Subgroups — #31801049

Order 17728.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22579967	C₂	1	2	yes	maximal, complemented
2	#22579950	C₂	1	1	yes	maximal, characteristic, complemented
3	#22579944	C₂	1	2	yes	maximal, complemented
4	#22579934	C₂	1	1	yes	maximal, characteristic, complemented
5	#22579899	C₂	1	1	yes	maximal, characteristic
6	#17984573	C₂²	1	2	yes	—
7	#17984572	—	2	2	no	—
8	D₂₇₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14555667	—	2	2	no	—
10	C₄×#11764308	C₂²	1	1	yes	characteristic
11	#17984562	C₂²	1	2	yes	—
12	Dic₁₁₀₈	C₄	1	1	yes	characteristic, complemented
13	#17984551	C₄	1	2	yes	complemented
14	C₈×D₂₇₇	C₂²	1	1	yes	characteristic
15	D₈×C₂₇₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11764308	—	2	1	no	—
17	#14555667	—	2	2	no	—
18	#14555667	D₄	1	2	yes	—
19	Dic₅₅₄	C₂×C₄	1	2	yes	—
20	C₂²⋊D₂₇₇	—	2	2	no	—
21	C₂×Dic₂₇₇	—	2	2	no	—
22	C₄×D₂₇₇	C₂³	1	1	yes	characteristic
23	#14555660	C₂×C₄	1	1	yes	characteristic
24	D₄×C₂₇₇	C₂×C₄	1	2	yes	—
25	C₈×C₂₇₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₇₇	—	2	2	no	—
27	C₂×D₂₇₇	#41	1	1	yes	characteristic
28	#11764308	—	4	1	no	—
29	C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
30	Dic₂₇₇	—	2	2	no	—
31	Dic₂₇₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₂₇₇	—	4	2	no	—
33	C₂×C₂₇₇	C₄×D₄	1	1	yes	characteristic
34	D₂₇₇	—	2	1	no	—
35	C₂₇₇	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	277	1	no	maximal, Sylow
37	#135	—	277	1	no	—
38	#131	—	277	2	no	—
39	#108	—	277	1	no	—
40	#104	—	277	2	no	—
41	#97	—	277	1	no	—
42	#41	—	277	2	no	—
43	Q₁₆	—	277	1	no	—
44	SD₁₆	—	277	2	no	—
45	D₈	#11764308	1	1	yes	characteristic, complemented
46	M₁₆	—	277	2	no	—
47	M₁₆	—	554	2	no	—
48	C₂×C₈	—	277	1	no	—
49	C₂×C₈	—	277	1	no	—
50	C₂×C₈	—	554	2	no	—
51	C₄²	—	277	1	no	—
52	Q₈	—	277	2	no	—
53	D₄	—	554	2	no	—
54	D₄	C₂×#11764308	1	2	yes	—
55	C₂×C₄	—	277	1	no	—
56	C₂×C₄	—	554	2	no	—
57	C₂×C₄	—	554	1	no	—
58	C₈	—	277	2	no	—
59	C₈	—	277	1	no	—
60	C₈	—	554	2	no	—
61	C₈	C₂×#11764308	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	—	277	1	no	—
64	C₄	—	277	1	no	—
65	C₄	—	554	2	no	—
66	C₄	—	1108	1	no	—
67	C₄	C₂²×#11764308	1	1	yes	characteristic
68	C₂	—	4	2	no	—
69	C₂	—	554	1	no	—
70	C₂	D₄×#11764308	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11764308
commutator	C₄×C₂₇₇	C₂²×C₄
Frattini	C₄	C₂²×#11764308
Fitting	D₈×C₂₇₇	C₄
socle	C₂×C₂₇₇	C₄×D₄
prime radical	#22579899	C₂
cosocle	C₄×D₂₇₇	C₂³
hypercenter	D₈	#11764308
hypocenter	C₂₇₇	#444
LCS	G ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #22579967	C₂	semidirect
2. #22579950	C₂	semidirect
3. #22579944	C₂	semidirect
4. #22579934	C₂	semidirect
12. Dic₁₁₀₈	C₄	semidirect
13. #17984551	C₄	semidirect
15. D₈×C₂₇₇	C₄	semidirect
35. C₂₇₇	#444	semidirect
45. D₈	#11764308	semidirect

Maximal subgroups

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

subgroup	index
#22579967	2
#22579950	2
#22579944	2
#22579934	2
#22579899	2
#444	277