Subgroups — C₂×#16861216 #22550639

Order 8128.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861216	C₂	1	8	yes	maximal, direct product factor
2	C₂×#14523232	C₂	1	3	yes	maximal, complemented
3	C₂²×C₁₂₇⋊C₈	C₂	1	3	yes	maximal, complemented
4	C₂×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#14523232	C₂²	1	12	yes	complemented
6	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₈	C₂²	1	3	yes	—
8	C₂×C₁₂₇⋊C₈	C₂²	1	12	yes	complemented
9	#41×C₁₂₇	C₄	1	4	yes	—
10	#41×C₁₂₇	C₂²	1	4	yes	—
11	C₂×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
12	C₂×D₄×C₁₂₇	C₄	1	3	yes	—
13	C₂²×C₄×C₁₂₇	C₂²	1	3	yes	—
14	C₁₂₇⋊C₈	C₂³	1	2	yes	—
15	C₁₂₇⋊C₈	C₂³	1	6	yes	—
16	C₂³×C₁₂₇	C₂×C₄	1	3	yes	—
17	Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
18	D₄×C₁₂₇	C₂×C₄	1	12	yes	—
19	C₂×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
20	C₂×C₄×C₁₂₇	C₂×C₄	1	6	yes	—
21	C₂×C₄×C₁₂₇	C₂³	1	6	yes	—
22	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂²×C₁₂₇	—	2	6	no	—
24	C₂²×C₁₂₇	C₂²×C₄	1	6	yes	—
25	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
28	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
29	C₂×C₁₂₇	—	2	6	no	—
30	C₂×C₁₂₇	#131	1	2	yes	complemented
31	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
32	C₁₂₇	C₂×#131	1	1	yes	characteristic, complemented, Sylow
33	C₂×#131	—	127	1	no	maximal, Sylow
34	C₂×#41	D₁₂₇	1	1	yes	characteristic
35	#131	—	127	8	no	—
36	C₂×M₁₆	—	127	3	no	—
37	C₂²×C₈	—	127	3	no	—
38	#41	Dic₁₂₇	1	4	yes	—
39	#41	C₂×D₁₂₇	1	4	yes	—
40	C₂×Q₈	Dic₁₂₇	1	1	yes	characteristic
41	C₂×D₄	Dic₁₂₇	1	3	yes	—
42	C₂²×C₄	C₂×D₁₂₇	1	3	yes	—
43	M₁₆	—	127	12	no	—
44	C₂×C₈	—	127	12	no	—
45	C₂×C₈	—	127	1	no	—
46	C₂×C₈	—	127	3	no	—
47	C₂³	C₂×Dic₁₂₇	1	3	yes	—
48	Q₈	C₂×Dic₁₂₇	1	4	yes	—
49	D₄	C₂×Dic₁₂₇	1	12	yes	—
50	C₂×C₄	C₂×Dic₁₂₇	1	3	yes	—
51	C₂×C₄	C₂×Dic₁₂₇	1	6	yes	—
52	C₂×C₄	C₂²×D₁₂₇	1	6	yes	—
53	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
54	C₈	—	127	2	no	—
55	C₈	—	127	6	no	—
56	C₂²	—	2	6	no	—
57	C₂²	C₂²×Dic₁₂₇	1	6	yes	—
58	C₂²	C₂²×Dic₁₂₇	1	1	yes	characteristic
59	C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
60	C₄	C₂²×Dic₁₂₇	1	6	yes	—
61	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
62	C₂	—	2	6	no	—
63	C₂	#16861216	1	2	yes	direct product factor
64	C₂	C₂³×Dic₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₄
Frattini	C₄	C₂³×D₁₂₇
Fitting	C₂×#41×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×C₄
hypercenter	C₂×#41	D₁₂₇
hypocenter	C₁₂₇	C₂×#131
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #16861216	C₂	direct (N × H)
2. C₂×#14523232	C₂	semidirect
3. C₂²×C₁₂₇⋊C₈	C₂	semidirect
5. #14523232	C₂²	semidirect
8. C₂×C₁₂₇⋊C₈	C₂²	semidirect
30. C₂×C₁₂₇	#131	semidirect
32. C₁₂₇	C₂×#131	semidirect
63. C₂	#16861216	direct (N × H)

Subgroups — C2×#16861216 #22550639

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16861216 #22550639