Group #30746282 — subgroups

Subgroups — #30746282

Order 17152.

330 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22559895	C₂	1	1	yes	maximal, characteristic
2	#22558501	C₂	1	1	yes	maximal, characteristic
3	#22558039	C₂	1	2	yes	maximal
4	#22558037	C₂	1	2	yes	maximal
5	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523954	C₂²	1	1	yes	characteristic
8	C₂×#14523953	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
10	#16863242	C₂²	1	1	yes	characteristic
11	#16863118	C₄	1	4	yes	complemented
12	#431×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#322×C₆₇	C₄	1	4	yes	complemented
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
17	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
19	#14523954	—	2	2	no	—
20	#14523953	—	2	4	no	—
21	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
22	C₈×Dic₆₇	—	2	4	no	—
23	C₆₇⋊#95	—	2	2	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	2	no	—
28	C₂²⋊C₈×C₆₇	—	2	2	no	—
29	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×Dic₆₇	—	2	2	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	2	4	no	—
36	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
37	C₄×Dic₆₇	—	2	2	no	—
38	C₄×Dic₆₇	M₁₆	1	4	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	#41	1	2	yes	—
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	—	2	2	no	—
43	C₂×C₈×C₆₇	—	2	4	no	—
44	C₂×C₈×C₆₇	C₄²	1	2	yes	—
45	C₂×C₈×C₆₇	C₄²	1	2	yes	—
46	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₆₇	#41	1	2	yes	—
49	C₂³×C₆₇	#117	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#117	1	2	yes	—
53	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#117	1	2	yes	—
55	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
57	C₂×Dic₆₇	—	2	4	no	—
58	C₂×Dic₆₇	—	4	4	no	—
59	C₈×C₆₇	—	2	2	no	—
60	C₈×C₆₇	—	2	2	no	—
61	C₈×C₆₇	—	4	4	no	—
62	C₆₇⋊C₈	—	2	4	no	—
63	C₆₇⋊C₈	—	4	4	no	—
64	C₂²×C₆₇	—	2	2	no	—
65	C₂²×C₆₇	#376	1	1	yes	characteristic
66	C₂²×C₆₇	#432	1	2	yes	—
67	C₄×C₆₇	—	2	2	no	—
68	C₄×C₆₇	—	2	2	no	—
69	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
71	C₄×C₆₇	#432	1	2	yes	—
72	Dic₆₇	—	4	4	no	—
73	#8140	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	—	2	2	no	—
75	C₂×C₆₇	#1745	1	1	yes	characteristic
76	C₂×C₆₇	#1747	1	2	yes	—
77	#1726	—	67	1	no	—
78	#1452	—	67	1	no	—
79	#1452	D₆₇	1	1	yes	characteristic
80	#1275	—	67	2	no	—
81	#1274	—	67	2	no	—
82	C₆₇	#8140	1	1	yes	characteristic, complemented, Sylow
83	#431	—	67	2	no	—
84	#431	C₂×D₆₇	1	2	yes	—
85	C₂×#97	—	67	2	no	—
86	C₂×#97	—	67	1	no	—
87	C₂×C₄×C₈	—	67	2	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	#376	—	67	1	no	—
90	#322	—	67	4	no	—
91	#322	Dic₆₇	1	4	yes	complemented
92	C₂²×C₈	—	67	2	no	—
93	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
95	C₂×C₄²	—	67	2	no	—
96	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	134	2	no	—
99	C₂²⋊C₈	—	2	2	no	—
100	C₂²⋊C₈	—	134	2	no	—
101	#97	—	134	4	no	—
102	#97	—	134	2	no	—
103	C₄×C₈	—	67	4	no	—
104	C₄×C₈	—	134	4	no	—
105	C₄×C₈	C₄×D₆₇	1	2	yes	—
106	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
107	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
108	#95	—	134	2	no	—
109	C₂²×C₄	—	134	2	no	—
110	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
111	C₂²×C₄	#11761628	1	2	yes	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	67	4	no	—
116	C₂×C₈	—	134	4	no	—
117	C₂×C₈	—	134	2	no	—
118	C₂×C₈	—	134	2	no	—
119	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
120	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
121	C₄²	—	67	4	no	—
122	C₄²	—	134	2	no	—
123	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
124	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
125	C₄²	#11761628	1	2	yes	—
126	C₂³	#14524000	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	268	4	no	—
129	C₂×C₄	—	134	4	no	—
130	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
131	C₂×C₄	#14524000	1	2	yes	—
132	C₂×C₄	#14524000	1	1	yes	characteristic
133	C₂×C₄	#14524002	1	2	yes	—
134	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
135	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	4	4	no	—
139	C₈	—	268	4	no	—
140	C₈	—	134	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#16863242	1	1	yes	characteristic
143	C₂²	#16863451	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	268	4	no	—
147	C₄	M₁₆×Dic₆₇	1	1	yes	characteristic
148	C₄	M₁₆×Dic₆₇	1	1	yes	characteristic
149	C₄	#16863451	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#22559957	1	1	yes	characteristic
152	C₂	#22559964	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559895

