Group #22565418 — subgroups

Subgroups — #22565418

Order 8576.

500 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂×C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#16863315	C₂	1	4	yes	maximal
3	#16863301	C₂	1	2	yes	maximal
4	#16863292	C₂	1	4	yes	maximal
5	#16863265	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₃₄	—	2	4	no	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
13	C₂×Dic₆₇⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
16	C₆₇⋊#95	C₂²	1	2	yes	—
17	C₆₇⋊#95	C₂²	1	8	yes	—
18	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₆₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	2	no	—
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×Dic₁₃₄	Q₈	1	4	yes	complemented
26	C₂²×Dic₆₇	C₂³	1	2	yes	—
27	C₂²×Dic₆₇	C₂³	1	2	yes	—
28	C₂²×Dic₆₇	C₂³	1	4	yes	—
29	C₄⋊Dic₆₇	—	2	2	no	—
30	C₄⋊Dic₆₇	Q₈	1	8	yes	complemented
31	Dic₆₇⋊C₄	—	2	4	no	—
32	Dic₆₇⋊C₄	—	4	8	no	—
33	C₄×Dic₆₇	—	2	4	no	—
34	C₄×Dic₆₇	—	2	4	no	—
35	C₂×Q₈×C₆₇	—	2	4	no	—
36	C₂×Q₈×C₆₇	—	4	4	no	—
37	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
38	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₄²×C₆₇	D₄	1	4	yes	—
44	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
45	Q₈×C₆₇	—	4	8	no	—
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	—	2	8	no	—
49	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₆₇	#41	1	2	yes	—
55	C₂×Dic₆₇	—	2	2	no	—
56	C₂×Dic₆₇	—	2	4	no	—
57	C₂×Dic₆₇	—	2	2	no	—
58	C₂×Dic₆₇	—	2	4	no	—
59	C₂×Dic₆₇	—	2	2	no	—
60	C₂×Dic₆₇	—	2	4	no	—
61	C₂×Dic₆₇	C₂×Q₈	1	8	yes	—
62	C₂×Dic₆₇	C₂×Q₈	1	4	yes	—
63	Dic₁₃₄	—	2	8	no	—
64	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
67	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
68	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
69	C₄×C₆₇	—	2	2	no	—
70	C₄×C₆₇	—	4	4	no	—
71	C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
72	C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
73	Dic₆₇	—	2	4	no	—
74	Dic₆₇	—	2	8	no	—
75	Dic₆₇	—	4	2	no	—
76	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₆₇	#535	1	1	yes	characteristic
79	C₂×C₆₇	#554	1	2	yes	—
80	C₂×C₆₇	Q₈²	1	2	yes	—
81	#2567	—	67	1	no	maximal, Sylow
82	C₆₇	#2567	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
84	C₂×C₄×Q₈	—	67	1	no	—
85	#395	—	67	4	no	—
86	#391	—	67	2	no	—
87	#389	—	67	4	no	—
88	#384	—	67	2	no	—
89	C₄×C₄⋊C₄	—	67	1	no	—
90	C₂²×Q₈	—	67	1	no	—
91	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₄×Q₈	—	134	4	no	—
94	C₂×C₄⋊C₄	—	67	8	no	—
95	C₂×C₄⋊C₄	—	67	2	no	—
96	C₂×C₄⋊C₄	—	67	2	no	—
97	C₂×C₄⋊C₄	—	67	1	no	—
98	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
99	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
100	C₂×C₄²	—	67	2	no	—
101	C₂×C₄²	—	67	2	no	—
102	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
103	#95	—	67	2	no	—
104	#95	—	67	8	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	4	no	—
107	C₂×Q₈	—	67	4	no	—
108	C₂×Q₈	—	134	2	no	—
109	C₂×Q₈	—	134	2	no	—
110	C₂²×C₄	—	67	2	no	—
111	C₂²×C₄	—	67	4	no	—
112	C₂²×C₄	—	67	2	no	—
113	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
114	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	268	8	no	—
120	C₄⋊C₄	—	67	8	no	—
121	C₄⋊C₄	—	134	4	no	—
122	C₄⋊C₄	—	134	2	no	—
123	C₄²	—	134	4	no	—
124	C₄²	—	134	4	no	—
125	C₄²	C₂²⋊D₆₇	1	4	yes	—
126	C₂³	C₂³×D₆₇	1	1	yes	characteristic
127	Q₈	—	4	8	no	—
128	Q₈	—	134	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	8	no	—
132	C₂×C₄	—	67	8	no	—
133	C₂×C₄	—	67	4	no	—
134	C₂×C₄	—	134	2	no	—
135	C₂×C₄	—	134	4	no	—
136	C₂×C₄	—	134	2	no	—
137	C₂×C₄	—	134	4	no	—
138	C₂×C₄	—	134	2	no	—
139	C₂×C₄	—	134	4	no	—
140	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
143	C₂×C₄	Q₈×D₆₇	1	2	yes	—
144	C₂×C₄	Q₈×D₆₇	1	4	yes	—
145	C₂×C₄	#11761629	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂²	C₂×Q₈×D₆₇	1	2	yes	—
148	C₂²	C₂×Q₈×D₆₇	1	1	yes	characteristic
149	C₂²	C₂×#11761629	1	1	yes	characteristic
150	C₂²	#14524088	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	268	2	no	—
154	C₄	—	134	8	no	—
155	C₄	—	134	4	no	—
156	C₄	#14524019	1	4	yes	—
157	C₄	#14524041	1	4	yes	—
158	C₂	C₂×#14524019	1	1	yes	characteristic
159	C₂	C₂×#14524041	1	1	yes	characteristic
160	C₂	#16863963	1	1	yes	characteristic
161	C₂	#16864107	1	2	yes	—
162	C₂	#16864126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863315

