Subgroups — C₂₃×#34400 #19182026

Order 5888.

1990 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2376	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2373	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2357	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2353	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2346	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2343	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2337	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2336	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2333	C₂	1	2	yes	maximal
11	C₂₃×#2332	C₂	1	2	yes	maximal, complemented
12	C₂₃×#2320	C₂	1	2	yes	maximal, complemented
13	C₂₃×#2318	C₂	1	2	yes	maximal, complemented
14	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#529	C₄	1	16	yes	complemented
17	C₂×C₂₃×#126	C₂²	1	2	yes	—
18	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#123	C₂²	1	2	yes	—
21	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂₃×#117	C₂²	1	2	yes	—
27	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#391	C₂²	1	4	yes	—
29	C₂₃×#390	C₂²	1	4	yes	—
30	C₂₃×#389	C₂²	1	4	yes	—
31	C₂₃×#389	C₂²	1	4	yes	—
32	C₂₃×#388	C₂²	1	4	yes	—
33	C₂₃×#388	C₂²	1	4	yes	—
34	C₂₃×#388	C₂²	1	4	yes	—
35	C₂₃×#388	C₂²	1	4	yes	—
36	C₂₃×#388	C₂²	1	4	yes	—
37	C₂₃×#388	C₂²	1	4	yes	—
38	C₂₃×#387	C₂²	1	4	yes	—
39	C₂₃×#387	C₂²	1	4	yes	—
40	C₂₃×#387	C₂²	1	4	yes	—
41	C₂₃×#387	C₂²	1	4	yes	—
42	C₂₃×#387	C₂²	1	4	yes	—
43	C₂₃×#387	C₂²	1	4	yes	—
44	C₂₃×#386	C₂²	1	4	yes	—
45	C₂₃×#386	C₂²	1	4	yes	—
46	C₂₃×#385	C₂²	1	4	yes	—
47	C₂₃×#385	C₂²	1	4	yes	—
48	C₂₃×#383	C₂²	1	2	yes	—
49	C₂₃×#383	C₂²	1	2	yes	—
50	C₂₃×#383	C₂²	1	2	yes	—
51	C₂₃×#382	C₂²	1	2	yes	—
52	C₂₃×#382	C₂²	1	2	yes	—
53	C₂₃×#382	C₂²	1	2	yes	—
54	C₂₃×#381	C₂²	1	2	yes	—
55	C₂₃×#381	C₂²	1	2	yes	—
56	C₂₃×#381	C₂²	1	2	yes	—
57	C₂₃×#380	C₂²	1	2	yes	—
58	C₂₃×#380	C₂²	1	2	yes	—
59	C₂₃×#380	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
64	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
65	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
67	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
68	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
69	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
71	C₂₃×#376	C₂²	1	1	yes	characteristic
72	C₂₃×#376	C₂²	1	1	yes	characteristic
73	C₂₃×#376	C₂²	1	2	yes	—
74	C₂₃×#376	C₂²	1	2	yes	—
75	C₂×C₂₃×#95	C₂²	1	2	yes	—
76	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
79	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
80	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
81	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂₃×#126	C₂×C₄	1	16	yes	complemented
85	C₂₃×#125	C₂×C₄	1	8	yes	complemented
86	C₂₃×#124	C₂×C₄	1	8	yes	—
87	C₂₃×#123	C₂×C₄	1	16	yes	complemented
88	C₂₃×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
89	C₂₃×#121	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
91	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
93	C₂₃×#117	C₂×C₄	1	16	yes	complemented
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	4	yes	—
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
116	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
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	2	yes	—
121	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
122	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
123	C₂×C₄²×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	1	yes	characteristic
128	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
129	C₂₃×#95	C₂³	1	2	yes	—
130	C₂₃×#95	C₂³	1	2	yes	—
131	C₂₃×#95	C₂³	1	4	yes	—
132	C₂₃×#95	C₂³	1	2	yes	—
133	C₂₃×#95	C₂³	1	4	yes	—
134	C₂₃×#95	C₂³	1	2	yes	—
135	C₂₃×#95	C₂³	1	4	yes	—
136	C₂₃×#95	C₂³	1	2	yes	—
137	C₂₃×#95	C₂³	1	4	yes	—
