₄	—	2	4	no	—
444	C₄×D₄	—	2	2	no	—
445	C₄×D₄	—	2	4	no	—
446	C₄×D₄	—	2	4	no	—
447	C₄×D₄	—	2	2	no	—
448	C₄×D₄	D₄×C₂₃	1	4	yes	—
449	C₄×D₄	D₄×C₂₃	1	2	yes	—
450	C₄×D₄	C₂³×C₂₃	1	4	yes	—
451	#117	—	2	2	no	—
452	#117	—	2	4	no	—
453	#117	C₂³×C₂₃	1	2	yes	—
454	#117	C₂³×C₂₃	1	4	yes	—
455	#117	C₂³×C₂₃	1	1	yes	characteristic
456	#117	C₂³×C₂₃	1	2	yes	—
457	C₂×C₄⋊C₄	—	2	2	no	—
458	C₂×C₄⋊C₄	—	2	2	no	—
459	C₂×C₄⋊C₄	—	2	4	no	—
460	C₂×C₄⋊C₄	—	2	2	no	—
461	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
462	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
463	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
464	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
465	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
466	#107	C₂³×C₂₃	1	4	yes	—
467	#107	C₂³×C₂₃	1	4	yes	—
468	#107	C₂³×C₂₃	1	4	yes	—
469	#107	C₂³×C₂₃	1	4	yes	—
470	#106	C₂³×C₂₃	1	4	yes	—
471	#106	C₂³×C₂₃	1	4	yes	—
472	#106	C₂³×C₂₃	1	4	yes	—
473	#106	C₂³×C₂₃	1	4	yes	—
474	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
475	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
476	#103	C₂³×C₂₃	1	2	yes	—
477	#103	C₂³×C₂₃	1	2	yes	—
478	#103	C₂³×C₂₃	1	2	yes	—
479	#103	C₂³×C₂₃	1	2	yes	—
480	#103	C₂³×C₂₃	1	4	yes	—
481	#103	C₂³×C₂₃	1	4	yes	—
482	#102	C₂³×C₂₃	1	2	yes	—
483	#102	C₂³×C₂₃	1	2	yes	—
484	#102	C₂³×C₂₃	1	2	yes	—
485	#102	C₂³×C₂₃	1	2	yes	—
486	#102	C₂³×C₂₃	1	4	yes	—
487	#102	C₂³×C₂₃	1	4	yes	—
488	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
489	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
490	#97	C₂³×C₂₃	1	2	yes	—
491	C₄×C₈	C₂³×C₂₃	1	2	yes	—
492	C₄×C₈	C₂³×C₂₃	1	2	yes	—
493	C₄×C₈	C₂³×C₂₃	1	2	yes	—
494	C₂₃	#60882	1	1	yes	characteristic, direct product factor, Sylow
495	#41	—	2	2	no	—
496	#41	—	2	2	no	—
497	#41	—	2	4	no	—
498	#41	—	2	4	no	—
499	C₂×Q₈	—	2	4	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
502	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	2	2	no	—
506	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
507	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
508	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	2	4	no	—
514	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
515	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
516	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
517	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
518	M₁₆	—	2	2	no	—
519	M₁₆	—	2	2	no	—
520	C₂×C₈	—	2	2	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
523	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
524	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
525	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	4	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	4	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
543	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
544	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
545	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
546	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
547	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
548	C₂²⋊C₄	—	2	4	no	—
549	C₂²⋊C₄	—	2	2	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	2	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	2	no	—
555	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
556	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
557	C₄²	—	2	4	no	—
558	C₄²	—	2	4	no	—
559	C₄²	—	2	2	no	—
560	C₄²	C₂×D₄×C₂₃	1	4	yes	—
561	C₄²	C₂×D₄×C₂₃	1	2	yes	—
562	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
563	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
564	C₄²	C₂⁴×C₂₃	1	2	yes	—
565	C₂³	—	2	2	no	—
566	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
567	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
568	Q₈	—	2	2	no	—
569	Q₈	—	2	2	no	—
570	Q₈	—	2	2	no	—
571	Q₈	—	8	4	no	—
572	Q₈	—	8	2	no	—
573	D₄	—	2	4	no	—
574	D₄	—	2	2	no	—
575	D₄	—	2	2	no	—
576	D₄	—	2	2	no	—
577	D₄	—	4	4	no	—
578	D₄	—	8	2	no	—
579	D₄	—	8	4	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	8	4	no	—
591	C₂×C₄	—	8	2	no	—
592	C₂×C₄	—	8	2	no	—
593	C₂×C₄	—	8	4	no	—
594	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
596	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
597	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
598	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
599	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
600	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
601	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
602	C₈	—	2	2	no	—
603	C₈	—	2	2	no	—
604	C₈	—	2	2	no	—
605	C₈	—	2	2	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	4	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	C₂₃×#536	1	2	yes	—
613	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
614	C₄	—	2	2	no	—
615	C₄	—	2	2	no	—
616	C₄	—	2	4	no	—
617	C₄	—	2	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	4	no	—
620	C₄	—	4	4	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	2	no	—
623	C₄	C₂₃×#536	1	2	yes	—
624	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
625	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
626	C₂	—	2	2	no	—
627	C₂	—	4	2	no	—
628	C₂	—	8	2	no	—
629	C₂	C₂×C₂₃×#536	1	1	yes	characteristic
630	C₂	C₂₃×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3512	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
2. C₂₃×#3504	C₂	semidirect
3. C₂₃×#3242	C₂	semidirect
4. C₂₃×#3241	C₂	semidirect
5. C₂₃×#3233	C₂	semidirect
6. C₂₃×#3232	C₂	semidirect
7. C₂₃×#3166	C₂	semidirect
8. C₂×C₂₃×#499	C₂	semidirect
9. C₂₃×#3142	C₂	semidirect
10. C₂₃×#3131	C₂	semidirect
11. C₂₃×#3099	C₂	semidirect
12. C₂₃×#3098	C₂	semidirect
13. C₂₃×#3084	C₂	semidirect
14. C₂₃×#3083	C₂	semidirect
15. C₂₃×#2922	C₂	semidirect
249. #60882	C₂₃	direct (N × H)
292. #3504	C₂×C₂₃	semidirect
293. #3242	C₂×C₂₃	semidirect
294. #3241	C₂×C₂₃	semidirect
295. #3233	C₂×C₂₃	semidirect
296. #3232	C₂×C₂₃	semidirect
297. #3166	C₂×C₂₃	semidirect
298. C₂×#499	C₂×C₂₃	semidirect
299. #3142	C₂×C₂₃	semidirect
300. #3131	C₂×C₂₃	semidirect
301. #3099	C₂×C₂₃	semidirect
302. #3098	C₂×C₂₃	semidirect
303. #3084	C₂×C₂₃	semidirect
304. #3083	C₂×C₂₃	semidirect
305. #2922	C₂×C₂₃	semidirect
494. C₂₃	#60882	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×#3512	2
C₂₃×#3504	2
C₂₃×#3242	2
C₂₃×#3241	2
C₂₃×#3233	2
C₂₃×#3232	2
C₂₃×#3166	2
C₂×C₂₃×#499	2
C₂₃×#3142	2
C₂₃×#3131	2
C₂₃×#3099	2
C₂₃×#3098	2
C₂₃×#3084	2
C₂₃×#3083	2
C₂₃×#2922	2
#60882	23

Subgroups — C23×#60882 #19208508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60882 #19208508