Group #22568854 — subgroups

Subgroups — #22568854

Order 8576.

400 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₄×#11761628	C₂	1	1	yes	maximal, characteristic
2	#16863458	C₂	1	4	yes	maximal, complemented
3	#16863447	C₂	1	4	yes	maximal, complemented
4	#16863424	C₂	1	2	yes	maximal, complemented
5	#16863364	C₂	1	2	yes	maximal
6	C₂×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761628	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₁₃₄	C₄	1	2	yes	—
11	C₄×D₂₆₈	C₄	1	2	yes	—
12	C₄×C₂²⋊D₆₇	C₄	1	4	yes	—
13	#14524002	C₂²	1	1	yes	characteristic
14	#14524000	C₂²	1	2	yes	—
15	C₄²×D₆₇	C₂²	1	2	yes	—
16	Dic₆₇⋊C₈	C₂²	1	8	yes	—
17	(C₂×D₆₇)⋊C₈	C₂²	1	8	yes	—
18	C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
19	C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
20	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#97×C₆₇	C₂²	1	2	yes	—
23	C₄×C₈×C₆₇	C₂²	1	2	yes	—
24	#11761628	—	2	4	no	—
25	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
26	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₆₇	C₂³	1	2	yes	—
30	C₂×C₆₇⋊C₈	—	2	4	no	—
31	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
32	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
33	Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
35	(C₂×D₆₇)⋊C₄	C₂×C₄	1	4	yes	—
36	C₄×Dic₆₇	C₂×C₄	1	4	yes	—
37	C₄×Dic₆₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	M₁₆×C₆₇	—	2	4	no	—
41	C₂×C₈×C₆₇	C₂³	1	4	yes	—
42	C₄²×C₆₇	C₂×C₄	1	2	yes	—
43	C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₆₇	#41	1	4	yes	—
52	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₆₇	—	2	4	no	—
54	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
55	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
56	D₂₆₈	—	2	2	no	—
57	C₄×D₆₇	—	2	8	no	—
58	C₄×D₆₇	—	2	4	no	—
59	Dic₁₃₄	—	2	2	no	—
60	C₈×C₆₇	—	2	4	no	—
61	C₆₇⋊C₈	—	2	4	no	—
62	C₂²×C₆₇	—	2	2	no	—
63	C₂²×C₆₇	#117	1	2	yes	—
64	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₆₇	—	2	2	no	—
66	C₂×D₆₇	—	2	4	no	—
67	C₄×C₆₇	—	2	2	no	—
68	C₄×C₆₇	#117	1	2	yes	—
69	C₄×C₆₇	#131	1	4	yes	—
70	C₄×C₆₇	C₂×#41	1	2	yes	—
71	Dic₆₇	—	2	2	no	—
72	Dic₆₇	—	2	4	no	—
73	C₂×C₆₇	—	2	2	no	—
74	C₂×C₆₇	C₂×#117	1	1	yes	characteristic
75	C₂×C₆₇	C₂×#131	1	2	yes	—
76	D₆₇	—	4	2	no	—
77	#2922	—	67	1	no	maximal, Sylow
78	C₆₇	#2922	1	1	yes	characteristic, complemented, Sylow
79	C₄×#41	—	67	1	no	—
80	#433	—	67	4	no	—
81	#431	—	67	4	no	—
82	#424	—	67	2	no	—
83	#408	—	67	2	no	—
84	C₄×M₁₆	D₆₇	1	1	yes	characteristic, complemented
85	C₂×C₄×C₈	—	67	1	no	—
86	C₂×#41	—	67	1	no	—
87	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
88	C₂²×C₈	—	67	2	no	—
89	C₄×Q₈	—	67	2	no	—
90	C₄×D₄	—	67	4	no	—
91	C₄×D₄	—	67	2	no	—
92	#117	—	67	1	no	—
93	#117	—	67	2	no	—
94	C₂×C₄²	—	67	2	no	—
95	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
96	C₄⋊C₈	—	67	8	no	—
97	C₂²⋊C₈	—	67	8	no	—
98	#97	C₂×D₆₇	1	2	yes	—
99	C₄×C₈	—	67	2	no	—
100	C₄×C₈	—	67	2	no	—
101	C₄×C₈	C₂×D₆₇	1	2	yes	—
102	#41	—	134	4	no	—
103	C₂×Q₈	—	67	1	no	—
104	C₂×D₄	—	67	2	no	—
105	C₂×D₄	—	67	1	no	—
106	C₂²×C₄	—	67	4	no	—
107	C₂²×C₄	—	67	2	no	—
108	C₂²×C₄	C₄×D₆₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	67	4	no	—
112	C₂×C₈	—	134	4	no	—
113	C₂×C₈	C₂²×D₆₇	1	4	yes	—
114	C₄⋊C₄	—	67	4	no	—
115	C₄⋊C₄	—	67	2	no	—
116	C₂²⋊C₄	—	67	2	no	—
117	C₂²⋊C₄	—	67	4	no	—
118	C₄²	—	67	4	no	—
119	C₄²	—	67	2	no	—
120	C₄²	C₄×D₆₇	1	2	yes	—
121	C₄²	C₂²×D₆₇	1	1	yes	characteristic
122	C₄²	C₂²×D₆₇	1	1	yes	characteristic
123	C₂³	—	67	2	no	—
124	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
125	Q₈	—	134	2	no	—
126	D₄	—	134	4	no	—
127	D₄	—	134	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	67	4	no	—
130	C₂×C₄	—	67	2	no	—
131	C₂×C₄	—	134	8	no	—
132	C₂×C₄	—	134	4	no	—
133	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
136	C₂×C₄	#11761628	1	4	yes	—
137	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	134	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	134	2	no	—
142	C₂²	—	134	4	no	—
143	C₂²	#14524000	1	2	yes	—
144	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	134	2	no	—
147	C₄	—	134	4	no	—
148	C₄	#14524000	1	2	yes	—
149	C₄	#14524051	1	4	yes	—
150	C₄	C₂×#11761628	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	268	2	no	—
153	C₂	C₂×#14524000	1	1	yes	characteristic
154	C₂	C₂×#14524051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11761628

