Group #22550475 — subgroups

Subgroups — #22550475

Order 8128.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#16861204	C₂	1	2	yes	maximal, complemented
2	#16861200	C₂	1	2	yes	maximal, complemented
3	#16861194	C₂	1	2	yes	maximal, complemented
4	#16861193	C₂	1	2	yes	maximal, complemented
5	#16861185	C₂	1	2	yes	maximal, complemented
6	#16861183	C₂	1	1	yes	maximal, characteristic
7	(C₂²⋊D₁₂₇)⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₂×Dic₁₂₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
12	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
16	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
18	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
19	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
20	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₁₂₇	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
29	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
31	C₂²×D₁₂₇	C₂³	1	2	yes	—
32	D₅₀₈	—	4	1	no	—
33	C₂²⋊D₁₂₇	—	2	4	no	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	—	4	1	no	—
36	C₂×Dic₁₂₇	C₂³	1	2	yes	—
37	C₂×Dic₁₂₇	C₂³	1	2	yes	—
38	C₂×Dic₁₂₇	C₂³	1	2	yes	—
39	C₄×D₁₂₇	—	4	2	no	—
40	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
41	Q₈×C₁₂₇	—	4	1	no	—
42	C₂×C₄×C₁₂₇	—	4	1	no	—
43	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₁₂₇	—	2	1	no	—
48	C₂²×C₁₂₇	—	2	1	no	—
49	C₂²×C₁₂₇	#41	1	2	yes	—
50	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₂₇	—	2	2	no	—
52	C₂×D₁₂₇	—	2	2	no	—
53	C₂×D₁₂₇	—	2	2	no	—
54	C₄×C₁₂₇	—	2	1	no	—
55	C₄×C₁₂₇	—	2	1	no	—
56	C₄×C₁₂₇	—	2	2	no	—
57	C₄×C₁₂₇	—	2	2	no	—
58	Dic₁₂₇	—	2	2	no	—
59	Dic₁₂₇	—	2	2	no	—
60	Dic₁₂₇	—	2	2	no	—
61	C₂×C₁₂₇	—	2	2	no	—
62	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
63	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
65	D₁₂₇	—	4	2	no	—
66	C₁₂₇	#539	1	1	yes	characteristic, complemented, Sylow
67	#539	—	127	1	no	maximal, Sylow
68	#126	—	127	2	no	—
69	#125	—	127	2	no	—
70	#123	—	127	2	no	—
71	#123	—	127	2	no	—
72	C₂²⋊Q₈	—	127	2	no	—
73	C₂²⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
74	#121	—	127	1	no	—
75	C₄×D₄	—	127	2	no	—
76	C₂×C₄⋊C₄	—	127	1	no	—
77	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
78	C₂×D₄	—	127	2	no	—
79	C₂×D₄	—	127	1	no	—
80	C₂²×C₄	—	127	1	no	—
81	C₂²×C₄	—	127	1	no	—
82	C₂²×C₄	—	127	2	no	—
83	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
84	C₄⋊C₄	—	127	2	no	—
85	C₄⋊C₄	—	127	2	no	—
86	C₄⋊C₄	—	127	2	no	—
87	C₄⋊C₄	—	127	1	no	—
88	C₄⋊C₄	—	127	2	no	—
89	C₄⋊C₄	—	127	2	no	—
90	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
91	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
92	C₂²⋊C₄	—	127	2	no	—
93	C₂²⋊C₄	—	127	2	no	—
94	C₂²⋊C₄	—	127	2	no	—
95	C₂²⋊C₄	—	127	2	no	—
96	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
97	C₄²	—	127	2	no	—
98	C₂³	—	127	2	no	—
99	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
100	Q₈	—	4	1	no	—
101	D₄	—	508	1	no	—
102	D₄	—	254	4	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	127	2	no	—
105	C₂×C₄	—	127	2	no	—
106	C₂×C₄	—	127	2	no	—
107	C₂×C₄	—	508	2	no	—
108	C₂×C₄	—	508	1	no	—
109	C₂×C₄	—	254	2	no	—
110	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
111	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
113	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	254	2	no	—
117	C₂²	—	254	2	no	—
118	C₂²	—	254	2	no	—
119	C₂²	#14523249	1	2	yes	—
120	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	254	2	no	—
126	C₄	—	254	2	no	—
127	C₄	—	254	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	508	2	no	—
130	C₂	C₂×#14523249	1	1	yes	characteristic
131	C₂	#16861254	1	1	yes	characteristic
132	C₂	#16861256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861204

C₂

yes

maximal, complemented

#16861200

C₂

yes

maximal, complemented

#16861194

C₂

yes

maximal, complemented

#16861193

C₂

yes

maximal, complemented

#16861185

C₂

yes

maximal, complemented

#16861183

C₂

yes

maximal, characteristic

(C₂²⋊D₁₂₇)⋊C₄

C₂

yes

maximal, complemented

C₂×Dic₁₂₇⋊C₄

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₂²×D₁₂₇

C₂³

yes

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

C₁₂₇

#539

yes

characteristic, complemented, Sylow

#539

—

127

maximal, Sylow

#126

—

127

—

#125

—

127

—

#123

—

127

—

#123

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

D₁₂₇

yes

characteristic, complemented

#121

—

127

—

C₄×D₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₄²

—

127

—

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

characteristic

100

Q₈

—

101

D₄

—

508

—

102

D₄

—

254

—

103

C₂×C₄

—

104

C₂×C₄

—

127

—

105

C₂×C₄

—

127

—

106

C₂×C₄

—

127

—

107

C₂×C₄

—

508

—

108

C₂×C₄

—

508

—

109

C₂×C₄

—

254

—

110

C₂×C₄

C₂²×D₁₂₇

yes

—

111

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

112

C₂×C₄

C₂²×D₁₂₇

yes

—

113

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

254

—

117

C₂²

—

254

—

118

C₂²

—

254

—

119

C₂²

#14523249

yes

—

120

C₂²

C₂³×D₁₂₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

254

—

126

C₄

—

254

—

127

C₄

—

254

—

128

C₂

—

129

C₂

—

508

—

130

C₂

C₂×#14523249

yes

characteristic

131

C₂

#16861254

yes

characteristic

132

C₂

#16861256

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₂²⋊Q₈×C₁₂₇	C₂
hypercenter	C₂²⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#539
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₂²⋊Q₈×C₁₂₇

C₂

hypercenter

C₂²⋊Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #16861204	C₂	semidirect
2. #16861200	C₂	semidirect
3. #16861194	C₂	semidirect
4. #16861193	C₂	semidirect
5. #16861185	C₂	semidirect
7. (C₂²⋊D₁₂₇)⋊C₄	C₂	semidirect
8. C₂×Dic₁₂₇⋊C₄	C₂	semidirect
9. C₂²⋊Q₈×C₁₂₇	C₂	semidirect
66. C₁₂₇	#539	semidirect
73. C₂²⋊Q₈	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861204

C₂

semidirect

2. #16861200

C₂

semidirect

3. #16861194

C₂

semidirect

4. #16861193

C₂

semidirect

5. #16861185

C₂

semidirect

7. (C₂²⋊D₁₂₇)⋊C₄

C₂

semidirect

8. C₂×Dic₁₂₇⋊C₄

C₂

semidirect

9. C₂²⋊Q₈×C₁₂₇

C₂

semidirect

66. C₁₂₇

#539

semidirect

73. C₂²⋊Q₈

D₁₂₇

semidirect

Subgroups — #22550475

Conjugacy classes of subgroups

Complements