Subgroups — #30641010

Order 16256.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550281	C₂	1	1	yes	maximal, characteristic
2	#22549703	C₂	1	4	yes	maximal
3	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#16861209	—	2	1	no	—
6	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
7	#16861143	—	2	1	no	—
8	#16861143	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
10	#107×C₁₂₇	—	2	1	no	—
11	#107×C₁₂₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₅₄	—	4	1	no	—
14	C₄⋊Dic₁₂₇	—	2	2	no	—
15	C₄⋊Dic₁₂₇	—	4	1	no	—
16	C₁₂₇⋊C₁₆	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₁₂₇	—	4	1	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	1	no	—
22	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
23	Dic₂₅₄	—	8	2	no	—
24	C₂×Dic₁₂₇	—	4	2	no	—
25	Q₈×C₁₂₇	—	8	2	no	—
26	C₂×C₄×C₁₂₇	—	4	2	no	—
27	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₂₇	Q₁₆	1	2	yes	—
31	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₂₇	—	8	2	no	—
34	C₄×C₁₂₇	SD₃₂	1	2	yes	—
35	C₄×C₁₂₇	SD₃₂	1	2	yes	—
36	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₁₂₇	—	8	2	no	—
39	C₂×C₁₂₇	#500	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
41	C₂×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
42	#2254	—	127	1	no	maximal, Sylow
43	C₁₂₇	#2254	1	1	yes	characteristic, complemented, Sylow
44	#500	—	127	1	no	—
45	#500	D₁₂₇	1	1	yes	characteristic
46	#367	—	127	4	no	—
47	C₄×C₁₆	—	127	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	254	1	no	—
50	C₂×C₁₆	—	127	2	no	—
51	#107	—	2	1	no	—
52	#107	—	127	2	no	—
53	#107	—	254	1	no	—
54	#107	C₂×D₁₂₇	1	2	yes	—
55	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	508	1	no	—
58	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	508	1	no	—
63	C₄⋊C₄	—	254	2	no	—
64	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
65	C₁₆	—	127	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	1016	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	508	2	no	—
70	C₂×C₄	#14523233	1	1	yes	characteristic
71	C₂×C₄	#14523233	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
73	C₈	#14523238	1	2	yes	—
74	C₈	Q₈×D₁₂₇	1	2	yes	—
75	C₂²	C₂×#14523233	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	1016	2	no	—
78	C₄	#16861152	1	2	yes	—
79	C₄	#16861153	1	2	yes	—
80	C₄	#16861207	1	1	yes	characteristic
81	C₄	C₂×#14523238	1	1	yes	characteristic
82	C₂	#22550279	1	1	yes	characteristic
83	C₂	C₂×#16861152	1	1	yes	characteristic
84	C₂	C₂×#16861153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523233
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#500×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#500	D₁₂₇
hypocenter	C₁₂₇	#2254
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. C₁₂₇⋊C₁₆	Q₈	semidirect
43. C₁₂₇	#2254	semidirect

Maximal subgroups

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

subgroup	index
#22550281	2
#22549703	2
C₄×C₁₂₇⋊C₁₆	2
#500×C₁₂₇	2
#2254	127