Group #30770107 — subgroups

Subgroups — #30770107

Order 17152.

212 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22562117	C₂	1	1	yes	maximal, characteristic
2	#22559576	C₂	1	2	yes	maximal
3	#22558283	C₂	1	1	yes	maximal, characteristic
4	#22558258	C₂	1	2	yes	maximal
5	C₆₇×#2247	C₂	1	1	yes	maximal, characteristic
6	#16863678	—	2	2	no	—
7	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₆₇⋊C₁₆	—	2	1	no	—
10	#16863171	—	2	1	no	—
11	#16863149	C₂²	1	2	yes	—
12	#16863142	—	2	2	no	—
13	C₂×Q₃₂×C₆₇	—	2	1	no	—
14	C₂×Q₃₂×C₆₇	—	2	1	no	—
15	#494×C₆₇	C₂²	1	2	yes	—
16	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
18	C₂×C₆₇⋊C₁₆	—	2	1	no	—
19	#14523974	—	4	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₆₇⋊C₈	—	2	1	no	—
23	#14523966	—	4	2	no	—
24	C₈×Dic₆₇	—	2	1	no	—
25	C₂×Q₁₆×C₆₇	—	2	2	no	—
26	C₂×Q₁₆×C₆₇	—	2	2	no	—
27	C₄⋊Q₈×C₆₇	—	2	2	no	—
28	Q₃₂×C₆₇	—	4	2	no	—
29	Q₃₂×C₆₇	—	4	2	no	—
30	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
37	Dic₆₇⋊C₄	—	2	1	no	—
38	C₄×Dic₆₇	—	2	1	no	—
39	C₆₇⋊C₁₆	—	4	1	no	—
40	C₂×Q₈×C₆₇	—	4	2	no	—
41	Q₁₆×C₆₇	—	4	2	no	—
42	Q₁₆×C₆₇	—	4	2	no	—
43	Q₁₆×C₆₇	—	8	2	no	—
44	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₆₇	—	4	2	no	—
47	C₄⋊C₄×C₆₇	—	4	2	no	—
48	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₆₇	—	2	1	no	—
50	C₁₆×C₆₇	SD₁₆	1	2	yes	—
51	Q₈×C₆₇	—	8	2	no	—
52	Q₈×C₆₇	—	8	2	no	—
53	C₂×C₄×C₆₇	—	8	2	no	—
54	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
57	C₂×Dic₆₇	—	2	1	no	—
58	C₂×Dic₆₇	—	2	1	no	—
59	Dic₁₃₄	—	2	2	no	—
60	C₈×C₆₇	—	2	1	no	—
61	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₆₇⋊C₈	—	4	1	no	—
64	C₆₇⋊C₈	—	8	2	no	—
65	C₂²×C₆₇	#466	1	1	yes	characteristic
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	—	16	2	no	—
68	C₄×C₆₇	#460	1	1	yes	characteristic
69	C₄×C₆₇	#472	1	1	yes	characteristic
70	C₄×C₆₇	#510	1	2	yes	—
71	Dic₆₇	—	4	1	no	—
72	Dic₆₇	—	4	1	no	—
73	#12443	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	#1668	1	1	yes	characteristic
75	C₂×C₆₇	#2219	1	1	yes	characteristic
76	C₂×C₆₇	#2222	1	1	yes	characteristic
77	#2247	D₆₇	1	1	yes	characteristic
78	#2183	—	67	1	no	—
79	#1670	—	67	2	no	—
80	#1372	—	67	1	no	—
81	#1362	—	67	2	no	—
82	C₆₇	#12443	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₃₂	—	2	1	no	—
84	C₂×Q₃₂	—	2	1	no	—
85	#494	C₂×D₆₇	1	2	yes	—
86	#475	—	134	2	no	—
87	C₈×Q₈	—	67	1	no	—
88	C₄⋊C₁₆	—	67	1	no	—
89	C₄⋊C₁₆	—	134	1	no	—
90	#346	—	134	1	no	—
91	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
92	#335	—	67	2	no	—
93	#332	—	134	2	no	—
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	2	2	no	—
96	C₄⋊Q₈	—	2	2	no	—
97	C₄×Q₈	—	67	1	no	—
98	Q₃₂	—	4	2	no	—
99	Q₃₂	—	4	2	no	—
100	C₂×C₁₆	—	134	1	no	—
101	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
103	#106	—	268	2	no	—
104	C₄⋊C₈	—	67	1	no	—
105	C₄⋊C₈	—	134	1	no	—
106	C₄⋊C₈	—	134	2	no	—
107	#103	—	268	2	no	—
108	C₄×C₈	—	134	1	no	—
109	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
110	C₂×Q₈	—	4	2	no	—
111	C₂×Q₈	—	67	1	no	—
112	Q₁₆	—	4	2	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	8	2	no	—
115	C₂×C₈	—	268	2	no	—
116	C₂×C₈	—	134	1	no	—
117	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
118	C₂×C₈	#11761627	1	1	yes	characteristic
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	67	1	no	—
122	C₄⋊C₄	—	134	1	no	—
123	C₄²	—	134	1	no	—
124	C₄²	D₄×D₆₇	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	268	1	no	—
127	C₁₆	#11761616	1	2	yes	—
128	Q₈	—	8	2	no	—
129	Q₈	—	8	2	no	—
130	Q₈	—	134	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	134	1	no	—
133	C₂×C₄	—	134	1	no	—
134	C₂×C₄	#14524014	1	1	yes	characteristic
135	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
136	C₂×C₄	#14524062	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	268	1	no	—
139	C₈	—	536	2	no	—
140	C₈	C₂×#11761616	1	1	yes	characteristic
141	C₈	#14524069	1	1	yes	characteristic
142	C₂²	#16863629	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	268	1	no	—
145	C₄	—	268	1	no	—
146	C₄	—	16	2	no	—
147	C₄	#16863593	1	1	yes	characteristic
148	C₄	#16863663	1	1	yes	characteristic
149	C₄	#16863838	1	2	yes	—
150	C₂	#22559566	1	1	yes	characteristic
151	C₂	#22562314	1	1	yes	characteristic
152	C₂	#22562331	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562117

