Subgroups — #30636770

Order 16256.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550249	C₂	1	1	yes	maximal, characteristic
2	#22549697	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523237	—	2	1	no	—
5	C₂×#14523237	—	2	1	no	—
6	#16861209	C₄	1	1	yes	characteristic
7	#16861206	C₄	1	1	yes	characteristic
8	#16861149	—	2	1	no	—
9	M₃₂×C₁₂₇	—	2	1	no	—
10	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₅₄	—	2	1	no	—
12	C₂×D₅₀₈	—	2	1	no	—
13	C₂×D₅₀₈	—	4	1	no	—
14	#14523237	—	4	1	no	—
15	#14523237	—	4	1	no	—
16	#14523237	—	4	1	no	—
17	#14523237	—	4	1	no	—
18	C₄⋊Dic₁₂₇	—	4	1	no	—
19	C₁₂₇⋊C₁₆	—	4	1	no	—
20	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
22	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₁₂₇	—	4	1	no	—
24	C₂²×D₁₂₇	—	4	1	no	—
25	Dic₂₅₄	—	4	1	no	—
26	Dic₂₅₄	—	4	1	no	—
27	D₅₀₈	—	4	1	no	—
28	D₅₀₈	—	4	1	no	—
29	D₅₀₈	—	8	2	no	—
30	C₂×Dic₁₂₇	—	4	1	no	—
31	C₂×C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₂₇	D₈	1	1	yes	characteristic
34	C₈×C₁₂₇	D₈	1	1	yes	characteristic
35	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
36	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
37	C₂²×C₁₂₇	#100	1	1	yes	characteristic
38	C₂×D₁₂₇	—	8	2	no	—
39	C₂×D₁₂₇	—	8	1	no	—
40	C₄×C₁₂₇	—	2	2	no	—
41	C₄×C₁₂₇	#102	1	1	yes	characteristic
42	C₄×C₁₂₇	#102	1	1	yes	characteristic
43	Dic₁₂₇	—	8	1	no	—
44	C₂×C₁₂₇	—	2	1	no	—
45	C₂×C₁₂₇	#331	1	1	yes	characteristic
46	D₁₂₇	—	16	1	no	—
47	#1359	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1359	1	1	yes	characteristic, complemented, Sylow
49	#492	—	127	1	no	—
50	#364	—	127	1	no	—
51	#364	D₁₂₇	1	1	yes	characteristic, complemented
52	C₂×SD₁₆	—	254	1	no	—
53	C₂×SD₁₆	—	254	1	no	—
54	C₄⋊Q₈	—	127	1	no	—
55	#127	—	127	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	254	1	no	—
58	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
59	C₂×Q₈	—	254	1	no	—
60	C₂×D₄	—	508	1	no	—
61	C₂×D₄	—	254	1	no	—
62	SD₁₆	—	508	1	no	—
63	SD₁₆	—	508	1	no	—
64	SD₁₆	—	508	1	no	—
65	SD₁₆	—	508	1	no	—
66	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
67	C₂×C₈	D₅₀₈	1	1	yes	characteristic
68	C₄⋊C₄	—	508	1	no	—
69	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	508	1	no	—
72	C₂³	—	508	1	no	—
73	Q₈	—	508	1	no	—
74	Q₈	—	508	1	no	—
75	D₄	—	1016	2	no	—
76	D₄	—	508	1	no	—
77	D₄	—	508	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	508	1	no	—
80	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
81	C₈	#14523233	1	1	yes	characteristic
82	C₈	D₁₀₁₆	1	1	yes	characteristic
83	C₈	#14523235	1	1	yes	characteristic
84	C₈	#14523237	1	1	yes	characteristic
85	C₂²	—	1016	2	no	—
86	C₂²	—	1016	1	no	—
87	C₂²	#16861125	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	1016	1	no	—
90	C₄	#16861129	1	1	yes	characteristic
91	C₄	#16861131	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2032	1	no	—
94	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	#16861206	C₄
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#364	D₁₂₇
hypocenter	C₁₂₇	#1359
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
2. #22549697	C₂	semidirect
3. #364×C₁₂₇	C₂	semidirect
48. C₁₂₇	#1359	semidirect
51. #364	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550249	2
#22549697	2
#364×C₁₂₇	2
#1359	127