Subgroups — #30636713

Order 16256.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550006	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
5	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₁₂₇⋊C₈)	C₄	1	1	yes	characteristic
7	Dic₁₂₇⋊C₈	—	2	1	no	—
8	#16861119	—	2	1	no	—
9	C₂×C₁₆×C₁₂₇	—	2	1	no	—
10	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₅₄	C₈	1	1	yes	characteristic
12	C₂×C₁₂₇⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₂₇	C₈	1	1	yes	characteristic
14	Dic₁₂₇⋊C₄	—	2	1	no	—
15	C₄×Dic₁₂₇	—	2	1	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	—	2	2	no	—
21	Dic₂₅₄	—	4	1	no	—
22	C₂×Dic₁₂₇	—	2	1	no	—
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₁₂₇⋊C₈	—	4	1	no	—
25	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
28	C₈×C₁₂₇	—	2	1	no	—
29	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
31	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₁₂₇	—	2	1	no	—
33	C₄×C₁₂₇	—	2	1	no	—
34	C₄×C₁₂₇	#103	1	1	yes	characteristic
35	C₄×C₁₂₇	#104	1	1	yes	characteristic
36	Dic₁₂₇	—	4	1	no	—
37	Dic₁₂₇	—	4	1	no	—
38	C₂×C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic
39	C₂×C₁₂₇	#349	1	1	yes	characteristic
40	C₂×C₁₂₇	#350	1	1	yes	characteristic
41	#1338	—	127	1	no	maximal, Sylow
42	C₁₂₇	#1338	1	1	yes	characteristic, complemented, Sylow
43	#446	—	127	1	no	—
44	C₄⋊C₁₆	—	127	1	no	—
45	#346	D₁₂₇	1	1	yes	characteristic
46	C₄×Q₈	—	127	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	254	1	no	—
49	C₄⋊C₈	—	127	1	no	—
50	C₄⋊C₈	—	254	1	no	—
51	#97	—	254	1	no	—
52	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂×Q₈	—	127	1	no	—
54	C₂×C₈	—	254	1	no	—
55	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
56	C₂×C₈	D₅₀₈	1	1	yes	characteristic
57	C₄⋊C₄	—	127	1	no	—
58	C₄⋊C₄	—	254	1	no	—
59	C₄²	—	254	1	no	—
60	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₁₆	—	508	1	no	—
63	Q₈	—	508	1	no	—
64	C₂×C₄	—	254	1	no	—
65	C₂×C₄	—	254	1	no	—
66	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
67	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
68	C₂×C₄	#14523231	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	508	1	no	—
71	C₈	#14523237	1	1	yes	characteristic
72	C₈	Dic₅₀₈	1	1	yes	characteristic
73	C₂²	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	508	1	no	—
77	C₄	—	508	1	no	—
78	C₄	#16861134	1	1	yes	characteristic
79	C₄	#16861137	1	1	yes	characteristic
80	C₂	#22549608	1	1	yes	characteristic
81	C₂	#22549657	1	1	yes	characteristic
82	C₂	#22549661	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₁₂₇)⋊C₄
commutator	C₂×C₄×C₁₂₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#346×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂²⋊C₈
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#346	D₁₂₇
hypocenter	C₁₂₇	#1338
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
42. C₁₂₇	#1338	semidirect

Maximal subgroups

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

subgroup	index
#22550006	2
C₄⋊(C₁₂₇⋊C₁₆)	2
#346×C₁₂₇	2
#1338	127