Subgroups — #15682408

Order 2944.

406 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67288	C₂	1	1	yes	maximal, characteristic
2	#13092545	C₂	1	2	yes	maximal, complemented
3	#13092490	C₂	1	2	yes	maximal, complemented
4	#13092488	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
6	#13092474	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092472	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092472	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092369	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₂₃	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₂₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
19	#10625097	C₄	1	1	yes	characteristic
20	C₂×#67262	C₂²	1	2	yes	—
21	C₂×C₈×D₂₃	C₂²	1	2	yes	—
22	C₄×D₉₂	C₄	1	1	yes	characteristic
23	C₄×D₉₂	C₂²	1	1	yes	characteristic
24	#10625043	C₄	1	1	yes	characteristic
25	#10625043	C₄	1	1	yes	characteristic
26	C₄²×D₂₃	C₂²	1	1	yes	characteristic
27	C₄²×D₂₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₄₆	C₄	1	1	yes	characteristic
29	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
30	C₂₃×#97	C₂²	1	1	yes	characteristic
31	C₂₃×#97	C₂²	1	1	yes	characteristic
32	C₂₃×#97	C₂²	1	2	yes	—
33	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
34	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
35	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
36	#10624997	C₂²	1	2	yes	—
37	Dic₂₃⋊C₈	C₂²	1	2	yes	—
38	Dic₂₃⋊C₈	C₂²	1	2	yes	—
39	C₈×Dic₂₃	C₂²	1	2	yes	—
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
41	#10624985	C₂²	1	1	yes	characteristic
42	#10624985	C₂²	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
48	#67288	—	2	4	no	—
49	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
56	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	C₂³	1	2	yes	—
59	C₂×C₈×C₂₃	C₂³	1	2	yes	—
60	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
66	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
72	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
73	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
74	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
75	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
81	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
83	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
84	#67262	—	2	4	no	—
85	C₈×D₂₃	—	2	4	no	—
86	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	4	1	no	—
92	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₂₃	—	2	2	no	—
99	C₂²⋊D₂₃	M₁₆	1	4	yes	complemented
100	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
106	D₉₂	—	2	2	no	—
107	C₄×D₂₃	—	2	2	no	—
108	C₄×D₂₃	—	2	2	no	—
109	C₄×D₂₃	—	2	2	no	—
110	C₄×D₂₃	—	2	2	no	—
111	C₄×D₂₃	—	4	1	no	—
112	C₄×D₂₃	—	4	1	no	—
113	Dic₄₆	—	2	2	no	—
114	C₈×C₂₃	—	2	2	no	—
115	C₈×C₂₃	#41	1	4	yes	complemented
116	C₂₃⋊C₈	—	2	2	no	—
117	C₂₃⋊C₈	—	2	2	no	—
118	#2970	—	23	1	no	maximal, Sylow
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
122	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×D₂₃	—	2	1	no	—
124	C₂×D₂₃	—	2	1	no	—
125	C₂×D₂₃	—	2	1	no	—
126	C₂×D₂₃	—	2	1	no	—
127	C₂×D₂₃	—	2	1	no	—
128	C₂×D₂₃	C₂×M₁₆	1	2	yes	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	C₂×#41	1	2	yes	—
135	Dic₂₃	—	2	1	no	—
136	Dic₂₃	—	2	1	no	—
137	Dic₂₃	—	2	1	no	—
138	Dic₂₃	—	2	1	no	—
139	Dic₂₃	—	2	1	no	—
140	Dic₂₃	C₂×M₁₆	1	2	yes	—
141	C₄×#41	—	23	1	no	—
142	#446	—	23	2	no	—
143	#435	—	23	2	no	—
144	#435	—	23	2	no	—
145	C₈×D₄	—	23	2	no	—
146	#432	—	23	1	no	—
147	#432	—	23	1	no	—
148	#432	—	23	1	no	—
149	C₄×M₁₆	—	23	1	no	—
150	C₂×#97	—	23	1	no	—
151	C₂×#97	D₂₃	1	1	yes	characteristic, complemented
152	C₂×C₂₃	—	2	2	no	—
153	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
154	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₂×C₂₃	#568	1	1	yes	characteristic
156	D₂₃	—	2	2	no	—
157	D₂₃	—	4	1	no	—
