Subgroups — #30636697

Order 16256.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549652	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
6	C₂×D₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₄×D₄×C₁₂₇	—	2	1	no	—
8	#107×C₁₂₇	C₄	1	1	yes	characteristic
9	#102×C₁₂₇	—	2	1	no	—
10	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₂₇⋊C₈	—	4	1	no	—
12	C₁₂₇⋊C₁₆	—	2	2	no	—
13	C₂×D₄×C₁₂₇	—	2	1	no	—
14	C₂²×C₄×C₁₂₇	—	4	1	no	—
15	D₈×C₁₂₇	—	2	1	no	—
16	D₈×C₁₂₇	C₈	1	2	yes	complemented
17	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₁₂₇	—	2	1	no	—
20	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
21	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
22	C₁₂₇⋊C₈	—	8	1	no	—
23	C₂³×C₁₂₇	—	4	1	no	—
24	D₄×C₁₂₇	—	2	2	no	—
25	D₄×C₁₂₇	—	4	1	no	—
26	C₂×C₄×C₁₂₇	—	4	1	no	—
27	C₂×C₄×C₁₂₇	—	8	2	no	—
28	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₂₇	—	2	1	no	—
32	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
34	C₂²×C₁₂₇	—	4	2	no	—
35	C₂²×C₁₂₇	—	8	1	no	—
36	C₂²×C₁₂₇	—	8	1	no	—
37	C₂²×C₁₂₇	#102	1	1	yes	characteristic
38	C₄×C₁₂₇	—	2	1	no	—
39	C₄×C₁₂₇	—	2	1	no	—
40	C₄×C₁₂₇	—	8	1	no	—
41	C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₁₂₇	#104	1	1	yes	characteristic
43	C₂×C₁₂₇	—	8	2	no	—
44	C₂×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₂₇	#360	1	1	yes	characteristic
46	C₂×C₁₂₇	#361	1	1	yes	characteristic
47	#1333	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1333	1	1	yes	characteristic, complemented, Sylow
49	C₄×D₈	D₁₂₇	1	1	yes	characteristic
50	#346	—	127	1	no	—
51	#334	—	127	1	no	—
52	C₂×D₈	Dic₁₂₇	1	1	yes	characteristic
53	C₄×D₄	—	2	1	no	—
54	C₂×C₁₆	—	254	1	no	—
55	#107	Dic₁₂₇	1	1	yes	characteristic
56	C₄⋊C₈	—	254	1	no	—
57	#102	—	2	1	no	—
58	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
59	C₂×D₄	—	2	1	no	—
60	C₂²×C₄	—	4	1	no	—
61	D₈	—	2	1	no	—
62	D₈	C₁₂₇⋊C₈	1	2	yes	complemented
63	C₂×C₈	—	508	1	no	—
64	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₂²⋊C₄	—	4	1	no	—
68	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
69	C₁₆	—	254	2	no	—
70	C₂³	—	4	1	no	—
71	D₄	—	2	2	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
76	C₂×C₄	#14523233	1	1	yes	characteristic
77	C₂×C₄	#14523236	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	1016	1	no	—
80	C₈	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
81	C₈	#14523232	1	1	yes	characteristic
82	C₂²	—	4	2	no	—
83	C₂²	—	8	1	no	—
84	C₂²	—	8	1	no	—
85	C₂²	#16861130	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	8	1	no	—
89	C₄	C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
90	C₄	#16861136	1	1	yes	characteristic
91	C₂	—	8	2	no	—
92	C₂	D₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
93	C₂	#22549686	1	1	yes	characteristic
94	C₂	#22549689	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861130
commutator	C₈×C₁₂₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	C₄×D₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#102
prime radical	C₂×D₈×C₁₂₇	C₄
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	C₄×D₈	D₁₂₇
hypocenter	C₁₂₇	#1333
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #22549652	C₂	semidirect
2. #22549627	C₂	semidirect
16. D₈×C₁₂₇	C₈	semidirect
48. C₁₂₇	#1333	semidirect
62. D₈	C₁₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22549652	2
#22549627	2
C₄×D₈×C₁₂₇	2
#1333	127