Subgroups — #31800359

Order 17728.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22579868	C₂	1	1	yes	maximal, characteristic
2	#22579803	C₂	1	2	yes	maximal, complemented
3	#22579800	C₂	1	2	yes	maximal
4	C₄×#14555660	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₅₄	C₂²	1	1	yes	characteristic
7	C₂×D₁₁₀₈	C₂²	1	1	yes	characteristic
8	C₂×#14555660	C₂²	1	2	yes	—
9	#17984540	—	2	2	no	—
10	C₄⋊C₄×C₂₇₇	—	2	2	no	—
11	C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
12	C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₂²×D₂₇₇	—	2	1	no	—
14	Dic₅₅₄	—	2	2	no	—
15	D₁₁₀₈	—	2	2	no	—
16	C₂×Dic₂₇₇	—	2	1	no	—
17	#14555660	—	2	2	no	—
18	C₂×C₄×C₂₇₇	—	2	2	no	—
19	C₂×C₄×C₂₇₇	D₄	1	2	yes	—
20	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
21	C₂²×C₂₇₇	C₂×D₄	1	1	yes	characteristic
22	C₂×D₂₇₇	—	4	1	no	—
23	C₂×D₂₇₇	—	4	2	no	—
24	C₄×C₂₇₇	—	2	2	no	—
25	C₄×C₂₇₇	—	4	2	no	—
26	C₄×C₂₇₇	#41	1	2	yes	—
27	Dic₂₇₇	—	4	1	no	—
28	C₂×C₂₇₇	#124	1	1	yes	characteristic
29	C₂×C₂₇₇	#135	1	2	yes	—
30	D₂₇₇	—	8	1	no	—
31	C₂₇₇	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	277	1	no	maximal, Sylow
33	#125	D₂₇₇	1	1	yes	characteristic, complemented
34	#124	—	277	1	no	—
35	#103	—	277	2	no	—
36	#102	—	277	2	no	—
37	C₄×C₈	—	277	1	no	—
38	C₂×Q₈	—	277	1	no	—
39	C₂×D₄	—	277	1	no	—
40	C₂×C₈	—	277	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
43	C₂²⋊C₄	—	554	2	no	—
44	C₄²	C₂×D₂₇₇	1	1	yes	characteristic
45	C₂³	—	554	1	no	—
46	Q₈	—	554	2	no	—
47	D₄	—	554	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	554	1	no	—
50	C₂×C₄	C₂²⋊D₂₇₇	1	2	yes	—
51	C₂×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
52	C₈	—	554	2	no	—
53	C₂²	—	1108	1	no	—
54	C₂²	—	1108	2	no	—
55	C₂²	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	1108	1	no	—
59	C₄	#17984564	1	2	yes	—
60	C₂	—	2216	1	no	—
61	C₂	#22579869	1	1	yes	characteristic
62	C₂	#22579900	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₇₇
commutator, cosocle	C₂×C₄×C₂₇₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₇
Fitting	#125×C₂₇₇	C₂
socle	C₂²×C₂₇₇	C₂×D₄
prime radical	C₂×D₁₁₀₈	C₂²
hypercenter	#125	D₂₇₇
hypocenter	C₂₇₇	#488
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #22579803	C₂	semidirect
4. C₄×#14555660	C₂	semidirect
5. #125×C₂₇₇	C₂	semidirect
31. C₂₇₇	#488	semidirect
33. #125	D₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
#22579868	2
#22579803	2
#22579800	2
C₄×#14555660	2
#125×C₂₇₇	2
#488	277