158	C₂×#41	—	23	1	no	—
159	C₂×M₁₆	—	23	2	no	—
160	C₂²×C₈	—	23	2	no	—
161	C₂²×C₈	C₂×D₂₃	1	2	yes	—
162	C₄×Q₈	—	23	1	no	—
163	C₄×Q₈	—	23	1	no	—
164	C₄×D₄	—	23	1	no	—
165	C₄×D₄	—	23	1	no	—
166	C₄×D₄	—	23	1	no	—
167	C₄×D₄	—	23	1	no	—
168	C₄×D₄	—	23	1	no	—
169	C₄×D₄	—	23	1	no	—
170	#117	—	23	1	no	—
171	#117	—	23	1	no	—
172	#117	—	23	1	no	—
173	C₂×C₄²	—	23	1	no	—
174	C₂×C₄²	—	23	1	no	—
175	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
176	C₄⋊C₈	—	23	2	no	—
177	C₄⋊C₈	—	23	2	no	—
178	C₄⋊C₈	—	23	2	no	—
179	C₂²⋊C₈	—	23	2	no	—
180	C₂²⋊C₈	—	23	2	no	—
181	C₂²⋊C₈	—	23	2	no	—
182	#97	—	23	2	no	—
183	#97	—	23	1	no	—
184	#97	—	23	1	no	—
185	#97	C₂×D₂₃	1	1	yes	characteristic
186	#97	C₂×D₂₃	1	1	yes	characteristic
187	#97	C₂×D₂₃	1	2	yes	—
188	C₄×C₈	—	23	2	no	—
189	C₂₃	#2970	1	1	yes	characteristic, complemented, Sylow
190	#41	—	46	4	no	—
191	C₂×Q₈	—	23	1	no	—
192	C₂×D₄	—	23	1	no	—
193	C₂×D₄	—	23	1	no	—
194	C₂×D₄	—	23	1	no	—
195	C₂²×C₄	—	23	1	no	—
196	C₂²×C₄	—	23	1	no	—
197	C₂²×C₄	—	23	1	no	—
198	C₂²×C₄	—	23	1	no	—
199	C₂²×C₄	—	23	1	no	—
200	C₂²×C₄	—	23	1	no	—
201	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
204	M₁₆	—	46	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	23	2	no	—
207	C₂×C₈	—	23	2	no	—
208	C₂×C₈	—	46	4	no	—
209	C₂×C₈	C₂²×D₂₃	1	2	yes	—
210	C₂×C₈	C₂²×D₂₃	1	2	yes	—
211	C₄⋊C₄	—	23	1	no	—
212	C₄⋊C₄	—	23	1	no	—
213	C₄⋊C₄	—	23	1	no	—
214	C₄⋊C₄	—	23	1	no	—
215	C₄⋊C₄	—	23	1	no	—
216	C₄⋊C₄	—	23	1	no	—
217	C₂²⋊C₄	—	23	1	no	—
218	C₂²⋊C₄	—	23	1	no	—
219	C₂²⋊C₄	—	23	1	no	—
220	C₂²⋊C₄	—	23	1	no	—
221	C₂²⋊C₄	—	23	1	no	—
222	C₂²⋊C₄	—	23	1	no	—
223	C₄²	—	23	1	no	—
224	C₄²	—	23	1	no	—
225	C₄²	—	23	1	no	—
226	C₄²	—	23	1	no	—
227	C₄²	—	23	1	no	—
228	C₄²	—	23	1	no	—
229	C₄²	C₄×D₂₃	1	1	yes	characteristic
230	C₄²	C₄×D₂₃	1	1	yes	characteristic
231	C₄²	C₂²×D₂₃	1	1	yes	characteristic
232	C₄²	C₂²×D₂₃	1	1	yes	characteristic
233	C₂³	—	23	1	no	—
234	C₂³	—	23	1	no	—
235	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
236	Q₈	—	46	2	no	—
237	D₄	—	23	4	no	—
238	D₄	—	46	2	no	—
239	D₄	—	46	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	23	1	no	—
244	C₂×C₄	—	23	1	no	—
245	C₂×C₄	—	23	1	no	—
246	C₂×C₄	—	23	1	no	—
247	C₂×C₄	—	23	1	no	—
248	C₂×C₄	—	23	1	no	—
249	C₂×C₄	—	46	2	no	—
250	C₂×C₄	—	46	2	no	—
251	C₂×C₄	—	46	2	no	—
252	C₂×C₄	—	46	2	no	—
253	C₂×C₄	—	92	1	no	—
254	C₂×C₄	—	92	1	no	—
255	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	46	2	no	—
263	C₈	—	46	2	no	—
264	C₈	#67288	1	4	yes	complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	23	2	no	—
268	C₂²	—	46	1	no	—
269	C₂²	—	46	1	no	—
270	C₂²	—	46	1	no	—
271	C₂²	—	46	1	no	—
272	C₂²	—	46	1	no	—
273	C₂²	M₁₆×D₂₃	1	2	yes	—
274	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	23	2	no	—
281	C₄	—	46	1	no	—
282	C₄	—	46	1	no	—
283	C₄	—	46	1	no	—
284	C₄	—	46	1	no	—
285	C₄	—	46	1	no	—
286	C₄	C₂×#67288	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	46	2	no	—
289	C₂	—	92	1	no	—
290	C₂	C₄×#67288	1	1	yes	characteristic
291	C₂	C₂×M₁₆×D₂₃	1	1	yes	characteristic
292	C₂	#13093183	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂×C₂₃×#97	C₂
prime radical	C₂×#67288	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#97	D₂₃
hypocenter	C₂₃	#2970
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #13092545	C₂	semidirect
3. #13092490	C₂	semidirect
4. #13092488	C₂	semidirect
5. C₈×C₂²⋊D₂₃	C₂	semidirect
6. #13092474	C₂	semidirect
7. #13092472	C₂	semidirect
8. #13092472	C₂	semidirect
9. #13092369	C₂	semidirect
10. #97×D₂₃	C₂	semidirect
11. C₂×C₂₃×#97	C₂	semidirect
99. C₂²⋊D₂₃	M₁₆	semidirect
115. C₈×C₂₃	#41	semidirect
151. C₂×#97	D₂₃	semidirect
189. C₂₃	#2970	semidirect
264. C₈	#67288	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#67288	2
#13092545	2
#13092490	2
#13092488	2
C₈×C₂²⋊D₂₃	2
#13092474	2
#13092472	2
#13092369	2
#97×D₂₃	2
C₂×C₂₃×#97	2
#2970	23