Subgroups — C₂₃×#36590 #19184216

Order 5888.

1666 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2861	C₂	1	2	yes	maximal
2	C₂₃×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2594	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2582	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2538	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#2525	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2514	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2513	C₂	1	2	yes	maximal
9	C₂₃×#2494	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2421	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2362	C₂	1	2	yes	maximal, complemented
12	C₂₃×#2355	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#2354	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#2347	C₂	1	2	yes	maximal, complemented
15	C₄×C₂₃×#125	C₂	1	2	yes	maximal, complemented
16	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
17	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₂₃×#125	C₂²	1	2	yes	—
19	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#117	C₂²	1	2	yes	—
27	C₂₃×#401	C₂²	1	4	yes	—
28	C₂₃×#400	C₂²	1	4	yes	—
29	C₂₃×#398	C₂²	1	4	yes	—
30	C₂₃×#397	C₂²	1	4	yes	—
31	C₂₃×#397	C₂²	1	4	yes	—
32	C₂₃×#396	C₂²	1	4	yes	—
33	C₂₃×#391	C₂²	1	2	yes	—
34	C₂₃×#391	C₂²	1	1	yes	characteristic
35	C₂₃×#391	C₂²	1	2	yes	—
36	C₂₃×#391	C₂²	1	2	yes	—
37	C₂₃×#390	C₂²	1	1	yes	characteristic
38	C₂₃×#390	C₂²	1	2	yes	—
39	C₂₃×#390	C₂²	1	2	yes	—
40	C₂₃×#389	C₂²	1	2	yes	—
41	C₂₃×#389	C₂²	1	2	yes	—
42	C₂₃×#389	C₂²	1	2	yes	—
43	C₂₃×#389	C₂²	1	2	yes	—
44	C₂₃×#389	C₂²	1	2	yes	—
45	C₂₃×#389	C₂²	1	2	yes	—
46	C₂₃×#389	C₂²	1	2	yes	—
47	C₂₃×#388	C₂²	1	4	yes	—
48	C₂₃×#388	C₂²	1	4	yes	—
49	C₂₃×#388	C₂²	1	4	yes	—
50	C₂₃×#388	C₂²	1	4	yes	—
51	C₂₃×#388	C₂²	1	2	yes	—
52	C₂₃×#388	C₂²	1	4	yes	—
53	C₂₃×#388	C₂²	1	4	yes	—
54	C₂₃×#388	C₂²	1	2	yes	—
55	C₂₃×#388	C₂²	1	4	yes	—
56	C₂₃×#387	C₂²	1	4	yes	—
57	C₂₃×#387	C₂²	1	4	yes	—
58	C₂₃×#387	C₂²	1	4	yes	—
59	C₂₃×#387	C₂²	1	4	yes	—
60	C₂₃×#384	C₂²	1	1	yes	characteristic
61	C₂₃×#384	C₂²	1	1	yes	characteristic
62	C₂₃×#383	C₂²	1	4	yes	—
63	C₂₃×#382	C₂²	1	2	yes	—
64	C₂₃×#382	C₂²	1	2	yes	—
65	C₂₃×#382	C₂²	1	1	yes	characteristic
66	C₂₃×#380	C₂²	1	2	yes	—
67	C₂₃×#380	C₂²	1	2	yes	—
68	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
69	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
74	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
77	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
78	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
81	C₄³×C₂₃	C₂²	1	1	yes	characteristic
82	C₄³×C₂₃	C₂²	1	2	yes	—
83	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
87	C₂₃×#125	—	2	8	no	—
88	C₂₃×#124	—	2	4	no	—
89	C₂₃×C₂²⋊Q₈	—	2	8	no	—
90	C₄×Q₈×C₂₃	—	2	4	no	—
91	C₄×Q₈×C₂₃	—	2	4	no	—
92	C₄×Q₈×C₂₃	—	2	4	no	—
93	C₄×Q₈×C₂₃	—	2	4	no	—
94	C₄×D₄×C₂₃	—	2	4	no	—
95	C₂₃×#117	—	2	8	no	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
124	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
127	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
128	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
130	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
