Group #22550581 — subgroups

Subgroups — #22550581

Order 8128.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#16861199	C₂	1	2	yes	maximal, complemented
3	#16861189	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861188	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861188	C₂	1	2	yes	maximal, complemented
6	#16861187	C₂	1	2	yes	maximal, complemented
7	#16861174	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊D₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₄×C₂²⋊D₁₂₇	C₂	1	2	yes	maximal
10	C₄⋊C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₅₄	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
18	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
27	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₁₂₇	C₂²	1	2	yes	—
29	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
32	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
33	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₂²×D₁₂₇	C₂³	1	1	yes	characteristic
36	Dic₂₅₄	—	2	2	no	—
37	Dic₂₅₄	—	4	2	no	—
38	C₂²⋊D₁₂₇	Q₈	1	4	yes	complemented
39	C₂×Dic₁₂₇	C₂³	1	2	yes	—
40	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₂₇	C₂³	1	2	yes	—
43	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
44	C₄×D₁₂₇	—	2	2	no	—
45	C₄×D₁₂₇	—	4	2	no	—
46	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	—	2	2	no	—
48	C₂×C₄×C₁₂₇	—	4	2	no	—
49	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
52	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₂²×C₁₂₇	—	2	1	no	—
54	C₂²×C₁₂₇	—	2	1	no	—
55	C₂²×C₁₂₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	C₂×D₁₂₇	—	2	1	no	—
58	C₂×D₁₂₇	—	2	1	no	—
59	C₂×D₁₂₇	C₂×Q₈	1	2	yes	—
60	C₄×C₁₂₇	—	2	1	no	—
61	C₄×C₁₂₇	—	2	2	no	—
62	C₄×C₁₂₇	—	2	2	no	—
63	C₄×C₁₂₇	#41	1	2	yes	—
64	Dic₁₂₇	—	2	1	no	—
65	Dic₁₂₇	—	2	2	no	—
66	Dic₁₂₇	—	2	2	no	—
67	Dic₁₂₇	—	2	1	no	—
68	Dic₁₂₇	C₂×Q₈	1	2	yes	—
69	C₂×C₁₂₇	—	2	2	no	—
70	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
72	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
73	D₁₂₇	—	2	2	no	—
74	C₁₂₇	#554	1	1	yes	characteristic, complemented, Sylow
75	#554	—	127	1	no	maximal, Sylow
76	C₄⋊Q₈	—	127	1	no	—
77	#125	—	127	2	no	—
78	C₂²⋊Q₈	—	127	2	no	—
79	C₂²⋊Q₈	—	127	1	no	—
80	C₂²⋊Q₈	—	127	2	no	—
81	C₂²⋊Q₈	—	127	1	no	—
82	C₄×Q₈	—	127	1	no	—
83	C₄×D₄	—	127	2	no	—
84	C₄×D₄	—	127	1	no	—
85	C₂×C₄⋊C₄	—	127	1	no	—
86	C₂×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic, complemented
87	C₂×Q₈	—	127	2	no	—
88	C₂×Q₈	—	127	1	no	—
89	C₂×D₄	—	127	1	no	—
90	C₂²×C₄	—	127	2	no	—
91	C₂²×C₄	—	127	1	no	—
92	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
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	—	127	2	no	—
99	C₄⋊C₄	—	127	1	no	—
100	C₄⋊C₄	—	127	2	no	—
101	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
102	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
103	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
104	C₂²⋊C₄	—	127	2	no	—
105	C₂²⋊C₄	—	127	1	no	—
106	C₂²⋊C₄	—	127	2	no	—
107	C₂²⋊C₄	—	127	1	no	—
108	C₄²	—	127	2	no	—
109	C₄²	—	127	1	no	—
110	C₂³	—	127	1	no	—
111	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
112	Q₈	—	508	2	no	—
113	Q₈	—	254	2	no	—
114	D₄	—	127	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	127	1	no	—
118	C₂×C₄	—	127	2	no	—
119	C₂×C₄	—	127	1	no	—
120	C₂×C₄	—	127	1	no	—
121	C₂×C₄	—	127	2	no	—
122	C₂×C₄	—	508	2	no	—
123	C₂×C₄	—	254	2	no	—
124	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
125	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
127	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	127	2	no	—
131	C₂²	—	254	1	no	—
132	C₂²	—	254	1	no	—
133	C₂²	Q₈×D₁₂₇	1	2	yes	—
134	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	127	2	no	—
139	C₄	—	254	1	no	—
140	C₄	—	254	2	no	—
141	C₄	—	254	2	no	—
142	C₄	—	254	1	no	—
143	C₄	#14523248	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	254	2	no	—
146	C₂	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
147	C₂	C₂×#14523248	1	1	yes	characteristic
148	C₂	#16861254	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861208

