Subgroups — #30636693

Order 16256.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₀₈	C₄	1	1	yes	characteristic
5	C₄×Dic₂₅₄	—	2	1	no	—
6	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
7	#16861143	C₄	1	1	yes	characteristic
8	#16861134	—	2	1	no	—
9	C₄⋊C₈×C₁₂₇	—	2	1	no	—
10	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₅₄	—	2	1	no	—
12	Dic₅₀₈	—	2	1	no	—
13	Dic₅₀₈	C₈	1	2	yes	complemented
14	C₄⋊Dic₁₂₇	—	2	1	no	—
15	Dic₁₂₇⋊C₄	—	4	1	no	—
16	C₄×Dic₁₂₇	—	4	1	no	—
17	C₁₂₇⋊C₁₆	—	2	2	no	—
18	C₂×C₈×C₁₂₇	—	4	1	no	—
19	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
22	Dic₂₅₄	—	2	2	no	—
23	Dic₂₅₄	—	4	1	no	—
24	C₂×Dic₁₂₇	—	4	1	no	—
25	C₂×Dic₁₂₇	—	4	1	no	—
26	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
29	C₈×C₁₂₇	—	2	1	no	—
30	C₈×C₁₂₇	—	8	1	no	—
31	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₂₇	#102	1	1	yes	characteristic
34	C₄×C₁₂₇	—	2	1	no	—
35	C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₂₇	#104	1	1	yes	characteristic
37	C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
38	C₄×C₁₂₇	Q₃₂	1	1	yes	characteristic
39	Dic₁₂₇	—	4	2	no	—
40	Dic₁₂₇	—	8	1	no	—
41	C₂×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₂₇	#358	1	1	yes	characteristic
43	C₂×C₁₂₇	#359	1	1	yes	characteristic
44	#1332	—	127	1	no	maximal, Sylow
45	C₁₂₇	#1332	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	127	1	no	—
47	C₄×C₁₆	—	127	1	no	—
48	#335	D₁₂₇	1	1	yes	characteristic
49	C₂×Q₁₆	—	127	1	no	—
50	C₄×Q₈	—	254	1	no	—
51	C₂×C₁₆	—	254	1	no	—
52	#107	—	127	1	no	—
53	C₄⋊C₈	—	2	1	no	—
54	#103	—	254	1	no	—
55	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂×Q₈	—	254	1	no	—
57	Q₁₆	—	127	2	no	—
58	Q₁₆	—	254	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
61	C₂×C₈	D₅₀₈	1	1	yes	characteristic
62	C₄⋊C₄	—	508	1	no	—
63	C₄⋊C₄	—	254	1	no	—
64	C₄²	—	508	1	no	—
65	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
66	C₁₆	—	254	2	no	—
67	Q₈	—	508	1	no	—
68	Q₈	—	254	2	no	—
69	C₂×C₄	—	508	1	no	—
70	C₂×C₄	—	508	1	no	—
71	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
72	C₂×C₄	#14523233	1	1	yes	characteristic
73	C₂×C₄	#14523235	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	8	1	no	—
76	C₈	C₈×D₁₂₇	1	1	yes	characteristic
77	C₈	#14523231	1	1	yes	characteristic
78	C₂²	#16861129	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	1016	1	no	—
81	C₄	—	508	2	no	—
82	C₄	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
83	C₄	#16861135	1	1	yes	characteristic
84	C₄	#16861153	1	1	yes	characteristic
85	C₄	#16861155	1	1	yes	characteristic
86	C₂	#22549603	1	1	yes	characteristic
87	C₂	#22549679	1	1	yes	characteristic
88	C₂	#22549682	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523233
commutator	C₈×C₁₂₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#335×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#102
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#335	D₁₂₇
hypocenter	C₁₂₇	#1332
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
13. Dic₅₀₈	C₈	semidirect
45. C₁₂₇	#1332	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₅₀₈	2
C₄×C₁₂₇⋊C₁₆	2
#335×C₁₂₇	2
#1332	127