Subgroups — #30636786

Order 16256.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550281	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861209	—	2	1	no	—
5	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
6	#16861143	—	2	1	no	—
7	#16861143	C₄	1	2	yes	—
8	C₄⋊C₈×C₁₂₇	—	2	1	no	—
9	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₅₄	—	4	1	no	—
11	C₄⋊Dic₁₂₇	—	2	2	no	—
12	C₄⋊Dic₁₂₇	—	4	1	no	—
13	C₁₂₇⋊C₁₆	—	4	1	no	—
14	C₂×C₈×C₁₂₇	—	4	1	no	—
15	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
18	Dic₂₅₄	—	8	1	no	—
19	Dic₂₅₄	—	8	1	no	—
20	C₂×Dic₁₂₇	—	4	2	no	—
21	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
24	C₈×C₁₂₇	—	2	1	no	—
25	C₈×C₁₂₇	—	8	1	no	—
26	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
27	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
28	C₂²×C₁₂₇	#102	1	1	yes	characteristic
29	C₄×C₁₂₇	—	2	1	no	—
30	C₄×C₁₂₇	#101	1	1	yes	characteristic
31	C₄×C₁₂₇	#103	1	1	yes	characteristic
32	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
33	C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
34	Dic₁₂₇	—	8	2	no	—
35	C₂×C₁₂₇	#332	1	1	yes	characteristic
36	C₂×C₁₂₇	#357	1	1	yes	characteristic
37	C₂×C₁₂₇	#362	1	1	yes	characteristic
38	#1364	—	127	1	no	maximal, Sylow
39	C₁₂₇	#1364	1	1	yes	characteristic, complemented, Sylow
40	#500	—	127	1	no	—
41	C₄⋊C₁₆	—	127	1	no	—
42	#335	D₁₂₇	1	1	yes	characteristic
43	C₄⋊Q₈	—	254	1	no	—
44	C₂×C₁₆	—	254	1	no	—
45	#107	—	127	2	no	—
46	#107	—	254	1	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
49	C₂×Q₈	—	508	1	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
52	C₂×C₈	D₅₀₈	1	1	yes	characteristic
53	C₄⋊C₄	—	508	1	no	—
54	C₄⋊C₄	—	254	2	no	—
55	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
56	C₁₆	—	508	1	no	—
57	Q₈	—	1016	1	no	—
58	Q₈	—	1016	1	no	—
59	C₂×C₄	—	508	2	no	—
60	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
61	C₂×C₄	#14523233	1	1	yes	characteristic
62	C₂×C₄	#14523235	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	8	1	no	—
65	C₈	#14523237	1	1	yes	characteristic
66	C₈	Dic₅₀₈	1	1	yes	characteristic
67	C₂²	#16861129	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	1016	2	no	—
70	C₄	#16861127	1	1	yes	characteristic
71	C₄	#16861134	1	1	yes	characteristic
72	C₄	#16861150	1	1	yes	characteristic
73	C₄	#16861152	1	1	yes	characteristic
74	C₂	#22549623	1	1	yes	characteristic
75	C₂	#22549676	1	1	yes	characteristic
76	C₂	#22549691	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861129
commutator	C₂×C₈×C₁₂₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#335×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#102
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#335	D₁₂₇
hypocenter	C₁₂₇	#1364
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
39. C₁₂₇	#1364	semidirect

Maximal subgroups

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

subgroup	index
#22550281	2
C₄⋊(C₁₂₇⋊C₁₆)	2
#335×C₁₂₇	2
#1364	127