131	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
132	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
133	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
134	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
135	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
138	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
140	C₂₃×#95	C₂³	1	2	yes	—
141	C₂₃×#95	C₂³	1	2	yes	—
142	C₂₃×#95	C₂³	1	2	yes	—
143	C₂₃×#95	C₂³	1	2	yes	—
144	C₂₃×#95	C₂³	1	2	yes	—
145	C₂₃×#95	C₂³	1	2	yes	—
146	C₂₃×#95	C₂³	1	2	yes	—
147	C₂₃×#95	C₂³	1	4	yes	—
148	C₂₃×#95	C₂³	1	4	yes	—
149	C₂₃×#95	C₂³	1	4	yes	—
150	C₂₃×#95	C₂³	1	4	yes	—
151	C₂₃×#95	C₂³	1	4	yes	—
152	C₂₃×#95	C₂³	1	4	yes	—
153	C₂₃×#95	C₂³	1	4	yes	—
154	C₂₃×#95	C₂³	1	4	yes	—
155	C₂₃×#95	C₂³	1	4	yes	—
156	C₂₃×#95	C₂³	1	4	yes	—
157	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
158	C₂×Q₈×C₂₃	—	2	2	no	—
159	C₂×Q₈×C₂₃	—	2	2	no	—
160	C₂×Q₈×C₂₃	—	2	2	no	—
161	C₂×Q₈×C₂₃	—	2	2	no	—
162	C₂×Q₈×C₂₃	—	2	2	no	—
163	C₂×Q₈×C₂₃	#41	1	4	yes	—
164	C₂×D₄×C₂₃	—	2	2	no	—
165	C₂×D₄×C₂₃	—	2	2	no	—
166	C₂×D₄×C₂₃	—	2	2	no	—
167	C₂²×C₄×C₂₃	—	2	4	no	—
168	C₂²×C₄×C₂₃	—	2	4	no	—
169	C₂²×C₄×C₂₃	—	2	4	no	—
170	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₂₃	—	2	4	no	—
187	C₄⋊C₄×C₂₃	—	2	2	no	—
188	C₄⋊C₄×C₂₃	—	2	4	no	—
189	C₄⋊C₄×C₂₃	—	2	4	no	—
190	C₄⋊C₄×C₂₃	—	2	4	no	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	2	no	—
193	C₄⋊C₄×C₂₃	—	2	2	no	—
194	C₄⋊C₄×C₂₃	—	2	4	no	—
195	C₄⋊C₄×C₂₃	—	2	4	no	—
196	C₄⋊C₄×C₂₃	—	2	8	no	—
197	C₄⋊C₄×C₂₃	—	2	8	no	—
198	C₄⋊C₄×C₂₃	—	2	8	no	—
199	C₄⋊C₄×C₂₃	—	2	4	no	—
200	C₄⋊C₄×C₂₃	—	2	4	no	—
201	C₄⋊C₄×C₂₃	—	4	2	no	—
202	C₄⋊C₄×C₂₃	—	4	2	no	—
203	C₄⋊C₄×C₂₃	—	4	4	no	—
204	C₄⋊C₄×C₂₃	#41	1	4	yes	—
205	C₂²⋊C₄×C₂₃	—	2	8	no	—
206	C₂²⋊C₄×C₂₃	—	2	4	no	—
207	C₂²⋊C₄×C₂₃	—	2	8	no	—
208	C₂²⋊C₄×C₂₃	—	2	4	no	—
209	C₂²⋊C₄×C₂₃	—	2	4	no	—
210	C₂²⋊C₄×C₂₃	—	4	2	no	—
211	C₂²⋊C₄×C₂₃	—	4	4	no	—
212	C₂²⋊C₄×C₂₃	—	4	2	no	—
213	C₂²⋊C₄×C₂₃	—	4	4	no	—
214	C₄²×C₂₃	—	2	8	no	—
215	C₄²×C₂₃	—	2	4	no	—
216	C₄²×C₂₃	—	2	4	no	—
217	C₄²×C₂₃	—	2	2	no	—
218	C₄²×C₂₃	—	2	2	no	—
219	C₄²×C₂₃	—	2	4	no	—
220	C₄²×C₂₃	—	2	4	no	—
221	C₄²×C₂₃	—	2	4	no	—
222	C₄²×C₂₃	—	2	2	no	—
223	C₄²×C₂₃	—	4	2	no	—
224	C₄²×C₂₃	—	4	1	no	—
225	C₄²×C₂₃	—	4	1	no	—
226	C₄²×C₂₃	—	4	4	no	—
227	C₄²×C₂₃	—	4	2	no	—
228	C₄²×C₂₃	#41	1	4	yes	—
229	C₄²×C₂₃	#41	1	4	yes	—
230	#36590	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂³×C₂₃	—	2	2	no	—
232	C₂³×C₂₃	—	2	2	no	—
233	C₂³×C₂₃	—	2	2	no	—
234	C₂³×C₂₃	—	2	2	no	—
235	C₂³×C₂₃	—	2	2	no	—
236	C₂³×C₂₃	—	2	2	no	—
237	C₂³×C₂₃	—	2	2	no	—
238	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
239	Q₈×C₂₃	—	2	8	no	—
240	Q₈×C₂₃	—	4	4	no	—
241	D₄×C₂₃	—	4	4	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	4	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	2	no	—
260	C₂×C₄×C₂₃	—	2	2	no	—
261	C₂×C₄×C₂₃	—	2	4	no	—
262	C₂×C₄×C₂₃	—	2	2	no	—
263	C₂×C₄×C₂₃	—	2	4	no	—
264	C₂×C₄×C₂₃	—	2	4	no	—
265	C₂×C₄×C₂₃	—	2	2	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	2	no	—
268	C₂×C₄×C₂₃	—	2	1	no	—
269	C₂×C₄×C₂₃	—	2	2	no	—
270	C₂×C₄×C₂₃	—	2	2	no	—
271	C₂×C₄×C₂₃	—	2	2	no	—
272	C₂×C₄×C₂₃	—	2	1	no	—
273	C₂×C₄×C₂₃	—	2	1	no	—
