Subgroups — #31801082

Order 17728.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22579968	C₂	1	8	yes	maximal, complemented
2	D₂₇₇⋊M₁₆	C₂	1	6	yes	maximal, complemented
3	C₂×#17984564	C₂	1	1	yes	maximal, characteristic
4	#17984573	C₂²	1	12	yes	complemented
5	#17984572	C₂²	1	4	yes	—
6	D₂₇₇⋊C₈	C₂²	1	12	yes	complemented
7	#17984564	C₄	1	4	yes	—
8	#17984564	C₂²	1	4	yes	—
9	C₂×D₁₁₀₈	C₄	1	3	yes	—
10	C₂×C₄×D₂₇₇	C₂²	1	3	yes	—
11	C₂×Q₈×C₂₇₇	C₄	1	1	yes	characteristic
12	#14555667	C₂³	1	8	yes	—
13	C₂²×D₂₇₇	C₂×C₄	1	3	yes	—
14	D₁₁₀₈	C₂×C₄	1	12	yes	—
15	C₂×Dic₂₇₇	C₂³	1	1	yes	characteristic
16	C₄×D₂₇₇	C₂×C₄	1	6	yes	—
17	C₄×D₂₇₇	C₂³	1	6	yes	—
18	Q₈×C₂₇₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₂₇₇	C₂×C₄	1	3	yes	—
20	C₂²×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₂₇₇	—	4	3	no	—
22	C₂×D₂₇₇	C₂²×C₄	1	6	yes	—
23	C₄×C₂₇₇	C₂²×C₄	1	6	yes	—
24	Dic₂₇₇	C₂²×C₄	1	1	yes	characteristic
25	Dic₂₇₇	C₂⁴	1	1	yes	characteristic
26	C₂×C₂₇₇	—	2	1	no	—
27	C₂×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
28	D₂₇₇	—	2	6	no	—
29	C₂₇₇	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	277	1	no	maximal, Sylow
31	C₂×#41	—	277	1	no	—
32	#131	—	277	8	no	—
33	C₂×M₁₆	—	277	6	no	—
34	#41	—	277	4	no	—
35	#41	—	277	4	no	—
36	C₂×Q₈	#11764308	1	1	yes	characteristic
37	C₂×D₄	—	277	3	no	—
38	C₂²×C₄	—	277	3	no	—
39	M₁₆	—	277	4	no	—
40	M₁₆	—	277	12	no	—
41	C₂×C₈	—	277	12	no	—
42	C₂³	—	277	3	no	—
43	Q₈	C₂×#11764308	1	4	yes	—
44	D₄	—	277	12	no	—
45	C₂×C₄	—	277	1	no	—
46	C₂×C₄	—	277	6	no	—
47	C₂×C₄	—	277	6	no	—
48	C₂×C₄	C₂×#11764308	1	3	yes	—
49	C₈	—	277	8	no	—
50	C₂²	—	277	6	no	—
51	C₂²	—	1108	3	no	—
52	C₂²	C₂²×#11764308	1	1	yes	characteristic
53	C₄	—	277	1	no	—
54	C₄	—	277	1	no	—
55	C₄	C₂²×#11764308	1	6	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	554	6	no	—
58	C₂	C₂³×#11764308	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11764308
commutator, socle	C₂×C₂₇₇	C₂³×C₄
Fitting	C₂×Q₈×C₂₇₇	C₄
prime radical	C₂×#17984564	C₂
cosocle	Dic₂₇₇	C₂⁴
hypercenter	C₂×Q₈	#11764308
hypocenter	C₂₇₇	#568
LCS	G ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #22579968	C₂	semidirect
2. D₂₇₇⋊M₁₆	C₂	semidirect
4. #17984573	C₂²	semidirect
6. D₂₇₇⋊C₈	C₂²	semidirect
29. C₂₇₇	#568	semidirect

Maximal subgroups

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

subgroup	index
#22579968	2
D₂₇₇⋊M₁₆	2
C₂×#17984564	2
#568	277