Subgroups — #13092998

Order 1472.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#67288	C₂	1	2	yes	maximal, complemented
2	C₂×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625101	C₂	1	4	yes	maximal, complemented
4	#10625068	C₂	1	4	yes	maximal, complemented
5	#10625066	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	—
9	#67288	—	2	8	no	—
10	C₂×D₉₂	C₂²	1	2	yes	—
11	C₂×D₉₂	C₂²	1	4	yes	—
12	C₂×C₄×D₂₃	C₂²	1	4	yes	—
13	C₂×Dic₄₆	C₂²	1	2	yes	—
14	C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
15	(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	—
16	C₄⋊Dic₂₃	C₂²	1	4	yes	—
17	C₂³×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×D₂₃	C₂³	1	4	yes	—
19	C₂×C₄×C₂₃	—	4	2	no	—
20	C₂×C₄×C₂₃	D₄	1	4	yes	complemented
21	C₂×C₄×C₂₃	C₂³	1	4	yes	—
22	C₂×C₄×C₂₃	C₂³	1	2	yes	—
23	C₂²⋊D₂₃	—	2	8	no	—
24	C₂×Dic₂₃	C₂³	1	4	yes	—
25	D₉₂	—	2	4	no	—
26	D₉₂	—	4	4	no	—
27	C₄×D₂₃	—	2	8	no	—
28	Dic₄₆	—	2	4	no	—
29	C₂²×C₂₃	—	2	1	no	—
30	C₂²×C₂₃	—	2	1	no	—
31	C₂²×C₂₃	C₂×D₄	1	2	yes	—
32	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
33	C₂×D₂₃	—	2	4	no	—
34	C₂×D₂₃	—	2	4	no	—
35	C₂×D₂₃	—	2	4	no	—
36	C₄×C₂₃	—	2	4	no	—
37	C₄×C₂₃	C₂×D₄	1	4	yes	—
38	Dic₂₃	—	2	4	no	—
39	#537	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	—	2	2	no	—
41	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
42	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
43	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
44	D₂₃	—	4	4	no	—
45	C₂×#41	—	23	2	no	—
46	C₂²⋊Q₈	—	23	4	no	—
47	#121	—	23	4	no	—
48	#121	—	23	4	no	—
49	C₂×C₄⋊C₄	D₂₃	1	1	yes	characteristic, complemented
50	C₂₃	#537	1	1	yes	characteristic, complemented, Sylow
51	#41	—	46	8	no	—
52	C₂×Q₈	—	23	2	no	—
53	C₂×D₄	—	23	4	no	—
54	C₂×D₄	—	23	2	no	—
55	C₂×D₄	—	23	4	no	—
56	C₂²×C₄	—	23	4	no	—
57	C₂²×C₄	C₂×D₂₃	1	2	yes	—
58	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
59	C₄⋊C₄	—	23	4	no	—
60	C₄⋊C₄	C₂×D₂₃	1	4	yes	—
61	C₂²⋊C₄	—	23	8	no	—
62	C₂³	—	23	4	no	—
63	C₂³	C₂²×D₂₃	1	1	yes	characteristic
64	Q₈	—	46	4	no	—
65	D₄	—	46	8	no	—
66	D₄	—	46	4	no	—
67	D₄	—	92	4	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	23	4	no	—
70	C₂×C₄	—	46	8	no	—
71	C₂×C₄	D₉₂	1	4	yes	complemented
72	C₂×C₄	C₂²×D₂₃	1	4	yes	—
73	C₂×C₄	C₂²×D₂₃	1	2	yes	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	46	4	no	—
77	C₂²	—	46	4	no	—
78	C₂²	—	46	4	no	—
79	C₂²	C₂×D₉₂	1	2	yes	—
80	C₂²	C₂³×D₂₃	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	46	4	no	—
83	C₄	C₂×D₉₂	1	4	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	92	4	no	—
86	C₂	C₂²×D₉₂	1	1	yes	characteristic
87	C₂	#10625155	1	1	yes	characteristic
88	C₂	#10625158	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂×C₄⋊C₄×C₂₃	C₂
hypercenter	C₂×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#537
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#67288	C₂	semidirect
2. C₂×C₄⋊C₄×C₂₃	C₂	semidirect
3. #10625101	C₂	semidirect
4. #10625068	C₂	semidirect
5. #10625066	C₂	semidirect
20. C₂×C₄×C₂₃	D₄	semidirect
49. C₂×C₄⋊C₄	D₂₃	semidirect
50. C₂₃	#537	semidirect
71. C₂×C₄	D₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#67288	2
C₂×C₄⋊C₄×C₂₃	2
#10625101	2
#10625068	2
#10625066	2
#537	23