Group #30864232 — subgroups

Subgroups — #30864232

Order 17152.

1266 conjugacy classes of subgroups; only the 327 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 327 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×D₄×C₆₇⋊C₈	C₂	1	6	yes	maximal, complemented
2	#22561008	C₂	1	6	yes	maximal, complemented
3	#22560363	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559896	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2397	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	—
7	D₄×C₆₇⋊C₈	—	2	24	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
9	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	12	yes	—
10	C₂×#14523954	C₂²	1	3	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
12	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	—
13	#399×C₆₇	C₄	1	4	yes	—
14	#392×C₆₇	C₄	1	4	yes	—
15	#376×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	—	2	24	no	—
17	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
18	C₂²×C₆₇⋊C₈	C₂³	1	3	yes	—
19	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
20	C₄⋊(C₆₇⋊C₈)	—	2	12	no	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	24	no	—
22	#14523954	—	2	6	no	—
23	C₄×C₆₇⋊C₈	D₄	1	12	yes	complemented
24	C₂²×D₄×C₆₇	C₂×C₄	1	6	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
26	C₄×D₄×C₆₇	—	2	12	no	—
27	C₄×D₄×C₆₇	—	2	12	no	—
28	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
29	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	12	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
31	#95×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₆₇⋊C₈	—	2	6	no	—
33	C₂×C₆₇⋊C₈	—	2	12	no	—
34	C₂×C₆₇⋊C₈	—	4	16	no	—
35	C₂×C₆₇⋊C₈	C₂×D₄	1	6	yes	—
36	C₂×C₆₇⋊C₈	C₂×D₄	1	6	yes	—
37	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
38	C₂×D₄×C₆₇	—	2	24	no	—
39	C₂×D₄×C₆₇	—	2	12	no	—
40	C₂²×C₄×C₆₇	—	2	12	no	—
41	C₂²×C₄×C₆₇	—	2	12	no	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
45	C₄⋊C₄×C₆₇	—	2	12	no	—
46	C₂²⋊C₄×C₆₇	—	2	24	no	—
47	C₄²×C₆₇	C₂×D₄	1	6	yes	—
48	C₄²×C₆₇	#41	1	6	yes	—
49	C₂³×C₆₇	—	2	12	no	—
50	C₂³×C₆₇	—	2	12	no	—
51	C₂³×C₆₇	#131	1	8	yes	—
52	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
53	D₄×C₆₇	—	2	48	no	—
54	C₂×C₄×C₆₇	—	2	12	no	—
55	C₂×C₄×C₆₇	—	2	12	no	—
56	C₂×C₄×C₆₇	—	4	4	no	—
57	C₂×C₄×C₆₇	—	4	12	no	—
58	C₂×C₄×C₆₇	C₄×D₄	1	12	yes	—
59	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
60	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
61	C₆₇⋊C₈	—	2	6	no	—
62	C₆₇⋊C₈	—	4	4	no	—
63	C₆₇⋊C₈	#127	1	4	yes	—
64	C₂²×C₆₇	—	2	24	no	—
65	C₂²×C₆₇	—	4	4	no	—
66	C₂²×C₆₇	—	4	12	no	—
