Subgroups — #30636913

Order 16256.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550308	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₂₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861231	—	2	1	no	—
5	Dic₁₀₁₆	C₄	1	1	yes	characteristic
6	#16861154	—	2	1	no	—
7	D₂₀₃₂	C₄	1	1	yes	characteristic
8	C₁₂₇⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	#108×C₁₂₇	—	2	1	no	—
11	#14523248	—	4	1	no	—
12	Dic₅₀₈	—	2	1	no	—
13	#14523237	—	4	1	no	—
14	D₁₀₁₆	—	2	1	no	—
15	M₁₆×C₁₂₇	—	4	1	no	—
16	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
19	Dic₂₅₄	—	4	1	no	—
20	D₅₀₈	—	4	1	no	—
21	C₂²⋊D₁₂₇	—	8	1	no	—
22	C₄×D₁₂₇	—	8	1	no	—
23	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
24	C₈×C₁₂₇	—	8	1	no	—
25	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₂₇	SD₃₂	1	1	yes	characteristic
28	C₂×D₁₂₇	—	8	1	no	—
29	C₄×C₁₂₇	#102	1	1	yes	characteristic
30	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
31	Dic₁₂₇	—	8	1	no	—
32	C₂×C₁₂₇	—	2	1	no	—
33	C₂×C₁₂₇	#357	1	1	yes	characteristic
34	D₁₂₇	—	16	1	no	—
35	#1417	—	127	1	no	maximal, Sylow
36	C₁₂₇	#1417	1	1	yes	characteristic, complemented, Sylow
37	#508	—	127	1	no	—
38	C₂×C₃₂	—	127	1	no	—
39	#368	D₁₂₇	1	1	yes	characteristic, complemented
40	#135	—	254	1	no	—
41	Q₃₂	—	127	1	no	—
42	SD₃₂	—	254	1	no	—
43	D₁₆	—	127	1	no	—
44	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
45	#108	—	2	1	no	—
46	C₃₂	—	254	1	no	—
47	#41	—	508	1	no	—
48	Q₁₆	—	254	1	no	—
49	SD₁₆	—	508	1	no	—
50	D₈	—	254	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
53	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
54	C₁₆	D₅₀₈	1	1	yes	characteristic
55	Q₈	—	508	1	no	—
56	D₄	—	1016	1	no	—
57	D₄	—	508	1	no	—
58	C₂×C₄	—	1016	1	no	—
59	C₂×C₄	#14523233	1	1	yes	characteristic
60	C₈	—	8	1	no	—
61	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
62	C₈	#14523235	1	1	yes	characteristic
63	C₂²	—	1016	1	no	—
64	C₂²	#16861152	1	1	yes	characteristic
65	C₄	—	1016	1	no	—
66	C₄	#16861129	1	1	yes	characteristic
67	C₄	#16861150	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	2032	1	no	—
70	C₂	#22549676	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16861150
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	#368×C₁₂₇	C₂
socle	C₂×C₁₂₇	#357
prime radical	#22550308	C₂
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	#368	D₁₂₇
hypocenter	C₁₂₇	#1417
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₂₇⋊C₃₂	C₂	semidirect
3. #368×C₁₂₇	C₂	semidirect
36. C₁₂₇	#1417	semidirect
39. #368	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550308	2
C₂×C₁₂₇⋊C₃₂	2
#368×C₁₂₇	2
#1417	127