Subgroups — #30641008

Order 16256.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550277	C₂	1	2	yes	maximal
2	#22549704	C₂	1	1	yes	maximal, characteristic
3	#22549701	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#367×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861200	—	2	2	no	—
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16861143	C₂²	1	2	yes	—
9	#16861140	—	2	2	no	—
10	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	#107×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
14	C₄⋊Dic₁₂₇	—	2	2	no	—
15	C₄⋊Dic₁₂₇	—	4	2	no	—
16	Dic₁₂₇⋊C₄	—	4	2	no	—
17	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
18	C₁₂₇⋊C₁₆	Q₈	1	2	yes	—
19	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	—	2	2	no	—
21	C₁₆×C₁₂₇	Q₈	1	2	yes	—
22	C₂×Dic₁₂₇	—	4	2	no	—
23	C₂×Dic₁₂₇	D₈	1	2	yes	—
24	C₁₂₇⋊C₈	Q₁₆	1	2	yes	—
25	C₂×C₄×C₁₂₇	—	4	2	no	—
26	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
27	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
30	C₄×C₁₂₇	—	8	2	no	—
31	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
32	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
33	Dic₁₂₇	—	2	2	no	—
34	Dic₁₂₇	—	8	2	no	—
35	C₂×C₁₂₇	#500	1	1	yes	characteristic
36	C₂×C₁₂₇	#508	1	2	yes	—
37	#2253	—	127	1	no	maximal, Sylow
38	C₁₂₇	#2253	1	1	yes	characteristic, complemented, Sylow
39	#499	—	127	2	no	—
40	#367	—	127	1	no	—
41	#367	D₁₂₇	1	1	yes	characteristic
42	#366	—	127	2	no	—
43	C₄×C₁₆	—	127	1	no	—
44	#125	—	254	2	no	—
45	C₂×C₁₆	—	127	1	no	—
46	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
47	#107	—	127	2	no	—
48	#107	C₂×D₁₂₇	1	2	yes	—
49	#106	—	254	2	no	—
50	C₄×C₈	—	127	1	no	—
51	C₂×C₈	—	127	1	no	—
52	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	508	2	no	—
55	C₄⋊C₄	—	508	2	no	—
56	C₄⋊C₄	—	254	2	no	—
57	C₄²	—	127	1	no	—
58	C₁₆	—	127	2	no	—
59	C₁₆	Dic₂₅₄	1	2	yes	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	127	2	no	—
62	C₂×C₄	—	508	2	no	—
63	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
64	C₈	—	127	2	no	—
65	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
66	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
67	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
68	C₄	—	8	2	no	—
69	C₄	—	1016	2	no	—
70	C₄	—	254	2	no	—
71	C₄	#16861208	1	1	yes	characteristic
72	C₄	Q₁₆×D₁₂₇	1	1	yes	characteristic
73	C₂	#22550280	1	1	yes	characteristic
74	C₂	#22550307	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#367×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#367	D₁₂₇
hypocenter	C₁₂₇	#2253
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
38. C₁₂₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#22550277	2
#22549704	2
#22549701	2
C₁₆×Dic₁₂₇	2
#367×C₁₂₇	2
#2253	127