C₂

yes

maximal, characteristic, complemented

#16861199

C₂

yes

maximal, complemented

#16861189

C₂

yes

maximal, characteristic, complemented

#16861188

C₂

yes

maximal, characteristic, complemented

#16861188

C₂

yes

maximal, complemented

#16861187

C₂

yes

maximal, complemented

#16861174

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₂₇

C₂

yes

maximal, characteristic

C₄×C₂²⋊D₁₂₇

C₂

yes

maximal

C₄⋊C₄×D₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×Dic₂₅₄

C₂²

yes

—

C₂×C₂²⋊D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

—

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

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂²⋊D₁₂₇

Q₈

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

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

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#41

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×Q₈

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

D₁₂₇

—

C₁₂₇

#554

yes

characteristic, complemented, Sylow

#554

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

#125

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

C₄×D₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

D₁₂₇

yes

characteristic, complemented

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

100

C₄⋊C₄

—

127

—

101

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

102

C₄⋊C₄

C₂×D₁₂₇

yes

—

103

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

104

C₂²⋊C₄

—

127

—

105

C₂²⋊C₄

—

127

—

106

C₂²⋊C₄

—

127

—

107

C₂²⋊C₄

—

127

—

108

C₄²

—

127

—

109

C₄²

—

127

—

110

C₂³

—

127

—

111

C₂³

C₂²×D₁₂₇

yes

characteristic

112

Q₈

—

508

—

113

Q₈

—

254

—

114

D₄

—

127

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

127

—

118

C₂×C₄

—

127

—

119

C₂×C₄

—

127

—

120

C₂×C₄

—

127

—

121

C₂×C₄

—

127

—

122

C₂×C₄

—

508

—

123

C₂×C₄

—

254

—

124

C₂×C₄

C₂²×D₁₂₇

yes

—

125

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

126

C₂×C₄

C₂²×D₁₂₇

yes

—

127

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

127

—

131

C₂²

—

254

—

132

C₂²

—

254

—

133

C₂²

Q₈×D₁₂₇

yes

—

134

C₂²

C₂³×D₁₂₇

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

127

—

139

C₄

—

254

—

140

C₄

—

254

—

141

C₄

—

254

—

142

C₄

—

254

—

143

C₄

#14523248

yes

—

144

C₂

—

145

C₂

—

254

—

146

C₂

C₂×Q₈×D₁₂₇

yes

characteristic

147

C₂

C₂×#14523248

yes

characteristic

148

C₂

#16861254

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₂×C₄⋊C₄×C₁₂₇	C₂
hypercenter	C₂×C₄⋊C₄	D₁₂₇
hypocenter	C₁₂₇	#554
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₂×C₄⋊C₄×C₁₂₇

C₂

hypercenter

C₂×C₄⋊C₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #16861208	C₂	semidirect
2. #16861199	C₂	semidirect
3. #16861189	C₂	semidirect
4. #16861188	C₂	semidirect
5. #16861188	C₂	semidirect
6. #16861187	C₂	semidirect
7. #16861174	C₂	semidirect
10. C₄⋊C₄×D₁₂₇	C₂	semidirect
11. C₂×C₄⋊C₄×C₁₂₇	C₂	semidirect
38. C₂²⋊D₁₂₇	Q₈	semidirect
74. C₁₂₇	#554	semidirect
86. C₂×C₄⋊C₄	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861208

C₂

semidirect

2. #16861199

C₂

semidirect

3. #16861189

C₂

semidirect

4. #16861188

C₂

semidirect

5. #16861188

C₂

semidirect

6. #16861187

C₂

semidirect

7. #16861174

C₂

semidirect

10. C₄⋊C₄×D₁₂₇

C₂

semidirect

11. C₂×C₄⋊C₄×C₁₂₇

C₂

semidirect

38. C₂²⋊D₁₂₇

Q₈

semidirect

74. C₁₂₇

#554

semidirect

86. C₂×C₄⋊C₄

D₁₂₇

semidirect

Subgroups — #22550581

Conjugacy classes of subgroups

Complements