67	C₂²×C₆₇	C₂×C₄×D₄	1	3	yes	—
68	C₂²×C₆₇	C₂×#131	1	4	yes	—
69	C₄×C₆₇	—	2	12	no	—
70	C₄×C₆₇	—	4	4	no	—
71	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
72	C₄×C₆₇	#532	1	3	yes	—
73	#21683	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	—	4	8	no	—
75	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
76	C₂×C₆₇	#2929	1	6	yes	—
77	C₂×C₈×D₄	—	67	6	no	—
78	#2397	D₆₇	1	1	yes	characteristic
79	#1955	—	67	6	no	—
80	#1843	—	67	1	no	—
81	#1726	—	67	1	no	—
82	C₆₇	#21683	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₈	—	67	4	no	—
84	C₂×C₄×D₄	C₂×D₆₇	1	6	yes	—
85	C₈×D₄	—	134	24	no	—
86	C₂×C₄⋊C₈	—	67	6	no	—
87	C₂×C₂²⋊C₈	—	67	12	no	—
88	C₂×#97	—	67	3	no	—
89	C₂×C₄×C₈	—	67	3	no	—
90	#399	Dic₆₇	1	4	yes	—
91	#392	Dic₆₇	1	4	yes	—
92	#376	C₂×D₆₇	1	1	yes	characteristic
93	C₂²×D₄	C₂×Dic₆₇	1	6	yes	—
94	C₂³×C₄	C₂²×D₆₇	1	4	yes	—
95	C₂²×C₈	—	67	1	no	—
96	C₂²×C₈	—	67	3	no	—
97	C₂²×C₈	—	67	4	no	—
98	C₂²×C₈	—	134	24	no	—
99	C₄×D₄	—	2	12	no	—
100	C₄×D₄	—	2	12	no	—
101	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
102	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	12	yes	—
103	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
104	C₄⋊C₈	—	134	12	no	—
105	C₂²⋊C₈	—	134	24	no	—
106	#97	—	134	6	no	—
107	C₄×C₈	—	67	12	no	—
108	#95	C₂×Dic₆₇	1	4	yes	—
109	C₂⁴	C₂²×Dic₆₇	1	4	yes	—
110	C₂×D₄	—	2	24	no	—
111	C₂×D₄	—	2	12	no	—
112	C₂²×C₄	—	2	12	no	—
113	C₂²×C₄	—	2	12	no	—
114	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
115	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
116	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
117	C₂×C₈	—	268	16	no	—
118	C₂×C₈	—	67	6	no	—
119	C₂×C₈	—	67	6	no	—
120	C₂×C₈	—	134	12	no	—
121	C₂×C₈	—	134	6	no	—
122	C₄⋊C₄	—	2	12	no	—
123	C₂²⋊C₄	—	2	24	no	—
124	C₄²	D₄×D₆₇	1	6	yes	—
125	C₄²	#11761627	1	6	yes	—
126	C₂³	—	2	12	no	—
127	C₂³	—	2	12	no	—
128	C₂³	#14524050	1	8	yes	—
129	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
130	D₄	—	2	48	no	—
131	C₂×C₄	—	2	12	no	—
132	C₂×C₄	—	2	12	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	12	no	—
135	C₂×C₄	D₄×Dic₆₇	1	12	yes	—
136	C₂×C₄	C₂×D₄×D₆₇	1	3	yes	—
137	C₂×C₄	C₂×#11761627	1	3	yes	—
138	C₈	—	268	4	no	—
139	C₈	—	67	4	no	—
140	C₈	—	134	6	no	—
141	C₂²	—	2	24	no	—
142	C₂²	—	4	12	no	—
143	C₂²	—	4	4	no	—
144	C₂²	C₂×D₄×Dic₆₇	1	3	yes	—
145	C₂²	C₂×#14524050	1	4	yes	—
146	C₄	—	2	12	no	—
147	C₄	—	4	4	no	—
148	C₄	#127×D₆₇	1	1	yes	characteristic
149	C₄	#16863944	1	3	yes	—
150	C₂	—	4	8	no	—
151	C₂	#127×Dic₆₇	1	1	yes	characteristic
152	C₂	#22568904	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×C₆₇⋊C₈