138	C₂₃×#95	C₂³	1	2	yes	—
139	C₂₃×#95	C₂³	1	4	yes	—
140	C₂₃×#95	C₂³	1	4	yes	—
141	C₂₃×#95	C₂³	1	4	yes	—
142	C₂₃×#95	C₂³	1	4	yes	—
143	C₂₃×#95	C₂³	1	4	yes	—
144	C₂₃×#95	C₂³	1	4	yes	—
145	C₂₃×#95	C₂³	1	4	yes	—
146	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
147	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
148	C₂×Q₈×C₂₃	—	4	2	no	—
149	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
150	C₂×D₄×C₂₃	—	4	2	no	—
151	C₂×D₄×C₂₃	—	4	2	no	—
152	C₂×D₄×C₂₃	—	4	2	no	—
153	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
154	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
155	C₂²×C₄×C₂₃	—	4	2	no	—
156	C₂²×C₄×C₂₃	—	4	2	no	—
157	C₂²×C₄×C₂₃	—	4	4	no	—
158	C₂²×C₄×C₂₃	—	4	2	no	—
159	C₂²×C₄×C₂₃	—	4	4	no	—
160	C₂²×C₄×C₂₃	—	4	4	no	—
161	C₂²×C₄×C₂₃	—	4	4	no	—
162	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
163	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
164	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
165	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
166	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
179	C₄⋊C₄×C₂₃	—	4	4	no	—
180	C₄⋊C₄×C₂₃	—	4	4	no	—
181	C₄⋊C₄×C₂₃	—	4	4	no	—
182	C₄⋊C₄×C₂₃	—	4	4	no	—
183	C₄⋊C₄×C₂₃	—	4	4	no	—
184	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
185	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
186	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
187	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
188	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
189	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
190	C₂²⋊C₄×C₂₃	—	4	4	no	—
191	C₂²⋊C₄×C₂₃	—	4	4	no	—
192	C₂²⋊C₄×C₂₃	—	4	4	no	—
193	C₂²⋊C₄×C₂₃	—	4	2	no	—
194	C₂²⋊C₄×C₂₃	—	4	2	no	—
195	C₂²⋊C₄×C₂₃	—	4	4	no	—
196	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
197	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
198	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
199	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
200	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
201	C₄²×C₂₃	—	4	2	no	—
202	C₄²×C₂₃	—	4	2	no	—
203	C₄²×C₂₃	—	4	2	no	—
204	C₄²×C₂₃	—	4	2	no	—
205	C₄²×C₂₃	—	4	4	no	—
206	C₄²×C₂₃	—	4	4	no	—
207	C₄²×C₂₃	—	4	4	no	—
208	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
209	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
210	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
211	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
212	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
213	#34400	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂³×C₂₃	—	2	2	no	—
215	C₂³×C₂₃	—	2	2	no	—
216	C₂³×C₂₃	—	2	2	no	—
217	C₂³×C₂₃	—	2	4	no	—
218	C₂³×C₂₃	—	2	4	no	—
219	C₂³×C₂₃	—	2	2	no	—
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
222	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
223	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
224	Q₈×C₂₃	—	4	4	no	—
225	D₄×C₂₃	—	4	4	no	—
226	D₄×C₂₃	—	4	8	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	4	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	2	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	4	no	—
238	C₂×C₄×C₂₃	—	2	4	no	—
239	C₂×C₄×C₂₃	—	2	4	no	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	2	4	no	—
242	C₂×C₄×C₂₃	—	2	4	no	—
243	C₂×C₄×C₂₃	—	2	4	no	—
244	C₂×C₄×C₂₃	—	2	4	no	—
245	C₂×C₄×C₂₃	—	2	4	no	—
246	C₂×C₄×C₂₃	—	2	4	no	—
247	C₂×C₄×C₂₃	—	2	4	no	—
248	C₂×C₄×C₂₃	—	2	4	no	—
249	C₂×C₄×C₂₃	—	4	4	no	—
250	C₂×C₄×C₂₃	—	4	8	no	—
251	C₂×C₄×C₂₃	—	4	4	no	—
252	C₂×C₄×C₂₃	—	4	4	no	—
253	C₂×C₄×C₂₃	—	8	8	no	—
254	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
255	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
256	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
257	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
