Subgroups — #30636768

Order 16256.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550256	C₂	1	1	yes	maximal, characteristic
2	#22549697	C₂	1	1	yes	maximal, characteristic
3	#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₀₈	—	2	1	no	—
5	C₂×Dic₅₀₈	—	2	1	no	—
6	#16861209	C₄	1	2	yes	—
7	#16861149	—	2	1	no	—
8	M₃₂×C₁₂₇	—	2	1	no	—
9	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₅₄	—	2	2	no	—
11	Dic₅₀₈	—	4	1	no	—
12	Dic₅₀₈	—	4	1	no	—
13	Dic₅₀₈	—	4	1	no	—
14	Dic₅₀₈	—	4	1	no	—
15	C₄⋊Dic₁₂₇	—	4	2	no	—
16	C₁₂₇⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₂₇	—	4	1	no	—
21	Dic₂₅₄	—	4	2	no	—
22	Dic₂₅₄	—	4	2	no	—
23	C₂×Dic₁₂₇	—	4	2	no	—
24	C₂×C₄×C₁₂₇	—	2	1	no	—
25	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₂₇	D₈	1	1	yes	characteristic
27	C₈×C₁₂₇	D₈	1	1	yes	characteristic
28	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
29	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
30	C₂²×C₁₂₇	#100	1	1	yes	characteristic
31	C₄×C₁₂₇	—	2	2	no	—
32	C₄×C₁₂₇	#102	1	1	yes	characteristic
33	C₄×C₁₂₇	#102	1	1	yes	characteristic
34	Dic₁₂₇	—	8	2	no	—
35	C₂×C₁₂₇	—	2	1	no	—
36	C₂×C₁₂₇	#331	1	1	yes	characteristic
37	#1358	—	127	1	no	maximal, Sylow
38	C₁₂₇	#1358	1	1	yes	characteristic, complemented, Sylow
39	#494	—	127	1	no	—
40	#364	—	127	1	no	—
41	#364	D₁₂₇	1	1	yes	characteristic
42	C₂×Q₁₆	—	254	1	no	—
43	C₂×Q₁₆	—	254	1	no	—
44	C₄⋊Q₈	—	127	2	no	—
45	M₃₂	—	2	1	no	—
46	M₃₂	—	254	1	no	—
47	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
48	C₂×Q₈	—	254	2	no	—
49	Q₁₆	—	508	1	no	—
50	Q₁₆	—	508	1	no	—
51	Q₁₆	—	508	1	no	—
52	Q₁₆	—	508	1	no	—
53	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
54	C₂×C₈	D₅₀₈	1	1	yes	characteristic
55	C₄⋊C₄	—	508	2	no	—
56	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
57	C₁₆	—	4	1	no	—
58	C₁₆	—	508	1	no	—
59	Q₈	—	508	2	no	—
60	Q₈	—	508	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	508	2	no	—
63	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
64	C₈	#14523233	1	1	yes	characteristic
65	C₈	D₁₀₁₆	1	1	yes	characteristic
66	C₈	#14523235	1	1	yes	characteristic
67	C₈	#14523237	1	1	yes	characteristic
68	C₂²	#16861125	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	1016	2	no	—
71	C₄	#16861129	1	1	yes	characteristic
72	C₄	#16861131	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#22549619	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549619
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#364×C₁₂₇	C₂
socle	C₂×C₁₂₇	#331
prime radical	C₂²×C₁₂₇	#100
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#364	D₁₂₇
hypocenter	C₁₂₇	#1358
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
38. C₁₂₇	#1358	semidirect

Maximal subgroups

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

subgroup	index
#22550256	2
#22549697	2
#364×C₁₂₇	2
#1358	127