Subgroups — #30636688

Order 16256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549652	C₂	1	1	yes	maximal, characteristic, complemented
3	#436×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
5	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
6	C₂×M₁₆×C₁₂₇	—	2	1	no	—
7	C₄×D₄×C₁₂₇	C₄	1	1	yes	characteristic
8	C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
10	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₁₂₇⋊C₁₆	—	2	2	no	—
12	C₁₂₇⋊C₁₆	—	4	1	no	—
13	C₂×D₄×C₁₂₇	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	—	2	1	no	—
15	M₁₆×C₁₂₇	—	4	1	no	—
16	M₁₆×C₁₂₇	—	4	1	no	—
17	C₂×C₈×C₁₂₇	—	2	1	no	—
18	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	C₈	1	1	yes	characteristic
21	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
22	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₁₂₇	—	2	1	no	—
24	D₄×C₁₂₇	—	4	1	no	—
25	C₂×C₄×C₁₂₇	—	2	1	no	—
26	C₂×C₄×C₁₂₇	—	4	1	no	—
27	C₂×C₄×C₁₂₇	—	4	1	no	—
28	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
31	C₈×C₁₂₇	—	2	1	no	—
32	C₈×C₁₂₇	—	4	1	no	—
33	C₈×C₁₂₇	D₈	1	1	yes	characteristic
34	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₂₇	—	4	1	no	—
36	C₂²×C₁₂₇	—	4	1	no	—
37	C₂²×C₁₂₇	—	4	1	no	—
38	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₂₇	—	2	1	no	—
40	C₄×C₁₂₇	—	2	1	no	—
41	C₄×C₁₂₇	—	4	1	no	—
42	C₄×C₁₂₇	#102	1	1	yes	characteristic
43	C₄×C₁₂₇	#104	1	1	yes	characteristic
44	C₂×C₁₂₇	—	8	1	no	—
45	C₂×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₁₂₇	#349	1	1	yes	characteristic
47	C₂×C₁₂₇	#350	1	1	yes	characteristic
48	#1330	—	127	1	no	maximal, Sylow
49	C₁₂₇	#1330	1	1	yes	characteristic, complemented, Sylow
50	#436	D₁₂₇	1	1	yes	characteristic
51	C₄⋊C₁₆	—	127	1	no	—
52	#346	—	127	1	no	—
53	C₂×M₁₆	—	2	1	no	—
54	C₄×D₄	Dic₁₂₇	1	1	yes	characteristic
55	C₂×C₁₆	—	254	1	no	—
56	C₂×C₁₆	—	254	1	no	—
57	C₄⋊C₈	Dic₁₂₇	1	1	yes	characteristic
58	C₂²⋊C₈	—	2	1	no	—
59	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂×D₄	C₁₂₇⋊C₈	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	M₁₆	—	4	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
67	C₄⋊C₄	C₁₂₇⋊C₈	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
70	C₁₆	—	508	1	no	—
71	C₁₆	—	254	2	no	—
72	C₂³	—	2	1	no	—
73	D₄	—	4	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
79	C₂×C₄	#14523232	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	4	1	no	—
82	C₈	#14523233	1	1	yes	characteristic
83	C₈	#14523236	1	1	yes	characteristic
84	C₂²	—	4	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	#16861130	1	1	yes	characteristic
92	C₄	#16861136	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	D₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
95	C₂	#22549658	1	1	yes	characteristic
96	C₂	#22549660	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#436×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₈
prime radical	C₂×D₄×C₁₂₇	C₈
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#436	D₁₂₇
hypocenter	C₁₂₇	#1330
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₁₂₇⋊C₁₆)	C₂	semidirect
2. #22549652	C₂	semidirect
49. C₁₂₇	#1330	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₂₇⋊C₁₆)	2
#22549652	2
#436×C₁₂₇	2
#1330	127