Subgroups — #30641004

Order 16256.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550280	C₂	1	2	yes	maximal
2	#22549702	C₂	1	1	yes	maximal, characteristic
3	#22549701	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#366×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861208	—	2	2	no	—
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16861143	C₂²	1	2	yes	—
9	#16861142	—	2	2	no	—
10	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	#107×C₁₂₇	C₂²	1	2	yes	—
13	C₂×Dic₂₅₄	—	4	2	no	—
14	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₂₇	—	2	2	no	—
16	Dic₁₂₇⋊C₄	—	4	2	no	—
17	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
18	C₁₂₇⋊C₁₆	Q₈	1	2	yes	complemented
19	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	—	2	2	no	—
21	C₁₆×C₁₂₇	Q₈	1	2	yes	complemented
22	Dic₂₅₄	—	8	4	no	—
23	C₂×Dic₁₂₇	—	4	2	no	—
24	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
25	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
26	C₁₂₇⋊C₈	Q₁₆	1	2	yes	—
27	C₂×C₄×C₁₂₇	—	4	2	no	—
28	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₂₇	—	8	2	no	—
33	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
34	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₁₂₇	—	8	2	no	—
36	Dic₁₂₇	D₁₆	1	2	yes	—
37	Dic₁₂₇	Q₃₂	1	2	yes	—
38	C₂×C₁₂₇	#500	1	1	yes	characteristic
39	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
41	#2252	—	127	1	no	maximal, Sylow
42	C₁₂₇	#2252	1	1	yes	characteristic, complemented, Sylow
43	#500	—	127	2	no	—
44	#366	—	127	1	no	—
45	#366	—	127	2	no	—
46	#366	D₁₂₇	1	1	yes	characteristic
47	C₄×C₁₆	—	127	1	no	—
48	C₄⋊Q₈	—	254	2	no	—
49	C₂×C₁₆	—	127	1	no	—
50	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
51	#107	—	127	2	no	—
52	#107	—	254	2	no	—
53	#107	C₂×D₁₂₇	1	2	yes	—
54	C₄×C₈	—	127	1	no	—
55	C₂×Q₈	—	508	2	no	—
56	C₂×C₈	—	127	1	no	—
57	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	508	2	no	—
60	C₄⋊C₄	—	254	2	no	—
61	C₄²	—	127	1	no	—
62	C₁₆	—	127	2	no	—
63	C₁₆	Dic₂₅₄	1	2	yes	complemented
64	Q₈	—	1016	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	127	1	no	—
67	C₂×C₄	—	127	1	no	—
68	C₂×C₄	—	508	2	no	—
69	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
70	C₈	—	127	2	no	—
71	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
72	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
73	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	—	127	2	no	—
76	C₄	—	127	2	no	—
77	C₄	—	1016	2	no	—
78	C₄	#16861208	1	1	yes	characteristic
79	C₄	Q₁₆×D₁₂₇	1	1	yes	characteristic
80	C₂	#22550280	1	1	yes	characteristic
81	C₂	D₁₆×D₁₂₇	1	1	yes	characteristic
82	C₂	Q₃₂×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#366×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#366	D₁₂₇
hypocenter	C₁₂₇	#2252
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
18. C₁₂₇⋊C₁₆	Q₈	semidirect
21. C₁₆×C₁₂₇	Q₈	semidirect
42. C₁₂₇	#2252	semidirect
63. C₁₆	Dic₂₅₄	semidirect

Maximal subgroups

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

subgroup	index
#22550280	2
#22549702	2
#22549701	2
C₁₆×Dic₁₂₇	2
#366×C₁₂₇	2
#2252	127