Group #22549838 — subgroups

Subgroups — #22549838

Order 8128.

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

normal subgroups only

This group has 190 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	C₂²×C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523231	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₁₂₇)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₁₂₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₂₇	—	2	4	no	—
9	C₂×C₄×D₁₂₇	C₄	1	1	yes	characteristic
10	C₂×C₄×D₁₂₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
13	#14523231	—	2	4	no	—
14	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
15	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	C₂²	1	2	yes	complemented
17	C₂²×D₁₂₇	—	2	2	no	—
18	C₂²×D₁₂₇	—	2	2	no	—
19	C₂²×D₁₂₇	—	2	2	no	—
20	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₂₇	—	2	2	no	—
23	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₂₇	—	2	2	no	—
26	C₄×D₁₂₇	—	2	2	no	—
27	C₄×D₁₂₇	—	2	2	no	—
28	C₄×D₁₂₇	D₄	1	4	yes	—
29	C₁₂₇⋊C₈	—	2	2	no	—
30	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	—	2	2	no	—
32	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₈×C₁₂₇	—	2	2	no	—
35	C₂²×C₁₂₇	—	2	1	no	—
36	C₂²×C₁₂₇	—	2	1	no	—
37	C₂²×C₁₂₇	M₁₆	1	2	yes	complemented
38	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₂₇	—	2	8	no	—
40	C₂×D₁₂₇	—	2	1	no	—
41	C₂×D₁₂₇	—	2	1	no	—
42	C₂×D₁₂₇	—	2	1	no	—
43	C₂×D₁₂₇	—	2	1	no	—
44	C₂×D₁₂₇	C₂²⋊C₄	1	2	yes	—
45	C₂×D₁₂₇	M₁₆	1	2	yes	complemented
46	C₄×C₁₂₇	—	2	1	no	—
47	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
48	Dic₁₂₇	—	2	1	no	—
49	Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₁₂₇	—	2	2	no	—
51	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
54	D₁₂₇	—	2	2	no	—
55	D₁₂₇	—	2	2	no	—
56	C₁₂₇	#407	1	1	yes	characteristic, complemented, Sylow
57	#407	—	127	1	no	maximal, Sylow
58	C₂³×C₄	—	127	1	no	—
59	C₂×M₁₆	—	127	2	no	—
60	C₂²⋊C₈	—	127	1	no	—
61	C₂²⋊C₈	—	127	2	no	—
62	C₂²⋊C₈	D₁₂₇	1	1	yes	characteristic, complemented
63	C₂⁴	—	127	1	no	—
64	C₂²×C₄	—	127	1	no	—
65	C₂²×C₄	—	127	1	no	—
66	C₂²×C₄	—	127	1	no	—
67	C₂²×C₄	—	127	1	no	—
68	C₂²×C₄	—	127	1	no	—
69	C₂²×C₄	—	254	4	no	—
70	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
71	M₁₆	—	254	4	no	—
72	C₂×C₈	—	127	2	no	—
73	C₂×C₈	C₂×D₁₂₇	1	2	yes	complemented
74	C₂³	—	127	1	no	—
75	C₂³	—	127	1	no	—
76	C₂³	—	254	2	no	—
77	C₂³	—	254	2	no	—
78	C₂³	—	254	2	no	—
79	C₂³	C₄×D₁₂₇	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	127	1	no	—
82	C₂×C₄	—	127	4	no	—
83	C₂×C₄	—	127	1	no	—
84	C₂×C₄	—	254	2	no	—
85	C₂×C₄	—	254	2	no	—
86	C₂×C₄	—	254	2	no	—
87	C₂×C₄	—	254	2	no	—
88	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	254	2	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	127	2	no	—
95	C₂²	—	127	2	no	—
96	C₂²	—	254	8	no	—
97	C₂²	—	254	1	no	—
98	C₂²	—	254	1	no	—
99	C₂²	—	254	1	no	—
100	C₂²	—	254	1	no	—
101	C₂²	#14523231	1	2	yes	complemented
102	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	127	2	no	—
105	C₄	—	254	1	no	—
106	C₄	D₄×D₁₂₇	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	254	2	no	—
109	C₂	—	254	2	no	—
110	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
111	C₂	M₁₆×D₁₂₇	1	1	yes	characteristic
112	C₂	C₂×#14523231	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523231

C₂

yes

maximal, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₂₇)⋊C₈

C₂

yes

maximal, complemented

C₂²⋊C₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₄

yes

characteristic

C₂²×Dic₁₂₇

C₄

yes

characteristic

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₄

yes

characteristic

C₂×C₄×D₁₂₇

C₄

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂²

yes

complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

M₁₆

yes

complemented

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²⋊C₄

yes

—

C₂×D₁₂₇

M₁₆

yes

complemented

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₂₇

C₂×M₁₆

yes

characteristic

C₂×C₁₂₇

C₂×M₁₆

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#407

yes

characteristic, complemented, Sylow

#407

—

127

maximal, Sylow

C₂³×C₄

—

127

—

C₂×M₁₆

—

127

—

C₂²⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

C₂²⋊C₈

D₁₂₇

yes

characteristic, complemented

C₂⁴

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

complemented

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₄×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

127

—

C₂²

—

127

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

100

C₂²

—

254

—

101

C₂²

#14523231

yes

complemented

102

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

103

C₄

—

104

C₄

—

127

—

105

C₄

—

254

—

106

C₄

D₄×D₁₂₇

yes

—

107

C₂

—

108

C₂

—

254

—

109

C₂

—

254

—

110

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

111

C₂

M₁₆×D₁₂₇

yes

characteristic

112

C₂

C₂×#14523231

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₂²⋊C₈×C₁₂₇	C₂
hypercenter	C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#407
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₄

Fitting

C₂²⋊C₈×C₁₂₇

C₂

hypercenter

C₂²⋊C₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#14523231	C₂	semidirect
3. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
4. (C₂×D₁₂₇)⋊C₈	C₂	semidirect
5. C₂²⋊C₈×C₁₂₇	C₂	semidirect
14. C₂×C₁₂₇⋊C₈	C₂²	semidirect
16. C₂×C₈×C₁₂₇	C₂²	semidirect
37. C₂²×C₁₂₇	M₁₆	semidirect
45. C₂×D₁₂₇	M₁₆	semidirect
56. C₁₂₇	#407	semidirect
62. C₂²⋊C₈	D₁₂₇	semidirect
73. C₂×C₈	C₂×D₁₂₇	semidirect
101. C₂²	#14523231	semidirect

H ≅ G/N

type

2. C₂×#14523231

C₂

semidirect

3. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

4. (C₂×D₁₂₇)⋊C₈

C₂

semidirect

5. C₂²⋊C₈×C₁₂₇

C₂

semidirect

14. C₂×C₁₂₇⋊C₈

C₂²

semidirect

16. C₂×C₈×C₁₂₇

C₂²

semidirect

37. C₂²×C₁₂₇

M₁₆

semidirect

45. C₂×D₁₂₇

M₁₆

semidirect

56. C₁₂₇

#407

semidirect

62. C₂²⋊C₈

D₁₂₇

semidirect

73. C₂×C₈

C₂×D₁₂₇

semidirect

101. C₂²

#14523231

semidirect

Subgroups — #22549838

Conjugacy classes of subgroups

Complements