C₂

yes

maximal, characteristic

#22559576

C₂

yes

maximal

#22558283

C₂

yes

maximal, characteristic

#22558258

C₂

yes

maximal

C₆₇×#2247

C₂

yes

maximal, characteristic

#16863678

—

C₈×Dic₁₃₄

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₁₆)

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

—

#16863171

—

#16863149

C₂²

yes

—

#16863142

—

C₂×Q₃₂×C₆₇

—

C₂×Q₃₂×C₆₇

—

#494×C₆₇

C₂²

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×Dic₁₃₄

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

—

#14523974

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

characteristic

Dic₆₇⋊C₈

—

#14523966

—

C₈×Dic₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

—

Q₃₂×C₆₇

—

Q₃₂×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

D₈

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₈

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

SD₁₆

yes

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×SD₁₆

yes

characteristic

C₈×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#466

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

—

—

—

—

C₆₇

#12443

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂×D₆₇

yes

—

#475

—

134

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

#346

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

#335

—

#332

—

134

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

100

C₂×C₁₆

—

134

—

101

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

103

#106

—

268

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

134

—

106

C₄⋊C₈

—

134

—

107

#103

—

268

—

108

C₄×C₈

—

134

—

109

C₄×C₈

C₂²×D₆₇

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

115

C₂×C₈

—

268

—

116

C₂×C₈

—

134

—

117

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

118

C₂×C₈

#11761627

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

134

—

123

C₄²

—

134

—

124

C₄²

D₄×D₆₇

yes

characteristic

125

C₁₆

—

126

C₁₆

—

268

—

127

C₁₆

#11761616

yes

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

134

—

131

C₂×C₄

—

132

C₂×C₄

—

134

—

133

C₂×C₄

—

134

—

134

C₂×C₄

#14524014

yes

characteristic

135

C₂×C₄

D₈×D₆₇

yes

characteristic

136

C₂×C₄

#14524062

yes

characteristic

137

C₈

—

138

C₈

—

268

—

139

C₈

—

536

—

140

C₈

C₂×#11761616

yes

characteristic

141

C₈

#14524069

yes

characteristic

142

C₂²

#16863629

yes

characteristic

143

C₄

—

144

C₄

—

268

—

145

C₄

—

268

—

146

C₄

—

147

C₄

#16863593

yes

characteristic

148

C₄

#16863663

yes

characteristic

149

C₄

#16863838

yes

—

150

C₂

#22559566

yes

characteristic

151

C₂

#22562314

yes

characteristic

152

C₂

#22562331

yes

characteristic

	subgroup	G/H
center	C₂²	#16863629
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2247	C₂
socle, prime radical	C₂²×C₆₇	#466
hypercenter	#2247	D₆₇
hypocenter	C₆₇	#12443
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16863629

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2247

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
82. C₆₇	#12443	semidirect

H ≅ G/N

type

82. C₆₇

#12443

semidirect

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

subgroup	index
#22562117	2
#22559576	2
#22558283	2
#22558258	2
C₆₇×#2247	2
#12443	67

subgroup

index

C₆₇×#2247

Subgroups — #30770107

Conjugacy classes of subgroups

Complements

Maximal subgroups