Subgroups — #22550353

Order 8128.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₅₄	C₂	1	4	yes	maximal, complemented
3	#16861174	C₂	1	4	yes	maximal, complemented
4	#16861168	C₂	1	1	yes	maximal, characteristic
5	#16861165	C₂	1	4	yes	maximal
6	#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₂₅₄	C₄	1	8	yes	complemented
8	C₂×Dic₂₅₄	C₂²	1	2	yes	—
9	C₂×Dic₂₅₄	C₂²	1	2	yes	—
10	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
12	Dic₁₂₇⋊C₄	C₂²	1	8	yes	—
13	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
14	C₄×Dic₁₂₇	C₂²	1	8	yes	—
15	C₄×Dic₁₂₇	C₂²	1	2	yes	—
16	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
19	C₄²×C₁₂₇	C₂²	1	2	yes	—
20	Dic₂₅₄	C₂×C₄	1	8	yes	—
21	Dic₂₅₄	C₂×C₄	1	8	yes	—
22	C₂×Dic₁₂₇	C₂×C₄	1	8	yes	—
23	C₂×Dic₁₂₇	C₂³	1	4	yes	—
24	C₂×Dic₁₂₇	C₂³	1	4	yes	—
25	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
28	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₂²×C₁₂₇	—	2	2	no	—
31	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
32	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
33	C₄×C₁₂₇	—	2	4	no	—
34	C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
35	C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
36	Dic₁₂₇	—	2	4	no	—
37	Dic₁₂₇	C₂²×C₄	1	8	yes	—
38	C₂×C₁₂₇	—	2	2	no	—
39	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
41	C₁₂₇	#519	1	1	yes	characteristic, complemented, Sylow
42	#519	—	127	1	no	maximal, Sylow
43	C₂²×Q₈	—	127	1	no	—
44	C₄×Q₈	—	127	4	no	—
45	C₄×Q₈	—	127	4	no	—
46	#117	—	127	1	no	—
47	#117	—	127	4	no	—
48	#117	D₁₂₇	1	1	yes	characteristic
49	C₂×Q₈	—	127	8	no	—
50	C₂×Q₈	—	127	2	no	—
51	C₂×Q₈	—	127	2	no	—
52	C₂²×C₄	—	127	2	no	—
53	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
54	C₄⋊C₄	—	127	8	no	—
55	C₄⋊C₄	—	127	2	no	—
56	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
57	C₂²⋊C₄	—	127	2	no	—
58	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
59	C₄²	—	127	8	no	—
60	C₄²	—	127	2	no	—
61	C₄²	C₂×D₁₂₇	1	2	yes	—
62	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
63	Q₈	—	127	8	no	—
64	Q₈	—	127	8	no	—
65	C₂×C₄	—	127	8	no	—
66	C₂×C₄	—	127	4	no	—
67	C₂×C₄	—	127	4	no	—
68	C₂×C₄	C₄×D₁₂₇	1	4	yes	—
69	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
70	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
74	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	127	8	no	—
77	C₄	—	254	4	no	—
78	C₄	C₂×C₄×D₁₂₇	1	2	yes	—
79	C₄	C₂×C₄×D₁₂₇	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
82	C₂	#16861256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₄
Fitting	#117×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂⁴
hypercenter	#117	D₁₂₇
hypocenter	C₁₂₇	#519
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₂₅₄	C₂	semidirect
3. #16861174	C₂	semidirect
7. C₂×Dic₂₅₄	C₄	semidirect
41. C₁₂₇	#519	semidirect