Subgroups — #30641013

Order 16256.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550278	C₂	1	1	yes	maximal, characteristic
2	#22549703	C₂	1	2	yes	maximal
3	#22549701	C₂	1	2	yes	maximal
4	#22549652	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861201	—	2	1	no	—
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
8	#16861143	C₂²	1	1	yes	characteristic
9	#16861143	C₂²	1	1	yes	characteristic
10	#16861141	—	2	1	no	—
11	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
12	#107×C₁₂₇	—	2	1	no	—
13	#107×C₁₂₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₂₇	—	2	1	no	—
16	C₄⋊Dic₁₂₇	—	2	1	no	—
17	Dic₁₂₇⋊C₄	—	4	1	no	—
18	Dic₁₂₇⋊C₄	—	4	1	no	—
19	C₁₂₇⋊C₁₆	Q₈	1	4	yes	complemented
20	C₂×Q₈×C₁₂₇	—	4	1	no	—
21	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₂₇	—	2	2	no	—
24	C₄⋊C₄×C₁₂₇	—	4	1	no	—
25	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×Dic₁₂₇	—	4	1	no	—
27	C₂×Dic₁₂₇	—	4	1	no	—
28	Q₈×C₁₂₇	—	8	2	no	—
29	C₂×C₄×C₁₂₇	—	4	2	no	—
30	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₂₇	Q₁₆	1	2	yes	—
34	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
35	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₂₇	—	2	1	no	—
37	C₄×C₁₂₇	—	2	1	no	—
38	C₄×C₁₂₇	—	8	2	no	—
39	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
40	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
41	Dic₁₂₇	—	8	1	no	—
42	Dic₁₂₇	—	8	1	no	—
43	C₂×C₁₂₇	#500	1	1	yes	characteristic
44	C₂×C₁₂₇	#509	1	1	yes	characteristic
45	C₂×C₁₂₇	#510	1	1	yes	characteristic
46	#2255	—	127	1	no	maximal, Sylow
47	C₁₂₇	#2255	1	1	yes	characteristic, complemented, Sylow
48	#500	D₁₂₇	1	1	yes	characteristic
49	#499	—	127	1	no	—
50	#367	—	127	2	no	—
51	#366	—	127	2	no	—
52	#346	—	127	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	#125	—	254	1	no	—
55	C₂×C₁₆	—	127	2	no	—
56	#107	—	2	1	no	—
57	#107	—	127	1	no	—
58	#107	—	127	1	no	—
59	#107	C₂×D₁₂₇	1	2	yes	—
60	#106	—	254	1	no	—
61	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
62	C₂×Q₈	—	4	1	no	—
63	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	508	1	no	—
68	C₄⋊C₄	—	508	1	no	—
69	C₄⋊C₄	—	254	1	no	—
70	C₄⋊C₄	—	254	1	no	—
71	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
72	C₁₆	—	127	4	no	—
73	Q₈	—	8	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	508	1	no	—
76	C₂×C₄	—	508	1	no	—
77	C₂×C₄	#14523233	1	1	yes	characteristic
78	C₂×C₄	#14523233	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
80	C₈	#14523238	1	2	yes	—
81	C₈	Q₈×D₁₂₇	1	2	yes	—
82	C₂²	C₂×#14523233	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	2	no	—
86	C₄	—	1016	1	no	—
87	C₄	—	1016	1	no	—
88	C₄	#16861207	1	1	yes	characteristic
89	C₄	C₂×#14523238	1	1	yes	characteristic
90	C₂	#22550279	1	1	yes	characteristic
91	C₂	#22550312	1	1	yes	characteristic
92	C₂	#22550317	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	#2255
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
19. C₁₂₇⋊C₁₆	Q₈	semidirect
47. C₁₂₇	#2255	semidirect

Maximal subgroups

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

subgroup	index
#22550278	2
#22549703	2
#22549701	2
#22549652	2
#500×C₁₂₇	2
#2255	127