Group #30749646 — subgroups

Subgroups — #30749646

Order 17152.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22560347	C₂	1	1	yes	maximal, characteristic
2	#22558967	C₂	1	2	yes	maximal
3	#22558964	C₂	1	2	yes	maximal
4	C₂×#16863117	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863161	—	2	2	no	—
10	#16863148	—	2	4	no	—
11	#16863146	—	2	4	no	—
12	#16863117	C₄	1	4	yes	—
13	#423×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#341×C₆₇	—	2	2	no	—
16	C₂×#11761612	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
18	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
19	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
24	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	D₄	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	2	no	—
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	Q₈	1	2	yes	—
32	#11761612	—	4	4	no	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	M₁₆×C₆₇	—	4	4	no	—
42	C₂×C₈×C₆₇	—	2	4	no	—
43	C₂×C₈×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
45	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#135	1	4	yes	—
59	C₈×C₆₇	—	2	4	no	—
60	C₈×C₆₇	—	8	2	no	—
61	C₆₇⋊C₈	—	2	4	no	—
62	C₆₇⋊C₈	—	2	4	no	—
63	C₆₇⋊C₈	—	8	2	no	—
64	C₂²×C₆₇	—	2	1	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	#384	1	1	yes	characteristic
67	C₂²×C₆₇	#499	1	2	yes	—
68	C₄×C₆₇	—	2	1	no	—
69	C₄×C₆₇	—	2	1	no	—
70	C₄×C₆₇	#427	1	2	yes	—
71	C₄×C₆₇	#430	1	2	yes	—
72	C₄×C₆₇	#430	1	2	yes	—
73	C₄×C₆₇	#495	1	2	yes	—
74	#8827	—	67	1	no	maximal, Sylow
75	C₂×C₆₇	—	2	2	no	—
76	C₂×C₆₇	#1846	1	1	yes	characteristic
77	C₂×C₆₇	#1849	1	1	yes	characteristic
78	C₂×C₆₇	#1849	1	1	yes	characteristic
79	#1837	—	67	1	no	—
80	#1837	D₆₇	1	1	yes	characteristic
81	#1567	—	67	2	no	—
82	#1566	—	67	2	no	—
83	C₂×#322	—	67	1	no	—
84	C₆₇	#8827	1	1	yes	characteristic, complemented, Sylow
85	#423	—	67	2	no	—
86	#423	C₂×D₆₇	1	2	yes	—
87	C₂×C₄×C₈	—	67	1	no	—
88	C₂×C₄×C₈	—	67	1	no	—
89	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
90	#341	—	2	2	no	—
91	#341	—	134	2	no	—
92	#335	—	134	4	no	—
93	#334	—	134	4	no	—
94	#322	—	67	4	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	134	2	no	—
97	C₂²×C₈	—	67	2	no	—
98	C₂²×C₈	—	67	2	no	—
99	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
100	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	134	2	no	—
104	C₄⋊C₈	—	134	2	no	—
105	C₄×C₈	—	67	2	no	—
106	C₄×C₈	—	67	2	no	—
107	C₄×C₈	—	67	2	no	—
108	C₄×C₈	—	67	2	no	—
109	C₄×C₈	Dic₁₃₄	1	2	yes	—
110	C₄×C₈	C₄×D₆₇	1	2	yes	—
111	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
112	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
113	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
114	M₁₆	—	4	4	no	—
115	M₁₆	—	268	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	268	2	no	—
119	C₂×C₈	—	268	2	no	—
120	C₂×C₈	—	268	2	no	—
121	C₂×C₈	—	67	4	no	—
122	C₂×C₈	—	67	4	no	—
123	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
124	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
125	C₄²	D₄×D₆₇	1	1	yes	characteristic
126	C₄²	D₄×D₆₇	1	1	yes	characteristic
127	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
128	C₂³	#14524041	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
132	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
133	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
134	C₂×C₄	#14524039	1	1	yes	characteristic
135	C₂×C₄	#14524042	1	1	yes	characteristic
136	C₂×C₄	#14524042	1	1	yes	characteristic
137	C₂×C₄	#14524063	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	8	2	no	—
140	C₈	—	536	2	no	—
141	C₈	—	134	4	no	—
142	C₈	—	134	4	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#16863265	1	1	yes	characteristic
146	C₂²	#16863795	1	2	yes	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#16863432	1	2	yes	—
150	C₄	#16863443	1	2	yes	—
151	C₄	#16863443	1	2	yes	—
152	C₄	#16863778	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#22560371	1	1	yes	characteristic
155	C₂	#22560382	1	1	yes	characteristic
156	C₂	#22560382	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560347

C₂

yes

maximal, characteristic

#22558967

C₂

yes

maximal

#22558964

C₂

yes

maximal

C₂×#16863117

C₂

yes

maximal, characteristic

C₆₇×#1837

C₂

yes

maximal, characteristic

#16863420

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

C₄

yes

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#341×C₆₇

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

Q₈

yes

—

C₂²×C₆₇⋊C₈

Q₈

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#499

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#427

yes

—

C₄×C₆₇

#430

yes

—

C₄×C₆₇

#430

yes

—

C₄×C₆₇

#495

yes

—

#8827

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

—

—

—

C₆₇

#8827

yes

characteristic, complemented, Sylow

#423

—

#423

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#341

—

#341

—

134

—

#335

—

134

—

#334

—

134

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₆₇

yes

—

100

C₂×C₄²

C₂²×D₆₇

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

134

—

104

C₄⋊C₈

—

134

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

Dic₁₃₄

yes

—

110

C₄×C₈

C₄×D₆₇

yes

—

111

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

112

C₂²×C₄

Q₈×D₆₇

yes

characteristic

113

C₂²×C₄

Q₈×D₆₇

yes

characteristic

114

M₁₆

—

115

M₁₆

—

268

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

268

—

119

C₂×C₈

—

268

—

120

C₂×C₈

—

268

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

Dic₆₇⋊C₄

yes

—

124

C₄²

C₂×C₄×D₆₇

yes

characteristic

125

C₄²

D₄×D₆₇

yes

characteristic

126

C₄²

D₄×D₆₇

yes

characteristic

127

C₄²

C₂×Dic₁₃₄

yes

characteristic

128

C₂³

#14524041

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

132

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

133

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

134

C₂×C₄

#14524039

yes

characteristic

135

C₂×C₄

#14524042

yes

characteristic

136

C₂×C₄

#14524042

yes

characteristic

137

C₂×C₄

#14524063

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

536

—

141

C₈

—

134

—

142

C₈

—

134

—

143

C₂²

—

144

C₂²

—

145

C₂²

#16863265

yes

characteristic

146

C₂²

#16863795

yes

—

147

C₄

—

148

C₄

—

149

C₄

#16863432

yes

—

150

C₄

#16863443

yes

—

151

C₄

#16863443

yes

—

152

C₄

#16863778

yes

—

153

C₂

—

154

C₂

#22560371

yes

characteristic

155

C₂

#22560382

yes

characteristic

156

C₂

#22560382

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1837	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1837	D₆₇
hypocenter	C₆₇	#8827
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂×C₄

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1837

C₂

socle

C₂²×C₆₇

#384

prime radical

C₂³×C₆₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
84. C₆₇	#8827	semidirect

H ≅ G/N

type

84. C₆₇

#8827

semidirect

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

subgroup	index
#22560347	2
#22558967	2
#22558964	2
C₂×#16863117	2
C₆₇×#1837	2
#8827	67

subgroup

index

C₆₇×#1837

Subgroups — #30749646

Conjugacy classes of subgroups

Complements

Maximal subgroups