Subgroups — #31800874

Order 17728.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22579959	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11764308	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₂₇₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11764308	—	2	1	no	—
5	C₄⋊#11764308	C₄	1	2	yes	complemented
6	C₂²⋊#11764308	—	2	1	no	—
7	C₄×#11764308	—	2	1	no	—
8	C₄×#11764308	—	2	1	no	—
9	C₄×#11764308	C₄	1	2	yes	complemented
10	C₂×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
11	#17984547	C₄	1	1	yes	characteristic, complemented
12	#17984546	—	2	1	no	—
13	C₈×D₂₇₇	—	2	1	no	—
14	M₁₆×C₂₇₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#11764308	—	2	4	no	—
16	C₂×#11764308	—	2	2	no	—
17	C₂×#11764308	—	2	2	no	—
18	C₂²×D₂₇₇	D₄	1	1	yes	characteristic
19	C₂×Dic₂₇₇	Q₈	1	1	yes	characteristic
20	C₄×D₂₇₇	C₂×C₄	1	1	yes	characteristic
21	C₄×D₂₇₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₂₇₇	D₄	1	1	yes	characteristic
23	C₄×D₂₇₇	D₄	1	1	yes	characteristic
24	#14555660	—	2	1	no	—
25	C₂×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
26	C₈×C₂₇₇	—	2	1	no	—
27	C₂²×C₂₇₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₂₇₇	—	2	2	no	—
29	C₂×D₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
31	#11764308	—	2	4	no	—
32	#11764308	—	4	2	no	—
33	C₄×C₂₇₇	C₄²	1	1	yes	characteristic
34	C₄×C₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
35	Dic₂₇₇	C₄⋊C₄	1	1	yes	characteristic
36	Dic₂₇₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₂₇₇	—	2	1	no	—
38	C₂×C₂₇₇	#95	1	1	yes	characteristic
39	D₂₇₇	—	2	1	no	—
40	D₂₇₇	#104	1	2	yes	—
41	C₂₇₇	#339	1	1	yes	characteristic, complemented, Sylow
42	#339	—	277	1	no	maximal, Sylow
43	C₂×M₁₆	—	277	1	no	—
44	#117	—	277	1	no	—
45	C₂×C₄²	—	277	1	no	—
46	C₂²×C₄	—	277	1	no	—
47	C₂²×C₄	—	554	1	no	—
48	M₁₆	—	277	1	no	—
49	M₁₆	—	554	1	no	—
50	M₁₆	#11764308	1	1	yes	characteristic, complemented
51	C₂×C₈	—	554	1	no	—
52	C₄⋊C₄	—	277	2	no	—
53	C₂²⋊C₄	—	554	1	no	—
54	C₄²	—	277	2	no	—
55	C₄²	—	554	1	no	—
56	C₄²	—	554	1	no	—
57	C₂³	—	277	1	no	—
58	C₂×C₄	—	277	1	no	—
59	C₂×C₄	—	277	1	no	—
60	C₂×C₄	—	277	1	no	—
61	C₂×C₄	—	277	1	no	—
62	C₂×C₄	—	277	1	no	—
63	C₂×C₄	—	554	2	no	—
64	C₂×C₄	—	554	4	no	—
65	C₂×C₄	—	554	2	no	—
66	C₂×C₄	C₂×#11764308	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	554	1	no	—
69	C₂²	—	277	1	no	—
70	C₂²	—	277	1	no	—
71	C₂²	—	554	2	no	—
72	C₂²	C₄⋊#11764308	1	1	yes	characteristic
73	C₄	—	277	1	no	—
74	C₄	—	277	1	no	—
75	C₄	—	554	4	no	—
76	C₄	—	1108	2	no	—
77	C₄	C₄×#11764308	1	1	yes	characteristic
78	C₄	C₂²⋊#11764308	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	—	277	2	no	—
81	C₂	—	554	1	no	—
82	C₂	(C₂×#11764308)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#11764308
commutator	C₄×C₂₇₇	C₄²
Frattini	C₂×C₄	C₂×#11764308
Fitting	M₁₆×C₂₇₇	C₄
socle	C₂×C₂₇₇	#95
prime radical	C₂²×D₂₇₇	D₄
cosocle	C₂×C₄×D₂₇₇	C₂²
hypercenter	M₁₆	#11764308
hypocenter	C₂₇₇	#339
LCS	G ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄⋊#11764308	C₄	semidirect
9. C₄×#11764308	C₄	semidirect
11. #17984547	C₄	semidirect
14. M₁₆×C₂₇₇	C₄	semidirect
41. C₂₇₇	#339	semidirect
50. M₁₆	#11764308	semidirect

Maximal subgroups

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

subgroup	index
#22579959	2
C₂×C₄×#11764308	2
M₁₆×D₂₇₇	2
#339	277