Subgroups — #30636775

Order 16256.

118 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	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₁₆	—	2	1	no	—
5	C₂×D₁₀₁₆	C₄	1	2	yes	—
6	#16861206	—	2	1	no	—
7	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
8	C₄⋊C₈×C₁₂₇	—	2	1	no	—
9	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×D₅₀₈	—	2	2	no	—
11	C₂×D₅₀₈	—	4	1	no	—
12	C₂×D₅₀₈	—	4	1	no	—
13	D₁₀₁₆	—	4	2	no	—
14	D₁₀₁₆	—	4	1	no	—
15	D₁₀₁₆	—	4	1	no	—
16	C₁₂₇⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₂₇	—	4	1	no	—
18	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
20	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂²×D₁₂₇	—	4	2	no	—
22	D₅₀₈	—	4	2	no	—
23	D₅₀₈	—	4	2	no	—
24	D₅₀₈	—	8	2	no	—
25	D₅₀₈	—	8	1	no	—
26	D₅₀₈	—	8	1	no	—
27	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₂₇	—	2	1	no	—
31	C₈×C₁₂₇	—	8	1	no	—
32	C₈×C₁₂₇	D₈	1	1	yes	characteristic
33	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
34	C₂²×C₁₂₇	#102	1	1	yes	characteristic
35	C₂×D₁₂₇	—	8	2	no	—
36	C₂×D₁₂₇	—	8	2	no	—
37	C₂×D₁₂₇	—	8	2	no	—
38	C₄×C₁₂₇	—	2	1	no	—
39	C₄×C₁₂₇	#100	1	1	yes	characteristic
40	C₄×C₁₂₇	#102	1	1	yes	characteristic
41	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
42	C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
43	C₂×C₁₂₇	#331	1	1	yes	characteristic
44	C₂×C₁₂₇	#357	1	1	yes	characteristic
45	C₂×C₁₂₇	#361	1	1	yes	characteristic
46	D₁₂₇	—	16	2	no	—
47	#1361	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1361	1	1	yes	characteristic, complemented, Sylow
49	#493	—	127	1	no	—
50	C₄⋊C₁₆	—	127	1	no	—
51	#335	D₁₂₇	1	1	yes	characteristic, complemented
52	C₂×D₈	—	127	2	no	—
53	C₂×D₈	—	254	1	no	—
54	#127	—	254	1	no	—
55	C₂×C₁₆	—	254	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
58	C₂×D₄	—	508	1	no	—
59	C₂×D₄	—	508	1	no	—
60	C₂×D₄	—	254	2	no	—
61	D₈	—	508	2	no	—
62	D₈	—	508	1	no	—
63	D₈	—	508	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
66	C₂×C₈	D₅₀₈	1	1	yes	characteristic
67	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₁₆	—	508	1	no	—
69	C₂³	—	508	2	no	—
70	D₄	—	1016	2	no	—
71	D₄	—	1016	1	no	—
72	D₄	—	1016	1	no	—
73	D₄	—	508	2	no	—
74	D₄	—	508	2	no	—
75	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	#14523233	1	1	yes	characteristic
77	C₂×C₄	#14523235	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	8	1	no	—
80	C₈	D₁₀₁₆	1	1	yes	characteristic
81	C₈	#14523237	1	1	yes	characteristic
82	C₂²	—	1016	2	no	—
83	C₂²	—	1016	2	no	—
84	C₂²	—	1016	2	no	—
85	C₂²	#16861129	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#16861125	1	1	yes	characteristic
88	C₄	#16861131	1	1	yes	characteristic
89	C₄	#16861150	1	1	yes	characteristic
90	C₄	#16861152	1	1	yes	characteristic
91	C₂	—	2032	2	no	—
92	C₂	#22549619	1	1	yes	characteristic
93	C₂	#22549676	1	1	yes	characteristic
94	C₂	#22549688	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861129
commutator	C₂×C₈×C₁₂₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#335×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#102
prime radical	#22550253	C₂
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#335	D₁₂₇
hypocenter	C₁₂₇	#1361
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₁₂₇⋊C₁₆)	C₂	semidirect
3. #335×C₁₂₇	C₂	semidirect
48. C₁₂₇	#1361	semidirect
51. #335	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
C₄⋊(C₁₂₇⋊C₁₆)	2
#335×C₁₂₇	2
#1361	127