274	C₂×C₄×C₂₃	—	2	2	no	—
275	C₂×C₄×C₂₃	—	2	4	no	—
276	C₂×C₄×C₂₃	—	2	2	no	—
277	C₂×C₄×C₂₃	—	2	4	no	—
278	C₂×C₄×C₂₃	—	2	2	no	—
279	C₂×C₄×C₂₃	—	4	8	no	—
280	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
281	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
282	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
283	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
284	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
285	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
286	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
287	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
288	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
289	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
290	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
291	#2861	C₂×C₂₃	1	2	yes	—
292	#2859	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#2594	C₂×C₂₃	1	4	yes	complemented
294	#2582	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#2538	C₂×C₂₃	1	1	yes	characteristic
296	#2525	C₂×C₂₃	1	4	yes	complemented
297	#2514	C₂×C₂₃	1	2	yes	complemented
298	#2513	C₂×C₂₃	1	2	yes	—
299	#2494	C₂×C₂₃	1	2	yes	complemented
300	#2421	C₂×C₂₃	1	2	yes	complemented
301	#2362	C₂×C₂₃	1	2	yes	complemented
302	#2355	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#2354	C₂×C₂₃	1	1	yes	characteristic
304	#2347	C₂×C₂₃	1	2	yes	complemented
305	C₄×#125	C₂×C₂₃	1	2	yes	complemented
306	C₄×#124	C₂×C₂₃	1	1	yes	characteristic
307	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
308	C₂²×C₂₃	—	4	2	no	—
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	—	4	2	no	—
311	C₂²×C₂₃	—	4	2	no	—
312	C₂²×C₂₃	—	4	2	no	—
313	C₂²×C₂₃	—	4	2	no	—
314	C₂²×C₂₃	—	4	2	no	—
315	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
316	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
317	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
318	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
319	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
320	C₂²×C₂₃	#585	1	1	yes	characteristic
321	C₂²×C₂₃	#585	1	1	yes	characteristic
322	C₄×C₂₃	—	2	4	no	—
323	C₄×C₂₃	—	2	2	no	—
324	C₄×C₂₃	—	2	2	no	—
325	C₄×C₂₃	—	2	2	no	—
326	C₄×C₂₃	—	2	4	no	—
327	C₄×C₂₃	—	4	2	no	—
328	C₄×C₂₃	—	4	2	no	—
329	C₄×C₂₃	—	4	4	no	—
330	C₄×C₂₃	—	4	4	no	—
331	C₄×C₂₃	—	4	2	no	—
332	C₄×C₂₃	—	4	2	no	—
333	C₄×C₂₃	—	4	1	no	—
334	C₄×C₂₃	—	4	2	no	—
335	C₄×C₂₃	—	4	1	no	—
336	C₄×C₂₃	#529	1	4	yes	—
337	C₄×C₂₃	#556	1	4	yes	—
338	C₂×#125	C₂²×C₂₃	1	2	yes	—
339	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
341	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#117	C₂²×C₂₃	1	2	yes	—
347	#401	C₂²×C₂₃	1	4	yes	—
348	#400	C₂²×C₂₃	1	4	yes	—
349	#398	C₂²×C₂₃	1	4	yes	—
350	#397	C₂²×C₂₃	1	4	yes	—
351	#397	C₂²×C₂₃	1	4	yes	—
352	#396	C₂²×C₂₃	1	4	yes	—
353	#391	C₂²×C₂₃	1	2	yes	—
354	#391	C₂²×C₂₃	1	2	yes	—
355	#391	C₂²×C₂₃	1	2	yes	—
356	#391	C₂²×C₂₃	1	1	yes	characteristic
357	#390	C₂²×C₂₃	1	2	yes	—
358	#390	C₂²×C₂₃	1	2	yes	—
359	#390	C₂²×C₂₃	1	1	yes	characteristic
360	#389	C₂²×C₂₃	1	2	yes	—
361	#389	C₂²×C₂₃	1	2	yes	—
362	#389	C₂²×C₂₃	1	2	yes	—
363	#389	C₂²×C₂₃	1	2	yes	—
364	#389	C₂²×C₂₃	1	2	yes	—
365	#389	C₂²×C₂₃	1	2	yes	—
