Group #22558721 — subgroups

Subgroups — #22558721

Order 8576.

228 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16863453	C₂	1	1	yes	maximal, characteristic
2	#16863242	C₂	1	1	yes	maximal, characteristic
3	#16863133	C₂	1	4	yes	maximal
4	#533×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₆₇	—	2	1	no	—
6	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#14523954	—	2	1	no	—
9	C₆₇⋊#95	—	2	1	no	—
10	C₆₇⋊#95	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
12	C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#125×C₆₇	—	2	1	no	—
14	C₂²⋊Q₈×C₆₇	—	2	1	no	—
15	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₄×Q₈×C₆₇	—	2	1	no	—
17	C₄×Q₈×C₆₇	C₄	1	2	yes	complemented
18	#117×C₆₇	—	2	1	no	—
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₆₇	—	2	2	no	—
21	C₂×C₆₇⋊C₈	—	2	2	no	—
22	C₄×Dic₆₇	—	2	4	no	—
23	C₂×Q₈×C₆₇	—	2	1	no	—
24	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₆₇	—	2	2	no	—
35	C₄²×C₆₇	—	2	2	no	—
36	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₆₇	—	4	2	no	—
42	Q₈×C₆₇	—	4	2	no	—
43	C₂×C₄×C₆₇	—	2	2	no	—
44	C₂×C₄×C₆₇	—	2	2	no	—
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	4	1	no	—
48	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₆₇	—	2	4	no	—
55	C₂×Dic₆₇	—	4	2	no	—
56	C₂×Dic₆₇	—	4	2	no	—
57	C₆₇⋊C₈	—	4	2	no	—
58	C₂²×C₆₇	—	2	1	no	—
59	C₂²×C₆₇	—	2	1	no	—
60	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₆₇	#137	1	2	yes	—
62	C₄×C₆₇	—	2	1	no	—
63	C₄×C₆₇	—	2	1	no	—
64	C₄×C₆₇	—	2	1	no	—
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	—	4	2	no	—
68	C₄×C₆₇	—	4	2	no	—
69	C₄×C₆₇	#104	1	2	yes	—
70	Dic₆₇	—	4	4	no	—
71	C₂×C₆₇	—	2	2	no	—
72	C₂×C₆₇	#410	1	1	yes	characteristic
73	C₂×C₆₇	#419	1	1	yes	characteristic
74	C₂×C₆₇	C₂×#104	1	1	yes	characteristic
75	#1506	—	67	1	no	maximal, Sylow
76	C₆₇	#1506	1	1	yes	characteristic, complemented, Sylow
77	#533	D₆₇	1	1	yes	characteristic
78	#432	—	67	1	no	—
79	#376	—	67	1	no	—
80	#328	—	67	4	no	—
81	C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic, complemented
82	C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic, complemented
83	#125	—	2	1	no	—
84	C₂²⋊Q₈	—	2	1	no	—
85	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
86	C₄×Q₈	—	2	1	no	—
87	C₄×Q₈	Dic₆₇	1	2	yes	complemented
88	#117	—	2	1	no	—
89	C₂×C₄²	—	134	1	no	—
90	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
91	C₄⋊C₈	—	134	1	no	—
92	C₂²⋊C₈	—	67	2	no	—
93	#97	—	134	1	no	—
94	#95	—	67	2	no	—
95	#95	—	134	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	C₂×Dic₆₇	1	2	yes	—
98	C₂²×C₄	—	134	2	no	—
99	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
101	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
102	C₂×C₈	—	134	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	134	4	no	—
112	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
113	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
114	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
115	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	268	2	no	—
125	C₂×C₄	—	268	2	no	—
126	C₂×C₄	—	134	4	no	—
127	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
133	C₈	—	268	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
137	C₂²	#14524074	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	4	2	no	—
144	C₄	—	4	2	no	—
145	C₄	—	268	4	no	—
146	C₄	#14523970	1	2	yes	—
147	C₂	—	2	2	no	—
148	C₂	#16863370	1	1	yes	characteristic
149	C₂	#16863405	1	1	yes	characteristic
150	C₂	C₂×#14523970	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863453

C₂

yes

maximal, characteristic

#16863242

C₂

yes

maximal, characteristic

#16863133

C₂

yes

maximal

#533×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523954

—

C₆₇⋊#95

—

C₆₇⋊#95

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₄

yes

characteristic, complemented

C₄⋊Q₈×C₆₇

C₄

yes

characteristic, complemented

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

C₂²

yes

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

C₄

yes

complemented

#117×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#104

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#533

D₆₇

yes

—

—

—

C₄⋊Q₈

Dic₆₇

yes

characteristic, complemented

C₄⋊Q₈

Dic₆₇

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₆₇

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₆₇

yes

complemented

#117

—

C₂×C₄²

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

#97

—

134

—

#95

—

#95

—

134

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

100

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

101

C₂²×C₄

C₂²×D₆₇

yes

characteristic

102

C₂×C₈

—

134

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×Dic₆₇

yes

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

134

—

112

C₄²

C₂×Dic₆₇

yes

characteristic

113

C₄²

C₂×Dic₆₇

yes

characteristic

114

C₄²

C₂²⋊D₆₇

yes

characteristic

115

C₄²

C₂²⋊D₆₇

yes

characteristic

116

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

268

—

125

C₂×C₄

—

268

—

126

C₂×C₄

—

134

—

127

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

128

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

129

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

130

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

131

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

133

C₈

—

268

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂×C₂²⋊Dic₆₇

yes

characteristic

137

C₂²

#14524074

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

268

—

146

C₄

#14523970

yes

—

147

C₂

—

148

C₂

#16863370

yes

characteristic

149

C₂

#16863405

yes

characteristic

150

C₂

C₂×#14523970

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#533×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
hypercenter	#533	D₆₇
hypocenter	C₆₇	#1506
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#533×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
11. C₄⋊Q₈×C₆₇	C₄	semidirect
12. C₄⋊Q₈×C₆₇	C₄	semidirect
17. C₄×Q₈×C₆₇	C₄	semidirect
76. C₆₇	#1506	semidirect
81. C₄⋊Q₈	Dic₆₇	semidirect
82. C₄⋊Q₈	Dic₆₇	semidirect
87. C₄×Q₈	Dic₆₇	semidirect

H ≅ G/N

type

11. C₄⋊Q₈×C₆₇

C₄

semidirect

12. C₄⋊Q₈×C₆₇

C₄

semidirect

17. C₄×Q₈×C₆₇

C₄

semidirect

76. C₆₇

#1506

semidirect

81. C₄⋊Q₈

Dic₆₇

semidirect

82. C₄⋊Q₈

Dic₆₇

semidirect

87. C₄×Q₈

Dic₆₇

semidirect

Subgroups — #22558721

Conjugacy classes of subgroups

Complements