C₂

yes

maximal

#16863301

C₂

yes

maximal

#16863292

C₂

yes

maximal

#16863265

C₂

yes

maximal

C₄×C₄⋊Dic₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic

C₄×Dic₁₃₄

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

Q₈

yes

complemented

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

Q₈

yes

complemented

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×Q₈

yes

—

C₂×Dic₆₇

C₂×Q₈

yes

—

Dic₁₃₄

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

—

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₄⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₇

C₂×C₄⋊Q₈

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

Q₈²

yes

—

#2567

—

maximal, Sylow

C₆₇

#2567

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₆₇

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×D₆₇

yes

characteristic

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

134

—

109

C₂×Q₈

—

134

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₆₇

yes

—

114

C₂²×C₄

C₂²×D₆₇

yes

characteristic

115

C₂²×C₄

C₂²×D₆₇

yes

characteristic

116

C₂²×C₄

C₂²×D₆₇

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

268

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

134

—

122

C₄⋊C₄

—

134

—

123

C₄²

—

134

—

124

C₄²

—

134

—

125

C₄²

C₂²⋊D₆₇

yes

—

126

C₂³

C₂³×D₆₇

yes

characteristic

127

Q₈

—

128

Q₈

—

134

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

134

—

135

C₂×C₄

—

134

—

136

C₂×C₄

—

134

—

137

C₂×C₄

—

134

—

138

C₂×C₄

—

134

—

139

C₂×C₄

—

134

—

140

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

142

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

143

C₂×C₄

Q₈×D₆₇

yes

—

144

C₂×C₄

Q₈×D₆₇

yes

—

145

C₂×C₄

#11761629

yes

—

146

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

147

C₂²

C₂×Q₈×D₆₇

yes

—

148

C₂²

C₂×Q₈×D₆₇

yes

characteristic

149

C₂²

C₂×#11761629

yes

characteristic

150

C₂²

#14524088

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

268

—

154

C₄

—

134

—

155

C₄

—

134

—

156

C₄

#14524019

yes

—

157

C₄

#14524041

yes

—

158

C₂

C₂×#14524019

yes

characteristic

159

C₂

C₂×#14524041

yes

characteristic

160

C₂

#16863963

yes

characteristic

161

C₂

#16864107

yes

—

162

C₂

#16864126

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#2567
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
25. C₂×Dic₁₃₄	Q₈	semidirect
30. C₄⋊Dic₆₇	Q₈	semidirect
82. C₆₇	#2567	semidirect

H ≅ G/N

type

25. C₂×Dic₁₃₄

Q₈

semidirect

30. C₄⋊Dic₆₇

Q₈

semidirect

82. C₆₇

#2567

semidirect

Subgroups — #22565418

Conjugacy classes of subgroups

Complements