Group #22550428 — subgroups

Subgroups — #22550428

Order 8128.

222 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16861208	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861207	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861199	C₂	1	2	yes	maximal, complemented
4	#16861188	C₂	1	1	yes	maximal, characteristic
5	#16861186	C₂	1	2	yes	maximal
6	Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#16861174	C₂	1	2	yes	maximal, complemented
8	#16861174	C₂	1	1	yes	maximal, characteristic, complemented
9	#16861165	C₂	1	2	yes	maximal
10	C₂×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₂₅₄	C₂²	1	2	yes	—
13	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
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	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
22	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
24	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₂₇	C₂²	1	2	yes	—
29	C₄×Dic₁₂₇	C₂²	1	2	yes	—
30	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
33	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
35	Dic₂₅₄	—	2	2	no	—
36	Dic₂₅₄	—	2	4	no	—
37	C₂×Dic₁₂₇	—	2	2	no	—
38	C₂×Dic₁₂₇	Q₈	1	4	yes	complemented
39	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₂₇	C₂³	1	2	yes	—
41	C₂×Dic₁₂₇	C₂³	1	2	yes	—
42	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
45	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
46	Q₈×C₁₂₇	—	2	2	no	—
47	C₂×C₄×C₁₂₇	—	2	2	no	—
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	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
52	C₂²×C₁₂₇	—	2	1	no	—
53	C₂²×C₁₂₇	—	2	1	no	—
54	C₂²×C₁₂₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
56	C₄×C₁₂₇	—	2	1	no	—
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	#41	1	2	yes	—
60	Dic₁₂₇	—	2	1	no	—
61	Dic₁₂₇	—	2	2	no	—
62	Dic₁₂₇	—	2	2	no	—
63	Dic₁₂₇	C₂×Q₈	1	2	yes	—
64	Dic₁₂₇	C₂×Q₈	1	2	yes	—
65	Dic₁₂₇	#41	1	2	yes	—
66	C₂×C₁₂₇	—	2	2	no	—
67	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
70	C₁₂₇	#533	1	1	yes	characteristic, complemented, Sylow
71	#533	—	127	1	no	maximal, Sylow
72	C₄⋊Q₈	—	127	1	no	—
73	C₄⋊Q₈	—	127	1	no	—
74	#125	—	127	2	no	—
75	C₂²⋊Q₈	—	127	1	no	—
76	C₂²⋊Q₈	—	127	2	no	—
77	C₂²⋊Q₈	D₁₂₇	1	1	yes	characteristic
78	C₄×Q₈	—	127	1	no	—
79	C₄×Q₈	—	127	1	no	—
80	C₄×Q₈	—	127	2	no	—
81	#117	—	127	2	no	—
82	C₂×C₄²	—	127	1	no	—
83	C₂×Q₈	—	127	2	no	—
84	C₂×Q₈	—	127	1	no	—
85	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
86	C₂²×C₄	—	127	1	no	—
87	C₂²×C₄	—	127	1	no	—
88	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
89	C₄⋊C₄	—	127	2	no	—
90	C₄⋊C₄	—	127	2	no	—
91	C₄⋊C₄	—	127	2	no	—
92	C₄⋊C₄	—	127	2	no	—
93	C₄⋊C₄	—	127	2	no	—
94	C₄⋊C₄	—	127	2	no	—
95	C₄⋊C₄	—	127	1	no	—
96	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
97	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
98	C₂²⋊C₄	—	127	2	no	—
99	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
100	C₄²	—	127	1	no	—
101	C₄²	—	127	1	no	—
102	C₄²	—	127	1	no	—
103	C₄²	—	127	1	no	—
104	C₄²	—	127	2	no	—
105	C₄²	—	127	2	no	—
106	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
107	Q₈	—	2	2	no	—
108	Q₈	—	254	4	no	—
109	Q₈	—	254	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	127	4	no	—
112	C₂×C₄	—	127	1	no	—
113	C₂×C₄	—	127	1	no	—
114	C₂×C₄	—	127	1	no	—
115	C₂×C₄	—	127	1	no	—
116	C₂×C₄	—	127	2	no	—
117	C₂×C₄	—	127	2	no	—
118	C₂×C₄	—	254	2	no	—
119	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
121	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
122	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	Q₈×D₁₂₇	1	2	yes	—
126	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	127	2	no	—
131	C₄	—	127	2	no	—
132	C₄	—	127	2	no	—
133	C₄	—	254	1	no	—
134	C₄	—	254	2	no	—
135	C₄	—	254	2	no	—
136	C₄	#14523247	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
139	C₂	#41×D₁₂₇	1	1	yes	characteristic
140	C₂	C₂×#14523247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861208

