Subgroups — C₂₃×#36601 #19184227

Order 5888.

1658 conjugacy classes of subgroups.

The subgroup lattice has 564 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×#2596	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2581	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2580	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#2573	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2512	C₂	1	2	yes	maximal
8	C₂₃×#2511	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2469	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2373	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2357	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#2356	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₂₃×#125	C₂	1	2	yes	maximal, complemented
14	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
15	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#126	C₂²	1	4	yes	—
17	C₂×C₂₃×#125	C₂²	1	2	yes	—
18	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#400	C₂²	1	4	yes	—
24	C₂₃×#400	C₂²	1	4	yes	—
25	C₂₃×#400	C₂²	1	2	yes	—
26	C₂₃×#399	C₂²	1	2	yes	—
27	C₂₃×#397	C₂²	1	4	yes	—
28	C₂₃×#396	C₂²	1	4	yes	—
29	C₂₃×#395	C₂²	1	2	yes	—
30	C₂₃×#393	C₂²	1	2	yes	—
31	C₂₃×#391	C₂²	1	1	yes	characteristic
32	C₂₃×#391	C₂²	1	2	yes	—
33	C₂₃×#391	C₂²	1	2	yes	—
34	C₂₃×#390	C₂²	1	1	yes	characteristic
35	C₂₃×#390	C₂²	1	2	yes	—
36	C₂₃×#390	C₂²	1	2	yes	—
37	C₂₃×#389	C₂²	1	4	yes	—
38	C₂₃×#389	C₂²	1	2	yes	—
39	C₂₃×#389	C₂²	1	2	yes	—
40	C₂₃×#389	C₂²	1	4	yes	—
41	C₂₃×#388	C₂²	1	4	yes	—
42	C₂₃×#388	C₂²	1	4	yes	—
43	C₂₃×#388	C₂²	1	4	yes	—
44	C₂₃×#388	C₂²	1	4	yes	—
45	C₂₃×#388	C₂²	1	4	yes	—
46	C₂₃×#388	C₂²	1	4	yes	—
47	C₂₃×#388	C₂²	1	4	yes	—
48	C₂₃×#387	C₂²	1	4	yes	—
49	C₂₃×#387	C₂²	1	4	yes	—
50	C₂₃×#387	C₂²	1	4	yes	—
51	C₂₃×#387	C₂²	1	4	yes	—
52	C₂₃×#387	C₂²	1	4	yes	—
53	C₂₃×#387	C₂²	1	4	yes	—
54	C₂₃×#385	C₂²	1	4	yes	—
55	C₂₃×#384	C₂²	1	2	yes	—
56	C₂₃×#383	C₂²	1	2	yes	—
57	C₂₃×#382	C₂²	1	1	yes	characteristic
58	C₂₃×#382	C₂²	1	1	yes	characteristic
59	C₂₃×#382	C₂²	1	1	yes	characteristic
60	C₂₃×#381	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
65	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
66	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
67	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
68	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
71	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
72	C₂₃×#376	C₂²	1	2	yes	—
73	C₄³×C₂₃	C₂²	1	1	yes	characteristic
74	C₄³×C₂₃	C₂²	1	2	yes	—
75	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
79	C₂₃×#126	—	2	16	no	—
80	C₂₃×#125	—	2	8	no	—
81	C₂₃×#124	—	2	4	no	—
82	C₄×Q₈×C₂₃	—	2	8	no	—
83	C₄×Q₈×C₂₃	—	2	4	no	—
84	C₄×Q₈×C₂₃	—	2	4	no	—
85	C₄×D₄×C₂₃	—	2	4	no	—
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
109	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
110	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
113	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
114	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
115	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
116	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
117	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
118	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
119	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
120	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
121	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
123	C₂₃×#95	C₂³	1	1	yes	characteristic
124	C₂₃×#95	C₂³	1	2	yes	—
125	C₂₃×#95	C₂³	1	4	yes	—
126	C₂₃×#95	C₂³	1	4	yes	—
127	C₂₃×#95	C₂³	1	2	yes	—
128	C₂₃×#95	C₂³	1	4	yes	—
129	C₂₃×#95	C₂³	1	1	yes	characteristic
130	C₂₃×#95	C₂³	1	2	yes	—
131	C₂₃×#95	C₂³	1	2	yes	—
132	C₂₃×#95	C₂³	1	4	yes	—
133	C₂₃×#95	C₂³	1	4	yes	—
134	C₂₃×#95	C₂³	1	4	yes	—
135	C₂₃×#95	C₂³	1	4	yes	—
136	C₂₃×#95	C₂³	1	2	yes	—
137	C₂₃×#95	C₂³	1	2	yes	—
