Group #22549948 — subgroups

Subgroups — #22549948

Order 8128.

124 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861120	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#14523232	—	2	2	no	—
9	C₂×C₁₂₇⋊C₈	—	2	2	no	—
10	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₁₂₇	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
20	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₁₂₇⋊C₈	—	2	1	no	—
22	C₁₂₇⋊C₈	—	2	1	no	—
23	C₁₂₇⋊C₈	—	2	1	no	—
24	C₁₂₇⋊C₈	D₄	1	2	yes	complemented
25	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	D₄×C₁₂₇	—	2	2	no	—
28	C₂×C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄×C₁₂₇	—	2	1	no	—
30	C₂×C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×C₁₂₇	—	2	1	no	—
38	C₂²×C₁₂₇	—	2	1	no	—
39	C₂²×C₁₂₇	—	2	1	no	—
40	C₂²×C₁₂₇	—	2	1	no	—
41	C₂²×C₁₂₇	—	2	1	no	—
42	C₂²×C₁₂₇	M₁₆	1	2	yes	complemented
43	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₁₂₇	—	2	1	no	—
45	C₄×C₁₂₇	—	2	1	no	—
46	C₄×C₁₂₇	—	2	1	no	—
47	C₄×C₁₂₇	—	2	1	no	—
48	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₄×C₁₂₇	#41	1	1	yes	characteristic
50	C₂×C₁₂₇	—	2	2	no	—
51	C₂×C₁₂₇	—	4	1	no	—
52	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
53	C₂×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₁₂₇	#131	1	1	yes	characteristic
55	C₁₂₇	#435	1	1	yes	characteristic, complemented, Sylow
56	#435	—	127	1	no	maximal, Sylow
57	C₂×M₁₆	—	127	1	no	—
58	C₂²×C₈	—	127	1	no	—
59	C₄×D₄	D₁₂₇	1	1	yes	characteristic
60	C₄⋊C₈	—	127	1	no	—
61	C₂²⋊C₈	—	127	1	no	—
62	C₂²⋊C₈	—	127	1	no	—
63	#97	—	127	1	no	—
64	C₂×D₄	Dic₁₂₇	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
66	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
67	M₁₆	—	254	2	no	—
68	C₂×C₈	—	127	1	no	—
69	C₂×C₈	—	127	1	no	—
70	C₂×C₈	—	127	1	no	—
71	C₂×C₈	—	127	1	no	—
72	C₂×C₈	—	254	2	no	—
73	C₄⋊C₄	Dic₁₂₇	1	1	yes	characteristic
74	C₂²⋊C₄	Dic₁₂₇	1	1	yes	characteristic
75	C₂²⋊C₄	Dic₁₂₇	1	1	yes	characteristic
76	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
77	C₂³	C₂×Dic₁₂₇	1	1	yes	characteristic
78	C₂³	C₂×Dic₁₂₇	1	1	yes	characteristic
79	D₄	—	2	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
86	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
87	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
89	C₈	—	127	2	no	—
90	C₈	—	254	1	no	—
91	C₈	—	254	1	no	—
92	C₈	—	254	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#14523232	1	2	yes	complemented
99	C₂²	C₂²×Dic₁₂₇	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	D₄×D₁₂₇	1	1	yes	characteristic
105	C₄	#14523247	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	4	1	no	—
108	C₂	D₄×Dic₁₂₇	1	1	yes	characteristic
109	C₂	C₂×#14523232	1	1	yes	characteristic
110	C₂	#16861216	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523232

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

#16861120

C₂

yes

maximal, characteristic, complemented

C₄×D₄×C₁₂₇

C₂

yes

maximal, characteristic

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₄

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₄

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₄

yes

characteristic

C₄²×C₁₂₇

C₂²

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

D₄

yes

complemented

C₂³×C₁₂₇

C₂×C₄

yes

characteristic

C₂³×C₁₂₇

C₂×C₄

yes

characteristic

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

M₁₆

yes

complemented

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₄×D₄

yes

characteristic

C₂×C₁₂₇

C₂×M₁₆

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#435

—

127

maximal, Sylow

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

C₄×D₄

D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

#97

—

127

—

C₂×D₄

Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₄⋊C₄

Dic₁₂₇

yes

characteristic

C₂²⋊C₄

Dic₁₂₇

yes

characteristic

C₂²⋊C₄

Dic₁₂₇

yes

characteristic

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

C₂×Dic₁₂₇

yes

characteristic

C₂³

C₂×Dic₁₂₇

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

—

254

—

C₈

—

254

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14523232

yes

complemented

C₂²

C₂²×Dic₁₂₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

D₄×D₁₂₇

yes

characteristic

105

C₄

#14523247

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

D₄×Dic₁₂₇

yes

characteristic

109

C₂

C₂×#14523232

yes

characteristic

110

C₂

#16861216

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₄×D₄×C₁₂₇	C₂
hypercenter	C₄×D₄	D₁₂₇
hypocenter	C₁₂₇	#435
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₄

Fitting

C₄×D₄×C₁₂₇

C₂

hypercenter

C₄×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#14523232	C₂	semidirect
2. C₂²×C₁₂₇⋊C₈	C₂	semidirect
3. C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
4. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
5. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
6. #16861120	C₂	semidirect
24. C₁₂₇⋊C₈	D₄	semidirect
42. C₂²×C₁₂₇	M₁₆	semidirect
55. C₁₂₇	#435	semidirect
98. C₂²	#14523232	semidirect

H ≅ G/N

type

1. C₂×#14523232

C₂

semidirect

2. C₂²×C₁₂₇⋊C₈

C₂

semidirect

3. C₄⋊(C₁₂₇⋊C₈)

C₂

semidirect

4. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

5. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

6. #16861120

C₂

semidirect

24. C₁₂₇⋊C₈

D₄

semidirect

42. C₂²×C₁₂₇

M₁₆

semidirect

55. C₁₂₇

#435

semidirect

98. C₂²

#14523232

semidirect

Subgroups — #22549948

Conjugacy classes of subgroups

Complements