C₂

yes

maximal, characteristic

#16863458

C₂

yes

maximal, complemented

#16863447

C₂

yes

maximal, complemented

#16863424

C₂

yes

maximal, complemented

#16863364

C₂

yes

maximal

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₄×Dic₁₃₄

C₄

yes

—

C₄×D₂₆₈

C₄

yes

—

C₄×C₂²⋊D₆₇

C₄

yes

—

#14524002

C₂²

yes

characteristic

#14524000

C₂²

yes

—

C₄²×D₆₇

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

(C₂×D₆₇)⋊C₈

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#97×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

#11761628

—

C₂×Dic₁₃₄

C₂×C₄

yes

characteristic

C₂×D₂₆₈

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₆₇

C₂×C₄

yes

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₆₇

C₂×C₄

yes

—

(C₂×D₆₇)⋊C₄

C₂×C₄

yes

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂²×D₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂²⋊D₆₇

—

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#117

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#117

yes

—

C₄×C₆₇

#131

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#117

yes

characteristic

C₂×C₆₇

C₂×#131

yes

—

D₆₇

—

#2922

—

maximal, Sylow

C₆₇

#2922

yes

characteristic, complemented, Sylow

C₄×#41

—

—

—

—

—

C₄×M₁₆

D₆₇

yes

characteristic, complemented

C₂×C₄×C₈

—

C₂×#41

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

C₂×D₆₇

yes

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₆₇

yes

—

102

#41

—

134

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₆₇

yes

—

109

C₂²×C₄

C₂²×D₆₇

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

134

—

113

C₂×C₈

C₂²×D₆₇

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₆₇

yes

—

121

C₄²

C₂²×D₆₇

yes

characteristic

122

C₄²

C₂²×D₆₇

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₄×D₆₇

yes

characteristic

125

Q₈

—

134

—

126

D₄

—

134

—

127

D₄

—

134

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

134

—

132

C₂×C₄

—

134

—

133

C₂×C₄

C₂×C₄×D₆₇

yes

—

134

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

135

C₂×C₄

C₂×C₄×D₆₇

yes

—

136

C₂×C₄

#11761628

yes

—

137

C₂×C₄

C₂³×D₆₇

yes

characteristic

138

C₈

—

139

C₈

—

134

—

140

C₂²

—

141

C₂²

—

134

—

142

C₂²

—

134

—

143

C₂²

#14524000

yes

—

144

C₂²

C₂²×C₄×D₆₇

yes

characteristic

145

C₄

—

146

C₄

—

134

—

147

C₄

—

134

—

148

C₄

#14524000

yes

—

149

C₄

#14524051

yes

—

150

C₄

C₂×#11761628

yes

—

151

C₂

—

152

C₂

—

268

—

153

C₂

C₂×#14524000

yes

characteristic

154

C₂

C₂×#14524051

yes

—

	subgroup	G/H
center	C₄²	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₄×M₁₆×C₆₇	C₂
hypercenter	C₄×M₁₆	D₆₇
hypocenter	C₆₇	#2922
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₄×M₁₆×C₆₇

C₂

hypercenter

C₄×M₁₆

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #16863458	C₂	semidirect
3. #16863447	C₂	semidirect
4. #16863424	C₂	semidirect
6. C₂×C₄×C₆₇⋊C₈	C₂	semidirect
7. C₄×M₁₆×C₆₇	C₂	semidirect
78. C₆₇	#2922	semidirect
84. C₄×M₁₆	D₆₇	semidirect

H ≅ G/N

type

2. #16863458

C₂

semidirect

3. #16863447

C₂

semidirect

4. #16863424

C₂

semidirect

6. C₂×C₄×C₆₇⋊C₈

C₂

semidirect

7. C₄×M₁₆×C₆₇

C₂

semidirect

78. C₆₇

#2922

semidirect

84. C₄×M₁₆

D₆₇

semidirect

Subgroups — #22568854

Conjugacy classes of subgroups

Complements