Group #30749591 — subgroups

Subgroups — #30749591

Order 17152.

362 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 179 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	#22560326	C₂	1	2	yes	maximal
2	C₂×#16863149	C₂	1	2	yes	maximal
3	C₂×#16863147	C₂	1	2	yes	maximal
4	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
6	#16863151	—	2	4	no	—
7	#16863149	—	2	4	no	—
8	#16863147	—	2	4	no	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₄³×C₆₇	C₂²	1	1	yes	characteristic
12	#336×C₆₇	C₄	1	4	yes	—
13	C₂²×C₆₇⋊C₈	—	2	4	no	—
14	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
15	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₆₇	D₄	1	2	yes	—
20	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
21	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₆₇	D₄	1	2	yes	—
23	C₄×C₈×C₆₇	D₄	1	2	yes	—
24	C₄×C₈×C₆₇	Q₈	1	2	yes	—
25	C₄×C₈×C₆₇	Q₈	1	2	yes	—
26	C₂×C₆₇⋊C₈	—	4	4	no	—
27	C₂×C₆₇⋊C₈	—	4	8	no	—
28	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
29	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈×C₆₇	—	2	4	no	—
33	C₂×C₈×C₆₇	—	2	4	no	—
34	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
36	C₄²×C₆₇	—	2	4	no	—
37	C₄²×C₆₇	—	2	4	no	—
38	C₄²×C₆₇	—	4	2	no	—
39	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	2	2	no	—
45	C₂×C₄×C₆₇	—	2	4	no	—
46	C₂×C₄×C₆₇	—	2	4	no	—
47	C₂×C₄×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#131	1	4	yes	—
55	C₂×C₄×C₆₇	#135	1	4	yes	—
56	C₈×C₆₇	—	2	4	no	—
57	C₈×C₆₇	—	2	4	no	—
58	C₆₇⋊C₈	—	8	4	no	—
59	C₂²×C₆₇	#384	1	1	yes	characteristic
60	C₂²×C₆₇	#424	1	2	yes	—
61	C₂²×C₆₇	#427	1	2	yes	—
62	C₂²×C₆₇	#430	1	2	yes	—
63	C₄×C₆₇	—	2	1	no	—
64	C₄×C₆₇	—	2	1	no	—
65	C₄×C₆₇	—	4	4	no	—
66	C₄×C₆₇	#424	1	2	yes	—
67	C₄×C₆₇	#424	1	2	yes	—
68	C₄×C₆₇	#495	1	2	yes	—
69	C₄×C₆₇	#499	1	2	yes	—
70	#8811	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	#1572	1	4	yes	—
72	C₂×C₆₇	#1843	1	1	yes	characteristic
73	C₂×C₆₇	#1846	1	1	yes	characteristic
74	C₂×C₆₇	#1849	1	1	yes	characteristic
75	#1831	—	67	2	no	—
76	#1744	D₆₇	1	1	yes	characteristic
77	C₂×#335	—	67	2	no	—
78	C₂×#334	—	67	2	no	—
79	C₆₇	#8811	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄⋊C₈	—	67	4	no	—
81	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
83	C₄³	C₂×D₆₇	1	1	yes	characteristic
84	#336	—	134	4	no	—
85	#336	Dic₆₇	1	4	yes	—
86	#335	—	134	4	no	—
87	#334	—	134	4	no	—
88	C₂²×C₈	—	134	4	no	—
89	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
90	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
91	C₂×C₄²	Dic₁₃₄	1	2	yes	—
92	C₂×C₄²	D₂₆₈	1	2	yes	—
93	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
94	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
95	C₄⋊C₈	—	134	4	no	—
96	C₄⋊C₈	—	134	4	no	—
97	C₄×C₈	Dic₁₃₄	1	2	yes	—
98	C₄×C₈	Dic₁₃₄	1	2	yes	—
99	C₄×C₈	D₂₆₈	1	2	yes	—
100	C₄×C₈	D₂₆₈	1	2	yes	—
101	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
102	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
103	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
104	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	268	4	no	—
108	C₂×C₈	—	268	8	no	—
109	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
110	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
111	C₄²	—	2	4	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	4	2	no	—
114	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
115	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
116	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
117	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
118	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
125	C₂×C₄	#14524040	1	1	yes	characteristic
126	C₂×C₄	#14524043	1	1	yes	characteristic
127	C₂×C₄	#14524043	1	1	yes	characteristic
128	C₂×C₄	#14524043	1	1	yes	characteristic
129	C₂×C₄	#14524050	1	4	yes	—
130	C₂×C₄	#14524065	1	4	yes	—
131	C₈	—	2	4	no	—
132	C₈	—	2	4	no	—
133	C₈	—	536	4	no	—
134	C₂²	#16863266	1	1	yes	characteristic
135	C₂²	#16863423	1	2	yes	—
136	C₂²	#16863434	1	2	yes	—
137	C₂²	#16863445	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	4	no	—
141	C₄	#16863423	1	2	yes	—
142	C₄	#16863423	1	2	yes	—
143	C₄	#16863780	1	2	yes	—
144	C₄	#16863798	1	2	yes	—
145	C₂	#22558989	1	4	yes	—
146	C₂	#22560363	1	1	yes	characteristic
147	C₂	#22560373	1	1	yes	characteristic
148	C₂	#22560384	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560326