138	C₂₃×#95	C₂³	1	4	yes	—
139	C₂₃×#95	C₂³	1	4	yes	—
140	C₂₃×#95	C₂³	1	4	yes	—
141	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
142	C₂×Q₈×C₂₃	—	2	4	no	—
143	C₂×Q₈×C₂₃	—	2	2	no	—
144	C₂×Q₈×C₂₃	—	2	2	no	—
145	C₂×Q₈×C₂₃	—	2	2	no	—
146	C₂×Q₈×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	2	2	no	—
149	C₂×D₄×C₂₃	—	2	2	no	—
150	C₂²×C₄×C₂₃	—	2	4	no	—
151	C₂²×C₄×C₂₃	—	4	4	no	—
152	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
166	C₄⋊C₄×C₂₃	—	2	2	no	—
167	C₄⋊C₄×C₂₃	—	2	4	no	—
168	C₄⋊C₄×C₂₃	—	2	4	no	—
169	C₄⋊C₄×C₂₃	—	2	4	no	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	2	no	—
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	8	no	—
174	C₄⋊C₄×C₂₃	—	2	8	no	—
175	C₄⋊C₄×C₂₃	—	2	4	no	—
176	C₄⋊C₄×C₂₃	—	2	4	no	—
177	C₄⋊C₄×C₂₃	—	2	8	no	—
178	C₄⋊C₄×C₂₃	—	2	8	no	—
179	C₄⋊C₄×C₂₃	—	4	4	no	—
180	C₄⋊C₄×C₂₃	—	4	4	no	—
181	C₄⋊C₄×C₂₃	#41	1	8	yes	—
182	C₂²⋊C₄×C₂₃	—	2	4	no	—
183	C₂²⋊C₄×C₂₃	—	2	8	no	—
184	C₂²⋊C₄×C₂₃	—	2	8	no	—
185	C₂²⋊C₄×C₂₃	—	2	8	no	—
186	C₂²⋊C₄×C₂₃	—	2	8	no	—
187	C₂²⋊C₄×C₂₃	—	4	4	no	—
188	C₂²⋊C₄×C₂₃	—	4	4	no	—
189	C₄²×C₂₃	—	2	4	no	—
190	C₄²×C₂₃	—	2	4	no	—
191	C₄²×C₂₃	—	2	4	no	—
192	C₄²×C₂₃	—	2	4	no	—
193	C₄²×C₂₃	—	2	4	no	—
194	C₄²×C₂₃	—	2	8	no	—
195	C₄²×C₂₃	—	2	2	no	—
196	C₄²×C₂₃	—	2	4	no	—
197	C₄²×C₂₃	—	4	2	no	—
198	C₄²×C₂₃	—	4	2	no	—
199	C₄²×C₂₃	—	4	2	no	—
200	C₄²×C₂₃	—	4	4	no	—
201	C₄²×C₂₃	#41	1	4	yes	—
202	C₄²×C₂₃	#41	1	4	yes	—
203	#36601	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₂₃	—	2	4	no	—
205	C₂³×C₂₃	—	2	2	no	—
206	C₂³×C₂₃	—	2	2	no	—
207	C₂³×C₂₃	—	2	2	no	—
208	C₂³×C₂₃	—	2	4	no	—
209	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
210	Q₈×C₂₃	—	2	8	no	—
211	Q₈×C₂₃	—	4	4	no	—
212	D₄×C₂₃	—	4	4	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	2	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	4	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	4	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	4	no	—
233	C₂×C₄×C₂₃	—	2	4	no	—
234	C₂×C₄×C₂₃	—	2	2	no	—
235	C₂×C₄×C₂₃	—	2	2	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	2	4	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	2	4	no	—
242	C₂×C₄×C₂₃	—	8	4	no	—
243	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
244	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
245	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
246	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
247	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
248	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
249	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
250	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
251	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
252	#2861	C₂×C₂₃	1	2	yes	—
253	#2859	C₂×C₂₃	1	1	yes	characteristic, complemented
254	#2596	C₂×C₂₃	1	4	yes	complemented
255	#2581	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#2580	C₂×C₂₃	1	1	yes	characteristic
257	#2573	C₂×C₂₃	1	4	yes	complemented
258	#2512	C₂×C₂₃	1	2	yes	—
259	#2511	C₂×C₂₃	1	2	yes	complemented
260	#2469	C₂×C₂₃	1	4	yes	complemented
261	#2373	C₂×C₂₃	1	4	yes	complemented
262	#2357	C₂×C₂₃	1	1	yes	characteristic
263	#2356	C₂×C₂₃	1	1	yes	characteristic, complemented
264	C₄×#125	C₂×C₂₃	1	2	yes	complemented
265	C₄×#124	C₂×C₂₃	1	1	yes	characteristic
266	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
267	C₂²×C₂₃	—	4	2	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	—	4	4	no	—
270	C₂²×C₂₃	—	4	2	no	—
271	C₂²×C₂₃	—	4	4	no	—
272	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
273	C₂²×C₂₃	C₂²×#41	1	2	yes	—
274	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
275	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
276	C₂²×C₂₃	#585	1	2	yes	—
277	C₄×C₂₃	—	2	4	no	—
278	C₄×C₂₃	—	2	2	no	—
279	C₄×C₂₃	—	2	4	no	—
280	C₄×C₂₃	—	2	2	no	—
281	C₄×C₂₃	—	2	4	no	—
282	C₄×C₂₃	—	2	4	no	—