258	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
259	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
260	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
261	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
262	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
263	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
264	#2376	C₂×C₂₃	1	4	yes	complemented
265	#2373	C₂×C₂₃	1	4	yes	complemented
266	#2357	C₂×C₂₃	1	2	yes	complemented
267	#2353	C₂×C₂₃	1	2	yes	complemented
268	#2346	C₂×C₂₃	1	2	yes	complemented
269	#2343	C₂×C₂₃	1	2	yes	complemented
270	#2337	C₂×C₂₃	1	2	yes	complemented
271	#2336	C₂×C₂₃	1	2	yes	complemented
272	#2333	C₂×C₂₃	1	2	yes	—
273	#2332	C₂×C₂₃	1	2	yes	complemented
274	#2320	C₂×C₂₃	1	2	yes	complemented
275	#2318	C₂×C₂₃	1	2	yes	complemented
276	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₂²×C₂₃	—	2	2	no	—
279	C₂²×C₂₃	—	2	2	no	—
280	C₂²×C₂₃	—	2	2	no	—
281	C₂²×C₂₃	—	2	4	no	—
282	C₂²×C₂₃	—	2	4	no	—
283	C₂²×C₂₃	—	2	4	no	—
284	C₂²×C₂₃	—	4	4	no	—
285	C₂²×C₂₃	—	4	4	no	—
286	C₂²×C₂₃	—	4	4	no	—
287	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
288	C₂²×C₂₃	#585	1	2	yes	—
289	C₂²×C₂₃	#585	1	2	yes	—
290	C₂²×C₂₃	#585	1	2	yes	—
291	C₄×C₂₃	—	2	2	no	—
292	C₄×C₂₃	—	2	2	no	—
293	C₄×C₂₃	—	2	2	no	—
294	C₄×C₂₃	—	2	4	no	—
295	C₄×C₂₃	—	2	2	no	—
296	C₄×C₂₃	—	2	2	no	—
297	C₄×C₂₃	—	2	2	no	—
298	C₄×C₂₃	—	2	4	no	—
299	C₄×C₂₃	—	2	4	no	—
300	C₄×C₂₃	—	4	4	no	—
301	C₄×C₂₃	—	4	4	no	—
302	C₄×C₂₃	—	4	4	no	—
303	C₄×C₂₃	—	4	4	no	—
304	#529	C₄×C₂₃	1	16	yes	complemented
305	C₂×#126	C₂²×C₂₃	1	2	yes	—
306	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×#123	C₂²×C₂₃	1	2	yes	—
309	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
314	C₂×#117	C₂²×C₂₃	1	2	yes	—
315	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
316	#391	C₂²×C₂₃	1	4	yes	—
317	#390	C₂²×C₂₃	1	4	yes	—
318	#389	C₂²×C₂₃	1	4	yes	—
319	#389	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	#388	C₂²×C₂₃	1	4	yes	—
324	#388	C₂²×C₂₃	1	4	yes	—
325	#388	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	#387	C₂²×C₂₃	1	4	yes	—
330	#387	C₂²×C₂₃	1	4	yes	—
331	#387	C₂²×C₂₃	1	4	yes	—
332	#386	C₂²×C₂₃	1	4	yes	—
333	#386	C₂²×C₂₃	1	4	yes	—
334	#385	C₂²×C₂₃	1	4	yes	—
335	#385	C₂²×C₂₃	1	4	yes	—
336	#383	C₂²×C₂₃	1	2	yes	—
337	#383	C₂²×C₂₃	1	2	yes	—
338	#383	C₂²×C₂₃	1	2	yes	—
339	#382	C₂²×C₂₃	1	2	yes	—
340	#382	C₂²×C₂₃	1	2	yes	—
341	#382	C₂²×C₂₃	1	2	yes	—
342	#381	C₂²×C₂₃	1	2	yes	—
343	#381	C₂²×C₂₃	1	2	yes	—
344	#381	C₂²×C₂₃	1	2	yes	—
345	#380	C₂²×C₂₃	1	2	yes	—
346	#380	C₂²×C₂₃	1	2	yes	—
347	#380	C₂²×C₂₃	1	2	yes	—
348	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
349	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
350	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
351	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
352	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
353	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
354	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
355	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
356	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
357	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
358	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
359	#376	C₂²×C₂₃	1	2	yes	—
360	#376	C₂²×C₂₃	1	1	yes	characteristic
361	#376	C₂²×C₂₃	1	1	yes	characteristic
362	#376	C₂²×C₂₃	1	2	yes	—
363	C₂×#95	C₂²×C₂₃	1	2	yes	—
364	C₂×C₂₃	—	4	2	no	—
365	C₂×C₂₃	—	4	4	no	—
366	C₂×C₂₃	#3428	1	1	yes	characteristic
367	C₂×C₂₃	#3428	1	1	yes	characteristic
368	C₂×C₂₃	#3428	1	1	yes	characteristic
369	C₂×C₂₃	#3566	1	4	yes	—
370	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
371	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
372	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
