Subgroups — #15682362

Order 2944.

412 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67292	C₂	1	1	yes	maximal, characteristic
2	#13092548	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092496	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092494	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092490	C₂	1	2	yes	maximal, complemented
6	#13092487	C₂	1	2	yes	maximal, complemented
7	#13092469	C₂	1	2	yes	maximal, complemented
8	#13092469	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092369	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67292	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	Q₈×Dic₂₃	C₄	1	1	yes	characteristic
15	C₂×#67262	C₂²	1	2	yes	—
16	C₂×#67262	C₂²	1	2	yes	—
17	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
19	#10625064	C₄	1	2	yes	—
20	#10625064	C₄	1	1	yes	characteristic
21	#10625063	C₄	1	2	yes	—
22	#10625063	C₄	1	1	yes	characteristic
23	C₄×D₉₂	C₂²	1	2	yes	—
24	C₄×D₉₂	C₂²	1	1	yes	characteristic
25	C₄²×D₂₃	C₂²	1	1	yes	characteristic
26	C₄²×D₂₃	C₂²	1	2	yes	—
27	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
28	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂₃×#97	C₂²	1	2	yes	—
30	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
34	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
35	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₃	C₂²	1	2	yes	—
37	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
38	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
39	#10624985	C₂²	1	2	yes	—
40	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	#67292	—	2	4	no	—
43	C₂×D₉₂	C₂×C₄	1	2	yes	—
44	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₂₃	C₂³	1	2	yes	—
51	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
55	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₂₃	C₂³	1	1	yes	characteristic
57	C₄²×C₂₃	C₂³	1	2	yes	—
58	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
62	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
65	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
66	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
70	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
71	#67262	—	2	4	no	—
72	#67262	—	2	4	no	—
73	C₈×D₂₃	—	2	2	no	—
74	C₈×D₂₃	—	2	2	no	—
75	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
76	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
77	Q₈×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
87	D₉₂	—	2	4	no	—
88	D₉₂	—	2	2	no	—
89	C₄×D₂₃	—	2	1	no	—
90	C₄×D₂₃	—	2	1	no	—
91	C₄×D₂₃	—	2	2	no	—
92	C₄×D₂₃	—	2	2	no	—
93	C₄×D₂₃	—	2	4	no	—
94	C₄×D₂₃	—	2	2	no	—
95	C₄×D₂₃	—	2	4	no	—
96	C₄×D₂₃	M₁₆	1	4	yes	complemented
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	—	2	2	no	—
99	C₈×C₂₃	#41	1	2	yes	complemented
100	C₂₃⋊C₈	—	2	1	no	—
101	C₂₃⋊C₈	—	2	2	no	—
102	C₂₃⋊C₈	#41	1	2	yes	complemented
103	#2965	—	23	1	no	maximal, Sylow
104	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×D₂₃	—	2	1	no	—
106	C₂×D₂₃	—	2	1	no	—
107	C₂×D₂₃	—	2	2	no	—
108	C₂×D₂₃	—	2	2	no	—
109	C₂×D₂₃	—	2	2	no	—
110	C₂×D₂₃	C₂×M₁₆	1	2	yes	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	2	no	—
114	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
115	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
117	Dic₂₃	—	2	1	no	—
118	Dic₂₃	—	2	2	no	—
119	Dic₂₃	C₂×M₁₆	1	2	yes	—
120	Dic₂₃	#131	1	2	yes	—
121	C₄×#41	—	23	1	no	—
122	#446	—	23	1	no	—
123	#446	D₂₃	1	1	yes	characteristic, complemented
124	#436	—	23	1	no	—
125	#436	—	23	1	no	—
126	#435	—	23	2	no	—
127	#435	—	23	2	no	—
128	#431	—	23	2	no	—
129	#431	—	23	1	no	—
130	C₄×M₁₆	—	23	2	no	—
131	C₂×C₄×C₈	—	23	1	no	—
132	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
133	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
134	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
