Subgroups — #30636897

Order 16256.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549675	C₂	1	2	yes	maximal, complemented
2	#492×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861128	—	2	2	no	—
4	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
5	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	#127×C₁₂₇	C₄	1	1	yes	characteristic
7	C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
8	#14523232	—	4	2	no	—
9	C₂×Q₈×C₁₂₇	D₄	1	2	yes	—
10	C₂×D₄×C₁₂₇	—	2	1	no	—
11	C₂×D₄×C₁₂₇	—	4	1	no	—
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	C₁₂₇⋊C₈	—	8	2	no	—
17	C₂³×C₁₂₇	—	4	1	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	—	2	2	no	—
23	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₈×C₁₂₇	—	2	2	no	—
25	C₂²×C₁₂₇	—	8	2	no	—
26	C₂²×C₁₂₇	—	8	1	no	—
27	C₂²×C₁₂₇	#99	1	1	yes	characteristic
28	C₄×C₁₂₇	—	2	2	no	—
29	C₄×C₁₂₇	—	2	1	no	—
30	C₄×C₁₂₇	—	4	2	no	—
31	C₂×C₁₂₇	—	2	1	no	—
32	C₂×C₁₂₇	—	16	1	no	—
33	C₂×C₁₂₇	#353	1	1	yes	characteristic
34	#1410	—	127	1	no	maximal, Sylow
35	C₁₂₇	#1410	1	1	yes	characteristic, complemented, Sylow
36	#492	D₁₂₇	1	1	yes	characteristic
37	#356	—	127	2	no	—
38	C₂×SD₁₆	—	2	2	no	—
39	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
40	#127	Dic₁₂₇	1	1	yes	characteristic
41	#101	—	254	2	no	—
42	C₄×C₈	Dic₁₂₇	1	1	yes	characteristic
43	C₂×Q₈	C₂²⋊D₁₂₇	1	2	yes	—
44	C₂×D₄	—	2	1	no	—
45	C₂×D₄	—	4	1	no	—
46	SD₁₆	—	4	4	no	—
47	M₁₆	—	508	2	no	—
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
51	C₂³	—	4	1	no	—
52	Q₈	—	4	2	no	—
53	D₄	—	4	2	no	—
54	D₄	—	8	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	1016	2	no	—
60	C₂²	—	8	2	no	—
61	C₂²	—	8	1	no	—
62	C₂²	#16861124	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₂	—	2	1	no	—
67	C₂	—	16	1	no	—
68	C₂	#22549669	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549669
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₂₇
Fitting	#492×C₁₂₇	C₂
socle	C₂×C₁₂₇	#353
prime radical	#127×C₁₂₇	C₄
cosocle	C₄⋊Q₈×C₁₂₇	C₂²
hypercenter	#492	D₁₂₇
hypocenter	C₁₂₇	#1410
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #22549675	C₂	semidirect
35. C₁₂₇	#1410	semidirect

Maximal subgroups

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

subgroup	index
#22549675	2
#492×C₁₂₇	2
#1410	127