C₂

yes

maximal, characteristic

#22558501

C₂

yes

maximal, characteristic

#22558039

C₂

yes

maximal

#22558037

C₂

yes

maximal

C₆₇×#1452

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×#14523954

C₂²

yes

characteristic

C₂×#14523953

C₄

yes

—

C₂×C₈×Dic₆₇

C₄

yes

—

#16863242

C₂²

yes

characteristic

#16863118

C₄

yes

complemented

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

#14523953

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₈×Dic₆₇

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

M₁₆

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

—

C₂×C₈×C₆₇

C₄²

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

#432

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×M₁₆

yes

characteristic

C₄×C₆₇

C₄×M₁₆

yes

characteristic

C₄×C₆₇

#432

yes

—

Dic₆₇

—

#8140

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1745

yes

characteristic

C₂×C₆₇

#1747

yes

—

#1726

—

#1452

—

#1452

D₆₇

yes

characteristic

#1275

—

#1274

—

C₆₇

#8140

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₆₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

—

Dic₆₇

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

134

—

101

#97

—

134

—

102

#97

—

134

—

103

C₄×C₈

—

104

C₄×C₈

—

134

—

105

C₄×C₈

C₄×D₆₇

yes

—

106

C₄×C₈

C₂×Dic₆₇

yes

—

107

C₄×C₈

C₂×Dic₆₇

yes

—

108

#95

—

134

—

109

C₂²×C₄

—

134

—

110

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

111

C₂²×C₄

#11761628

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

134

—

117

C₂×C₈

—

134

—

118

C₂×C₈

—

134

—

119

C₂×C₈

C₄×Dic₆₇

yes

—

120

C₂×C₈

C₄×Dic₆₇

yes

—

121

C₄²

—

122

C₄²

—

134

—

123

C₄²

C₂×C₄×D₆₇

yes

characteristic

124

C₄²

C₂²×Dic₆₇

yes

characteristic

125

C₄²

#11761628

yes

—

126

C₂³

#14524000

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

268

—

129

C₂×C₄

—

134

—

130

C₂×C₄

C₂×C₄×Dic₆₇

yes

characteristic

131

C₂×C₄

#14524000

yes

—

132

C₂×C₄

#14524000

yes

characteristic

133

C₂×C₄

#14524002

yes

—

134

C₂×C₄

M₁₆×D₆₇

yes

—

135

C₂×C₄

M₁₆×D₆₇

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

268

—

140

C₈

—

134

—

141

C₂²

—

142

C₂²

#16863242

yes

characteristic

143

C₂²

#16863451

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

268

—

147

C₄

M₁₆×Dic₆₇

yes

characteristic

148

C₄

M₁₆×Dic₆₇

yes

characteristic

149

C₄

#16863451

yes

—

150

C₂

—

151

C₂

#22559957

yes

characteristic

152

C₂

#22559964

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	#376
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#8140
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #1452

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₆₇

commutator

C₂×C₄×C₆₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1452

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #1452

N	H ≅ G/N	type
11. #16863118	C₄	semidirect
14. #322×C₆₇	C₄	semidirect
82. C₆₇	#8140	semidirect
91. #322	Dic₆₇	semidirect

H ≅ G/N

type

11. #16863118

C₄

semidirect

14. #322×C₆₇

C₄

semidirect

82. C₆₇

#8140

semidirect

91. #322

Dic₆₇

semidirect

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

subgroup	index
#22559895	2
#22558501	2
#22558039	2
#22558037	2
C₆₇×#1452	2
#8140	67

subgroup

index

C₆₇×#1452

Subgroups — #30746282

Conjugacy classes of subgroups

Complements

Maximal subgroups