283	C₄×C₂₃	—	4	2	no	—
284	C₄×C₂₃	—	4	2	no	—
285	C₄×C₂₃	—	4	4	no	—
286	C₄×C₂₃	—	4	4	no	—
287	C₄×C₂₃	—	4	2	no	—
288	C₄×C₂₃	—	4	2	no	—
289	C₄×C₂₃	—	4	2	no	—
290	C₄×C₂₃	#556	1	4	yes	—
291	C₂×#126	C₂²×C₂₃	1	4	yes	—
292	C₂×#125	C₂²×C₂₃	1	2	yes	—
293	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
294	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
297	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
298	#400	C₂²×C₂₃	1	4	yes	—
299	#400	C₂²×C₂₃	1	4	yes	—
300	#400	C₂²×C₂₃	1	2	yes	—
301	#399	C₂²×C₂₃	1	2	yes	—
302	#397	C₂²×C₂₃	1	4	yes	—
303	#396	C₂²×C₂₃	1	4	yes	—
304	#395	C₂²×C₂₃	1	2	yes	—
305	#393	C₂²×C₂₃	1	2	yes	—
306	#391	C₂²×C₂₃	1	2	yes	—
307	#391	C₂²×C₂₃	1	2	yes	—
308	#391	C₂²×C₂₃	1	1	yes	characteristic
309	#390	C₂²×C₂₃	1	2	yes	—
310	#390	C₂²×C₂₃	1	2	yes	—
311	#390	C₂²×C₂₃	1	1	yes	characteristic
312	#389	C₂²×C₂₃	1	2	yes	—
313	#389	C₂²×C₂₃	1	2	yes	—
314	#389	C₂²×C₂₃	1	4	yes	—
315	#389	C₂²×C₂₃	1	4	yes	—
316	#388	C₂²×C₂₃	1	4	yes	—
317	#388	C₂²×C₂₃	1	4	yes	—
318	#388	C₂²×C₂₃	1	4	yes	—
319	#388	C₂²×C₂₃	1	4	yes	—
320	#388	C₂²×C₂₃	1	4	yes	—
321	#388	C₂²×C₂₃	1	4	yes	—
322	#388	C₂²×C₂₃	1	4	yes	—
323	#387	C₂²×C₂₃	1	4	yes	—
324	#387	C₂²×C₂₃	1	4	yes	—
325	#387	C₂²×C₂₃	1	4	yes	—
326	#387	C₂²×C₂₃	1	4	yes	—
327	#387	C₂²×C₂₃	1	4	yes	—
328	#387	C₂²×C₂₃	1	4	yes	—
329	#385	C₂²×C₂₃	1	4	yes	—
330	#384	C₂²×C₂₃	1	2	yes	—
331	#383	C₂²×C₂₃	1	2	yes	—
332	#382	C₂²×C₂₃	1	1	yes	characteristic
333	#382	C₂²×C₂₃	1	1	yes	characteristic
334	#382	C₂²×C₂₃	1	1	yes	characteristic
335	#381	C₂²×C₂₃	1	2	yes	—
336	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
337	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
338	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
339	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
340	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
341	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
342	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
343	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
344	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
345	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
346	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
347	#376	C₂²×C₂₃	1	2	yes	—
348	C₄³	C₂²×C₂₃	1	2	yes	—
349	C₄³	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×C₂₃	—	8	2	no	—
351	C₂×C₂₃	#3444	1	1	yes	characteristic
352	C₂×C₂₃	C₂×#556	1	1	yes	characteristic
353	C₂×C₂₃	#3480	1	2	yes	—
354	C₂×C₂₃	#3510	1	1	yes	characteristic
355	C₂×C₂₃	#3515	1	2	yes	—
356	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
357	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
360	#126	—	2	16	no	—
361	#125	—	2	8	no	—
362	#124	—	2	4	no	—
363	C₄×Q₈	—	2	8	no	—
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×D₄	—	2	4	no	—
367	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
375	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
378	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
379	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
380	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
381	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
390	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
391	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
392	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
393	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
394	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
395	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
396	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
397	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
399	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
400	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
401	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
402	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
404	#95	C₂³×C₂₃	1	1	yes	characteristic