366	#389	C₂²×C₂₃	1	2	yes	—
367	#388	C₂²×C₂₃	1	2	yes	—
368	#388	C₂²×C₂₃	1	4	yes	—
369	#388	C₂²×C₂₃	1	4	yes	—
370	#388	C₂²×C₂₃	1	4	yes	—
371	#388	C₂²×C₂₃	1	4	yes	—
372	#388	C₂²×C₂₃	1	4	yes	—
373	#388	C₂²×C₂₃	1	4	yes	—
374	#388	C₂²×C₂₃	1	4	yes	—
375	#388	C₂²×C₂₃	1	2	yes	—
376	#387	C₂²×C₂₃	1	4	yes	—
377	#387	C₂²×C₂₃	1	4	yes	—
378	#387	C₂²×C₂₃	1	4	yes	—
379	#387	C₂²×C₂₃	1	4	yes	—
380	#384	C₂²×C₂₃	1	1	yes	characteristic
381	#384	C₂²×C₂₃	1	1	yes	characteristic
382	#383	C₂²×C₂₃	1	4	yes	—
383	#382	C₂²×C₂₃	1	1	yes	characteristic
384	#382	C₂²×C₂₃	1	2	yes	—
385	#382	C₂²×C₂₃	1	2	yes	—
386	#380	C₂²×C₂₃	1	2	yes	—
387	#380	C₂²×C₂₃	1	2	yes	—
388	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
389	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
390	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
391	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
392	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
393	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
394	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
395	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
396	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
397	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
398	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
399	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
400	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
401	C₄³	C₂²×C₂₃	1	2	yes	—
402	C₄³	C₂²×C₂₃	1	1	yes	characteristic
403	C₂×C₂₃	—	8	2	no	—
404	C₂×C₂₃	C₂×#529	1	1	yes	characteristic
405	C₂×C₂₃	C₂×#556	1	1	yes	characteristic
406	C₂×C₂₃	#3480	1	1	yes	characteristic
407	C₂×C₂₃	#3480	1	1	yes	characteristic
408	C₂×C₂₃	#3512	1	1	yes	characteristic
409	C₂×C₂₃	#3515	1	1	yes	characteristic
410	C₂×C₂₃	#3518	1	1	yes	characteristic
411	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
412	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
413	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
415	#125	—	2	8	no	—
416	#124	—	2	4	no	—
417	C₂²⋊Q₈	—	2	8	no	—
418	C₄×Q₈	—	2	4	no	—
419	C₄×Q₈	—	2	4	no	—
420	C₄×Q₈	—	2	4	no	—
421	C₄×Q₈	—	2	4	no	—
422	C₄×D₄	—	2	4	no	—
423	#117	—	2	8	no	—
424	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
430	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
431	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
432	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
433	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
434	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
435	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
436	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
438	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
439	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
440	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
441	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
442	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
443	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
452	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
453	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
454	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
