Subgroups — #30638346

Order 16256.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550279	C₂	1	4	yes	maximal
2	#22550256	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₀₈	—	2	2	no	—
6	#16861209	C₂²	1	2	yes	—
7	#16861142	—	2	8	no	—
8	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
10	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₅₄	—	2	2	no	—
13	Dic₅₀₈	—	4	4	no	—
14	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
15	C₄⋊Dic₁₂₇	—	2	4	no	—
16	C₂×Q₈×C₁₂₇	—	2	2	no	—
17	Q₁₆×C₁₂₇	—	4	4	no	—
18	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₂₇	—	2	4	no	—
20	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
21	Dic₂₅₄	—	4	4	no	—
22	C₂×Dic₁₂₇	—	4	2	no	—
23	C₁₂₇⋊C₈	Q₁₆	1	8	yes	complemented
24	Q₈×C₁₂₇	—	4	4	no	—
25	C₂×C₄×C₁₂₇	—	4	2	no	—
26	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
28	C₈×C₁₂₇	D₈	1	4	yes	—
29	C₂²×C₁₂₇	#127	1	1	yes	characteristic
30	C₄×C₁₂₇	—	8	2	no	—
31	C₄×C₁₂₇	C₂×D₈	1	2	yes	—
32	C₄×C₁₂₇	C₂×Q₁₆	1	4	yes	—
33	Dic₁₂₇	—	8	2	no	—
34	C₂×C₁₂₇	#493	1	1	yes	characteristic
35	C₂×C₁₂₇	#494	1	2	yes	—
36	#1713	—	127	1	no	maximal, Sylow
37	C₁₂₇	#1713	1	1	yes	characteristic, complemented, Sylow
38	#500	—	127	4	no	—
39	#494	—	127	1	no	—
40	#494	D₁₂₇	1	1	yes	characteristic
41	C₈²	—	127	1	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₂×Q₁₆	—	254	2	no	—
44	C₄⋊Q₈	—	127	2	no	—
45	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
46	#107	—	254	8	no	—
47	C₄×C₈	—	127	2	no	—
48	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	254	2	no	—
51	Q₁₆	—	4	4	no	—
52	Q₁₆	—	508	4	no	—
53	C₂×C₈	—	127	4	no	—
54	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	254	4	no	—
57	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	508	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	508	2	no	—
62	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
64	C₈	—	127	8	no	—
65	C₈	#14523233	1	4	yes	—
66	C₂²	#16861205	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	1016	2	no	—
69	C₄	C₂×#14523233	1	2	yes	—
70	C₄	Q₁₆×D₁₂₇	1	4	yes	—
71	C₂	#22550252	1	1	yes	characteristic
72	C₂	#22550254	1	2	yes	—

	subgroup	G/H
center	C₂²	#16861205
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#494×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#127
hypercenter	#494	D₁₂₇
hypocenter	C₁₂₇	#1713
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
23. C₁₂₇⋊C₈	Q₁₆	semidirect
37. C₁₂₇	#1713	semidirect

Maximal subgroups

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

subgroup	index
#22550279	2
#22550256	2
C₈×C₁₂₇⋊C₈	2
#494×C₁₂₇	2
#1713	127