Subgroups — #13092421

Order 1472.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#67267	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624991	C₂	1	4	yes	maximal, complemented
5	C₂³×D₂₃	—	2	1	no	—
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×D₉₂	—	2	2	no	—
8	C₂×D₉₂	—	2	1	no	—
9	C₂×D₉₂	C₄	1	4	yes	complemented
10	C₂×D₉₂	C₂²	1	2	yes	—
11	C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
12	C₂²⋊C₄×C₂₃	—	2	1	no	—
13	C₄²×C₂₃	—	2	1	no	—
14	#67267	—	2	2	no	—
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	complemented
16	C₂³×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×D₂₃	—	2	4	no	—
18	C₂²×D₂₃	—	2	2	no	—
19	C₂²×D₂₃	—	4	4	no	—
20	C₂×C₄×C₂₃	—	2	2	no	—
21	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₃	D₄	1	2	yes	—
23	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	D₉₂	—	2	4	no	—
26	D₉₂	—	2	2	no	—
27	D₉₂	C₂×C₄	1	4	yes	—
28	C₂₃⋊C₈	—	2	2	no	—
29	C₂²×C₂₃	—	2	2	no	—
30	C₂²×C₂₃	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×D₂₃	—	2	4	no	—
33	C₂×D₂₃	—	4	8	no	—
34	C₂×D₂₃	—	4	4	no	—
35	C₄×C₂₃	—	4	2	no	—
36	C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	#418	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	—	2	2	no	—
41	C₂×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
43	D₂₃	—	4	4	no	—
44	C₂²×D₄	—	23	1	no	—
45	C₂×M₁₆	—	23	1	no	—
46	#117	D₂₃	1	1	yes	characteristic, complemented
47	#102	—	23	4	no	—
48	C₂₃	#418	1	1	yes	characteristic, complemented, Sylow
49	C₂⁴	—	46	1	no	—
50	C₂×D₄	—	23	4	no	—
51	C₂×D₄	—	23	2	no	—
52	C₂×D₄	—	46	2	no	—
53	C₂×D₄	—	46	1	no	—
54	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
55	M₁₆	—	46	2	no	—
56	C₂×C₈	—	23	2	no	—
57	C₄⋊C₄	C₂×D₂₃	1	2	yes	complemented
58	C₂²⋊C₄	—	2	1	no	—
59	C₄²	—	2	1	no	—
60	C₂³	—	46	4	no	—
61	C₂³	—	46	2	no	—
62	C₂³	—	92	4	no	—
63	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
64	D₄	—	23	4	no	—
65	D₄	—	46	4	no	—
66	D₄	—	46	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₄×D₂₃	1	2	yes	—
69	C₂×C₄	D₉₂	1	2	yes	—
70	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
72	C₈	—	46	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	—	46	4	no	—
75	C₂²	—	92	8	no	—
76	C₂²	—	92	4	no	—
77	C₂²	(C₂×D₂₃)⋊C₄	1	2	yes	—
78	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
81	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
82	C₄	C₂×D₉₂	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	92	4	no	—
85	C₂	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
86	C₂	#10625120	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#117	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂²×D₉₂	C₂
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	#117	D₂₃
hypocenter	C₂₃	#418
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₂₃×#117	C₂	semidirect
3. C₂×#67267	C₂	semidirect
4. #10624991	C₂	semidirect
9. C₂×D₉₂	C₄	semidirect
11. C₄⋊C₄×C₂₃	C₂²	semidirect
15. C₂×C₂₃⋊C₈	C₂²	semidirect
46. #117	D₂₃	semidirect
48. C₂₃	#418	semidirect
57. C₄⋊C₄	C₂×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₉₂	2
C₂₃×#117	2
C₂×#67267	2
#10624991	2
#418	23