456	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
457	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
458	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
459	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
460	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
461	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
462	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
463	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
464	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
465	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
466	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
467	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
468	#95	C₂³×C₂₃	1	2	yes	—
469	#95	C₂³×C₂₃	1	2	yes	—
470	#95	C₂³×C₂₃	1	2	yes	—
471	#95	C₂³×C₂₃	1	2	yes	—
472	#95	C₂³×C₂₃	1	2	yes	—
473	#95	C₂³×C₂₃	1	2	yes	—
474	#95	C₂³×C₂₃	1	2	yes	—
475	#95	C₂³×C₂₃	1	4	yes	—
476	#95	C₂³×C₂₃	1	4	yes	—
477	#95	C₂³×C₂₃	1	4	yes	—
478	#95	C₂³×C₂₃	1	4	yes	—
479	#95	C₂³×C₂₃	1	4	yes	—
480	#95	C₂³×C₂₃	1	4	yes	—
481	#95	C₂³×C₂₃	1	4	yes	—
482	#95	C₂³×C₂₃	1	4	yes	—
483	#95	C₂³×C₂₃	1	4	yes	—
484	#95	C₂³×C₂₃	1	4	yes	—
485	C₂₃	#36590	1	1	yes	characteristic, direct product factor, Sylow
486	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
487	C₂×Q₈	—	2	2	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	2	no	—
490	C₂×Q₈	—	2	2	no	—
491	C₂×Q₈	—	2	2	no	—
492	C₂×Q₈	#41×C₂₃	1	4	yes	—
493	C₂×D₄	—	2	2	no	—
494	C₂×D₄	—	2	2	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂²×C₄	—	2	4	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
503	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
505	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
507	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
508	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
509	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
510	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
512	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
513	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
514	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
515	C₄⋊C₄	—	2	4	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	4	no	—
518	C₄⋊C₄	—	2	4	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	2	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	8	no	—
526	C₄⋊C₄	—	2	8	no	—
527	C₄⋊C₄	—	2	8	no	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	4	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	—	4	2	no	—
532	C₄⋊C₄	—	4	4	no	—
533	C₄⋊C₄	#41×C₂₃	1	4	yes	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	8	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	4	2	no	—
540	C₂²⋊C₄	—	4	4	no	—
541	C₂²⋊C₄	—	4	2	no	—
542	C₂²⋊C₄	—	4	4	no	—
543	C₄²	—	2	8	no	—
544	C₄²	—	2	4	no	—
545	C₄²	—	2	4	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	2	2	no	—
548	C₄²	—	2	4	no	—
549	C₄²	—	2	4	no	—
550	C₄²	—	2	4	no	—
551	C₄²	—	2	2	no	—
552	C₄²	—	4	2	no	—
553	C₄²	—	4	1	no	—
554	C₄²	—	4	1	no	—
555	C₄²	—	4	4	no	—
556	C₄²	—	4	2	no	—
557	C₄²	#41×C₂₃	1	4	yes	—
558	C₄²	#41×C₂₃	1	4	yes	—
559	C₂³	—	2	2	no	—
560	C₂³	—	2	2	no	—
561	C₂³	—	2	2	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	2	no	—
