Subgroups — #30636708

Order 16256.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
5	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₁₂₇⋊C₈)	C₄	1	1	yes	characteristic
7	Dic₁₂₇⋊C₈	—	2	1	no	—
8	C₈×Dic₁₂₇	—	2	1	no	—
9	C₂×C₁₆×C₁₂₇	—	2	1	no	—
10	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₅₄	C₈	1	1	yes	characteristic
12	C₂×C₁₂₇⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₂₇	C₈	1	1	yes	characteristic
14	Dic₁₂₇⋊C₄	—	2	1	no	—
15	C₄×Dic₁₂₇	—	2	1	no	—
16	C₁₂₇⋊C₁₆	—	2	2	no	—
17	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₂₇	—	4	1	no	—
21	Dic₂₅₄	—	2	1	no	—
22	Dic₂₅₄	C₁₆	1	2	yes	complemented
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₂×Dic₁₂₇	—	2	1	no	—
25	C₁₂₇⋊C₈	—	4	1	no	—
26	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
29	C₈×C₁₂₇	—	2	1	no	—
30	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
32	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₂₇	—	2	1	no	—
34	C₄×C₁₂₇	#103	1	1	yes	characteristic
35	C₄×C₁₂₇	#104	1	1	yes	characteristic
36	C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₂₇	M₃₂	1	1	yes	characteristic
38	Dic₁₂₇	—	2	2	no	—
39	Dic₁₂₇	—	4	1	no	—
40	C₂×C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₂₇	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₂₇	#350	1	1	yes	characteristic
43	Q₈⋊C₁₆	—	127	1	no	maximal, Sylow
44	C₁₂₇	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₈×Q₈	—	127	1	no	—
46	C₄⋊C₁₆	D₁₂₇	1	1	yes	characteristic
47	C₄×C₁₆	—	127	1	no	—
48	C₄×Q₈	—	127	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	254	1	no	—
51	C₄⋊C₈	—	127	1	no	—
52	C₄⋊C₈	—	254	1	no	—
53	C₄×C₈	—	254	1	no	—
54	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
55	C₂×Q₈	—	127	1	no	—
56	C₂×C₈	—	254	1	no	—
57	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
58	C₂×C₈	D₅₀₈	1	1	yes	characteristic
59	C₄⋊C₄	—	127	1	no	—
60	C₄⋊C₄	—	254	1	no	—
61	C₄²	—	254	1	no	—
62	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
63	C₁₆	—	4	1	no	—
64	C₁₆	—	254	2	no	—
65	Q₈	—	127	2	no	—
66	Q₈	—	254	1	no	—
67	C₂×C₄	—	254	1	no	—
68	C₂×C₄	—	254	1	no	—
69	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
70	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
71	C₂×C₄	#14523231	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	508	1	no	—
74	C₈	#14523236	1	1	yes	characteristic
75	C₈	#14523238	1	1	yes	characteristic
76	C₂²	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	508	1	no	—
79	C₄	—	254	2	no	—
80	C₄	#16861132	1	1	yes	characteristic
81	C₄	#16861135	1	1	yes	characteristic
82	C₄	C₁₆×D₁₂₇	1	1	yes	characteristic
83	C₄	#16861148	1	1	yes	characteristic
84	C₂	#22549606	1	1	yes	characteristic
85	C₂	(C₂×D₁₂₇)⋊C₁₆	1	1	yes	characteristic
86	C₂	#22549661	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	C₄⋊C₁₆×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂²⋊C₈
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	C₄⋊C₁₆	D₁₂₇
hypocenter	C₁₂₇	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
22. Dic₂₅₄	C₁₆	semidirect
44. C₁₂₇	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₅₄	2
C₄×C₁₂₇⋊C₁₆	2
C₄⋊C₁₆×C₁₂₇	2
Q₈⋊C₁₆	127