Subgroups — #30636895

Order 16256.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549669	C₂	1	2	yes	maximal
2	#492×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861124	—	2	2	no	—
4	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
5	C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic, complemented
6	#127×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊Dic₁₂₇	—	4	2	no	—
9	C₂×Q₈×C₁₂₇	—	2	1	no	—
10	C₂×D₄×C₁₂₇	—	4	1	no	—
11	C₂×D₄×C₁₂₇	D₄	1	2	yes	—
12	SD₁₆×C₁₂₇	—	4	4	no	—
13	C₂×C₈×C₁₂₇	—	2	1	no	—
14	C₄⋊C₄×C₁₂₇	—	4	1	no	—
15	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₁₂₇	—	8	2	no	—
17	C₂³×C₁₂₇	—	2	2	no	—
18	Q₈×C₁₂₇	—	4	2	no	—
19	D₄×C₁₂₇	—	4	2	no	—
20	D₄×C₁₂₇	—	8	2	no	—
21	C₂×C₄×C₁₂₇	—	2	1	no	—
22	C₂×C₄×C₁₂₇	—	4	1	no	—
23	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₈×C₁₂₇	—	2	2	no	—
25	C₂²×C₁₂₇	—	4	2	no	—
26	C₂²×C₁₂₇	—	8	2	no	—
27	C₂²×C₁₂₇	#99	1	1	yes	characteristic
28	C₄×C₁₂₇	—	2	2	no	—
29	C₄×C₁₂₇	—	2	1	no	—
30	C₄×C₁₂₇	—	8	1	no	—
31	Dic₁₂₇	—	16	2	no	—
32	C₂×C₁₂₇	—	2	1	no	—
33	C₂×C₁₂₇	—	8	2	no	—
34	C₂×C₁₂₇	#353	1	1	yes	characteristic
35	#1409	—	127	1	no	maximal, Sylow
36	C₁₂₇	#1409	1	1	yes	characteristic, complemented, Sylow
37	#492	D₁₂₇	1	1	yes	characteristic
38	#353	—	127	2	no	—
39	C₂×SD₁₆	—	2	2	no	—
40	C₄⋊Q₈	Dic₁₂₇	1	1	yes	characteristic, complemented
41	#127	C₂×D₁₂₇	1	1	yes	characteristic
42	#99	—	254	2	no	—
43	C₄×C₈	Dic₁₂₇	1	1	yes	characteristic, complemented
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	4	1	no	—
46	C₂×D₄	C₂²⋊D₁₂₇	1	2	yes	—
47	SD₁₆	—	4	4	no	—
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	508	2	no	—
51	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
52	C₂³	—	2	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	4	2	no	—
55	D₄	—	8	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	1016	2	no	—
59	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	8	2	no	—
63	C₂²	#16861124	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	2032	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	8	2	no	—
70	C₂	#22549669	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549669
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	#127	C₂×D₁₂₇
Fitting	#492×C₁₂₇	C₂
socle	C₂×C₁₂₇	#353
prime radical, cosocle	#127×C₁₂₇	C₂²
hypercenter	#492	D₁₂₇
hypocenter	C₁₂₇	#1409
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #492

Complements

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

N	H ≅ G/N	type
5. C₄⋊Q₈×C₁₂₇	C₄	semidirect
7. C₄×C₈×C₁₂₇	C₄	semidirect
36. C₁₂₇	#1409	semidirect
40. C₄⋊Q₈	Dic₁₂₇	semidirect
43. C₄×C₈	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
#492×C₁₂₇	2
#1409	127