Group #22558331 — subgroups

Subgroups — #22558331

Order 8576.

158 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16863380	C₂	1	1	yes	maximal, characteristic
2	#16863363	C₂	1	1	yes	maximal, characteristic
3	#413×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	—	2	1	no	—
5	C₂×#11761612	C₄	1	1	yes	characteristic
6	C₂²×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	#14523962	—	2	1	no	—
8	#14523960	C₄	1	2	yes	—
9	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
11	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₆₇	—	2	1	no	—
13	#101×C₆₇	—	2	1	no	—
14	#100×C₆₇	C₄	1	2	yes	—
15	#11761612	—	2	2	no	—
16	#11761612	—	4	1	no	—
17	#11761612	—	4	1	no	—
18	C₂×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₆₇⋊C₈	—	2	1	no	—
20	C₂×C₆₇⋊C₈	—	4	1	no	—
21	C₂×C₆₇⋊C₈	—	4	1	no	—
22	#41×C₆₇	—	4	1	no	—
23	#41×C₆₇	—	4	1	no	—
24	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₆₇	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
31	M₁₆×C₆₇	—	2	2	no	—
32	M₁₆×C₆₇	—	4	1	no	—
33	C₂×C₈×C₆₇	—	4	1	no	—
34	C₂³×C₆₇	C₄²	1	1	yes	characteristic
35	C₂³×C₆₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂³×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	Q₈×C₆₇	—	4	1	no	—
38	D₄×C₆₇	—	4	1	no	—
39	D₄×C₆₇	—	4	1	no	—
40	D₄×C₆₇	—	4	1	no	—
41	C₂×C₄×C₆₇	—	2	1	no	—
42	C₂×C₄×C₆₇	—	2	1	no	—
43	C₂×C₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	2	1	no	—
46	C₂×C₄×C₆₇	—	2	1	no	—
47	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
51	C₈×C₆₇	—	4	2	no	—
52	C₆₇⋊C₈	—	4	2	no	—
53	C₆₇⋊C₈	—	4	1	no	—
54	C₆₇⋊C₈	—	4	1	no	—
55	C₂²×C₆₇	—	2	1	no	—
56	C₂²×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	—	4	1	no	—
59	C₂²×C₆₇	—	4	1	no	—
60	C₂²×C₆₇	—	4	1	no	—
61	C₂²×C₆₇	#95	1	1	yes	characteristic
62	C₄×C₆₇	—	2	1	no	—
63	C₄×C₆₇	—	2	1	no	—
64	C₄×C₆₇	—	2	1	no	—
65	C₄×C₆₇	#99	1	1	yes	characteristic
66	C₄×C₆₇	#99	1	1	yes	characteristic
67	C₂×C₆₇	—	2	1	no	—
68	C₂×C₆₇	—	4	1	no	—
69	C₂×C₆₇	—	4	1	no	—
70	C₂×C₆₇	—	4	1	no	—
71	C₂×C₆₇	#342	1	1	yes	characteristic
72	#1392	—	67	1	no	maximal, Sylow
73	C₆₇	#1392	1	1	yes	characteristic, complemented, Sylow
74	#413	—	67	1	no	—
75	#413	D₆₇	1	1	yes	characteristic
76	#408	—	67	1	no	—
77	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
78	C₂×M₁₆	—	2	1	no	—
79	C₂×M₁₆	—	67	1	no	—
80	C₂×M₁₆	—	134	1	no	—
81	C₂²×C₈	—	67	1	no	—
82	#101	—	2	1	no	—
83	#101	—	134	1	no	—
84	#100	—	67	2	no	—
85	#100	Dic₆₇	1	2	yes	—
86	C₂²⋊C₈	—	134	1	no	—
87	C₂²⋊C₈	—	134	1	no	—
88	#41	—	4	1	no	—
89	#41	—	4	1	no	—
90	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
91	C₂×D₄	Dic₁₃₄	1	1	yes	characteristic
92	C₂×D₄	C₄×D₆₇	1	1	yes	characteristic
93	C₂×D₄	C₂×Dic₆₇	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
95	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	—	4	1	no	—
99	M₁₆	—	268	1	no	—
100	M₁₆	—	268	1	no	—
101	M₁₆	—	134	2	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	268	1	no	—
104	C₂×C₈	—	268	1	no	—
105	C₂×C₈	—	134	1	no	—
106	C₂×C₈	—	134	1	no	—
107	C₂³	C₄×Dic₆₇	1	1	yes	characteristic
108	C₂³	Dic₆₇⋊C₄	1	1	yes	characteristic
109	C₂³	C₄⋊Dic₆₇	1	1	yes	characteristic
110	Q₈	—	4	1	no	—
111	D₄	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	4	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
123	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
124	C₈	—	4	2	no	—
125	C₈	—	268	2	no	—
126	C₈	—	268	1	no	—
127	C₈	—	268	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	4	1	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	C₆₇⋊#95	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	#14523957	1	1	yes	characteristic
139	C₄	#14523957	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	4	1	no	—
142	C₂	—	4	1	no	—
143	C₂	—	4	1	no	—
144	C₂	#16863162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863380