373	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
374	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
375	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
376	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
378	#126	C₂×C₄×C₂₃	1	16	yes	complemented
379	#125	C₂×C₄×C₂₃	1	8	yes	complemented
380	#124	C₂×C₄×C₂₃	1	8	yes	—
381	#123	C₂×C₄×C₂₃	1	16	yes	complemented
382	C₂²⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
383	#121	C₂×C₄×C₂₃	1	8	yes	—
384	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
385	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
386	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
387	#117	C₂×C₄×C₂₃	1	16	yes	complemented
388	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
410	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
411	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
412	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
413	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
414	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
415	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
416	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
417	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
418	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
420	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
423	#95	C₂³×C₂₃	1	2	yes	—
424	#95	C₂³×C₂₃	1	2	yes	—
425	#95	C₂³×C₂₃	1	4	yes	—
426	#95	C₂³×C₂₃	1	2	yes	—
427	#95	C₂³×C₂₃	1	4	yes	—
428	#95	C₂³×C₂₃	1	2	yes	—
429	#95	C₂³×C₂₃	1	4	yes	—
430	#95	C₂³×C₂₃	1	2	yes	—
431	#95	C₂³×C₂₃	1	4	yes	—
432	#95	C₂³×C₂₃	1	2	yes	—
433	#95	C₂³×C₂₃	1	4	yes	—
434	#95	C₂³×C₂₃	1	4	yes	—
435	#95	C₂³×C₂₃	1	4	yes	—
436	#95	C₂³×C₂₃	1	4	yes	—
437	#95	C₂³×C₂₃	1	4	yes	—
438	#95	C₂³×C₂₃	1	4	yes	—
439	#95	C₂³×C₂₃	1	4	yes	—
440	C₂₃	#34400	1	1	yes	characteristic, direct product factor, Sylow
441	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
442	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
443	C₂×Q₈	—	4	2	no	—
444	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	4	2	no	—
447	C₂×D₄	—	4	2	no	—
448	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
449	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	4	no	—
453	C₂²×C₄	—	4	2	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	4	no	—
457	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
458	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
459	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
460	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
461	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
462	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
463	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
465	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
467	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
468	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
471	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
473	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
474	C₄⋊C₄	—	4	4	no	—
475	C₄⋊C₄	—	4	4	no	—
476	C₄⋊C₄	—	4	4	no	—
477	C₄⋊C₄	—	4	4	no	—
478	C₄⋊C₄	—	4	4	no	—
479	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
480	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
481	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
482	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
483	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
484	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
485	C₂²⋊C₄	—	4	4	no	—
486	C₂²⋊C₄	—	4	4	no	—
487	C₂²⋊C₄	—	4	4	no	—
488	C₂²⋊C₄	—	4	2	no	—
489	C₂²⋊C₄	—	4	2	no	—
490	C₂²⋊C₄	—	4	4	no	—
491	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
492	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
