Subgroups — #30638337

Order 16256.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550253	C₂	1	1	yes	maximal, characteristic
2	#22550224	C₂	1	4	yes	maximal, complemented
3	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₁₆	—	2	2	no	—
6	#16861206	C₂²	1	2	yes	—
7	#16861129	—	2	8	no	—
8	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
10	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×D₅₀₈	—	2	2	no	—
13	C₂×D₅₀₈	—	2	4	no	—
14	D₁₀₁₆	—	4	4	no	—
15	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
16	C₂×Q₈×C₁₂₇	—	2	2	no	—
17	Q₁₆×C₁₂₇	—	4	4	no	—
18	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₂₇	—	2	4	no	—
20	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₂²×D₁₂₇	—	4	2	no	—
22	D₅₀₈	—	4	4	no	—
23	D₅₀₈	—	4	8	no	—
24	C₁₂₇⋊C₈	—	2	4	no	—
25	Q₈×C₁₂₇	—	4	4	no	—
26	C₂×C₄×C₁₂₇	—	4	2	no	—
27	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₂₇	D₈	1	4	yes	—
30	C₂²×C₁₂₇	#127	1	1	yes	characteristic
31	C₂×D₁₂₇	—	8	2	no	—
32	C₂×D₁₂₇	—	8	4	no	—
33	C₄×C₁₂₇	—	8	2	no	—
34	C₄×C₁₂₇	C₂×D₈	1	2	yes	—
35	C₄×C₁₂₇	#135	1	4	yes	—
36	C₂×C₁₂₇	#493	1	1	yes	characteristic
37	C₂×C₁₂₇	#495	1	2	yes	—
38	D₁₂₇	—	16	2	no	—
39	#1709	—	127	1	no	maximal, Sylow
40	C₁₂₇	#1709	1	1	yes	characteristic, complemented, Sylow
41	#494	D₁₂₇	1	1	yes	characteristic, complemented
42	#493	—	127	1	no	—
43	#486	—	127	4	no	—
44	C₈²	—	127	1	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	254	2	no	—
47	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
48	#127	—	127	2	no	—
49	#102	—	254	8	no	—
50	C₄×C₈	—	127	2	no	—
51	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	254	4	no	—
54	C₂×D₄	—	254	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	508	4	no	—
57	C₂×C₈	—	127	4	no	—
58	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
61	C₂³	—	508	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	508	4	no	—
64	D₄	—	508	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₈	—	254	4	no	—
69	C₈	#14523233	1	4	yes	—
70	C₂²	—	1016	2	no	—
71	C₂²	—	1016	4	no	—
72	C₂²	#16861205	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#14523233	1	2	yes	—
75	C₄	#16861232	1	4	yes	—
76	C₂	—	2032	2	no	—
77	C₂	#22550252	1	1	yes	characteristic
78	C₂	#22550261	1	2	yes	—

	subgroup	G/H
center	C₂²	#16861205
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#494×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
prime radical	#22550253	C₂
hypercenter	#494	D₁₂₇
hypocenter	C₁₂₇	#1709
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #22550224	C₂	semidirect
3. C₈×C₁₂₇⋊C₈	C₂	semidirect
4. #494×C₁₂₇	C₂	semidirect
40. C₁₂₇	#1709	semidirect
41. #494	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550253	2
#22550224	2
C₈×C₁₂₇⋊C₈	2
#494×C₁₂₇	2
#1709	127