Group #30770192 — subgroups

Subgroups — #30770192

Order 17152.

192 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22559610	C₂	1	2	yes	maximal
3	#22558281	C₂	1	1	yes	maximal, characteristic
4	#22558267	C₂	1	2	yes	maximal
5	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic
6	#16863688	—	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	C₁₆×Dic₆₇	—	2	1	no	—
11	#16863149	C₂²	1	2	yes	—
12	#16863145	—	2	2	no	—
13	#500×C₆₇	C₂²	1	2	yes	—
14	#366×C₆₇	—	2	1	no	—
15	#366×C₆₇	—	2	1	no	—
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	#14523976	—	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	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
28	#107×C₆₇	—	2	2	no	—
29	#107×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
32	C₂×C₆₇⋊C₈	—	2	1	no	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
35	Dic₆₇⋊C₄	—	2	1	no	—
36	C₄×Dic₆₇	—	2	1	no	—
37	C₆₇⋊C₁₆	—	4	1	no	—
38	C₂×Q₈×C₆₇	—	4	2	no	—
39	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₆₇	—	4	2	no	—
42	C₄⋊C₄×C₆₇	—	4	2	no	—
43	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₆₇	—	2	1	no	—
45	C₁₆×C₆₇	Q₁₆	1	2	yes	—
46	Q₈×C₆₇	—	16	2	no	—
47	C₂×C₄×C₆₇	—	8	2	no	—
48	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
51	C₂×Dic₆₇	—	2	1	no	—
52	C₂×Dic₆₇	—	2	1	no	—
53	Dic₁₃₄	—	2	1	no	—
54	Dic₁₃₄	Q₃₂	1	2	yes	—
55	C₈×C₆₇	—	2	1	no	—
56	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₆₇	#137	1	1	yes	characteristic
58	C₆₇⋊C₈	—	4	1	no	—
59	C₆₇⋊C₈	—	8	2	no	—
60	C₂²×C₆₇	#466	1	1	yes	characteristic
61	C₄×C₆₇	—	2	1	no	—
62	C₄×C₆₇	—	16	2	no	—
63	C₄×C₆₇	#462	1	1	yes	characteristic
64	C₄×C₆₇	#473	1	1	yes	characteristic
65	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
66	C₄×C₆₇	#508	1	1	yes	characteristic
67	Dic₆₇	—	2	2	no	—
68	Dic₆₇	—	4	1	no	—
69	#12458	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	#1674	1	1	yes	characteristic
71	C₂×C₆₇	#2216	1	1	yes	characteristic
72	C₂×C₆₇	#2221	1	1	yes	characteristic
73	#2252	D₆₇	1	1	yes	characteristic
74	Q₈×C₁₆	—	67	1	no	—
75	#1676	—	67	2	no	—
76	#1371	—	67	1	no	—
77	#1365	—	67	2	no	—
78	C₆₇	#12458	1	1	yes	characteristic, complemented, Sylow
79	#500	C₂×D₆₇	1	2	yes	—
80	#477	—	134	2	no	—
81	C₈×Q₈	—	67	1	no	—
82	#366	—	2	1	no	—
83	#366	—	2	1	no	—
84	C₄⋊C₁₆	—	67	1	no	—
85	C₄⋊C₁₆	—	134	1	no	—
86	C₄×C₁₆	—	134	1	no	—
87	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
88	#335	—	67	2	no	—
89	#333	—	134	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	67	1	no	—
92	C₂×C₁₆	—	134	1	no	—
93	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	268	2	no	—
98	C₄⋊C₈	—	67	1	no	—
99	C₄⋊C₈	—	134	1	no	—
100	C₄⋊C₈	—	134	2	no	—
101	#103	—	268	2	no	—
102	C₄×C₈	—	134	1	no	—
103	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	67	1	no	—
106	C₂×C₈	—	268	2	no	—
107	C₂×C₈	—	134	1	no	—
108	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
109	C₂×C₈	#11761627	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	67	1	no	—
113	C₄⋊C₄	—	134	1	no	—
114	C₄²	—	134	1	no	—
115	C₄²	D₄×D₆₇	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	268	1	no	—
118	C₁₆	#11761618	1	2	yes	—
119	Q₈	—	16	2	no	—
120	Q₈	—	67	2	no	—
121	Q₈	—	134	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	134	1	no	—
124	C₂×C₄	—	134	1	no	—
125	C₂×C₄	#14524014	1	1	yes	characteristic
126	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
127	C₂×C₄	#14524062	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	268	1	no	—
130	C₈	—	536	2	no	—
131	C₈	C₂×#11761618	1	1	yes	characteristic
132	C₈	#14524074	1	1	yes	characteristic
133	C₂²	#16863629	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	268	1	no	—
136	C₄	—	16	2	no	—
137	C₄	—	134	2	no	—
138	C₄	#16863603	1	1	yes	characteristic
139	C₄	#16863668	1	1	yes	characteristic
140	C₄	Q₃₂×D₆₇	1	1	yes	characteristic
141	C₄	#16863829	1	1	yes	characteristic
142	C₂	#22559600	1	1	yes	characteristic
143	C₂	#22562297	1	1	yes	characteristic
144	C₂	#22562326	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₃₄