493	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
494	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
495	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
496	C₄²	—	4	2	no	—
497	C₄²	—	4	2	no	—
498	C₄²	—	4	2	no	—
499	C₄²	—	4	2	no	—
500	C₄²	—	4	4	no	—
501	C₄²	—	4	4	no	—
502	C₄²	—	4	4	no	—
503	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
504	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
505	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
506	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
507	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	2	4	no	—
513	C₂³	—	2	2	no	—
514	C₂³	—	2	2	no	—
515	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
516	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
517	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
518	Q₈	—	4	4	no	—
519	D₄	—	4	4	no	—
520	D₄	—	4	8	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	4	4	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	8	8	no	—
548	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
549	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
550	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
551	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
552	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
553	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
554	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
555	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
556	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
567	C₂²	C₂₃×#585	1	2	yes	—
568	C₂²	C₂₃×#585	1	2	yes	—
569	C₂²	C₂₃×#585	1	2	yes	—
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	4	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	4	no	—
578	C₄	—	2	4	no	—
579	C₄	—	4	4	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	4	no	—
583	C₂	—	4	2	no	—
584	C₂	—	4	4	no	—
585	C₂	C₂₃×#3428	1	1	yes	characteristic
586	C₂	C₂₃×#3428	1	1	yes	characteristic
587	C₂	C₂₃×#3428	1	1	yes	characteristic
588	C₂	C₂₃×#3566	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2376	C₂	semidirect
3. C₂₃×#2373	C₂	semidirect
4. C₂₃×#2357	C₂	semidirect
5. C₂₃×#2353	C₂	semidirect
6. C₂₃×#2346	C₂	semidirect
7. C₂₃×#2343	C₂	semidirect
8. C₂₃×#2337	C₂	semidirect
9. C₂₃×#2336	C₂	semidirect
11. C₂₃×#2332	C₂	semidirect
12. C₂₃×#2320	C₂	semidirect
13. C₂₃×#2318	C₂	semidirect
14. C₂₃×#2283	C₂	semidirect
15. C₂₃×#2283	C₂	semidirect
16. C₂₃×#529	C₄	semidirect
84. C₂₃×#126	C₂×C₄	semidirect
85. C₂₃×#125	C₂×C₄	semidirect
87. C₂₃×#123	C₂×C₄	semidirect
88. C₂₃×C₂²⋊Q₈	C₂×C₄	semidirect
90. C₄×Q₈×C₂₃	C₂×C₄	semidirect
92. C₄×D₄×C₂₃	C₂×C₄	semidirect
93. C₂₃×#117	C₂×C₄	semidirect
116. C₂×C₄²×C₂₃	C₂×C₄	semidirect
213. #34400	C₂₃	direct (N × H)
264. #2376	C₂×C₂₃	semidirect
265. #2373	C₂×C₂₃	semidirect
266. #2357	C₂×C₂₃	semidirect
267. #2353	C₂×C₂₃	semidirect
268. #2346	C₂×C₂₃	semidirect
269. #2343	C₂×C₂₃	semidirect
270. #2337	C₂×C₂₃	semidirect
271. #2336	C₂×C₂₃	semidirect
273. #2332	C₂×C₂₃	semidirect
274. #2320	C₂×C₂₃	semidirect
275. #2318	C₂×C₂₃	semidirect
276. #2283	C₂×C₂₃	semidirect
277. #2283	C₂×C₂₃	semidirect
304. #529	C₄×C₂₃	semidirect
378. #126	C₂×C₄×C₂₃	semidirect
379. #125	C₂×C₄×C₂₃	semidirect
381. #123	C₂×C₄×C₂₃	semidirect
382. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
384. C₄×Q₈	C₂×C₄×C₂₃	semidirect
386. C₄×D₄	C₂×C₄×C₂₃	semidirect
387. #117	C₂×C₄×C₂₃	semidirect
410. C₂×C₄²	C₂×C₄×C₂₃	semidirect
440. C₂₃	#34400	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×#529	2
C₂₃×#2376	2
C₂₃×#2373	2
C₂₃×#2357	2
C₂₃×#2353	2
C₂₃×#2346	2
C₂₃×#2343	2
C₂₃×#2337	2
C₂₃×#2336	2
C₂₃×#2333	2
C₂₃×#2332	2
C₂₃×#2320	2
C₂₃×#2318	2
C₂₃×#2283	2
#34400	23

Subgroups — C23×#34400 #19182026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34400 #19182026