C₂

yes

maximal

C₂×#16863149

C₂

yes

maximal

C₂×#16863147

C₂

yes

maximal

C₆₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#16863151

—

#16863149

—

#16863147

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

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

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#424

yes

—

C₂²×C₆₇

#427

yes

—

C₂²×C₆₇

#430

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#424

yes

—

C₄×C₆₇

#424

yes

—

C₄×C₆₇

#495

yes

—

C₄×C₆₇

#499

yes

—

#8811

—

maximal, Sylow

C₂×C₆₇

#1572

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

C₂×#335

—

C₂×#334

—

C₆₇

#8811

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄³

C₂×D₆₇

yes

characteristic

#336

—

134

—

#336

Dic₆₇

yes

—

#335

—

134

—

#334

—

134

—

C₂²×C₈

—

134

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂×C₄²

Dic₁₃₄

yes

—

C₂×C₄²

D₂₆₈

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

134

—

C₄⋊C₈

—

134

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

D₂₆₈

yes

—

100

C₄×C₈

D₂₆₈

yes

—

101

C₂²×C₄

C₄⋊Dic₆₇

yes

—

102

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

103

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

104

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

268

—

108

C₂×C₈

—

268

—

109

C₂×C₈

C₄⋊Dic₆₇

yes

—

110

C₂×C₈

C₄⋊Dic₆₇

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×D₂₆₈

yes

characteristic

115

C₄²

C₂×D₂₆₈

yes

characteristic

116

C₄²

C₂×Dic₁₃₄

yes

characteristic

117

C₄²

C₂×Dic₁₃₄

yes

characteristic

118

C₂³

C₂×C₄⋊Dic₆₇

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

124

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

125

C₂×C₄

#14524040

yes

characteristic

126

C₂×C₄

#14524043

yes

characteristic

127

C₂×C₄

#14524043

yes

characteristic

128

C₂×C₄

#14524043

yes

characteristic

129

C₂×C₄

#14524050

yes

—

130

C₂×C₄

#14524065

yes

—

131

C₈

—

132

C₈

—

133

C₈

—

536

—

134

C₂²

#16863266

yes

characteristic

135

C₂²

#16863423

yes

—

136

C₂²

#16863434

yes

—

137

C₂²

#16863445

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#16863423

yes

—

142

C₄

#16863423

yes

—

143

C₄

#16863780

yes

—

144

C₄

#16863798

yes

—

145

C₂

#22558989

yes

—

146

C₂

#22560363

yes

characteristic

147

C₂

#22560373

yes

characteristic

148

C₂

#22560384

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1744	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#8811
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₆₈

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1744

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
79. C₆₇	#8811	semidirect

H ≅ G/N

type

79. C₆₇

#8811

semidirect

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

subgroup	index
#22560326	2
C₂×#16863149	2
C₂×#16863147	2
C₆₇×#1744	2
#8811	67

subgroup

index

C₆₇×#1744

Subgroups — #30749591

Conjugacy classes of subgroups

Complements

Maximal subgroups