C₂

yes

maximal, characteristic

#22559610

C₂

yes

maximal

#22558281

C₂

yes

maximal, characteristic

#22558267

C₂

yes

maximal

C₆₇×#2252

C₂

yes

maximal, characteristic

#16863688

—

C₈×Dic₁₃₄

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₁₆)

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

—

C₁₆×Dic₆₇

—

#16863149

C₂²

yes

—

#16863145

—

#500×C₆₇

C₂²

yes

—

#366×C₆₇

—

#366×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×Dic₁₃₄

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

—

#14523976

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

characteristic

Dic₆₇⋊C₈

—

#14523966

—

C₈×Dic₆₇

—

C₄⋊Q₈×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

#107×C₆₇

—

#107×C₆₇

—

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

—

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

Q₁₆

yes

—

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

—

Dic₁₃₄

Q₃₂

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₁₆

yes

characteristic

C₈×C₆₇

#137

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#466

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×Q₃₂

yes

characteristic

C₄×C₆₇

#508

yes

characteristic

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

characteristic

Q₈×C₁₆

—

#1676

—

#1371

—

#1365

—

C₆₇

#12458

yes

characteristic, complemented, Sylow

#500

C₂×D₆₇

yes

—

#477

—

134

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

C₄×C₁₆

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

#335

—

#333

—

134

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

134

—

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₇

yes

—

—

—

268

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

100

C₄⋊C₈

—

134

—

101

#103

—

268

—

102

C₄×C₈

—

134

—

103

C₄×C₈

C₂²×D₆₇

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×C₈

—

268

—

107

C₂×C₈

—

134

—

108

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

109

C₂×C₈

#11761627

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

134

—

114

C₄²

—

134

—

115

C₄²

D₄×D₆₇

yes

characteristic

116

C₁₆

—

117

C₁₆

—

268

—

118

C₁₆

#11761618

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

134

—

122

C₂×C₄

—

123

C₂×C₄

—

134

—

124

C₂×C₄

—

134

—

125

C₂×C₄

#14524014

yes

characteristic

126

C₂×C₄

D₈×D₆₇

yes

characteristic

127

C₂×C₄

#14524062

yes

characteristic

128

C₈

—

129

C₈

—

268

—

130

C₈

—

536

—

131

C₈

C₂×#11761618

yes

characteristic

132

C₈

#14524074

yes

characteristic

133

C₂²

#16863629

yes

characteristic

134

C₄

—

135

C₄

—

268

—

136

C₄

—

137

C₄

—

134

—

138

C₄

#16863603

yes

characteristic

139

C₄

#16863668

yes

characteristic

140

C₄

Q₃₂×D₆₇

yes

characteristic

141

C₄

#16863829

yes

characteristic

142

C₂

#22559600

yes

characteristic

143

C₂

#22562297

yes

characteristic

144

C₂

#22562326

yes

characteristic

	subgroup	G/H
center	C₂²	#16863629
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2252	C₂
socle, prime radical	C₂²×C₆₇	#466
hypercenter	#2252	D₆₇
hypocenter	C₆₇	#12458
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16863629

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2252

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
78. C₆₇	#12458	semidirect

H ≅ G/N

type

78. C₆₇

#12458

semidirect

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

subgroup	index
C₁₆×Dic₁₃₄	2
#22559610	2
#22558281	2
#22558267	2
C₆₇×#2252	2
#12458	67

subgroup

index

C₁₆×Dic₁₃₄

C₆₇×#2252

Subgroups — #30770192

Conjugacy classes of subgroups

Complements

Maximal subgroups