Subgroups — #15686802

Order 2944.

996 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093048	C₂	1	4	yes	maximal, complemented
2	#13093045	C₂	1	8	yes	maximal, complemented
3	#13093043	C₂	1	4	yes	maximal
4	#13092974	C₂	1	4	yes	maximal, complemented
5	#13092970	C₂	1	8	yes	maximal, complemented
6	C₂×#10625097	C₂	1	2	yes	maximal
7	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#117	C₂²	1	4	yes	—
10	C₂₃×#117	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	—
15	#10625097	C₂²	1	4	yes	—
16	#10625097	C₂²	1	4	yes	—
17	#10625097	C₂²	1	4	yes	—
18	#10625097	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
20	#10625095	C₂²	1	16	yes	—
21	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
22	#10625061	C₂²	1	16	yes	complemented
23	#10625059	C₂²	1	16	yes	complemented
24	#10625049	C₂²	1	32	yes	—
25	#10625041	C₂²	1	8	yes	complemented
26	#10625040	C₂²	1	8	yes	complemented
27	C₄×Dic₄₆	C₂²	1	16	yes	complemented
28	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	Q₈	1	8	yes	complemented
30	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₂₃	C₂³	1	8	yes	—
36	C₂×Dic₄₆	C₂³	1	16	yes	—
37	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
38	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
39	C₄²×C₂₃	C₂³	1	4	yes	—
40	C₄²×C₂₃	C₂³	1	4	yes	—
41	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
42	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
43	C₄⋊Dic₂₃	C₂³	1	16	yes	—
44	C₄⋊Dic₂₃	C₂³	1	8	yes	—
45	C₄⋊Dic₂₃	C₂³	1	8	yes	—
46	Dic₂₃⋊C₄	C₂³	1	32	yes	—
47	C₄×Dic₂₃	C₂³	1	8	yes	—
48	C₄×Dic₂₃	C₂³	1	8	yes	—
49	C₂³×C₂₃	—	4	2	no	—
50	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
51	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂³×C₂₃	C₂⁴	1	2	yes	—
53	C₂×C₄×C₂₃	—	4	4	no	—
54	C₂×C₄×C₂₃	C₂×Q₈	1	8	yes	—
55	C₂×C₄×C₂₃	C₂×Q₈	1	8	yes	—
56	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
58	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
59	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
60	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₂₃	—	4	8	no	—
63	C₂×Dic₂₃	C₂⁴	1	16	yes	—
64	Dic₄₆	—	4	16	no	—
65	#3458	—	23	1	no	maximal, Sylow
66	C₂²×C₂₃	—	2	4	no	—
67	C₂²×C₂₃	—	2	2	no	—
68	C₂²×C₂₃	—	4	4	no	—
69	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
70	C₂²×C₂₃	C₂²×Q₈	1	4	yes	—
71	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
72	C₄×C₂₃	—	2	8	no	—
73	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
74	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
75	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
76	Dic₂₃	—	2	16	no	—
77	#544	—	23	8	no	—
78	#544	—	23	4	no	—
79	#543	—	23	4	no	—
80	#533	—	23	8	no	—
81	#533	—	23	4	no	—
82	C₂×#117	—	23	2	no	—
83	C₂×#117	D₂₃	1	1	yes	characteristic
84	C₂×C₂₃	—	2	4	no	—
85	C₂×C₂₃	—	2	2	no	—
86	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
87	C₂×C₂₃	#585	1	2	yes	—
88	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
89	C₄⋊Q₈	—	23	16	no	—
90	C₄⋊Q₈	—	23	8	no	—
91	#125	—	23	16	no	—
92	#125	—	23	8	no	—
93	C₂²⋊Q₈	—	23	16	no	—
94	C₂²⋊Q₈	—	23	32	no	—
95	C₄×Q₈	—	23	16	no	—
96	#117	—	23	4	no	—
97	#117	—	23	4	no	—
98	#117	—	23	4	no	—
99	#117	—	23	4	no	—
100	#117	C₂×D₂₃	1	4	yes	—
101	#117	C₂×D₂₃	1	4	yes	—
102	C₂×C₄⋊C₄	—	23	4	no	—
103	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
104	C₂×C₂²⋊C₄	—	23	4	no	—
105	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
106	C₂×C₄²	—	23	4	no	—
107	C₂×C₄²	C₂×D₂₃	1	2	yes	—
108	C₂₃	#3458	1	1	yes	characteristic, complemented, Sylow
109	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
110	C₂×Q₈	—	23	16	no	—
111	C₂²×C₄	—	23	8	no	—
112	C₂²×C₄	Dic₄₆	1	8	yes	complemented
113	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
114	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
115	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
118	C₄⋊C₄	—	23	8	no	—
119	C₄⋊C₄	—	23	8	no	—
120	C₄⋊C₄	—	23	32	no	—
121	C₄⋊C₄	—	23	16	no	—
122	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
123	C₂²⋊C₄	—	23	8	no	—
124	C₂²⋊C₄	—	23	8	no	—
125	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
126	C₄²	—	23	8	no	—
127	C₄²	—	23	8	no	—
128	C₄²	C₂²×D₂₃	1	4	yes	—
129	C₄²	C₂²×D₂₃	1	4	yes	—
130	C₂³	—	4	2	no	—
131	C₂³	C₂×Dic₄₆	1	4	yes	—
132	C₂³	C₂³×D₂₃	1	1	yes	characteristic
133	C₂³	C₂³×D₂₃	1	2	yes	—
134	Q₈	—	92	16	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	23	16	no	—
137	C₂×C₄	—	92	8	no	—
138	C₂×C₄	C₂×Dic₄₆	1	8	yes	—
139	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
140	C₂×C₄	C₂×Dic₄₆	1	8	yes	—
141	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
142	C₂×C₄	C₂³×D₂₃	1	8	yes	—
143	C₂×C₄	C₂³×D₂₃	1	2	yes	—
144	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	C₂²×Dic₄₆	1	2	yes	—
150	C₂²	C₂²×Dic₄₆	1	4	yes	—
151	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	46	16	no	—
154	C₄	C₂²×Dic₄₆	1	4	yes	—
155	C₄	C₂²×Dic₄₆	1	2	yes	—
156	C₄	C₂²×Dic₄₆	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	2	4	no	—
159	C₂	C₂³×Dic₄₆	1	1	yes	characteristic
160	C₂	#13093253	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×#117	C₂
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#3458
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #13093048	C₂	semidirect
2. #13093045	C₂	semidirect
4. #13092974	C₂	semidirect
5. #13092970	C₂	semidirect
22. #10625061	C₂²	semidirect
23. #10625059	C₂²	semidirect
25. #10625041	C₂²	semidirect
26. #10625040	C₂²	semidirect
27. C₄×Dic₄₆	C₂²	semidirect
29. C₂²×C₄×C₂₃	Q₈	semidirect
108. C₂₃	#3458	semidirect
112. C₂²×C₄	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
#13093048	2
#13093045	2
#13093043	2
#13092974	2
#13092970	2
C₂×#10625097	2
C₂×C₂₃×#117	2
#3458	23