Group #22550506 — subgroups

Subgroups — #22550506

Order 8128.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#16861209	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861208	C₂	1	2	yes	maximal, complemented
3	#16861208	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861200	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861199	C₂	1	2	yes	maximal, complemented
6	#16861189	C₂	1	2	yes	maximal
7	#16861187	C₂	1	2	yes	maximal
8	#16861167	C₂	1	1	yes	maximal, characteristic
9	#16861166	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic
11	#125×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₅₄	C₂²	1	2	yes	—
13	C₂×Dic₂₅₄	C₂²	1	2	yes	—
14	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
18	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
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	Dic₁₂₇⋊C₄	C₂²	1	1	yes	characteristic
25	(C₂×D₁₂₇)⋊C₄	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	2	yes	—
29	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
31	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
32	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₂²×D₁₂₇	C₂³	1	1	yes	characteristic
36	Dic₂₅₄	—	4	2	no	—
37	Dic₂₅₄	—	4	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	—	4	2	no	—
44	C₄×D₁₂₇	Q₈	1	4	yes	complemented
45	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₂₇	—	2	1	no	—
52	C₂×D₁₂₇	—	2	1	no	—
53	C₂×D₁₂₇	C₂×Q₈	1	2	yes	—
54	C₄×C₁₂₇	—	2	2	no	—
55	C₄×C₁₂₇	—	2	1	no	—
56	C₄×C₁₂₇	—	2	1	no	—
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
59	Dic₁₂₇	—	2	2	no	—
60	Dic₁₂₇	—	2	2	no	—
61	Dic₁₂₇	—	2	1	no	—
62	Dic₁₂₇	—	2	1	no	—
63	Dic₁₂₇	C₂×Q₈	1	2	yes	—
64	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
67	D₁₂₇	—	2	2	no	—
68	C₁₂₇	#544	1	1	yes	characteristic, complemented, Sylow
69	#544	—	127	1	no	maximal, Sylow
70	C₄⋊Q₈	—	127	2	no	—
71	C₄⋊Q₈	—	127	1	no	—
72	C₄⋊Q₈	—	127	1	no	—
73	#125	—	127	2	no	—
74	#125	—	127	1	no	—
75	#125	D₁₂₇	1	1	yes	characteristic, complemented
76	C₂²⋊Q₈	—	127	2	no	—
77	C₂²⋊Q₈	—	127	2	no	—
78	#117	—	127	1	no	—
79	#117	—	127	1	no	—
80	C₂×C₄⋊C₄	—	127	1	no	—
81	C₂×Q₈	—	127	2	no	—
82	C₂×Q₈	—	127	2	no	—
83	C₂²×C₄	—	127	1	no	—
84	C₂²×C₄	—	127	2	no	—
85	C₄⋊C₄	—	127	2	no	—
86	C₄⋊C₄	—	127	1	no	—
87	C₄⋊C₄	—	127	1	no	—
88	C₄⋊C₄	—	127	2	no	—
89	C₄⋊C₄	—	127	2	no	—
90	C₄⋊C₄	—	127	1	no	—
91	C₄⋊C₄	—	127	1	no	—
92	C₄⋊C₄	—	127	2	no	—
93	C₄⋊C₄	—	127	2	no	—
94	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
95	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
96	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
97	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
98	C₂²⋊C₄	—	127	1	no	—
99	C₂²⋊C₄	—	127	2	no	—
100	C₂²⋊C₄	—	127	1	no	—
101	C₄²	—	127	2	no	—
102	C₄²	—	127	1	no	—
103	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
104	C₂³	—	127	1	no	—
105	Q₈	—	508	2	no	—
106	Q₈	—	508	2	no	—
107	C₂×C₄	—	127	1	no	—
108	C₂×C₄	—	127	4	no	—
109	C₂×C₄	—	127	1	no	—
110	C₂×C₄	—	127	1	no	—
111	C₂×C₄	—	127	2	no	—
112	C₂×C₄	—	127	2	no	—
113	C₂×C₄	—	508	2	no	—
114	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
115	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
118	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
119	C₂²	—	127	2	no	—
120	C₂²	—	254	1	no	—
121	C₂²	—	254	1	no	—
122	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	127	2	no	—
128	C₄	—	254	2	no	—
129	C₄	—	254	1	no	—
130	C₄	—	254	2	no	—
131	C₄	—	254	1	no	—
132	C₄	Q₈×D₁₂₇	1	2	yes	—
133	C₂	—	254	2	no	—
134	C₂	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
135	C₂	#16861255	1	1	yes	characteristic
136	C₂	#16861256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861209

C₂

yes

maximal, characteristic, complemented

#16861208

C₂

yes

maximal, complemented

#16861208

C₂

yes

maximal, characteristic, complemented

#16861200

C₂

yes

maximal, characteristic, complemented

#16861199

C₂

yes

maximal, complemented

#16861189

C₂

yes

maximal

#16861187

C₂

yes

maximal

#16861167

C₂

yes

maximal, characteristic

#16861166

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₂₇

C₂

yes

maximal, characteristic

#125×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

—

C₂×Dic₂₅₄

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

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

Q₈

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×Q₈

yes

—

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

C₁₂₇

#544

yes

characteristic, complemented, Sylow

#544

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

#125

—

127

—

#125

—

127

—

#125

D₁₂₇

yes

characteristic, complemented

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

#117

—

127

—

#117

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂²×C₄

—

127

—

C₂²×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

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

100

C₂²⋊C₄

—

127

—

101

C₄²

—

127

—

102

C₄²

—

127

—

103

C₄²

C₂×D₁₂₇

yes

characteristic

104

C₂³

—

127

—

105

Q₈

—

508

—

106

Q₈

—

508

—

107

C₂×C₄

—

127

—

108

C₂×C₄

—

127

—

109

C₂×C₄

—

127

—

110

C₂×C₄

—

127

—

111

C₂×C₄

—

127

—

112

C₂×C₄

—

127

—

113

C₂×C₄

—

508

—

114

C₂×C₄

C₂²×D₁₂₇

yes

—

115

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

116

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

117

C₂×C₄

C₂²×D₁₂₇

yes

—

118

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

119

C₂²

—

127

—

120

C₂²

—

254

—

121

C₂²

—

254

—

122

C₂²

C₂³×D₁₂₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

127

—

128

C₄

—

254

—

129

C₄

—

254

—

130

C₄

—

254

—

131

C₄

—

254

—

132

C₄

Q₈×D₁₂₇

yes

—

133

C₂

—

254

—

134

C₂

C₂×Q₈×D₁₂₇

yes

characteristic

135

C₂

#16861255

yes

characteristic

136

C₂

#16861256

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#125×C₁₂₇	C₂
hypercenter	#125	D₁₂₇
hypocenter	C₁₂₇	#544
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

#125×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #16861209	C₂	semidirect
2. #16861208	C₂	semidirect
3. #16861208	C₂	semidirect
4. #16861200	C₂	semidirect
5. #16861199	C₂	semidirect
11. #125×C₁₂₇	C₂	semidirect
44. C₄×D₁₂₇	Q₈	semidirect
68. C₁₂₇	#544	semidirect
75. #125	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861209

C₂

semidirect

2. #16861208

C₂

semidirect

3. #16861208

C₂

semidirect

4. #16861200

C₂

semidirect

5. #16861199

C₂

semidirect

11. #125×C₁₂₇

C₂

semidirect

44. C₄×D₁₂₇

Q₈

semidirect

68. C₁₂₇

#544

semidirect

75. #125

D₁₂₇

semidirect

Subgroups — #22550506

Conjugacy classes of subgroups

Complements