Subgroups — #31800856

Order 17728.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22579937	C₂	1	1	yes	maximal, characteristic
2	(C₂×#11764308)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#22579856	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11764308	—	2	1	no	—
5	C₂×#14555667	—	2	1	no	—
6	C₂×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₂₇₇	C₄	1	1	yes	characteristic, complemented
8	#17984542	—	2	1	no	—
9	#17984540	—	2	1	no	—
10	C₂×Q₈×C₂₇₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#11764308	—	2	2	no	—
12	C₂×#11764308	—	4	1	no	—
13	C₂×#11764308	—	4	1	no	—
14	#14555667	—	4	1	no	—
15	C₂²×D₂₇₇	D₄	1	1	yes	characteristic
16	C₂×Dic₂₇₇	—	2	1	no	—
17	C₂×Dic₂₇₇	D₄	1	1	yes	characteristic
18	C₄×D₂₇₇	—	4	1	no	—
19	Q₈×C₂₇₇	—	4	1	no	—
20	C₂×C₄×C₂₇₇	—	2	1	no	—
21	C₂×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₂₇₇	—	2	1	no	—
24	C₂×D₂₇₇	—	2	1	no	—
25	C₂×D₂₇₇	SD₁₆	1	1	yes	characteristic
26	C₂×D₂₇₇	Q₁₆	1	1	yes	characteristic
27	#11764308	—	4	2	no	—
28	C₄×C₂₇₇	—	2	1	no	—
29	C₄×C₂₇₇	—	4	1	no	—
30	Dic₂₇₇	—	2	1	no	—
31	Dic₂₇₇	—	4	1	no	—
32	C₂×C₂₇₇	#99	1	1	yes	characteristic
33	C₂×C₂₇₇	#103	1	1	yes	characteristic
34	C₂×C₂₇₇	#104	1	1	yes	characteristic
35	D₂₇₇	—	2	1	no	—
36	D₂₇₇	—	2	1	no	—
37	C₂₇₇	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	277	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	277	1	no	—
40	C₂²⋊C₈	—	277	1	no	—
41	#95	—	277	1	no	—
42	C₂×Q₈	#11764308	1	1	yes	characteristic, complemented
43	C₂²×C₄	—	277	1	no	—
44	C₂²×C₄	—	554	1	no	—
45	C₂×C₈	—	554	1	no	—
46	C₄⋊C₄	—	277	1	no	—
47	C₄⋊C₄	—	554	1	no	—
48	C₂²⋊C₄	—	554	1	no	—
49	C₂³	—	277	1	no	—
50	Q₈	—	4	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	277	1	no	—
53	C₂×C₄	—	554	2	no	—
54	C₂×C₄	—	554	1	no	—
55	C₂×C₄	—	1108	1	no	—
56	C₂×C₄	—	1108	1	no	—
57	C₂×C₄	—	1108	1	no	—
58	C₂×C₄	C₂×#11764308	1	1	yes	characteristic
59	C₈	—	1108	1	no	—
60	C₂²	—	277	1	no	—
61	C₂²	—	277	1	no	—
62	C₂²	—	554	1	no	—
63	C₂²	—	554	1	no	—
64	C₂²	C₂²⋊#11764308	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	554	1	no	—
68	C₄	—	1108	2	no	—
69	C₄	—	1108	1	no	—
70	C₂	—	554	1	no	—
71	C₂	—	554	1	no	—
72	C₂	#22579938	1	1	yes	characteristic
73	C₂	#22579943	1	1	yes	characteristic
74	C₂	#22579945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11764308
commutator	C₂×C₄×C₂₇₇	C₂×C₄
Frattini	C₂×C₄	C₂×#11764308
Fitting	C₂×Q₈×C₂₇₇	C₄
socle	C₂²×C₂₇₇	C₂²⋊C₄
prime radical	C₂²×D₂₇₇	D₄
cosocle	C₂×C₄×D₂₇₇	C₂²
hypercenter	C₂×Q₈	#11764308
hypocenter	C₂₇₇	#328
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
7. C₄⋊Dic₂₇₇	C₄	semidirect
10. C₂×Q₈×C₂₇₇	C₄	semidirect
37. C₂₇₇	#328	semidirect
42. C₂×Q₈	#11764308	semidirect

Maximal subgroups

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

subgroup	index
#22579937	2
(C₂×#11764308)⋊C₄	2
#22579856	2
#328	277