566	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
567	Q₈	—	2	8	no	—
568	Q₈	—	4	4	no	—
569	D₄	—	4	4	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	4	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	4	8	no	—
608	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
609	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
610	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
611	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
612	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
613	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
614	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
615	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
616	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
617	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
618	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	2	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	2	no	—
626	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
627	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
628	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
629	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
630	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
631	C₂²	C₂₃×#585	1	1	yes	characteristic
632	C₂²	C₂₃×#585	1	1	yes	characteristic
633	C₄	—	2	4	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	2	no	—
636	C₄	—	2	2	no	—
637	C₄	—	2	4	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	4	no	—
641	C₄	—	4	4	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	2	no	—
644	C₄	—	4	1	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	1	no	—
647	C₄	C₂₃×#529	1	4	yes	—
648	C₄	C₂₃×#556	1	4	yes	—
649	C₂	—	8	2	no	—
650	C₂	C₂×C₂₃×#529	1	1	yes	characteristic
651	C₂	C₂×C₂₃×#556	1	1	yes	characteristic
652	C₂	C₂₃×#3480	1	1	yes	characteristic
653	C₂	C₂₃×#3480	1	1	yes	characteristic
654	C₂	C₂₃×#3512	1	1	yes	characteristic
655	C₂	C₂₃×#3515	1	1	yes	characteristic
656	C₂	C₂₃×#3518	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2859	C₂	semidirect
3. C₂₃×#2594	C₂	semidirect
4. C₂₃×#2582	C₂	semidirect
6. C₂₃×#2525	C₂	semidirect
7. C₂₃×#2514	C₂	semidirect
9. C₂₃×#2494	C₂	semidirect
10. C₂₃×#2421	C₂	semidirect
11. C₂₃×#2362	C₂	semidirect
12. C₂₃×#2355	C₂	semidirect
14. C₂₃×#2347	C₂	semidirect
15. C₄×C₂₃×#125	C₂	semidirect
17. C₄²×Q₈×C₂₃	C₂	semidirect
230. #36590	C₂₃	direct (N × H)
292. #2859	C₂×C₂₃	semidirect
293. #2594	C₂×C₂₃	semidirect
294. #2582	C₂×C₂₃	semidirect
296. #2525	C₂×C₂₃	semidirect
297. #2514	C₂×C₂₃	semidirect
299. #2494	C₂×C₂₃	semidirect
300. #2421	C₂×C₂₃	semidirect
301. #2362	C₂×C₂₃	semidirect
302. #2355	C₂×C₂₃	semidirect
304. #2347	C₂×C₂₃	semidirect
305. C₄×#125	C₂×C₂₃	semidirect
307. C₄²×Q₈	C₂×C₂₃	semidirect
485. C₂₃	#36590	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×#2861	2
C₂₃×#2859	2
C₂₃×#2594	2
C₂₃×#2582	2
C₂₃×#2538	2
C₂₃×#2525	2
C₂₃×#2514	2
C₂₃×#2513	2
C₂₃×#2494	2
C₂₃×#2421	2
C₂₃×#2362	2
C₂₃×#2355	2
C₂₃×#2354	2
C₂₃×#2347	2
C₄×C₂₃×#125	2
C₄×C₂₃×#124	2
C₄²×Q₈×C₂₃	2
#36590	23

Subgroups — C23×#36590 #19184216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36590 #19184216