Subgroups — #30640606

Order 16256.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
2	#22549704	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549677	C₂	1	2	yes	maximal, complemented
4	#22549651	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	D₄×Dic₁₂₇	—	2	2	no	—
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16861143	C₂²	1	2	yes	—
9	#16861130	—	2	2	no	—
10	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₁₂₇	C₂²	1	2	yes	—
12	D₁₆×C₁₂₇	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₂₇	—	4	2	no	—
15	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
16	C₄⋊Dic₁₂₇	—	2	2	no	—
17	C₂²⋊Dic₁₂₇	—	4	2	no	—
18	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
19	C₁₂₇⋊C₁₆	—	2	1	no	—
20	C₂×D₄×C₁₂₇	—	2	2	no	—
21	D₈×C₁₂₇	—	2	2	no	—
22	D₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
23	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	complemented
25	C₂×Dic₁₂₇	—	4	2	no	—
26	C₂×Dic₁₂₇	—	8	4	no	—
27	C₂×Dic₁₂₇	D₈	1	2	yes	—
28	C₁₂₇⋊C₈	—	2	1	no	—
29	C₂³×C₁₂₇	—	4	2	no	—
30	D₄×C₁₂₇	—	2	4	no	—
31	D₄×C₁₂₇	—	4	2	no	—
32	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₂₇	#41	1	1	yes	characteristic
35	C₂²×C₁₂₇	—	4	4	no	—
36	C₂²×C₁₂₇	—	8	2	no	—
37	C₂²×C₁₂₇	—	8	2	no	—
38	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
39	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₂₇	#135	1	1	yes	characteristic
41	Dic₁₂₇	—	2	2	no	—
42	Dic₁₂₇	—	8	2	no	—
43	C₂×C₁₂₇	—	8	4	no	—
44	C₂×C₁₂₇	C₄×D₈	1	1	yes	characteristic
45	C₂×C₁₂₇	#509	1	2	yes	—
46	#2177	—	127	1	no	maximal, Sylow
47	C₁₂₇	#2177	1	1	yes	characteristic, complemented, Sylow
48	C₂×D₁₆	D₁₂₇	1	1	yes	characteristic
49	C₄×D₈	—	127	2	no	—
50	#367	—	127	1	no	—
51	#357	—	127	2	no	—
52	#346	—	127	1	no	—
53	C₂×D₈	C₂×D₁₂₇	1	2	yes	—
54	C₄×D₄	—	254	2	no	—
55	D₁₆	Dic₁₂₇	1	4	yes	complemented
56	C₂×C₁₆	—	127	1	no	—
57	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
58	#107	—	127	2	no	—
59	#102	—	254	2	no	—
60	C₄×C₈	—	127	1	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂²×C₄	—	508	2	no	—
63	D₈	—	2	2	no	—
64	D₈	C₂×Dic₁₂₇	1	4	yes	complemented
65	C₂×C₈	—	127	1	no	—
66	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
67	C₄⋊C₄	—	254	2	no	—
68	C₂²⋊C₄	—	508	2	no	—
69	C₄²	—	127	1	no	—
70	C₁₆	—	254	1	no	—
71	C₁₆	C₂×Dic₁₂₇	1	2	yes	complemented
72	C₂³	—	4	2	no	—
73	D₄	—	2	4	no	—
74	D₄	—	4	2	no	—
75	C₂×C₄	—	127	2	no	—
76	C₂×C₄	—	1016	4	no	—
77	C₂×C₄	—	508	2	no	—
78	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
79	C₈	—	254	1	no	—
80	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
81	C₈	#14523247	1	1	yes	characteristic
82	C₂²	—	4	4	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	8	2	no	—
85	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
86	C₄	—	1016	2	no	—
87	C₄	—	254	2	no	—
88	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
89	C₄	#16861228	1	1	yes	characteristic
90	C₂	—	8	4	no	—
91	C₂	D₈×Dic₁₂₇	1	1	yes	characteristic
92	C₂	#22550310	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting, prime radical	C₂×D₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂×D₁₆	D₁₂₇
hypocenter	C₁₂₇	#2177
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. D₈×Dic₁₂₇	C₂	semidirect
2. #22549704	C₂	semidirect
3. #22549677	C₂	semidirect
4. #22549651	C₂	semidirect
12. D₁₆×C₁₂₇	C₄	semidirect
22. D₈×C₁₂₇	C₂×C₄	semidirect
24. C₁₆×C₁₂₇	C₂×C₄	semidirect
47. C₁₂₇	#2177	semidirect
55. D₁₆	Dic₁₂₇	semidirect
64. D₈	C₂×Dic₁₂₇	semidirect
71. C₁₆	C₂×Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
D₈×Dic₁₂₇	2
#22549704	2
#22549677	2
#22549651	2
C₂×D₁₆×C₁₂₇	2
#2177	127