Group #22550565 — subgroups

Subgroups — #22550565

Order 8128.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16861202	C₂	1	2	yes	maximal, complemented
2	#16861201	C₂	1	2	yes	maximal, complemented
3	#16861200	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861191	C₂	1	2	yes	maximal
5	#16861189	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861188	C₂	1	1	yes	maximal, characteristic, complemented
7	#16861174	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊D₁₂₇	C₂	1	1	yes	maximal, characteristic
9	#16861166	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
16	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
18	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
19	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
20	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
21	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
22	Dic₁₂₇⋊C₄	C₂²	1	1	yes	characteristic
23	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
24	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
25	C₂²⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
26	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₁₂₇)⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₂₇	C₂²	1	2	yes	—
30	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
34	C₄²×C₁₂₇	C₂²	1	2	yes	—
35	C₂²×D₁₂₇	C₂³	1	1	yes	characteristic
36	Dic₂₅₄	—	2	2	no	—
37	C₂²⋊D₁₂₇	—	2	2	no	—
38	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₂₇	C₂³	1	2	yes	—
40	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₂₇	C₂³	1	2	yes	—
42	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
43	C₄×D₁₂₇	—	2	4	no	—
44	C₄×D₁₂₇	—	2	2	no	—
45	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	—	2	2	no	—
47	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₂²×C₁₂₇	—	2	1	no	—
52	C₂²×C₁₂₇	—	2	1	no	—
53	C₂²×C₁₂₇	—	2	1	no	—
54	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
55	C₂×D₁₂₇	—	2	1	no	—
56	C₂×D₁₂₇	—	2	1	no	—
57	C₂×D₁₂₇	#41	1	2	yes	—
58	C₄×C₁₂₇	—	2	1	no	—
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	—	2	2	no	—
61	C₄×C₁₂₇	#41	1	2	yes	—
62	Dic₁₂₇	—	2	2	no	—
63	Dic₁₂₇	—	2	1	no	—
64	Dic₁₂₇	—	2	1	no	—
65	Dic₁₂₇	—	2	2	no	—
66	Dic₁₂₇	—	2	1	no	—
67	C₂×C₁₂₇	—	4	1	no	—
68	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
71	D₁₂₇	—	2	2	no	—
72	C₁₂₇	#552	1	1	yes	characteristic, complemented, Sylow
73	#552	—	127	1	no	maximal, Sylow
74	#126	—	127	2	no	—
75	#125	—	127	1	no	—
76	#125	—	127	2	no	—
77	#123	—	127	2	no	—
78	C₂²⋊Q₈	—	127	1	no	—
79	C₂²⋊Q₈	—	127	1	no	—
80	C₄×Q₈	—	127	1	no	—
81	C₄×D₄	—	127	1	no	—
82	#117	—	127	2	no	—
83	#117	D₁₂₇	1	1	yes	characteristic, complemented
84	C₂×C₄⋊C₄	—	127	1	no	—
85	C₂×Q₈	—	127	1	no	—
86	C₂×D₄	—	127	1	no	—
87	C₂²×C₄	—	127	2	no	—
88	C₂²×C₄	—	127	1	no	—
89	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
90	C₄⋊C₄	—	127	2	no	—
91	C₄⋊C₄	—	127	2	no	—
92	C₄⋊C₄	—	127	2	no	—
93	C₄⋊C₄	—	127	1	no	—
94	C₄⋊C₄	—	127	2	no	—
95	C₄⋊C₄	—	127	1	no	—
96	C₄⋊C₄	—	127	2	no	—
97	C₄⋊C₄	—	127	2	no	—
98	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
100	C₂²⋊C₄	—	127	1	no	—
101	C₂²⋊C₄	—	127	2	no	—
102	C₂²⋊C₄	—	127	1	no	—
103	C₂²⋊C₄	—	127	2	no	—
104	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
105	C₄²	—	127	2	no	—
106	C₄²	—	127	1	no	—
107	C₄²	C₂×D₁₂₇	1	2	yes	—
108	C₂³	—	127	1	no	—
109	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
110	Q₈	—	254	2	no	—
111	D₄	—	254	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	127	1	no	—
114	C₂×C₄	—	127	2	no	—
115	C₂×C₄	—	127	1	no	—
116	C₂×C₄	—	127	2	no	—
117	C₂×C₄	—	127	1	no	—
118	C₂×C₄	—	254	4	no	—
119	C₂×C₄	—	254	2	no	—
120	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
121	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
123	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	127	2	no	—
128	C₂²	—	254	1	no	—
129	C₂²	—	254	1	no	—
130	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	254	2	no	—
135	C₄	—	254	1	no	—
136	C₄	—	254	1	no	—
137	C₄	—	254	2	no	—
138	C₄	—	254	1	no	—
139	C₄	#14523248	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	254	2	no	—
142	C₂	#41×D₁₂₇	1	1	yes	characteristic
143	C₂	C₂×#14523248	1	1	yes	characteristic
144	C₂	#16861256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861202

