Subgroups — #22550081

Order 8128.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523236	C₂	1	2	yes	maximal
2	#16861209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861132	C₂	1	2	yes	maximal, complemented
5	C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₅₄	C₂²	1	2	yes	—
7	#14523236	—	2	4	no	—
8	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₂₇	—	2	1	no	—
10	C₄⋊Dic₁₂₇	—	2	1	no	—
11	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	—	2	1	no	—
13	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
15	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	Dic₂₅₄	—	2	2	no	—
17	Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₁₂₇	—	2	2	no	—
19	C₁₂₇⋊C₈	—	2	2	no	—
20	C₂³×C₁₂₇	—	2	1	no	—
21	D₄×C₁₂₇	—	2	1	no	—
22	D₄×C₁₂₇	D₄	1	2	yes	—
23	C₂×C₄×C₁₂₇	—	2	1	no	—
24	C₂×C₄×C₁₂₇	—	4	2	no	—
25	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂²×C₁₂₇	—	2	2	no	—
29	C₂²×C₁₂₇	—	4	1	no	—
30	C₂²×C₁₂₇	—	4	1	no	—
31	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₂₇	—	2	1	no	—
33	C₄×C₁₂₇	—	4	1	no	—
34	C₄×C₁₂₇	SD₁₆	1	2	yes	complemented
35	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₂₇	#41	1	1	yes	characteristic
37	Dic₁₂₇	—	4	2	no	—
38	C₂×C₁₂₇	—	4	2	no	—
39	C₂×C₁₂₇	#121	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₁₂₇	#137	1	1	yes	characteristic
42	C₁₂₇	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	127	1	no	maximal, Sylow
44	C₂×SD₁₆	—	127	2	no	—
45	C₄⋊Q₈	—	127	1	no	—
46	C₄×D₄	D₁₂₇	1	1	yes	characteristic
47	C₄⋊C₈	—	127	1	no	—
48	#103	—	127	2	no	—
49	C₂×Q₈	—	127	2	no	—
50	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	SD₁₆	—	254	4	no	—
53	C₂×C₈	—	127	2	no	—
54	C₄⋊C₄	—	254	1	no	—
55	C₄⋊C₄	—	254	1	no	—
56	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
59	C₂³	—	2	1	no	—
60	Q₈	—	254	2	no	—
61	Q₈	—	254	2	no	—
62	D₄	—	2	1	no	—
63	D₄	D₅₀₈	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	254	2	no	—
67	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
70	C₈	—	254	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	508	2	no	—
78	C₄	#14523236	1	2	yes	complemented
79	C₄	C₂×D₅₀₈	1	1	yes	characteristic
80	C₄	#14523248	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	#16861183	1	1	yes	characteristic
83	C₂	C₂×#14523236	1	1	yes	characteristic
84	C₂	#16861239	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₄×D₄×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₄×D₄	D₁₂₇
hypocenter	C₁₂₇	#461
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #16861209	C₂	semidirect
3. C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
4. #16861132	C₂	semidirect
34. C₄×C₁₂₇	SD₁₆	semidirect
42. C₁₂₇	#461	semidirect
78. C₄	#14523236	semidirect