Subgroups — #31801042

Order 17728.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11764308	C₂	1	2	yes	maximal
2	#22579948	C₂	1	1	yes	maximal, characteristic
3	#22579943	C₂	1	2	yes	maximal
4	#22579934	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₂₇₇	C₂	1	1	yes	maximal, characteristic
6	D₂₇₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊#11764308	—	2	2	no	—
8	C₄⋊#11764308	C₂²	1	2	yes	—
9	C₄×#11764308	—	2	2	no	—
10	C₄×#11764308	C₂²	1	1	yes	characteristic
11	Q₈×D₂₇₇	C₂²	1	2	yes	—
12	Dic₁₁₀₈	C₄	1	1	yes	characteristic, complemented
13	#17984553	C₄	1	2	yes	complemented
14	C₈×D₂₇₇	C₂²	1	1	yes	characteristic
15	Q₁₆×C₂₇₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11764308	—	2	2	no	—
17	C₂×#11764308	D₄	1	2	yes	—
18	#14555667	—	2	1	no	—
19	Dic₅₅₄	—	2	2	no	—
20	Dic₅₅₄	C₂×C₄	1	2	yes	—
21	C₄×D₂₇₇	—	2	2	no	—
22	C₄×D₂₇₇	C₂³	1	1	yes	characteristic
23	#14555660	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₂₇₇	C₂×C₄	1	2	yes	—
25	C₈×C₂₇₇	C₂×C₄	1	1	yes	characteristic
26	C₂×D₂₇₇	C₂×D₄	1	1	yes	characteristic
27	#11764308	—	2	2	no	—
28	#11764308	—	4	2	no	—
29	C₄×C₂₇₇	—	2	2	no	—
30	C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
31	Dic₂₇₇	—	2	2	no	—
32	Dic₂₇₇	#41	1	1	yes	characteristic
33	C₂×C₂₇₇	C₄×D₄	1	1	yes	characteristic
34	D₂₇₇	#137	1	2	yes	—
35	C₂₇₇	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	277	1	no	maximal, Sylow
37	C₂×Q₁₆	—	277	1	no	—
38	C₄×Q₈	—	277	2	no	—
39	#106	—	277	1	no	—
40	#103	—	277	2	no	—
41	#97	—	277	1	no	—
42	C₂×Q₈	—	277	2	no	—
43	Q₁₆	—	277	2	no	—
44	Q₁₆	—	277	1	no	—
45	Q₁₆	#11764308	1	1	yes	characteristic, complemented
46	C₂×C₈	—	277	1	no	—
47	C₂×C₈	—	277	1	no	—
48	C₄⋊C₄	—	277	2	no	—
49	C₄⋊C₄	—	554	2	no	—
50	C₄²	—	277	1	no	—
51	C₄²	—	554	2	no	—
52	Q₈	—	277	2	no	—
53	Q₈	—	554	2	no	—
54	Q₈	C₂×#11764308	1	2	yes	—
55	C₂×C₄	—	277	1	no	—
56	C₂×C₄	—	277	2	no	—
57	C₂×C₄	—	554	2	no	—
58	C₂×C₄	—	554	2	no	—
59	C₈	—	277	1	no	—
60	C₈	—	554	1	no	—
61	C₈	C₂×#11764308	1	1	yes	characteristic
62	C₂²	—	277	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	277	1	no	—
65	C₄	—	554	2	no	—
66	C₄	—	554	2	no	—
67	C₄	—	1108	2	no	—
68	C₄	C₂²×#11764308	1	1	yes	characteristic
69	C₂	—	277	2	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	Q₁₆×C₂₇₇	C₄
socle	C₂×C₂₇₇	C₄×D₄
prime radical	C₂×D₂₇₇	C₂×D₄
cosocle	C₄×D₂₇₇	C₂³
hypercenter	Q₁₆	#11764308
hypocenter	C₂₇₇	#441
LCS	G ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₁₁₀₈	C₄	semidirect
13. #17984553	C₄	semidirect
15. Q₁₆×C₂₇₇	C₄	semidirect
35. C₂₇₇	#441	semidirect
45. Q₁₆	#11764308	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11764308	2
#22579948	2
#22579943	2
#22579934	2
Q₁₆×D₂₇₇	2
#441	277