Subgroups — #30640977

Order 16256.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550256	C₂	1	1	yes	maximal, characteristic
2	#22549679	C₂	1	4	yes	maximal
3	#22549652	C₂	1	1	yes	maximal, characteristic
4	#499×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₀₈	—	2	1	no	—
6	C₂×Dic₅₀₈	C₂²	1	2	yes	—
7	#16861209	—	2	1	no	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
9	#125×C₁₂₇	—	2	1	no	—
10	#107×C₁₂₇	C₂²	1	2	yes	—
11	#106×C₁₂₇	—	2	1	no	—
12	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₅₄	—	2	2	no	—
14	Dic₅₀₈	—	2	4	no	—
15	Dic₅₀₈	—	4	2	no	—
16	C₄⋊Dic₁₂₇	—	4	2	no	—
17	C₁₂₇⋊C₁₆	—	2	2	no	—
18	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	—	2	2	no	—
21	C₄⋊C₄×C₁₂₇	—	4	2	no	—
22	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
23	Dic₂₅₄	—	4	2	no	—
24	Dic₂₅₄	—	4	2	no	—
25	C₂×Dic₁₂₇	—	4	2	no	—
26	C₂×C₄×C₁₂₇	—	4	2	no	—
27	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
28	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₂₇	SD₁₆	1	2	yes	—
30	C₈×C₁₂₇	#41	1	2	yes	—
31	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₂₇	—	2	2	no	—
33	C₄×C₁₂₇	—	8	2	no	—
34	C₄×C₁₂₇	#124	1	1	yes	characteristic
35	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
36	Dic₁₂₇	—	8	2	no	—
37	C₂×C₁₂₇	#486	1	1	yes	characteristic
38	C₂×C₁₂₇	#510	1	2	yes	—
39	#2245	—	127	1	no	maximal, Sylow
40	C₁₂₇	#2245	1	1	yes	characteristic, complemented, Sylow
41	#499	D₁₂₇	1	1	yes	characteristic
42	#494	—	127	1	no	—
43	#358	—	127	4	no	—
44	#346	—	127	1	no	—
45	C₂×Q₁₆	—	127	2	no	—
46	C₂×Q₁₆	—	254	1	no	—
47	C₄⋊Q₈	—	254	1	no	—
48	#125	—	2	1	no	—
49	C₂×C₁₆	—	127	2	no	—
50	#107	C₂×D₁₂₇	1	2	yes	—
51	#106	—	2	1	no	—
52	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂×Q₈	—	254	2	no	—
54	Q₁₆	—	508	2	no	—
55	Q₁₆	—	254	4	no	—
56	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	508	2	no	—
61	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
62	C₁₆	—	254	2	no	—
63	Q₈	—	508	2	no	—
64	Q₈	—	508	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	508	2	no	—
67	C₂×C₄	#14523233	1	2	yes	—
68	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
69	C₈	#14523235	1	2	yes	—
70	C₈	#14523249	1	2	yes	—
71	C₂²	C₂×#14523233	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	1016	2	no	—
75	C₄	#16861198	1	1	yes	characteristic
76	C₄	C₂×#14523235	1	1	yes	characteristic
77	C₂	#22550224	1	1	yes	characteristic
78	C₂	#22550316	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523233
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#499×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#499	D₁₂₇
hypocenter	C₁₂₇	#2245
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
40. C₁₂₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#22550256	2
#22549679	2
#22549652	2
#499×C₁₂₇	2
#2245	127