Subgroups — #31800853

Order 17728.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22579937	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂×#11764308)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#22579851	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11764308	—	2	1	no	—
5	C₂×#14555667	—	2	1	no	—
6	C₂×C₂²⋊D₂₇₇	—	2	1	no	—
7	C₂×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₇₇	C₄	1	1	yes	characteristic, complemented
9	C₂²⋊Dic₂₇₇	—	2	1	no	—
10	C₂×D₄×C₂₇₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#11764308	—	2	2	no	—
12	C₂×#11764308	—	4	1	no	—
13	C₂×#11764308	—	4	1	no	—
14	#14555667	—	4	1	no	—
15	C₂²×D₂₇₇	D₄	1	1	yes	characteristic
16	C₂²⋊D₂₇₇	—	4	1	no	—
17	C₂²⋊D₂₇₇	—	4	1	no	—
18	C₂×Dic₂₇₇	—	2	1	no	—
19	C₂×Dic₂₇₇	D₄	1	1	yes	characteristic
20	C₄×D₂₇₇	—	4	1	no	—
21	C₂³×C₂₇₇	—	2	1	no	—
22	D₄×C₂₇₇	—	4	1	no	—
23	C₂×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₂₇₇	—	4	1	no	—
25	C₂²×C₂₇₇	—	4	1	no	—
26	C₂²×C₂₇₇	—	4	1	no	—
27	C₂²×C₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×D₂₇₇	—	2	1	no	—
29	C₂×D₂₇₇	—	2	1	no	—
30	C₂×D₂₇₇	D₈	1	1	yes	characteristic
31	C₂×D₂₇₇	SD₁₆	1	1	yes	characteristic
32	#11764308	—	4	2	no	—
33	C₄×C₂₇₇	—	2	1	no	—
34	Dic₂₇₇	—	2	1	no	—
35	Dic₂₇₇	—	4	1	no	—
36	C₂×C₂₇₇	—	8	1	no	—
37	C₂×C₂₇₇	#99	1	1	yes	characteristic
38	C₂×C₂₇₇	#102	1	1	yes	characteristic
39	C₂×C₂₇₇	#104	1	1	yes	characteristic
40	D₂₇₇	—	2	1	no	—
41	D₂₇₇	—	2	1	no	—
42	C₂₇₇	#327	1	1	yes	characteristic, complemented, Sylow
43	#327	—	277	1	no	maximal, Sylow
44	#121	—	277	1	no	—
45	C₂²⋊C₈	—	277	1	no	—
46	#95	—	277	1	no	—
47	C₂×D₄	—	554	1	no	—
48	C₂×D₄	#11764308	1	1	yes	characteristic, complemented
49	C₂²×C₄	—	277	1	no	—
50	C₂²×C₄	—	554	1	no	—
51	C₂×C₈	—	554	1	no	—
52	C₄⋊C₄	—	277	1	no	—
53	C₂²⋊C₄	—	554	1	no	—
54	C₂³	—	2	1	no	—
55	C₂³	—	277	1	no	—
56	D₄	—	4	1	no	—
57	D₄	—	1108	1	no	—
58	D₄	—	1108	1	no	—
59	C₂×C₄	—	277	1	no	—
60	C₂×C₄	—	554	2	no	—
61	C₂×C₄	—	554	1	no	—
62	C₂×C₄	—	1108	1	no	—
63	C₂×C₄	—	1108	1	no	—
64	C₂×C₄	—	1108	1	no	—
65	C₂×C₄	C₂×#11764308	1	1	yes	characteristic
66	C₈	—	1108	1	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	277	1	no	—
71	C₂²	—	277	1	no	—
72	C₂²	—	554	1	no	—
73	C₂²	—	554	1	no	—
74	C₂²	C₂²⋊#11764308	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	554	1	no	—
77	C₄	—	1108	2	no	—
78	C₄	—	1108	1	no	—
79	C₂	—	8	1	no	—
80	C₂	—	554	1	no	—
81	C₂	—	554	1	no	—
82	C₂	#22579938	1	1	yes	characteristic
83	C₂	#22579942	1	1	yes	characteristic
84	C₂	#22579944	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11764308
commutator	C₂×C₄×C₂₇₇	C₂×C₄
Frattini	C₂×C₄	C₂×#11764308
Fitting	C₂×D₄×C₂₇₇	C₄
socle	C₂²×C₂₇₇	C₂²⋊C₄
prime radical	#22579851	C₂
cosocle	C₂×C₄×D₂₇₇	C₂²
hypercenter	C₂×D₄	#11764308
hypocenter	C₂₇₇	#327
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #22579937	C₂	semidirect
2. (C₂×#11764308)⋊C₄	C₂	semidirect
8. C₄⋊Dic₂₇₇	C₄	semidirect
10. C₂×D₄×C₂₇₇	C₄	semidirect
42. C₂₇₇	#327	semidirect
48. C₂×D₄	#11764308	semidirect

Maximal subgroups

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

subgroup	index
#22579937	2
(C₂×#11764308)⋊C₄	2
#22579851	2
#327	277