Group #22568570 — subgroups

Subgroups — #22568570

Order 8576.

472 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14524034	C₂	1	2	yes	maximal
2	#16863295	C₂	1	4	yes	maximal
3	#16863292	C₂	1	4	yes	maximal
4	#16863266	C₂	1	1	yes	maximal, characteristic
5	#16863261	C₂	1	2	yes	maximal
6	C₄²×Dic₆₇	C₂	1	1	yes	maximal, characteristic
7	#384×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524034	—	2	8	no	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
12	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
14	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
17	C₆₇⋊#95	C₂²	1	8	yes	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×Dic₆₇	C₂³	1	4	yes	—
22	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₆₇	C₂³	1	2	yes	—
25	C₄⋊Dic₆₇	—	2	4	no	—
26	C₄⋊Dic₆₇	—	2	8	no	—
27	C₄⋊Dic₆₇	—	2	8	no	—
28	Dic₆₇⋊C₄	—	2	8	no	—
29	C₄×Dic₆₇	—	2	4	no	—
30	C₄×Dic₆₇	—	2	4	no	—
31	C₄×Dic₆₇	D₄	1	4	yes	—
32	C₄×Dic₆₇	Q₈	1	4	yes	—
33	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
36	C₄⋊C₄×C₆₇	—	2	4	no	—
37	C₄⋊C₄×C₆₇	—	2	8	no	—
38	C₄²×C₆₇	Q₈	1	4	yes	—
39	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
40	C₂×C₄×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	—	2	8	no	—
42	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
45	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₆₇	#41	1	4	yes	—
47	C₂×C₄×C₆₇	#41	1	4	yes	—
48	C₂×Dic₆₇	—	2	2	no	—
49	C₂×Dic₆₇	—	2	2	no	—
50	C₂×Dic₆₇	—	2	2	no	—
51	C₂×Dic₆₇	—	2	2	no	—
52	C₂×Dic₆₇	—	2	4	no	—
53	C₂×Dic₆₇	—	2	8	no	—
54	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
55	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
56	C₂×Dic₆₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
58	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
59	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
60	C₂²×C₆₇	C₂×#41	1	2	yes	—
61	C₂²×C₆₇	C₂×#41	1	2	yes	—
62	C₄×C₆₇	—	4	4	no	—
63	C₄×C₆₇	#125	1	8	yes	complemented
64	C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
65	Dic₆₇	—	2	2	no	—
66	Dic₆₇	—	2	2	no	—
67	Dic₆₇	—	2	4	no	—
68	Dic₆₇	—	4	4	no	—
69	C₂×C₆₇	C₂×#125	1	2	yes	—
70	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₆₇	#532	1	1	yes	characteristic
72	C₂×C₆₇	#532	1	1	yes	characteristic
73	C₂×C₆₇	#533	1	2	yes	—
74	#2859	—	67	1	no	maximal, Sylow
75	C₆₇	#2859	1	1	yes	characteristic, complemented, Sylow
76	C₂×#125	—	67	2	no	—
77	#389	—	67	4	no	—
78	#389	—	67	4	no	—
79	#384	—	67	1	no	—
80	#384	D₆₇	1	1	yes	characteristic
81	#382	—	67	2	no	—
82	C₄³	—	67	1	no	—
83	#125	—	134	8	no	—
84	C₂×C₄⋊C₄	—	67	4	no	—
85	C₂×C₄⋊C₄	—	67	2	no	—
86	C₂×C₄⋊C₄	—	67	4	no	—
87	C₂×C₄⋊C₄	—	67	4	no	—
88	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
89	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
90	C₂×C₄²	—	67	1	no	—
91	C₂×C₄²	—	67	1	no	—
92	C₂×C₄²	—	67	2	no	—
93	C₂×C₄²	—	67	2	no	—
94	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
95	#95	—	67	8	no	—
96	C₂²×C₄	—	67	2	no	—
97	C₂²×C₄	—	67	1	no	—
98	C₂²×C₄	—	67	1	no	—
99	C₂²×C₄	—	67	4	no	—
100	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
101	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	8	no	—
105	C₄⋊C₄	—	134	8	no	—
106	C₄⋊C₄	—	134	8	no	—
107	C₄⋊C₄	—	134	4	no	—
108	C₄⋊C₄	—	134	8	no	—
109	C₄²	—	67	4	no	—
110	C₄²	—	67	4	no	—
111	C₄²	—	134	4	no	—
112	C₄²	—	134	4	no	—
113	C₄²	Dic₁₃₄	1	4	yes	—
114	C₂³	C₂³×D₆₇	1	1	yes	characteristic
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	8	no	—
117	C₂×C₄	—	67	2	no	—
118	C₂×C₄	—	67	2	no	—
119	C₂×C₄	—	67	4	no	—
120	C₂×C₄	—	134	4	no	—
121	C₂×C₄	—	134	8	no	—
122	C₂×C₄	—	134	2	no	—
123	C₂×C₄	—	134	2	no	—
124	C₂×C₄	—	134	2	no	—
125	C₂×C₄	—	134	2	no	—
126	C₂×C₄	D₄×D₆₇	1	2	yes	—
127	C₂×C₄	Q₈×D₆₇	1	2	yes	—
128	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
129	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
130	C₂×C₄	#11761627	1	4	yes	—
131	C₂×C₄	#11761629	1	4	yes	—
132	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
133	C₂²	C₂×Q₈×D₆₇	1	1	yes	characteristic
134	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
135	C₂²	C₂×#11761627	1	2	yes	—
136	C₂²	C₂×#11761629	1	2	yes	—
137	C₄	—	4	4	no	—
138	C₄	—	268	4	no	—
139	C₄	—	134	2	no	—
140	C₄	—	134	2	no	—
141	C₄	—	134	4	no	—
142	C₄	#14524034	1	8	yes	complemented
143	C₄	#14524042	1	4	yes	—
144	C₂	C₂×#14524034	1	2	yes	—
145	C₂	C₂×#14524042	1	1	yes	characteristic
146	C₂	#16863944	1	1	yes	characteristic
147	C₂	#16863947	1	1	yes	characteristic
148	C₂	#16863951	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524034