C₂

yes

maximal, characteristic, complemented

#16861207

C₂

yes

maximal, characteristic, complemented

#16861199

C₂

yes

maximal, complemented

#16861188

C₂

yes

maximal, characteristic

#16861186

C₂

yes

maximal

Q₈×Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861174

C₂

yes

maximal, complemented

#16861174

C₂

yes

maximal, characteristic, complemented

#16861165

C₂

yes

maximal

C₂×C₄×Dic₁₂₇

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

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

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

Q₈

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₂×Dic₁₂₇

C₂³

yes

characteristic

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

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#41

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×Q₈

yes

—

Dic₁₂₇

C₂×Q₈

yes

—

Dic₁₂₇

#41

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₁₂₇

#533

yes

characteristic, complemented, Sylow

#533

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

#125

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

D₁₂₇

yes

characteristic

C₄×Q₈

—

127

—

C₄×Q₈

—

127

—

C₄×Q₈

—

127

—

#117

—

127

—

C₂×C₄²

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

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

C₂×D₁₂₇

yes

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

100

C₄²

—

127

—

101

C₄²

—

127

—

102

C₄²

—

127

—

103

C₄²

—

127

—

104

C₄²

—

127

—

105

C₄²

—

127

—

106

C₂³

C₂²×D₁₂₇

yes

characteristic

107

Q₈

—

108

Q₈

—

254

—

109

Q₈

—

254

—

110

C₂×C₄

—

111

C₂×C₄

—

127

—

112

C₂×C₄

—

127

—

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

C₂²×D₁₂₇

yes

characteristic

120

C₂×C₄

C₂²×D₁₂₇

yes

—

121

C₂×C₄

C₂²×D₁₂₇

yes

—

122

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

Q₈×D₁₂₇

yes

—

126

C₂²

C₂³×D₁₂₇

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

127

—

131

C₄

—

127

—

132

C₄

—

127

—

133

C₄

—

254

—

134

C₄

—

254

—

135

C₄

—

254

—

136

C₄

#14523247

yes

—

137

C₂

—

138

C₂

C₂×Q₈×D₁₂₇

yes

characteristic

139

C₂

#41×D₁₂₇

yes

characteristic

140

C₂

C₂×#14523247

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	#533
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. #16861208	C₂	semidirect
2. #16861207	C₂	semidirect
3. #16861199	C₂	semidirect
6. Q₈×Dic₁₂₇	C₂	semidirect
7. #16861174	C₂	semidirect
8. #16861174	C₂	semidirect
38. C₂×Dic₁₂₇	Q₈	semidirect
70. C₁₂₇	#533	semidirect

H ≅ G/N

type

1. #16861208

C₂

semidirect

2. #16861207

C₂

semidirect

3. #16861199

C₂

semidirect

6. Q₈×Dic₁₂₇

C₂

semidirect

7. #16861174

C₂

semidirect

8. #16861174

C₂

semidirect

38. C₂×Dic₁₂₇

Q₈

semidirect

70. C₁₂₇

#533

semidirect

Subgroups — #22550428

Conjugacy classes of subgroups

Complements