Subgroups — #30636898

Order 16256.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550255	C₂	1	1	yes	maximal, characteristic
2	#22549675	C₂	1	1	yes	maximal, characteristic
3	#356×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523238	—	2	1	no	—
5	C₂×#14523238	—	2	1	no	—
6	#16861209	C₄	1	1	yes	characteristic
7	#16861128	—	2	1	no	—
8	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	#101×C₁₂₇	—	2	1	no	—
11	C₂×Dic₂₅₄	—	2	1	no	—
12	#14523238	—	4	2	no	—
13	#14523238	—	4	2	no	—
14	#14523232	—	4	1	no	—
15	C₂×C₁₂₇⋊C₈	—	2	1	no	—
16	C₄⋊Dic₁₂₇	—	4	1	no	—
17	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
19	M₁₆×C₁₂₇	—	4	1	no	—
20	C₄⋊C₄×C₁₂₇	—	4	1	no	—
21	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	Dic₂₅₄	—	4	1	no	—
23	Dic₂₅₄	—	4	1	no	—
24	C₂×Dic₁₂₇	—	4	1	no	—
25	C₁₂₇⋊C₈	—	2	2	no	—
26	C₁₂₇⋊C₈	—	8	1	no	—
27	Q₈×C₁₂₇	—	4	1	no	—
28	Q₈×C₁₂₇	—	4	1	no	—
29	C₂×C₄×C₁₂₇	—	2	1	no	—
30	C₂×C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₂₇	—	8	1	no	—
34	C₂²×C₁₂₇	#99	1	1	yes	characteristic
35	C₄×C₁₂₇	—	2	2	no	—
36	C₄×C₁₂₇	—	2	1	no	—
37	C₄×C₁₂₇	—	4	1	no	—
38	C₄×C₁₂₇	—	4	1	no	—
39	Dic₁₂₇	—	8	1	no	—
40	C₂×C₁₂₇	—	2	1	no	—
41	C₂×C₁₂₇	#353	1	1	yes	characteristic
42	#1411	—	127	1	no	maximal, Sylow
43	C₁₂₇	#1411	1	1	yes	characteristic, complemented, Sylow
44	#494	—	127	1	no	—
45	#356	—	127	1	no	—
46	#356	D₁₂₇	1	1	yes	characteristic
47	C₂×Q₁₆	—	254	1	no	—
48	C₂×Q₁₆	—	254	1	no	—
49	C₄⋊Q₈	—	127	1	no	—
50	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
51	#101	—	2	1	no	—
52	#101	—	254	1	no	—
53	C₄×C₈	—	127	1	no	—
54	C₂×Q₈	—	254	1	no	—
55	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
56	C₂×Q₈	D₅₀₈	1	1	yes	characteristic
57	Q₁₆	—	508	2	no	—
58	Q₁₆	—	508	2	no	—
59	M₁₆	—	4	1	no	—
60	M₁₆	—	508	1	no	—
61	C₂×C₈	—	254	1	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	508	1	no	—
64	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
65	Q₈	—	4	1	no	—
66	Q₈	—	4	1	no	—
67	Q₈	—	508	1	no	—
68	Q₈	—	508	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	508	1	no	—
73	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
74	C₈	—	8	1	no	—
75	C₈	—	1016	1	no	—
76	C₈	—	254	2	no	—
77	C₂²	#16861123	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	1016	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#22549668	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549668
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₂₇
Fitting	#356×C₁₂₇	C₂
socle	C₂×C₁₂₇	#353
prime radical	C₂²×C₁₂₇	#99
cosocle	C₄⋊Q₈×C₁₂₇	C₂²
hypercenter	#356	D₁₂₇
hypocenter	C₁₂₇	#1411
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

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

Maximal subgroups

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

subgroup	index
#22550255	2
#22549675	2
#356×C₁₂₇	2
#1411	127