Subgroups — C₂²×#16861149 #30651826

Order 16256.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861149	C₂	1	12	yes	maximal, direct product factor
2	C₂²×C₁₂₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861149	C₂²	1	16	yes	direct product factor
5	C₂×C₁₂₇⋊C₁₆	C₂²	1	12	yes	complemented
6	C₂³×C₄×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₁₂₇	C₄	1	6	yes	—
9	C₂²×C₈×C₁₂₇	C₂²	1	6	yes	—
10	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₁₂₇⋊C₁₆	C₂³	1	8	yes	complemented
12	C₂⁴×C₁₂₇	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₁₂₇	C₈	1	6	yes	—
14	C₂²×C₄×C₁₂₇	C₈	1	1	yes	characteristic
15	C₂²×C₄×C₁₂₇	C₂×C₄	1	6	yes	—
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₂₇	C₂×C₄	1	12	yes	—
18	C₂×C₈×C₁₂₇	C₂×C₄	1	6	yes	—
19	C₂×C₈×C₁₂₇	C₂×C₄	1	3	yes	—
20	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
21	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
22	C₂³×C₁₂₇	—	2	4	no	—
23	C₂³×C₁₂₇	C₂×C₈	1	6	yes	—
24	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₂₇	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₁₂₇	C₂×C₈	1	6	yes	—
27	C₂×C₄×C₁₂₇	C₂×C₈	1	3	yes	—
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
30	C₈×C₁₂₇	C₂²×C₄	1	3	yes	—
31	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
32	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
33	C₂²×C₁₂₇	—	2	12	no	—
34	C₂²×C₁₂₇	M₃₂	1	4	yes	complemented
35	C₂²×C₁₂₇	C₂²×C₈	1	4	yes	—
36	C₂²×C₁₂₇	C₂²×C₈	1	3	yes	—
37	C₄×C₁₂₇	C₂²×C₈	1	3	yes	—
38	C₄×C₁₂₇	C₂²×C₈	1	4	yes	—
39	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₁₂₇	—	2	4	no	—
41	C₂×C₁₂₇	C₂×M₃₂	1	6	yes	complemented
42	C₂×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
43	C₂²×M₃₂	—	127	1	no	maximal, Sylow
44	C₁₂₇	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
45	C₂³×C₈	D₁₂₇	1	1	yes	characteristic
46	C₂×M₃₂	—	127	12	no	—
47	C₂²×C₁₆	—	127	2	no	—
48	C₂³×C₄	Dic₁₂₇	1	1	yes	characteristic
49	C₂²×C₈	Dic₁₂₇	1	1	yes	characteristic
50	C₂²×C₈	Dic₁₂₇	1	6	yes	—
51	C₂²×C₈	C₂×D₁₂₇	1	6	yes	—
52	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
53	M₃₂	—	127	16	no	—
54	C₂×C₁₆	—	127	12	no	—
55	C₂⁴	C₁₂₇⋊C₈	1	1	yes	characteristic
56	C₂²×C₄	C₁₂₇⋊C₈	1	6	yes	—
57	C₂²×C₄	C₁₂₇⋊C₈	1	1	yes	characteristic
58	C₂²×C₄	C₂×Dic₁₂₇	1	6	yes	—
59	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
60	C₂×C₈	C₂×Dic₁₂₇	1	6	yes	—
61	C₂×C₈	C₂×Dic₁₂₇	1	12	yes	—
62	C₂×C₈	C₂×Dic₁₂₇	1	3	yes	—
63	C₂×C₈	C₂²×D₁₂₇	1	3	yes	—
64	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
65	C₁₆	—	127	8	no	—
66	C₂³	—	2	4	no	—
67	C₂³	C₂×C₁₂₇⋊C₈	1	6	yes	—
68	C₂³	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
69	C₂×C₄	C₂×C₁₂₇⋊C₈	1	12	yes	—
70	C₂×C₄	C₂×C₁₂₇⋊C₈	1	6	yes	—
71	C₂×C₄	C₂×C₁₂₇⋊C₈	1	3	yes	—
72	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
73	C₂×C₄	C₂²×Dic₁₂₇	1	3	yes	—
74	C₈	C₂²×Dic₁₂₇	1	4	yes	—
75	C₈	C₂²×Dic₁₂₇	1	3	yes	—
76	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
77	C₂²	—	2	12	no	—
78	C₂²	#16861149	1	4	yes	direct product factor
79	C₂²	C₂²×C₁₂₇⋊C₈	1	4	yes	—
80	C₂²	C₂²×C₁₂₇⋊C₈	1	3	yes	—
81	C₄	C₂²×C₁₂₇⋊C₈	1	3	yes	—
82	C₄	C₂²×C₁₂₇⋊C₈	1	4	yes	—
83	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#16861149	1	6	yes	direct product factor
86	C₂	C₂³×C₁₂₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₈
Frattini	C₈	C₂³×D₁₂₇
Fitting	C₂³×C₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×C₈
prime radical	C₂⁴×C₁₂₇	C₈
cosocle	C₈×C₁₂₇	C₂⁴
hypercenter	C₂³×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂²×M₃₂
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16861149	C₂	direct (N × H)
2. C₂²×C₁₂₇⋊C₁₆	C₂	semidirect
4. #16861149	C₂²	direct (N × H)
5. C₂×C₁₂₇⋊C₁₆	C₂²	semidirect
11. C₁₂₇⋊C₁₆	C₂³	semidirect
34. C₂²×C₁₂₇	M₃₂	semidirect
41. C₂×C₁₂₇	C₂×M₃₂	semidirect
44. C₁₂₇	C₂²×M₃₂	semidirect
78. C₂²	#16861149	direct (N × H)
85. C₂	C₂×#16861149	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×#16861149	2
C₂²×C₁₂₇⋊C₁₆	2
C₂³×C₈×C₁₂₇	2
C₂²×M₃₂	127

Subgroups — C22×#16861149 #30651826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16861149 #30651826