C₂

yes

maximal

#16863295

C₂

yes

maximal

#16863292

C₂

yes

maximal

#16863266

C₂

yes

maximal, characteristic

#16863261

C₂

yes

maximal

C₄²×Dic₆₇

C₂

yes

maximal, characteristic

#384×C₆₇

C₂

yes

maximal, characteristic

#14524034

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

C₆₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

Q₈

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×Q₈

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

C₂×#41

yes

—

C₄×C₆₇

—

C₄×C₆₇

#125

yes

complemented

C₄×C₆₇

C₄⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×#125

yes

—

C₂×C₆₇

C₂×C₄⋊Q₈

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#2859

—

maximal, Sylow

C₆₇

#2859

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic

#382

—

C₄³

—

#125

—

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

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂²×D₆₇

yes

—

101

C₂²×C₄

C₂²×D₆₇

yes

characteristic

102

C₂²×C₄

C₂²×D₆₇

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄⋊C₄

—

134

—

108

C₄⋊C₄

—

134

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

134

—

112

C₄²

—

134

—

113

C₄²

Dic₁₃₄

yes

—

114

C₂³

C₂³×D₆₇

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

134

—

121

C₂×C₄

—

134

—

122

C₂×C₄

—

134

—

123

C₂×C₄

—

134

—

124

C₂×C₄

—

134

—

125

C₂×C₄

—

134

—

126

C₂×C₄

D₄×D₆₇

yes

—

127

C₂×C₄

Q₈×D₆₇

yes

—

128

C₂×C₄

C₂×Dic₁₃₄

yes

—

129

C₂×C₄

C₂×Dic₁₃₄

yes

—

130

C₂×C₄

#11761627

yes

—

131

C₂×C₄

#11761629

yes

—

132

C₂²

C₂×D₄×D₆₇

yes

characteristic

133

C₂²

C₂×Q₈×D₆₇

yes

characteristic

134

C₂²

C₂²×Dic₁₃₄

yes

characteristic

135

C₂²

C₂×#11761627

yes

—

136

C₂²

C₂×#11761629

yes

—

137

C₄

—

138

C₄

—

268

—

139

C₄

—

134

—

140

C₄

—

134

—

141

C₄

—

134

—

142

C₄

#14524034

yes

complemented

143

C₄

#14524042

yes

—

144

C₂

C₂×#14524034

yes

—

145

C₂

C₂×#14524042

yes

characteristic

146

C₂

#16863944

yes

characteristic

147

C₂

#16863947

yes

characteristic

148

C₂

#16863951

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#384×C₆₇	C₂
hypercenter	#384	D₆₇
hypocenter	C₆₇	#2859
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#384×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
63. C₄×C₆₇	#125	semidirect
75. C₆₇	#2859	semidirect
142. C₄	#14524034	semidirect

H ≅ G/N

type

63. C₄×C₆₇

#125

semidirect

75. C₆₇

#2859

semidirect

142. C₄

#14524034

semidirect

Subgroups — #22568570

Conjugacy classes of subgroups

Complements