Subgroups — #30638642

Order 16256.

612 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	1	6	yes	maximal, complemented
2	C₂²×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
4	C₂²⋊(C₁₂₇⋊C₈)	—	2	12	no	—
5	C₂×#41×C₁₂₇	—	2	12	no	—
6	C₂²×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×D₄×C₁₂₇	C₄	1	3	yes	—
8	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
9	C₂×C₁₂₇⋊C₈	—	2	12	no	—
10	C₂⁴×C₁₂₇	C₂×C₄	1	3	yes	—
11	#41×C₁₂₇	—	2	32	no	—
12	C₂×Q₈×C₁₂₇	—	2	6	no	—
13	C₂×D₄×C₁₂₇	—	2	12	no	—
14	C₂×D₄×C₁₂₇	—	2	6	no	—
15	C₂²×C₄×C₁₂₇	—	2	12	no	—
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₁₂₇	D₄	1	12	yes	—
18	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₁₂₇⋊C₈	—	4	4	no	—
20	C₂³×C₁₂₇	—	2	6	no	—
21	C₂³×C₁₂₇	—	2	12	no	—
22	C₂³×C₁₂₇	C₂²⋊C₄	1	6	yes	—
23	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
24	Q₈×C₁₂₇	—	2	8	no	—
25	D₄×C₁₂₇	—	2	24	no	—
26	C₂×C₄×C₁₂₇	—	2	6	no	—
27	C₂×C₄×C₁₂₇	—	2	24	no	—
28	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	6	yes	—
29	C₂×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
30	C₂²×C₁₂₇	—	2	6	no	—
31	C₂²×C₁₂₇	—	2	12	no	—
32	C₂²×C₁₂₇	—	4	12	no	—
33	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	3	yes	—
34	C₂²×C₁₂₇	#131	1	4	yes	—
35	C₄×C₁₂₇	—	2	6	no	—
36	C₄×C₁₂₇	#120	1	4	yes	—
37	C₂×C₁₂₇	—	4	6	no	—
38	C₂×C₁₂₇	#379	1	1	yes	characteristic
39	C₂×C₁₂₇	#408	1	6	yes	—
40	#1780	—	127	1	no	maximal, Sylow
41	C₁₂₇	#1780	1	1	yes	characteristic, complemented, Sylow
42	C₂²×#41	D₁₂₇	1	1	yes	characteristic
43	C₂×C₂²⋊C₈	—	127	6	no	—
44	C₂×#41	—	2	12	no	—
45	C₂²×Q₈	Dic₁₂₇	1	1	yes	characteristic
46	C₂²×D₄	Dic₁₂₇	1	3	yes	—
47	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
48	C₂²×C₈	—	127	4	no	—
49	C₂²⋊C₈	—	254	12	no	—
50	C₂⁴	C₂×Dic₁₂₇	1	3	yes	—
51	#41	—	2	32	no	—
52	C₂×Q₈	—	2	6	no	—
53	C₂×D₄	—	2	12	no	—
54	C₂×D₄	—	2	6	no	—
55	C₂²×C₄	—	2	12	no	—
56	C₂²×C₄	C₂×Dic₁₂₇	1	3	yes	—
57	C₂²×C₄	C₂²⋊D₁₂₇	1	12	yes	—
58	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
59	C₂×C₈	—	254	12	no	—
60	C₂³	—	2	12	no	—
61	C₂³	—	2	6	no	—
62	C₂³	C₂²⋊Dic₁₂₇	1	6	yes	—
63	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
64	Q₈	—	2	8	no	—
65	D₄	—	2	24	no	—
66	C₂×C₄	—	2	6	no	—
67	C₂×C₄	—	2	24	no	—
68	C₂×C₄	C₂²⋊Dic₁₂₇	1	6	yes	—
69	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	6	yes	—
70	C₈	—	508	4	no	—
71	C₂²	—	2	12	no	—
72	C₂²	—	2	6	no	—
73	C₂²	—	4	12	no	—
74	C₂²	C₂×C₂²⋊Dic₁₂₇	1	3	yes	—
75	C₂²	#16861216	1	4	yes	—
76	C₄	—	2	6	no	—
77	C₄	#16861178	1	4	yes	—
78	C₂	—	4	6	no	—
79	C₂	#22549732	1	1	yes	characteristic
80	C₂	#22549843	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator, socle	C₂³×C₁₂₇	C₂²×C₄
Fitting, prime radical	C₂²×#41×C₁₂₇	C₂
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×#41	D₁₂₇
hypocenter	C₁₂₇	#1780
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
41. C₁₂₇	#1780	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₁₂₇⋊C₈)	2
C₂²×#41×C₁₂₇	2
#1780	127