C₂

yes

maximal, complemented

#16861201

C₂

yes

maximal, complemented

#16861200

C₂

yes

maximal, characteristic, complemented

#16861191

C₂

yes

maximal

#16861189

C₂

yes

maximal, characteristic, complemented

#16861188

C₂

yes

maximal, characteristic, complemented

#16861174

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₂₇

C₂

yes

maximal, characteristic

#16861166

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#117×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

—

C₂²×D₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

#41

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#41

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

C₁₂₇

#552

yes

characteristic, complemented, Sylow

#552

—

127

maximal, Sylow

#126

—

127

—

#125

—

127

—

#125

—

127

—

#123

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

#117

—

127

—

#117

D₁₂₇

yes

characteristic, complemented

C₂×C₄⋊C₄

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

—

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

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

100

C₂²⋊C₄

—

127

—

101

C₂²⋊C₄

—

127

—

102

C₂²⋊C₄

—

127

—

103

C₂²⋊C₄

—

127

—

104

C₂²⋊C₄

C₂×D₁₂₇

yes

—

105

C₄²

—

127

—

106

C₄²

—

127

—

107

C₄²

C₂×D₁₂₇

yes

—

108

C₂³

—

127

—

109

C₂³

C₂²×D₁₂₇

yes

characteristic

110

Q₈

—

254

—

111

D₄

—

254

—

112

C₂×C₄

—

113

C₂×C₄

—

127

—

114

C₂×C₄

—

127

—

115

C₂×C₄

—

127

—

116

C₂×C₄

—

127

—

117

C₂×C₄

—

127

—

118

C₂×C₄

—

254

—

119

C₂×C₄

—

254

—

120

C₂×C₄

C₂²×D₁₂₇

yes

—

121

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

122

C₂×C₄

C₂²×D₁₂₇

yes

—

123

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

127

—

128

C₂²

—

254

—

129

C₂²

—

254

—

130

C₂²

C₂³×D₁₂₇

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

254

—

135

C₄

—

254

—

136

C₄

—

254

—

137

C₄

—

254

—

138

C₄

—

254

—

139

C₄

#14523248

yes

—

140

C₂

—

141

C₂

—

254

—

142

C₂

#41×D₁₂₇

yes

characteristic

143

C₂

C₂×#14523248

yes

characteristic

144

C₂

#16861256

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#117×C₁₂₇	C₂
hypercenter	#117	D₁₂₇
hypocenter	C₁₂₇	#552
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

#117×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #16861202	C₂	semidirect
2. #16861201	C₂	semidirect
3. #16861200	C₂	semidirect
5. #16861189	C₂	semidirect
6. #16861188	C₂	semidirect
7. #16861174	C₂	semidirect
9. #16861166	C₂	semidirect
10. C₄⋊C₄×D₁₂₇	C₂	semidirect
11. #117×C₁₂₇	C₂	semidirect
72. C₁₂₇	#552	semidirect
83. #117	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861202

C₂

semidirect

2. #16861201

C₂

semidirect

3. #16861200

C₂

semidirect

5. #16861189

C₂

semidirect

6. #16861188

C₂

semidirect

7. #16861174

C₂

semidirect

9. #16861166

C₂

semidirect

10. C₄⋊C₄×D₁₂₇

C₂

semidirect

11. #117×C₁₂₇

C₂

semidirect

72. C₁₂₇

#552

semidirect

83. #117

D₁₂₇

semidirect

Subgroups — #22550565

Conjugacy classes of subgroups

Complements