C₂

yes

maximal, complemented

#22561008

C₂

yes

maximal, complemented

#22560363

C₂

yes

maximal, characteristic, complemented

#22559896

C₂

yes

maximal, characteristic, complemented

C₆₇×#2397

C₂

yes

maximal, characteristic

C₂³×C₆₇⋊C₈

C₂²

yes

—

D₄×C₆₇⋊C₈

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×#14523954

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄×D₄×C₆₇

C₂²

yes

—

#399×C₆₇

C₄

yes

—

#392×C₆₇

C₄

yes

—

#376×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

C₂²×C₆₇⋊C₈

C₂³

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

D₄

yes

complemented

C₂²×D₄×C₆₇

C₂×C₄

yes

—

C₂³×C₄×C₆₇

C₂³

yes

—

C₄×D₄×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

—

#95×C₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₂⁴×C₆₇

C₂²×C₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#131

yes

—

C₂³×C₆₇

C₂³×C₄

yes

characteristic

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₂×C₄×C₆₇

C₂×#41

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#127

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄×D₄

yes

—

C₂²×C₆₇

C₂×#131

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#127

yes

characteristic

C₄×C₆₇

#532

yes

—

#21683

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₄×#127

yes

characteristic

C₂×C₆₇

#2929

yes

—

C₂×C₈×D₄

—

#2397

D₆₇

yes

—

—

—

C₆₇

#21683

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×C₄×D₄

C₂×D₆₇

yes

—

C₈×D₄

—

134

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#399

Dic₆₇

yes

—

#392

Dic₆₇

yes

—

#376

C₂×D₆₇

yes

characteristic

C₂²×D₄

C₂×Dic₆₇

yes

—

C₂³×C₄

C₂²×D₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

134

—

C₄×D₄

—

100

C₄×D₄

—

101

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

102

C₂×C₂²⋊C₄

C₂×Dic₆₇

yes

—

103

C₂×C₄²

C₂²×D₆₇

yes

—

104

C₄⋊C₈

—

134

—

105

C₂²⋊C₈

—

134

—

106

#97

—

134

—

107

C₄×C₈

—

108

#95

C₂×Dic₆₇

yes

—

109

C₂⁴

C₂²×Dic₆₇

yes

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²×Dic₆₇

yes

—

115

C₂²×C₄

C₂²×Dic₆₇

yes

—

116

C₂²×C₄

C₂³×D₆₇

yes

characteristic

117

C₂×C₈

—

268

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

134

—

121

C₂×C₈

—

134

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

D₄×D₆₇

yes

—

125

C₄²

#11761627

yes

—

126

C₂³

—

127

C₂³

—

128

C₂³

#14524050

yes

—

129

C₂³

C₂³×Dic₆₇

yes

characteristic

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

D₄×Dic₆₇

yes

—

136

C₂×C₄

C₂×D₄×D₆₇

yes

—

137

C₂×C₄

C₂×#11761627

yes

—

138

C₈

—

268

—

139

C₈

—

140

C₈

—

134

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×D₄×Dic₆₇

yes

—

145

C₂²

C₂×#14524050

yes

—

146

C₄

—

147

C₄

—

148

C₄

#127×D₆₇

yes

characteristic

149

C₄

#16863944

yes

—

150

C₂

—

151

C₂

#127×Dic₆₇

yes

characteristic

152

C₂

#22568904

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting, prime radical	C₆₇×#2397	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2397	D₆₇
hypocenter	C₆₇	#21683
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2397

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂³×C₄

Fitting, prime radical

C₆₇×#2397

C₂

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₄ ◁ #2397

N	H ≅ G/N	type
1. C₂×D₄×C₆₇⋊C₈	C₂	semidirect
2. #22561008	C₂	semidirect
3. #22560363	C₂	semidirect
4. #22559896	C₂	semidirect
23. C₄×C₆₇⋊C₈	D₄	semidirect
82. C₆₇	#21683	semidirect

H ≅ G/N

type

1. C₂×D₄×C₆₇⋊C₈

C₂

semidirect

2. #22561008

C₂

semidirect

3. #22560363

C₂

semidirect

4. #22559896

C₂

semidirect

23. C₄×C₆₇⋊C₈

D₄

semidirect

82. C₆₇

#21683

semidirect

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

subgroup	index
C₂×D₄×C₆₇⋊C₈	2
#22561008	2
#22560363	2
#22559896	2
C₆₇×#2397	2
#21683	67

subgroup

index

C₂×D₄×C₆₇⋊C₈

C₆₇×#2397

Subgroups — #30864232

Conjugacy classes of subgroups

Complements

Maximal subgroups