Subgroups — #30636901

Order 16256.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549669	C₂	1	2	yes	maximal
2	#490×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861124	—	2	2	no	—
4	#127×C₁₂₇	C₂²	1	1	yes	characteristic
5	#125×C₁₂₇	C₄	1	1	yes	characteristic, complemented
6	#102×C₁₂₇	—	2	2	no	—
7	#97×C₁₂₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊Dic₁₂₇	—	4	2	no	—
9	C₂×D₄×C₁₂₇	—	4	1	no	—
10	C₂×D₄×C₁₂₇	D₄	1	2	yes	—
11	C₂×C₈×C₁₂₇	—	2	1	no	—
12	C₄⋊C₄×C₁₂₇	—	2	1	no	—
13	C₄⋊C₄×C₁₂₇	—	4	1	no	—
14	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₁₂₇	—	8	2	no	—
16	C₂³×C₁₂₇	—	2	2	no	—
17	D₄×C₁₂₇	—	4	2	no	—
18	D₄×C₁₂₇	—	8	2	no	—
19	C₂×C₄×C₁₂₇	—	2	1	no	—
20	C₂×C₄×C₁₂₇	—	4	1	no	—
21	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₂₇	—	4	1	no	—
23	C₂²×C₁₂₇	—	4	2	no	—
24	C₂²×C₁₂₇	—	8	2	no	—
25	C₂²×C₁₂₇	#99	1	1	yes	characteristic
26	C₄×C₁₂₇	—	2	1	no	—
27	C₄×C₁₂₇	—	4	1	no	—
28	C₄×C₁₂₇	—	8	1	no	—
29	Dic₁₂₇	—	16	2	no	—
30	C₂×C₁₂₇	—	2	1	no	—
31	C₂×C₁₂₇	—	8	2	no	—
32	C₂×C₁₂₇	#354	1	1	yes	characteristic
33	#1412	—	127	1	no	maximal, Sylow
34	C₁₂₇	#1412	1	1	yes	characteristic, complemented, Sylow
35	#490	D₁₂₇	1	1	yes	characteristic
36	#353	—	127	2	no	—
37	#127	C₂×D₁₂₇	1	1	yes	characteristic
38	#125	Dic₁₂₇	1	1	yes	characteristic, complemented
39	#102	—	2	2	no	—
40	#99	—	254	2	no	—
41	#97	Dic₁₂₇	1	1	yes	characteristic, complemented
42	C₂×D₄	—	4	1	no	—
43	C₂×D₄	C₂²⋊D₁₂₇	1	2	yes	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	508	2	no	—
48	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	D₄	—	4	2	no	—
51	D₄	—	8	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	1016	2	no	—
55	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₂²	—	4	2	no	—
58	C₂²	—	8	2	no	—
59	C₂²	#16861124	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	2032	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	8	2	no	—
66	C₂	#22549671	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549671
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	#127	C₂×D₁₂₇
Fitting	#490×C₁₂₇	C₂
socle	C₂×C₁₂₇	#354
prime radical, cosocle	#127×C₁₂₇	C₂²
hypercenter	#490	D₁₂₇
hypocenter	C₁₂₇	#1412
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #490

Complements

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

N	H ≅ G/N	type
5. #125×C₁₂₇	C₄	semidirect
7. #97×C₁₂₇	C₄	semidirect
34. C₁₂₇	#1412	semidirect
38. #125	Dic₁₂₇	semidirect
41. #97	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549669	2
#490×C₁₂₇	2
#1412	127