Subgroups — C₄×#16861120 #30638415

Order 16256.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861120	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₂₇⋊C₈	C₂	1	2	yes	maximal
3	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
5	#16861120	C₄	1	16	yes	direct product factor
6	C₄×C₁₂₇⋊C₈	C₄	1	8	yes	complemented
7	C₂×C₄²×C₁₂₇	C₄	1	4	yes	—
8	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	—
11	C₂×C₁₂₇⋊C₈	C₂×C₄	1	16	yes	—
12	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
15	C₄²×C₁₂₇	—	2	8	no	—
16	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
17	C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
18	C₁₂₇⋊C₈	C₄²	1	16	yes	complemented
19	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	—	2	8	no	—
21	C₂×C₄×C₁₂₇	C₄²	1	8	yes	—
22	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
23	C₂×C₄×C₁₂₇	M₁₆	1	8	yes	—
24	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
25	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
26	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
27	C₂²×C₁₂₇	C₂×C₄²	1	2	yes	—
28	C₂²×C₁₂₇	C₂×M₁₆	1	4	yes	—
29	C₄×C₁₂₇	—	2	8	no	—
30	C₄×C₁₂₇	#97	1	8	yes	complemented
31	C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
32	C₂×C₁₂₇	C₄³	1	1	yes	characteristic
33	C₂×C₁₂₇	C₂×#97	1	2	yes	—
34	C₂×C₁₂₇	C₄×M₁₆	1	4	yes	—
35	C₄×#97	—	127	1	no	maximal, Sylow
36	C₁₂₇	C₄×#97	1	1	yes	characteristic, complemented, Sylow
37	C₂×#97	—	127	4	no	—
38	C₂×C₄×C₈	—	127	2	no	—
39	C₄³	D₁₂₇	1	1	yes	characteristic
40	C₂²×C₈	—	127	4	no	—
41	C₂×C₄²	Dic₁₂₇	1	4	yes	—
42	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
43	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
44	#97	—	127	16	no	—
45	C₄×C₈	—	127	8	no	—
46	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
47	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
48	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
49	C₂×C₈	—	127	8	no	—
50	C₂×C₈	—	127	16	no	—
51	C₄²	—	2	8	no	—
52	C₄²	C₄×D₁₂₇	1	4	yes	—
53	C₄²	C₄×D₁₂₇	1	8	yes	—
54	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
55	C₂×C₄	—	2	8	no	—
56	C₂×C₄	C₄×Dic₁₂₇	1	8	yes	—
57	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
58	C₂×C₄	#14523232	1	8	yes	—
59	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
60	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
61	C₈	—	127	16	no	—
62	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
63	C₂²	C₂×C₄×Dic₁₂₇	1	2	yes	—
64	C₂²	C₂×#14523232	1	4	yes	—
65	C₄	—	2	8	no	—
66	C₄	#16861120	1	8	yes	direct product factor
67	C₄	C₄²×D₁₂₇	1	4	yes	—
68	C₂	C₄²×Dic₁₂₇	1	1	yes	characteristic
69	C₂	C₂×#16861120	1	2	yes	—
70	C₂	C₄×#14523232	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₄³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₄³×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₄³	D₁₂₇
hypocenter	C₁₂₇	C₄×#97
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. #16861120	C₄	direct (N × H)
6. C₄×C₁₂₇⋊C₈	C₄	semidirect
18. C₁₂₇⋊C₈	C₄²	semidirect
30. C₄×C₁₂₇	#97	semidirect
36. C₁₂₇	C₄×#97	semidirect
66. C₄	#16861120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16861120	2
C₂×C₄×C₁₂₇⋊C₈	2
C₄³×C₁₂₇	2
C₄×#97	127

Subgroups — C4×#16861120 #30638415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16861120 #30638415