C₂

yes

maximal, characteristic

#16863363

C₂

yes

maximal, characteristic

#413×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

C₂×#11761612

C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₄

yes

characteristic

#14523962

—

#14523960

C₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₂×#41×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

—

#101×C₆₇

—

#100×C₆₇

C₄

yes

—

#11761612

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

C₂×C₄

yes

characteristic

C₂×D₄×C₆₇

C₂×C₄

yes

characteristic

C₂×D₄×C₆₇

Q₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂³×C₆₇

C₄²

yes

characteristic

C₂³×C₆₇

C₄⋊C₄

yes

characteristic

C₂³×C₆₇

C₄⋊C₄

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×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

C₄⋊C₄

yes

characteristic

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#95

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#413

—

#413

D₆₇

yes

characteristic

#408

—

C₂×#41

C₂×D₆₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

—

#101

—

#101

—

134

—

#100

—

#100

Dic₆₇

yes

—

C₂²⋊C₈

—

134

—

C₂²⋊C₈

—

134

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

C₂×D₄

Dic₁₃₄

yes

characteristic

C₂×D₄

C₄×D₆₇

yes

characteristic

C₂×D₄

C₂×Dic₆₇

yes

characteristic

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

268

—

100

M₁₆

—

268

—

101

M₁₆

—

134

—

102

C₂×C₈

—

103

C₂×C₈

—

268

—

104

C₂×C₈

—

268

—

105

C₂×C₈

—

134

—

106

C₂×C₈

—

134

—

107

C₂³

C₄×Dic₆₇

yes

characteristic

108

C₂³

Dic₆₇⋊C₄

yes

characteristic

109

C₂³

C₄⋊Dic₆₇

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

123

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

124

C₈

—

125

C₈

—

268

—

126

C₈

—

268

—

127

C₈

—

268

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₆₇⋊#95

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#14523957

yes

characteristic

139

C₄

#14523957

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#16863162

yes

characteristic

	subgroup	G/H
center	C₄	#14523957
commutator	C₂³×C₆₇	C₄²
Frattini	C₂×#41	C₂×D₆₇
Fitting	#413×C₆₇	C₂
socle	C₂×C₆₇	#342
hypercenter	#413	D₆₇
hypocenter	C₆₇	#1392
LCS	G ▷ C₂³×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

subgroup

G/H

center

C₄

#14523957

commutator

C₂³×C₆₇

C₄²

Frattini

C₂×#41

C₂×D₆₇

Fitting

#413×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

N	H ≅ G/N	type
73. C₆₇	#1392	semidirect

H ≅ G/N

type

73. C₆₇

#1392

semidirect

Subgroups — #22558331

Conjugacy classes of subgroups

Complements