Subgroups — C₂²×#16861148 #30651825

Order 16256.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861148	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×D₁₂₇	C₄	1	1	yes	characteristic
6	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₂₇	C₄	1	6	yes	—
8	C₂×C₈×D₁₂₇	C₂²	1	6	yes	—
9	#16861148	C₂²	1	16	yes	direct product factor
10	C₂×C₁₂₇⋊C₁₆	C₂²	1	6	yes	complemented
11	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₂₇	C₂²	1	6	yes	complemented
13	C₂³×D₁₂₇	C₈	1	1	yes	characteristic
14	C₂²×Dic₁₂₇	C₈	1	1	yes	characteristic
15	C₂×C₄×D₁₂₇	C₈	1	6	yes	—
16	C₂×C₄×D₁₂₇	C₂×C₄	1	6	yes	—
17	C₂×C₁₂₇⋊C₈	C₂×C₄	1	6	yes	—
18	C₈×D₁₂₇	C₂×C₄	1	12	yes	—
19	C₈×D₁₂₇	C₂³	1	4	yes	—
20	C₁₂₇⋊C₁₆	C₂³	1	4	yes	complemented
21	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂×C₄	1	3	yes	—
23	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
24	C₁₆×C₁₂₇	C₂³	1	4	yes	complemented
25	C₂²×D₁₂₇	—	2	4	no	—
26	C₂²×D₁₂₇	C₂×C₈	1	6	yes	—
27	C₂×Dic₁₂₇	C₂×C₈	1	6	yes	—
28	C₄×D₁₂₇	C₂×C₈	1	12	yes	—
29	C₄×D₁₂₇	C₂²×C₄	1	4	yes	—
30	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
31	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂×C₈	1	3	yes	—
33	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
34	C₈×C₁₂₇	C₂²×C₄	1	3	yes	—
35	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
36	C₂²×C₁₂₇	M₃₂	1	4	yes	complemented
37	C₂²×C₁₂₇	C₂²×C₈	1	3	yes	—
38	C₂×D₁₂₇	—	2	12	no	—
39	C₂×D₁₂₇	C₂²×C₈	1	4	yes	—
40	C₄×C₁₂₇	C₂²×C₈	1	3	yes	—
41	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
42	Dic₁₂₇	C₂²×C₈	1	4	yes	—
43	C₂×C₁₂₇	C₂×M₃₂	1	6	yes	complemented
44	C₂×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
45	D₁₂₇	—	2	4	no	—
46	C₂²×M₃₂	—	127	1	no	maximal, Sylow
47	C₁₂₇	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
48	C₂³×C₈	—	127	1	no	—
49	C₂×M₃₂	—	127	12	no	—
50	C₂²×C₁₆	—	127	1	no	—
51	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic, complemented
52	C₂³×C₄	—	127	1	no	—
53	C₂²×C₈	—	127	1	no	—
54	C₂²×C₈	—	127	6	no	—
55	C₂²×C₈	—	127	6	no	—
56	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
57	M₃₂	—	127	16	no	—
58	C₂×C₁₆	—	127	6	no	—
59	C₂×C₁₆	C₂×D₁₂₇	1	6	yes	complemented
60	C₂⁴	—	127	1	no	—
61	C₂²×C₄	—	127	1	no	—
62	C₂²×C₄	—	127	6	no	—
63	C₂²×C₄	—	127	6	no	—
64	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
65	C₂×C₈	—	127	6	no	—
66	C₂×C₈	—	127	12	no	—
67	C₂×C₈	—	127	4	no	—
68	C₂×C₈	C₄×D₁₂₇	1	3	yes	—
69	C₂×C₈	C₂²×D₁₂₇	1	3	yes	—
70	C₁₆	—	127	4	no	—
71	C₁₆	C₂²×D₁₂₇	1	4	yes	complemented
72	C₂³	—	127	6	no	—
73	C₂³	—	254	4	no	—
74	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
75	C₂×C₄	—	127	6	no	—
76	C₂×C₄	—	127	12	no	—
77	C₂×C₄	—	127	4	no	—
78	C₂×C₄	C₈×D₁₂₇	1	3	yes	—
79	C₂×C₄	C₂×C₄×D₁₂₇	1	3	yes	—
80	C₈	—	127	4	no	—
81	C₈	C₂×C₄×D₁₂₇	1	3	yes	—
82	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
83	C₂²	—	127	4	no	—
84	C₂²	—	254	12	no	—
85	C₂²	#16861148	1	4	yes	direct product factor
86	C₂²	C₂×C₈×D₁₂₇	1	3	yes	—
87	C₄	—	127	4	no	—
88	C₄	C₂×C₈×D₁₂₇	1	3	yes	—
89	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
90	C₂	—	254	4	no	—
91	C₂	C₂×#16861148	1	6	yes	direct product factor
92	C₂	C₂²×C₈×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₈
Frattini	C₈	C₂³×D₁₂₇
Fitting	C₂²×C₁₆×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×C₈
prime radical	C₂³×D₁₂₇	C₈
cosocle	C₈×C₁₂₇	C₂⁴
hypercenter	C₂²×C₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂²×M₃₂
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16861148	C₂	direct (N × H)
3. C₂²×C₁₂₇⋊C₁₆	C₂	semidirect
4. C₂²×C₁₆×C₁₂₇	C₂	semidirect
9. #16861148	C₂²	direct (N × H)
10. C₂×C₁₂₇⋊C₁₆	C₂²	semidirect
12. C₂×C₁₆×C₁₂₇	C₂²	semidirect
20. C₁₂₇⋊C₁₆	C₂³	semidirect
24. C₁₆×C₁₂₇	C₂³	semidirect
36. C₂²×C₁₂₇	M₃₂	semidirect
43. C₂×C₁₂₇	C₂×M₃₂	semidirect
47. C₁₂₇	C₂²×M₃₂	semidirect
51. C₂²×C₁₆	D₁₂₇	semidirect
59. C₂×C₁₆	C₂×D₁₂₇	semidirect
71. C₁₆	C₂²×D₁₂₇	semidirect
85. C₂²	#16861148	direct (N × H)
91. C₂	C₂×#16861148	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₁₂₇	2
C₂×#16861148	2
C₂²×C₁₂₇⋊C₁₆	2
C₂²×C₁₆×C₁₂₇	2
C₂²×M₃₂	127

Subgroups — C22×#16861148 #30651825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16861148 #30651825