Subgroups — #30636869

Order 16256.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861148	C₂	1	1	yes	maximal, characteristic
2	#22549710	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₁₂₇	C₄	1	1	yes	characteristic
5	#16861148	—	2	1	no	—
6	#16861148	—	2	1	no	—
7	C₂×C₁₂₇⋊C₁₆	C₄	1	1	yes	characteristic
8	C₁₂₇⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₂₇	—	2	1	no	—
11	C₂×C₄×D₁₂₇	C₈	1	1	yes	characteristic
12	C₂×C₁₂₇⋊C₈	C₈	1	1	yes	characteristic
13	C₈×D₁₂₇	—	2	1	no	—
14	C₈×D₁₂₇	—	2	1	no	—
15	C₁₂₇⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
18	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂²×D₁₂₇	C₁₆	1	1	yes	characteristic
20	C₂×Dic₁₂₇	C₁₆	1	1	yes	characteristic
21	C₄×D₁₂₇	—	2	1	no	—
22	C₄×D₁₂₇	—	2	1	no	—
23	C₁₂₇⋊C₈	—	2	1	no	—
24	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
28	C₂×D₁₂₇	—	2	1	no	—
29	C₂×D₁₂₇	—	4	1	no	—
30	C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₂₇	M₃₂	1	1	yes	characteristic
32	Dic₁₂₇	—	2	1	no	—
33	C₂×C₁₂₇	—	2	1	no	—
34	C₂×C₁₂₇	C₂²⋊C₁₆	1	1	yes	characteristic
35	D₁₂₇	—	4	1	no	—
36	#1400	—	127	1	no	maximal, Sylow
37	C₁₂₇	#1400	1	1	yes	characteristic, complemented, Sylow
38	C₂×M₃₂	—	127	1	no	—
39	M₆₄	—	127	1	no	—
40	M₆₄	D₁₂₇	1	1	yes	characteristic, complemented
41	C₂²×C₈	—	127	1	no	—
42	M₃₂	—	254	1	no	—
43	M₃₂	—	254	1	no	—
44	C₂×C₁₆	—	127	1	no	—
45	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
46	C₃₂	—	2	1	no	—
47	C₃₂	—	254	1	no	—
48	C₂²×C₄	—	127	1	no	—
49	C₂×C₈	—	127	1	no	—
50	C₂×C₈	—	254	1	no	—
51	C₂×C₈	—	254	1	no	—
52	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
53	C₁₆	—	254	1	no	—
54	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
55	C₁₆	D₅₀₈	1	1	yes	characteristic
56	C₂³	—	127	1	no	—
57	C₂×C₄	—	127	1	no	—
58	C₂×C₄	—	254	1	no	—
59	C₂×C₄	—	254	1	no	—
60	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
61	C₈	—	254	1	no	—
62	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
63	C₈	#14523231	1	1	yes	characteristic
64	C₂²	—	508	1	no	—
65	C₂²	—	254	1	no	—
66	C₂²	C₁₆×D₁₂₇	1	1	yes	characteristic
67	C₄	—	254	1	no	—
68	C₄	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
69	C₄	#16861148	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	508	1	no	—
72	C₂	(C₂×D₁₂₇)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₈	(C₂×D₁₂₇)⋊C₄
commutator	C₂²×C₁₂₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	M₆₄×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²⋊C₁₆
prime radical	C₂²×D₁₂₇	C₁₆
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	M₆₄	D₁₂₇
hypocenter	C₁₂₇	#1400
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #22549710	C₂	semidirect
3. M₆₄×C₁₂₇	C₂	semidirect
37. C₁₂₇	#1400	semidirect
40. M₆₄	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861148	2
#22549710	2
M₆₄×C₁₂₇	2
#1400	127