135	D₂₃	—	2	2	no	—
136	D₂₃	—	4	2	no	—
137	C₂×#41	—	23	1	no	—
138	C₂×M₁₆	—	23	2	no	—
139	C₂×M₁₆	—	23	2	no	—
140	C₂²×C₈	—	23	1	no	—
141	C₂²×C₈	—	23	1	no	—
142	C₄×Q₈	—	23	1	no	—
143	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
144	C₄×D₄	—	23	2	no	—
145	C₄×D₄	—	23	1	no	—
146	C₄×D₄	—	23	2	no	—
147	C₄×D₄	—	23	1	no	—
148	#117	—	23	1	no	—
149	#117	—	23	2	no	—
150	C₂×C₄²	—	23	1	no	—
151	C₂×C₄²	—	23	2	no	—
152	C₄⋊C₈	—	23	2	no	—
153	C₄⋊C₈	—	23	1	no	—
154	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
155	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
156	C₂²⋊C₈	—	23	2	no	—
157	C₂²⋊C₈	—	23	2	no	—
158	C₂²⋊C₈	—	23	2	no	—
159	#97	—	23	2	no	—
160	#97	C₂×D₂₃	1	2	yes	—
161	C₄×C₈	—	23	1	no	—
162	C₄×C₈	—	23	1	no	—
163	C₄×C₈	—	23	2	no	—
164	C₄×C₈	—	23	1	no	—
165	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
166	C₂₃	#2965	1	1	yes	characteristic, complemented, Sylow
167	#41	—	46	4	no	—
168	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
169	C₂×D₄	—	23	2	no	—
170	C₂×D₄	—	23	1	no	—
171	C₂²×C₄	—	23	1	no	—
172	C₂²×C₄	—	23	2	no	—
173	C₂²×C₄	—	23	1	no	—
174	C₂²×C₄	—	23	2	no	—
175	C₂²×C₄	—	23	1	no	—
176	C₂²×C₄	—	23	2	no	—
177	M₁₆	—	46	4	no	—
178	M₁₆	—	46	4	no	—
179	C₂×C₈	—	23	1	no	—
180	C₂×C₈	—	23	1	no	—
181	C₂×C₈	—	23	2	no	—
182	C₂×C₈	—	46	2	no	—
183	C₂×C₈	—	46	2	no	—
184	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
185	C₂×C₈	C₂²×D₂₃	1	2	yes	—
186	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
187	C₄⋊C₄	—	23	2	no	—
188	C₄⋊C₄	—	23	1	no	—
189	C₄⋊C₄	C₄×D₂₃	1	2	yes	—
190	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
191	C₂²⋊C₄	—	23	2	no	—
192	C₂²⋊C₄	—	23	2	no	—
193	C₂²⋊C₄	—	23	2	no	—
194	C₄²	—	23	1	no	—
195	C₄²	—	23	2	no	—
196	C₄²	—	23	2	no	—
197	C₄²	—	23	1	no	—
198	C₄²	—	23	1	no	—
199	C₄²	C₂²×D₂₃	1	1	yes	characteristic
200	C₄²	C₂²×D₂₃	1	2	yes	—
201	C₂³	—	23	1	no	—
202	C₂³	—	23	2	no	—
203	Q₈	—	2	2	no	—
204	D₄	—	46	4	no	—
205	D₄	—	46	2	no	—
206	C₂×C₄	—	23	1	no	—
207	C₂×C₄	—	23	1	no	—
208	C₂×C₄	—	23	4	no	—
209	C₂×C₄	—	23	1	no	—
210	C₂×C₄	—	23	2	no	—
211	C₂×C₄	—	46	4	no	—
212	C₂×C₄	—	46	2	no	—
213	C₂×C₄	—	46	4	no	—
214	C₂×C₄	—	46	1	no	—
215	C₂×C₄	—	46	2	no	—
216	C₂×C₄	—	46	1	no	—
217	C₂×C₄	—	46	2	no	—
218	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	23	2	no	—
226	C₈	—	46	1	no	—
227	C₈	—	46	2	no	—
228	C₈	#67292	1	2	yes	complemented
229	C₂²	—	23	2	no	—
230	C₂²	—	46	1	no	—
231	C₂²	—	46	1	no	—
232	C₂²	—	46	2	no	—
233	C₂²	—	46	2	no	—
234	C₂²	—	46	2	no	—
235	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	2	no	—
239	C₄	—	23	2	no	—
240	C₄	—	23	2	no	—
241	C₄	—	46	1	no	—
242	C₄	—	46	2	no	—
243	C₄	M₁₆×D₂₃	1	2	yes	—
244	C₄	C₂×#67292	1	1	yes	characteristic
245	C₄	#41×D₂₃	1	1	yes	characteristic
246	C₂	—	46	2	no	—
247	C₂	—	92	2	no	—
248	C₂	C₄×#67292	1	1	yes	characteristic
249	C₂	C₂×M₁₆×D₂₃	1	1	yes	characteristic
250	C₂	#131×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#446	C₂
prime radical	C₂×#67292	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#446	D₂₃
hypocenter	C₂₃	#2965
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #13092548	C₂	semidirect
3. #13092496	C₂	semidirect
4. #13092494	C₂	semidirect
5. #13092490	C₂	semidirect
6. #13092487	C₂	semidirect
7. #13092469	C₂	semidirect
8. #13092469	C₂	semidirect
9. #13092369	C₂	semidirect
10. C₄×C₈×D₂₃	C₂	semidirect
11. C₂₃×#446	C₂	semidirect
96. C₄×D₂₃	M₁₆	semidirect
99. C₈×C₂₃	#41	semidirect
102. C₂₃⋊C₈	#41	semidirect
123. #446	D₂₃	semidirect
166. C₂₃	#2965	semidirect
228. C₈	#67292	semidirect

Maximal subgroups

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

subgroup	index
C₄×#67292	2
#13092548	2
#13092496	2
#13092494	2
#13092490	2
#13092487	2
#13092469	2
#13092369	2
C₄×C₈×D₂₃	2
C₂₃×#446	2
#2965	23