Subgroups — #37551521

Order 23264.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₉₀₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₇₂₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₇₂₇)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₂₇	C₂²	1	1	yes	characteristic
9	C₂²×D₇₂₇	C₂²	1	1	yes	characteristic
10	D₂₉₀₈	—	2	2	no	—
11	C₂²⋊D₇₂₇	—	2	2	no	—
12	C₂²⋊D₇₂₇	—	2	2	no	—
13	C₂×Dic₇₂₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₇₂₇	C₂²	1	1	yes	characteristic
15	C₄×D₇₂₇	—	2	2	no	—
16	C₂³×C₇₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
19	C₂²×C₇₂₇	—	2	1	no	—
20	C₂²×C₇₂₇	—	2	1	no	—
21	C₂²×C₇₂₇	—	2	1	no	—
22	C₂²×C₇₂₇	C₂³	1	1	yes	characteristic
23	C₂×D₇₂₇	—	2	1	no	—
24	C₂×D₇₂₇	—	2	1	no	—
25	C₂×D₇₂₇	—	2	1	no	—
26	C₂×D₇₂₇	—	2	1	no	—
27	C₂×D₇₂₇	—	2	1	no	—
28	C₂×D₇₂₇	D₄	1	2	yes	complemented
29	C₄×C₇₂₇	—	2	1	no	—
30	C₄×C₇₂₇	—	2	1	no	—
31	Dic₇₂₇	—	2	1	no	—
32	Dic₇₂₇	D₄	1	2	yes	complemented
33	C₂×C₇₂₇	—	4	1	no	—
34	C₂×C₇₂₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₇₂₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₇₂₇	#41	1	1	yes	characteristic
37	D₇₂₇	—	2	2	no	—
38	D₇₂₇	—	4	1	no	—
39	C₇₂₇	#121	1	1	yes	characteristic, complemented, Sylow
40	#121	—	727	1	no	maximal, Sylow
41	C₂×D₄	—	727	1	no	—
42	C₂×D₄	—	727	1	no	—
43	C₂×D₄	—	727	1	no	—
44	C₂²×C₄	—	727	1	no	—
45	C₄⋊C₄	—	727	1	no	—
46	C₂²⋊C₄	—	727	1	no	—
47	C₂²⋊C₄	D₇₂₇	1	1	yes	characteristic, complemented
48	C₂³	—	727	1	no	—
49	C₂³	—	727	1	no	—
50	C₂³	C₂×D₇₂₇	1	1	yes	characteristic
51	D₄	—	1454	2	no	—
52	D₄	—	1454	2	no	—
53	D₄	—	1454	2	no	—
54	C₂×C₄	—	1454	2	no	—
55	C₂×C₄	—	727	1	no	—
56	C₂×C₄	—	727	1	no	—
57	C₂×C₄	C₂×D₇₂₇	1	1	yes	characteristic
58	C₂×C₄	C₂×D₇₂₇	1	1	yes	characteristic
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	1454	1	no	—
63	C₂²	—	1454	1	no	—
64	C₂²	—	1454	1	no	—
65	C₂²	—	1454	1	no	—
66	C₂²	—	1454	1	no	—
67	C₂²	—	727	2	no	—
68	C₂²	C₂²×D₇₂₇	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	1454	1	no	—
72	C₄	—	727	2	no	—
73	C₂	—	4	1	no	—
74	C₂	—	2908	1	no	—
75	C₂	—	1454	2	no	—
76	C₂	D₄×D₇₂₇	1	1	yes	characteristic
77	C₂	D₄×D₇₂₇	1	1	yes	characteristic
78	C₂	#26042740	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇₂₇
commutator, socle, cosocle	C₂²×C₇₂₇	C₂³
Fitting	C₂²⋊C₄×C₇₂₇	C₂
hypercenter	C₂²⋊C₄	D₇₂₇
hypocenter	C₇₂₇	#121
LCS	G ▷ C₂²×C₇₂₇ ▷ C₇₂₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₉₀₈	C₂	semidirect
2. C₂×C₂²⋊D₇₂₇	C₂	semidirect
3. C₂×C₂²⋊D₇₂₇	C₂	semidirect
4. C₂×C₄×D₇₂₇	C₂	semidirect
5. Dic₇₂₇⋊C₄	C₂	semidirect
6. (C₂×D₇₂₇)⋊C₄	C₂	semidirect
7. C₂²⋊C₄×C₇₂₇	C₂	semidirect
28. C₂×D₇₂₇	D₄	semidirect
32. Dic₇₂₇	D₄	semidirect
39. C₇₂₇	#121	semidirect
47. C₂²⋊C₄	D₇₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₉₀₈	2
C₂×C₂²⋊D₇₂₇	2
C₂×C₄×D₇₂₇	2
Dic₇₂₇⋊C₄	2
(C₂×D₇₂₇)⋊C₄	2
C₂²⋊C₄×C₇₂₇	2
#121	727