Group #30743099 — subgroups

Subgroups — #30743099

Order 17152.

204 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22562012	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523979	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523979	—	2	1	no	—
5	C₄×#11761612	—	2	1	no	—
6	C₄⋊(C₆₇⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	#16863172	—	2	1	no	—
9	#16863161	C₄	1	2	yes	—
10	#432×C₆₇	—	2	1	no	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#335×C₆₇	—	2	1	no	—
13	#334×C₆₇	C₄	1	2	yes	—
14	C₂×#11761612	—	2	2	no	—
15	C₂×C₆₇⋊C₁₆	—	2	2	no	—
16	#14523979	—	2	4	no	—
17	#14523954	—	4	1	no	—
18	C₄×C₆₇⋊C₈	—	4	1	no	—
19	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	—	2	2	no	—
23	C₂²⋊C₈×C₆₇	—	4	1	no	—
24	#97×C₆₇	—	4	1	no	—
25	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
29	#11761612	—	2	4	no	—
30	#11761612	—	4	2	no	—
31	C₂×C₆₇⋊C₈	—	4	2	no	—
32	C₆₇⋊C₁₆	—	4	2	no	—
33	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
36	C₂×C₈×C₆₇	—	2	2	no	—
37	C₂×C₈×C₆₇	—	4	1	no	—
38	C₂×C₈×C₆₇	—	4	2	no	—
39	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₆₇	D₈	1	1	yes	characteristic
46	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
47	C₂³×C₆₇	#103	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	1	no	—
49	C₂×C₄×C₆₇	—	2	1	no	—
50	C₂×C₄×C₆₇	—	4	1	no	—
51	C₂×C₄×C₆₇	—	4	1	no	—
52	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
58	C₈×C₆₇	—	2	1	no	—
59	C₈×C₆₇	—	2	1	no	—
60	C₈×C₆₇	—	2	1	no	—
61	C₈×C₆₇	—	8	2	no	—
62	C₈×C₆₇	#104	1	2	yes	—
63	C₆₇⋊C₈	—	4	4	no	—
64	C₂²×C₆₇	—	2	1	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	—	2	1	no	—
67	C₂²×C₆₇	#340	1	1	yes	characteristic
68	C₄×C₆₇	—	2	1	no	—
69	C₄×C₆₇	—	2	1	no	—
70	C₄×C₆₇	—	2	1	no	—
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	#339	1	1	yes	characteristic
74	C₄×C₆₇	#344	1	1	yes	characteristic
75	#7303	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	—	4	1	no	—
77	C₂×C₆₇	#1303	1	1	yes	characteristic
78	C₂×C₆₇	#1385	1	1	yes	characteristic
79	C₂×C₆₇	#1389	1	1	yes	characteristic
80	#2163	—	67	1	no	—
81	C₄×#108	—	67	1	no	—
82	#1568	D₆₇	1	1	yes	characteristic
83	C₆₇	#7303	1	1	yes	characteristic, complemented, Sylow
84	#432	—	2	1	no	—
85	C₂×#108	—	134	1	no	—
86	C₄×M₁₆	—	134	1	no	—
87	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
88	C₄⋊C₁₆	—	67	2	no	—
89	C₂²⋊C₁₆	—	134	1	no	—
90	#346	—	134	1	no	—
91	#341	—	67	2	no	—
92	#335	—	2	1	no	—
93	#334	Dic₆₇	1	2	yes	—
94	C₂×M₁₆	—	134	2	no	—
95	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
96	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
98	C₂×C₁₆	—	134	2	no	—
99	#108	—	134	4	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	#97	—	4	1	no	—
103	#97	—	268	1	no	—
104	C₄×C₈	—	268	1	no	—
105	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
106	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
109	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	#11761616	1	1	yes	characteristic
111	C₂²×C₄	#11761618	1	1	yes	characteristic
112	M₁₆	—	268	2	no	—
113	M₁₆	—	134	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	268	2	no	—
118	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
119	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
120	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
122	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
123	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
124	C₄²	#11761613	1	1	yes	characteristic
125	C₄²	#11761615	1	1	yes	characteristic
126	C₁₆	—	268	2	no	—
127	C₂³	#14523966	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
133	C₂×C₄	#14523963	1	1	yes	characteristic
134	C₂×C₄	#14523964	1	1	yes	characteristic
135	C₂×C₄	#14523967	1	1	yes	characteristic
136	C₂×C₄	#14523974	1	1	yes	characteristic
137	C₂×C₄	#14523976	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	268	4	no	—
143	C₈	#14523969	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#16863158	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#16863157	1	1	yes	characteristic
154	C₄	#16863167	1	1	yes	characteristic
155	C₂	—	4	1	no	—
156	C₂	#22558110	1	1	yes	characteristic
157	C₂	#22558312	1	1	yes	characteristic
158	C₂	#22558325	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562012

