Subgroups — C₂²×#16861120 #30647134

Order 16256.

680 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₂₇⋊C₈	C₂	1	2	yes	maximal
2	C₂×#16861120	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₂₇⋊C₈	C₄	1	16	yes	complemented
5	C₂²×C₁₂₇⋊C₈	C₂²	1	12	yes	—
6	#16861120	C₂²	1	16	yes	direct product factor
7	C₂³×C₄×C₁₂₇	C₄	1	2	yes	—
8	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₂₇	C₄	1	6	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	6	yes	—
11	C₂×C₁₂₇⋊C₈	C₂×C₄	1	48	yes	complemented
12	C₂×C₁₂₇⋊C₈	C₂³	1	8	yes	—
13	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	—	2	8	no	—
15	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
19	C₄²×C₁₂₇	C₂×C₄	1	12	yes	—
20	C₄²×C₁₂₇	C₂³	1	4	yes	—
21	C₁₂₇⋊C₈	C₂²×C₄	1	16	yes	complemented
22	C₂³×C₁₂₇	C₄²	1	4	yes	—
23	C₂³×C₁₂₇	M₁₆	1	8	yes	—
24	C₂³×C₁₂₇	C₂²×C₄	1	3	yes	—
25	C₂×C₄×C₁₂₇	—	2	24	no	—
26	C₂×C₄×C₁₂₇	C₄²	1	12	yes	—
27	C₂×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
29	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
30	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
31	C₂²×C₁₂₇	#97	1	16	yes	complemented
32	C₂²×C₁₂₇	C₂×C₄²	1	6	yes	—
33	C₂²×C₁₂₇	C₂×M₁₆	1	12	yes	—
34	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₁₂₇	—	2	8	no	—
36	C₄×C₁₂₇	C₂×C₄²	1	6	yes	—
37	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
38	C₂×C₁₂₇	C₂×#97	1	12	yes	complemented
39	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂²×M₁₆	1	2	yes	—
41	C₂²×#97	—	127	1	no	maximal, Sylow
42	C₁₂₇	C₂²×#97	1	1	yes	characteristic, complemented, Sylow
43	C₂³×C₈	—	127	2	no	—
44	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
45	C₂×#97	—	127	12	no	—
46	C₂³×C₄	Dic₁₂₇	1	2	yes	—
47	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
48	C₂²×C₈	—	127	16	no	—
49	C₂²×C₈	—	127	12	no	—
50	C₂×C₄²	Dic₁₂₇	1	6	yes	—
51	C₂×C₄²	C₂×D₁₂₇	1	6	yes	—
52	#97	—	127	16	no	—
53	C₂⁴	C₂×Dic₁₂₇	1	1	yes	characteristic
54	C₂²×C₄	—	2	8	no	—
55	C₂²×C₄	C₄×D₁₂₇	1	8	yes	—
56	C₂²×C₄	C₂×Dic₁₂₇	1	12	yes	—
57	C₂²×C₄	C₂×Dic₁₂₇	1	3	yes	—
58	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
59	C₂×C₈	—	127	48	no	—
60	C₂×C₈	—	127	8	no	—
61	C₄²	C₂×Dic₁₂₇	1	12	yes	—
62	C₄²	C₂²×D₁₂₇	1	4	yes	—
63	C₂³	C₄×Dic₁₂₇	1	4	yes	—
64	C₂³	#14523232	1	8	yes	—
65	C₂³	C₂²×Dic₁₂₇	1	3	yes	—
66	C₂×C₄	—	2	24	no	—
67	C₂×C₄	C₄×Dic₁₂₇	1	12	yes	—
68	C₂×C₄	C₂×C₄×D₁₂₇	1	12	yes	—
69	C₂×C₄	C₂²×Dic₁₂₇	1	8	yes	—
70	C₂×C₄	C₂²×Dic₁₂₇	1	3	yes	—
71	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
72	C₈	—	127	16	no	—
73	C₂²	#16861120	1	16	yes	direct product factor
74	C₂²	C₂×C₄×Dic₁₂₇	1	6	yes	—
75	C₂²	C₂×#14523232	1	12	yes	—
76	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
77	C₄	—	2	8	no	—
78	C₄	C₂×C₄×Dic₁₂₇	1	6	yes	—
79	C₄	C₂²×C₄×D₁₂₇	1	2	yes	—
80	C₂	C₂×#16861120	1	12	yes	direct product factor
81	C₂	C₂²×C₄×Dic₁₂₇	1	1	yes	characteristic
82	C₂	C₂²×#14523232	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₂²×#97
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#16861120	C₂	direct (N × H)
4. C₂²×C₁₂₇⋊C₈	C₄	semidirect
6. #16861120	C₂²	direct (N × H)
11. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
21. C₁₂₇⋊C₈	C₂²×C₄	semidirect
31. C₂²×C₁₂₇	#97	semidirect
38. C₂×C₁₂₇	C₂×#97	semidirect
42. C₁₂₇	C₂²×#97	semidirect
73. C₂²	#16861120	direct (N × H)
80. C₂	C₂×#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
C₂³×C₁₂₇⋊C₈	2
C₂×#16861120	2
C₂²×C₄²×C₁₂₇	2
C₂²×#97	127

Subgroups — C22×#16861120 #30647134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16861120 #30647134