Subgroups — #31800029

Order 17728.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984547	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14555660	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14555660	C₂	1	4	yes	maximal, complemented
4	#117×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	#17984547	—	2	2	no	—
6	C₂×#14555660	—	2	2	no	—
7	C₂×#14555660	C₂²	1	2	yes	—
8	C₂×#14555660	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₇₇	C₄	1	2	yes	—
11	C₂²⋊C₄×C₂₇₇	C₄	1	2	yes	—
12	C₄²×C₂₇₇	C₂²	1	2	yes	—
13	#14555660	—	2	2	no	—
14	#14555660	—	2	2	no	—
15	C₂³×C₂₇₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₇₇	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₇₇	D₄	1	2	yes	—
19	C₂×C₄×C₂₇₇	Q₈	1	2	yes	—
20	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
21	C₂²×C₂₇₇	—	2	2	no	—
22	C₂²×C₂₇₇	C₄⋊C₄	1	2	yes	—
23	C₂²×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₇₇	—	2	4	no	—
25	C₄×C₂₇₇	C₄⋊C₄	1	2	yes	—
26	C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₂₇₇	—	2	2	no	—
29	C₂×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₂₇₇	#131	1	2	yes	—
31	C₂₇₇	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	277	1	no	maximal, Sylow
33	C₂×M₁₆	—	277	1	no	—
34	C₂²×C₈	—	277	1	no	—
35	#117	D₂₇₇	1	1	yes	characteristic
36	C₄⋊C₈	—	277	4	no	—
37	C₂²×C₄	C₂×D₂₇₇	1	1	yes	characteristic
38	M₁₆	—	554	2	no	—
39	C₂×C₈	—	277	2	no	—
40	C₂×C₈	—	277	2	no	—
41	C₂×C₈	—	554	2	no	—
42	C₄⋊C₄	Dic₂₇₇	1	2	yes	—
43	C₂²⋊C₄	Dic₂₇₇	1	2	yes	—
44	C₄²	C₂×D₂₇₇	1	2	yes	—
45	C₂³	C₂×Dic₂₇₇	1	1	yes	characteristic
46	C₂×C₄	C₂×Dic₂₇₇	1	4	yes	—
47	C₂×C₄	C₂×Dic₂₇₇	1	1	yes	characteristic
48	C₂×C₄	D₁₁₀₈	1	2	yes	—
49	C₂×C₄	Dic₅₅₄	1	2	yes	—
50	C₂×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
51	C₈	—	554	2	no	—
52	C₈	—	554	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊Dic₂₇₇	1	2	yes	—
55	C₂²	C₂²×Dic₂₇₇	1	1	yes	characteristic
56	C₄	—	2	4	no	—
57	C₄	C₄⋊Dic₂₇₇	1	2	yes	—
58	C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
59	C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₂₇₇	1	1	yes	characteristic
62	C₂	#22579887	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₇₇
commutator, socle	C₂²×C₂₇₇	C₂²×C₄
Fitting	#117×C₂₇₇	C₂
prime radical	C₂³×C₂₇₇	C₂×C₄
cosocle	C₂×C₄×C₂₇₇	C₂³
hypercenter	#117	D₂₇₇
hypocenter	C₂₇₇	#424
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊#14555660	C₂	semidirect
31. C₂₇₇	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984547	2
C₂²×#14555660	2
C₄⋊#14555660	2
#117×C₂₇₇	2
#424	277