C₂

yes

maximal, characteristic

C₄×#14523979

C₂

yes

maximal, characteristic

C₆₇×#1568

C₂

yes

maximal, characteristic

C₂×#14523979

—

C₄×#11761612

—

C₄⋊(C₆₇⋊C₁₆)

C₄

yes

—

C₂²⋊(C₆₇⋊C₁₆)

—

#16863172

—

#16863161

C₄

yes

—

#432×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#335×C₆₇

—

#334×C₆₇

C₄

yes

—

C₂×#11761612

—

C₂×C₆₇⋊C₁₆

—

#14523979

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

Q₈

yes

characteristic

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

Q₁₆

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₄²×C₆₇

C₂²⋊C₄

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

characteristic

C₄²×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

SD₁₆

yes

characteristic

C₂³×C₆₇

#103

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#104

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#340

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

C₆₇

#7303

yes

characteristic, complemented, Sylow

#432

—

C₂×#108

—

134

—

C₄×M₁₆

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

134

—

#346

—

134

—

#341

—

#335

—

#334

Dic₆₇

yes

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

D₂₆₈

yes

characteristic

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

C₂×C₁₆

—

134

—

#108

—

134

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

#97

—

268

—

104

C₄×C₈

—

268

—

105

C₄×C₈

Dic₁₃₄

yes

characteristic

106

C₄×C₈

C₄×D₆₇

yes

characteristic

107

C₄×C₈

C₂×Dic₆₇

yes

characteristic

108

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

109

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

110

C₂²×C₄

#11761616

yes

characteristic

111

C₂²×C₄

#11761618

yes

characteristic

112

M₁₆

—

268

—

113

M₁₆

—

134

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

268

—

118

C₂×C₈

C₄×Dic₆₇

yes

characteristic

119

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

120

C₂×C₈

Dic₆₇⋊C₄

yes

characteristic

121

C₂×C₈

C₄⋊Dic₆₇

yes

characteristic

122

C₄²

C₂²⋊Dic₆₇

yes

characteristic

123

C₄²

Dic₆₇⋊C₄

yes

characteristic

124

C₄²

#11761613

yes

characteristic

125

C₄²

#11761615

yes

characteristic

126

C₁₆

—

268

—

127

C₂³

#14523966

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₆₇⋊#95

yes

characteristic

133

C₂×C₄

#14523963

yes

characteristic

134

C₂×C₄

#14523964

yes

characteristic

135

C₂×C₄

#14523967

yes

characteristic

136

C₂×C₄

#14523974

yes

characteristic

137

C₂×C₄

#14523976

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

268

—

143

C₈

#14523969

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#16863158

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#16863157

yes

characteristic

154

C₄

#16863167

yes

characteristic

155

C₂

—

156

C₂

#22558110

yes

characteristic

157

C₂

#22558312

yes

characteristic

158

C₂

#22558325

yes

characteristic

	subgroup	G/H
center	C₂²	#16863158
commutator	C₂×C₈×C₆₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₆₇×#1568	C₂
socle	C₂²×C₆₇	#340
prime radical	C₂³×C₆₇	#103
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#7303
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

subgroup

G/H

center

C₂²

#16863158

commutator

C₂×C₈×C₆₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₇

Fitting

C₆₇×#1568

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄×C₈×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

N	H ≅ G/N	type
83. C₆₇	#7303	semidirect

H ≅ G/N

type

83. C₆₇

#7303

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22562012	2
C₄×#14523979	2
C₆₇×#1568	2
#7303	67

subgroup

index

#22562012

C₄×#14523979

C₆₇×#1568

#7303

Subgroups — #30743099

Conjugacy classes of subgroups

Complements

Maximal subgroups