Subgroups — #30636899

Order 16256.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549675	C₂	1	2	yes	maximal
2	#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861128	—	2	2	no	—
4	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
5	C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
6	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
8	#14523232	—	4	2	no	—
9	C₂×Q₈×C₁₂₇	—	2	1	no	—
10	C₂×Q₈×C₁₂₇	D₄	1	2	yes	—
11	Q₁₆×C₁₂₇	—	4	4	no	—
12	C₂×C₈×C₁₂₇	—	2	1	no	—
13	C₄⋊C₄×C₁₂₇	—	4	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	Q₈×C₁₂₇	—	4	2	no	—
18	Q₈×C₁₂₇	—	4	2	no	—
19	C₂×C₄×C₁₂₇	—	2	1	no	—
20	C₂×C₄×C₁₂₇	—	2	2	no	—
21	C₂×C₄×C₁₂₇	—	4	1	no	—
22	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₂₇	—	2	2	no	—
24	C₂²×C₁₂₇	#99	1	1	yes	characteristic
25	C₄×C₁₂₇	—	2	2	no	—
26	C₄×C₁₂₇	—	2	1	no	—
27	C₄×C₁₂₇	—	4	2	no	—
28	C₄×C₁₂₇	—	8	1	no	—
29	C₂×C₁₂₇	—	2	1	no	—
30	C₂×C₁₂₇	#353	1	1	yes	characteristic
31	#1411	—	127	1	no	maximal, Sylow
32	C₁₂₇	#1411	1	1	yes	characteristic, complemented, Sylow
33	#494	D₁₂₇	1	1	yes	characteristic
34	#356	—	127	2	no	—
35	C₂×Q₁₆	—	2	2	no	—
36	C₄⋊Q₈	Dic₁₂₇	1	1	yes	characteristic
37	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
38	#101	—	254	2	no	—
39	C₄×C₈	Dic₁₂₇	1	1	yes	characteristic
40	C₂×Q₈	—	2	1	no	—
41	C₂×Q₈	C₂²⋊D₁₂₇	1	2	yes	—
42	Q₁₆	—	4	4	no	—
43	M₁₆	—	508	2	no	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	1016	2	no	—
56	C₂²	#16861124	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	2	1	no	—
59	C₄	—	4	2	no	—
60	C₄	—	8	1	no	—
61	C₂	—	2	1	no	—
62	C₂	#22549669	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549669
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₂₇
Fitting	#494×C₁₂₇	C₂
socle	C₂×C₁₂₇	#353
prime radical	C₂²×C₁₂₇	#99
cosocle	C₄⋊Q₈×C₁₂₇	C₂²
hypercenter	#494	D₁₂₇
hypocenter	C₁₂₇	#1411
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #494

Complements

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

N	H ≅ G/N	type
32. C₁₂₇	#1411	semidirect

Maximal subgroups

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

subgroup	index
#22549675	2
#494×C₁₂₇	2
#1411	127