Group #22566184 — subgroups

Subgroups — #22566184

Order 8576.

360 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16863338	C₂	1	4	yes	maximal
2	#16863330	C₂	1	4	yes	maximal
3	#16863294	C₂	1	2	yes	maximal
4	#16863294	C₂	1	2	yes	maximal
5	#16863266	C₂	1	1	yes	maximal, characteristic
6	#16863262	C₂	1	1	yes	maximal, characteristic
7	#376×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
11	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
12	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
13	C₆₇⋊#95	C₂²	1	4	yes	—
14	C₆₇⋊#95	C₂²	1	4	yes	—
15	C₆₇⋊#95	C₂²	1	4	yes	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#95×C₆₇	C₂²	1	2	yes	—
20	#95×C₆₇	C₂²	1	2	yes	—
21	C₂²×Dic₆₇	C₂³	1	4	yes	—
22	C₂²×Dic₆₇	C₂³	1	4	yes	—
23	C₄⋊Dic₆₇	—	2	4	no	—
24	C₄⋊Dic₆₇	—	2	4	no	—
25	C₄⋊Dic₆₇	—	4	4	no	—
26	Dic₆₇⋊C₄	—	4	4	no	—
27	Dic₆₇⋊C₄	—	4	4	no	—
28	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₄²×C₆₇	—	2	2	no	—
34	C₄²×C₆₇	—	2	2	no	—
35	C₄²×C₆₇	Q₈	1	4	yes	—
36	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	2	1	no	—
40	C₂×C₄×C₆₇	—	2	1	no	—
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	—	2	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	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₆₇	#41	1	2	yes	—
49	C₂×C₄×C₆₇	#41	1	2	yes	—
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	—	2	4	no	—
52	C₂×Dic₆₇	—	2	4	no	—
53	C₂×Dic₆₇	—	2	4	no	—
54	C₂×Dic₆₇	—	2	4	no	—
55	C₂×Dic₆₇	—	2	4	no	—
56	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
57	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
58	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
59	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
60	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
61	C₄×C₆₇	—	2	4	no	—
62	C₄×C₆₇	—	4	1	no	—
63	C₄×C₆₇	—	4	1	no	—
64	C₄×C₆₇	—	4	2	no	—
65	C₄×C₆₇	#125	1	4	yes	—
66	Dic₆₇	—	4	4	no	—
67	Dic₆₇	—	4	4	no	—
68	C₂×C₆₇	C₂×#125	1	1	yes	characteristic
69	C₂×C₆₇	#540	1	1	yes	characteristic
70	C₂×C₆₇	#541	1	1	yes	characteristic
71	C₂×C₆₇	#542	1	2	yes	—
72	C₂×C₆₇	#544	1	2	yes	—
73	#2647	—	67	1	no	maximal, Sylow
74	C₆₇	#2647	1	1	yes	characteristic, complemented, Sylow
75	#400	—	67	4	no	—
76	#398	—	67	4	no	—
77	#389	—	67	2	no	—
78	#389	—	67	2	no	—
79	#384	—	67	1	no	—
80	#382	—	67	1	no	—
81	#376	D₆₇	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	67	4	no	—
83	C₂×C₄⋊C₄	—	67	4	no	—
84	C₂×C₄⋊C₄	—	67	4	no	—
85	C₂×C₄⋊C₄	—	67	2	no	—
86	C₂×C₄⋊C₄	—	67	2	no	—
87	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
90	#95	—	67	4	no	—
91	#95	—	67	4	no	—
92	#95	—	67	4	no	—
93	#95	C₂×D₆₇	1	2	yes	—
94	#95	C₂×D₆₇	1	2	yes	—
95	C₂²×C₄	—	67	4	no	—
96	C₂²×C₄	—	67	4	no	—
97	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
101	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
102	C₄⋊C₄	—	268	4	no	—
103	C₄⋊C₄	—	268	4	no	—
104	C₄⋊C₄	—	268	4	no	—
105	C₄⋊C₄	—	134	4	no	—
106	C₄⋊C₄	—	134	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	Dic₁₃₄	1	4	yes	—
110	C₂³	C₂³×D₆₇	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	134	4	no	—
121	C₂×C₄	—	134	4	no	—
122	C₂×C₄	—	134	4	no	—
123	C₂×C₄	—	134	4	no	—
124	C₂×C₄	—	134	4	no	—
125	C₂×C₄	—	134	4	no	—
126	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
127	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
128	C₂×C₄	#11761628	1	2	yes	—
129	C₂×C₄	#11761628	1	2	yes	—
130	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
131	C₂²	C₂×#11761628	1	1	yes	characteristic
132	C₂²	C₂×#11761628	1	1	yes	characteristic
133	C₂²	#14524089	1	2	yes	—
134	C₂²	#14524090	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	4	1	no	—
137	C₄	—	4	1	no	—
138	C₄	—	4	2	no	—
139	C₄	—	268	4	no	—
140	C₄	—	268	4	no	—
141	C₄	#14524035	1	4	yes	—
142	C₂	C₂×#14524035	1	1	yes	characteristic
143	C₂	#16864001	1	1	yes	characteristic
144	C₂	#16864008	1	1	yes	characteristic
145	C₂	#16864017	1	2	yes	—
146	C₂	#16864024	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863338

C₂

yes

maximal

#16863330

C₂

yes

maximal

#16863294

C₂

yes

maximal

#16863294

C₂

yes

maximal

#16863266

C₂

yes

maximal, characteristic

#16863262

C₂

yes

maximal, characteristic

#376×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₈

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

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

—

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#125

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#544

yes

—

#2647

—

maximal, Sylow

C₆₇

#2647

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₂×C₄²

C₂×D₆₇

yes

—

—

—

C₂×D₆₇

yes

—

#95

C₂×D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

100

C₂²×C₄

C₂²×D₆₇

yes

—

101

C₂²×C₄

C₂²×D₆₇

yes

characteristic

102

C₄⋊C₄

—

268

—

103

C₄⋊C₄

—

268

—

104

C₄⋊C₄

—

268

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₁₃₄

yes

—

110

C₂³

C₂³×D₆₇

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

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

C₂×Dic₁₃₄

yes

—

127

C₂×C₄

C₂×Dic₁₃₄

yes

—

128

C₂×C₄

#11761628

yes

—

129

C₂×C₄

#11761628

yes

—

130

C₂²

C₂²×Dic₁₃₄

yes

characteristic

131

C₂²

C₂×#11761628

yes

characteristic

132

C₂²

C₂×#11761628

yes

characteristic

133

C₂²

#14524089

yes

—

134

C₂²

#14524090

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

268

—

140

C₄

—

268

—

141

C₄

#14524035

yes

—

142

C₂

C₂×#14524035

yes

characteristic

143

C₂

#16864001

yes

characteristic

144

C₂

#16864008

yes

characteristic

145

C₂

#16864017

yes

—

146

C₂

#16864024

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#376×C₆₇	C₂
hypercenter	#376	D₆₇
hypocenter	C₆₇	#2647
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#376×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
74. C₆₇	#2647	semidirect

H ≅ G/N

type

74. C₆₇

#2647

semidirect

Subgroups — #22566184

Conjugacy classes of subgroups

Complements