Subgroups — C₂²×#14523238 #22550652

Order 8128.

258 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	C₂×#14523238	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₅₄	—	2	3	no	—
6	C₂×Dic₂₅₄	C₂²	1	6	yes	—
7	C₂²×Dic₁₂₇	—	2	1	no	—
8	#14523238	C₂²	1	16	yes	direct product factor
9	C₂×C₁₂₇⋊C₈	C₂²	1	6	yes	—
10	C₂×Q₈×C₁₂₇	—	2	3	no	—
11	C₂×Q₈×C₁₂₇	C₂²	1	6	yes	—
12	C₂²×C₄×C₁₂₇	—	2	1	no	—
13	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	Dic₂₅₄	—	2	6	no	—
15	Dic₂₅₄	C₂³	1	4	yes	—
16	C₂×Dic₁₂₇	—	2	6	no	—
17	C₁₂₇⋊C₈	C₂³	1	4	yes	—
18	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
19	Q₈×C₁₂₇	—	2	6	no	—
20	Q₈×C₁₂₇	C₂³	1	4	yes	—
21	C₂×C₄×C₁₂₇	—	2	6	no	—
22	C₂×C₄×C₁₂₇	D₄	1	3	yes	—
23	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
24	C₂²×C₁₂₇	Q₁₆	1	4	yes	complemented
25	C₂²×C₁₂₇	C₂×D₄	1	3	yes	—
26	C₄×C₁₂₇	—	2	4	no	—
27	C₄×C₁₂₇	C₂×D₄	1	3	yes	—
28	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
29	Dic₁₂₇	—	2	4	no	—
30	C₂×C₁₂₇	C₂×Q₁₆	1	6	yes	complemented
31	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
32	C₁₂₇	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
33	C₂²×Q₁₆	—	127	1	no	maximal, Sylow
34	C₂²×Q₈	—	127	1	no	—
35	C₂²×Q₈	D₁₂₇	1	1	yes	characteristic
36	C₂×Q₁₆	—	127	12	no	—
37	C₂²×C₈	—	127	1	no	—
38	C₂×Q₈	—	2	3	no	—
39	C₂×Q₈	—	127	6	no	—
40	C₂×Q₈	—	254	3	no	—
41	C₂×Q₈	C₂×D₁₂₇	1	6	yes	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	—	254	1	no	—
44	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
45	Q₁₆	—	127	16	no	—
46	C₂×C₈	—	127	6	no	—
47	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
48	Q₈	—	2	6	no	—
49	Q₈	—	127	4	no	—
50	Q₈	—	254	6	no	—
51	Q₈	C₂²×D₁₂₇	1	4	yes	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	254	6	no	—
54	C₂×C₄	C₂²⋊D₁₂₇	1	3	yes	—
55	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
56	C₈	—	127	4	no	—
57	C₂²	#14523238	1	4	yes	direct product factor
58	C₂²	C₂×C₂²⋊D₁₂₇	1	3	yes	—
59	C₄	—	2	4	no	—
60	C₄	—	254	4	no	—
61	C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
62	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
63	C₂	C₂×#14523238	1	6	yes	direct product factor
64	C₂	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂⁴
Frattini	C₄	C₂³×D₁₂₇
Fitting	C₂²×Q₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	D₄
hypercenter	C₂²×Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂²×Q₁₆
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#14523238	C₂	direct (N × H)
8. #14523238	C₂²	direct (N × H)
24. C₂²×C₁₂₇	Q₁₆	semidirect
30. C₂×C₁₂₇	C₂×Q₁₆	semidirect
32. C₁₂₇	C₂²×Q₁₆	semidirect
57. C₂²	#14523238	direct (N × H)
63. C₂	C₂×#14523238	direct (N × H)

Subgroups — C22×#14523238 #22550652

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14523238 #22550652