405	#95	C₂³×C₂₃	1	2	yes	—
406	#95	C₂³×C₂₃	1	4	yes	—
407	#95	C₂³×C₂₃	1	4	yes	—
408	#95	C₂³×C₂₃	1	2	yes	—
409	#95	C₂³×C₂₃	1	4	yes	—
410	#95	C₂³×C₂₃	1	1	yes	characteristic
411	#95	C₂³×C₂₃	1	2	yes	—
412	#95	C₂³×C₂₃	1	2	yes	—
413	#95	C₂³×C₂₃	1	4	yes	—
414	#95	C₂³×C₂₃	1	4	yes	—
415	#95	C₂³×C₂₃	1	4	yes	—
416	#95	C₂³×C₂₃	1	4	yes	—
417	#95	C₂³×C₂₃	1	2	yes	—
418	#95	C₂³×C₂₃	1	2	yes	—
419	#95	C₂³×C₂₃	1	4	yes	—
420	#95	C₂³×C₂₃	1	4	yes	—
421	#95	C₂³×C₂₃	1	4	yes	—
422	C₂₃	#36601	1	1	yes	characteristic, direct product factor, Sylow
423	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
424	C₂×Q₈	—	2	4	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
437	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
445	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	8	no	—
460	C₄⋊C₄	—	2	8	no	—
461	C₄⋊C₄	—	4	4	no	—
462	C₄⋊C₄	—	4	4	no	—
463	C₄⋊C₄	#41×C₂₃	1	8	yes	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	4	4	no	—
470	C₂²⋊C₄	—	4	4	no	—
471	C₄²	—	2	4	no	—
472	C₄²	—	2	4	no	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	4	no	—
475	C₄²	—	2	4	no	—
476	C₄²	—	2	8	no	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	4	no	—
479	C₄²	—	4	2	no	—
480	C₄²	—	4	2	no	—
481	C₄²	—	4	2	no	—
482	C₄²	—	4	4	no	—
483	C₄²	#41×C₂₃	1	4	yes	—
484	C₄²	#41×C₂₃	1	4	yes	—
485	C₂³	—	2	4	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	4	no	—
490	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
491	Q₈	—	2	8	no	—
492	Q₈	—	4	4	no	—
493	D₄	—	4	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	8	4	no	—
524	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
525	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
526	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
527	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
528	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
529	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
530	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
531	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
532	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	4	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	4	no	—
538	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
539	C₂²	C₂²×#41×C₂₃	1	2	yes	—
540	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
541	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
542	C₂²	C₂₃×#585	1	2	yes	—
543	C₄	—	2	4	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	4	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	4	no	—
548	C₄	—	2	4	no	—
549	C₄	—	4	2	no	—
550	C₄	—	4	2	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	2	no	—
556	C₄	C₂₃×#556	1	4	yes	—
557	C₂	—	8	2	no	—
558	C₂	C₂₃×#3444	1	1	yes	characteristic
559	C₂	C₂×C₂₃×#556	1	1	yes	characteristic
560	C₂	C₂₃×#3480	1	2	yes	—
561	C₂	C₂₃×#3510	1	1	yes	characteristic
562	C₂	C₂₃×#3515	1	2	yes	—

	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×#2596	C₂	semidirect
4. C₂₃×#2581	C₂	semidirect
6. C₂₃×#2573	C₂	semidirect
8. C₂₃×#2511	C₂	semidirect
9. C₂₃×#2469	C₂	semidirect
10. C₂₃×#2373	C₂	semidirect
12. C₂₃×#2356	C₂	semidirect
13. C₄×C₂₃×#125	C₂	semidirect
15. C₄²×Q₈×C₂₃	C₂	semidirect
203. #36601	C₂₃	direct (N × H)
253. #2859	C₂×C₂₃	semidirect
254. #2596	C₂×C₂₃	semidirect
255. #2581	C₂×C₂₃	semidirect
257. #2573	C₂×C₂₃	semidirect
259. #2511	C₂×C₂₃	semidirect
260. #2469	C₂×C₂₃	semidirect
261. #2373	C₂×C₂₃	semidirect
263. #2356	C₂×C₂₃	semidirect
264. C₄×#125	C₂×C₂₃	semidirect
266. C₄²×Q₈	C₂×C₂₃	semidirect
422. C₂₃	#36601	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×#2596	2
C₂₃×#2581	2
C₂₃×#2580	2
C₂₃×#2573	2
C₂₃×#2512	2
C₂₃×#2511	2
C₂₃×#2469	2
C₂₃×#2373	2
C₂₃×#2357	2
C₂₃×#2356	2
C₄×C₂₃×#125	2
C₄×C₂₃×#124	2
C₄²×Q₈×C₂₃	2
#36601	23

Subgroups — C23×#36601 #19184227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36601 #19184227