Group #30770190 — subgroups

Subgroups — #30770190

Order 17152.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22562484	C₂	1	1	yes	maximal, characteristic
2	#22559603	C₂	1	2	yes	maximal
3	#22558281	C₂	1	1	yes	maximal, characteristic
4	#22558263	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	2	yes	—
7	#16863606	—	2	2	no	—
8	#16863222	—	2	1	no	—
9	#16863222	—	2	1	no	—
10	#16863149	C₂²	1	2	yes	—
11	#16863141	—	2	2	no	—
12	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	—	2	1	no	—
14	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₇	—	2	1	no	—
16	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#11761618	—	4	2	no	—
18	#14524043	—	2	2	no	—
19	#14523977	—	2	2	no	—
20	#14523977	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	#14523966	—	4	2	no	—
23	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₇	—	2	1	no	—
25	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	1	no	—
28	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	—	2	1	no	—
30	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₃₄	—	4	2	no	—
32	#11761618	—	8	4	no	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₄⋊Dic₆₇	—	4	2	no	—
36	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
42	C₄²×C₆₇	—	2	1	no	—
43	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₆₇	—	2	1	no	—
45	C₁₆×C₆₇	—	4	1	no	—
46	C₁₆×C₆₇	Q₁₆	1	2	yes	complemented
47	Q₈×C₆₇	—	2	1	no	—
48	Q₈×C₆₇	D₁₆	1	2	yes	—
49	C₂×C₄×C₆₇	—	2	1	no	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
54	C₂×Dic₆₇	—	8	2	no	—
55	Dic₁₃₄	—	8	2	no	—
56	Dic₁₃₄	—	8	2	no	—
57	C₈×C₆₇	—	2	1	no	—
58	C₈×C₆₇	—	4	1	no	—
59	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₆₇	#137	1	1	yes	characteristic
61	C₆₇⋊C₈	—	8	2	no	—
62	C₂²×C₆₇	#466	1	1	yes	characteristic
63	C₄×C₆₇	—	2	2	no	—
64	C₄×C₆₇	—	2	1	no	—
65	C₄×C₆₇	—	4	1	no	—
66	C₄×C₆₇	#462	1	1	yes	characteristic
67	C₄×C₆₇	#473	1	1	yes	characteristic
68	C₄×C₆₇	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₆₇	#508	1	1	yes	characteristic
70	Dic₆₇	—	16	2	no	—
71	#12457	—	67	1	no	maximal, Sylow
72	C₂×C₆₇	#1674	1	1	yes	characteristic
73	C₂×C₆₇	#2215	1	1	yes	characteristic
74	C₂×C₆₇	#2222	1	1	yes	characteristic
75	#2252	—	67	1	no	—
76	Q₈×C₁₆	D₆₇	1	1	yes	characteristic
77	#1674	—	67	2	no	—
78	#1371	—	67	1	no	—
79	#1364	—	67	2	no	—
80	C₆₇	#12457	1	1	yes	characteristic, complemented, Sylow
81	#500	—	67	2	no	—
82	#462	—	134	2	no	—
83	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
84	#366	—	134	1	no	—
85	#366	—	134	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
90	#335	—	67	2	no	—
91	#332	—	134	2	no	—
92	C₂×Q₁₆	—	268	2	no	—
93	C₄⋊Q₈	—	134	2	no	—
94	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
98	#107	—	134	2	no	—
99	#107	—	134	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	134	2	no	—
102	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
103	#103	—	268	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
106	C₂×Q₈	—	268	2	no	—
107	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
108	Q₁₆	—	536	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	268	2	no	—
111	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
112	C₂×C₈	#11761628	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	268	2	no	—
115	C₄⋊C₄	—	268	2	no	—
116	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#11761618	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	536	2	no	—
124	Q₈	—	536	2	no	—
125	Q₈	D₁₀₇₂	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	536	2	no	—
129	C₂×C₄	#14524017	1	1	yes	characteristic
130	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
131	C₂×C₄	#14524065	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	536	2	no	—
135	C₈	C₂×#11761618	1	1	yes	characteristic
136	C₈	#14524073	1	1	yes	characteristic
137	C₂²	#16863632	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	1072	2	no	—
142	C₄	#16863606	1	1	yes	characteristic
143	C₄	#16863671	1	1	yes	characteristic
144	C₄	C₂×D₁₀₇₂	1	1	yes	characteristic
145	C₄	#16863828	1	1	yes	characteristic
146	C₂	#22559603	1	1	yes	characteristic
147	C₂	#22562295	1	1	yes	characteristic
148	C₂	#22562334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562484

C₂

yes

maximal, characteristic

#22559603

C₂

yes

maximal

#22558281

C₂

yes

maximal, characteristic

#22558263

C₂

yes

maximal

Q₈×C₁₆×C₆₇

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

—

—

—

C₂²

yes

—

#16863141

—

C₈×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₄⋊C₁₆×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

—

—

—

—

C₄⋊(C₆₇⋊C₈)

—

#14523966

—

C₄×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

#11761618

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

D₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

Q₁₆

yes

complemented

Q₈×C₆₇

—

Q₈×C₆₇

D₁₆

yes

—

C₂×C₄×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

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₁₆

yes

characteristic

C₈×C₆₇

#137

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

#466

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×D₁₆

yes

characteristic

C₄×C₆₇

#508

yes

characteristic

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

Q₈×C₁₆

D₆₇

yes

—

—

—

C₆₇

#12457

yes

characteristic, complemented, Sylow

#500

—

#462

—

134

—

C₈×Q₈

C₂×D₆₇

yes

characteristic

#366

—

134

—

#366

—

134

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₆₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

#335

—

#332

—

134

—

C₂×Q₁₆

—

268

—

C₄⋊Q₈

—

134

—

C₄×Q₈

D₂₆₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

#107

—

134

—

#107

—

134

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

134

—

102

C₄⋊C₈

D₂₆₈

yes

characteristic

103

#103

—

268

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₆₇

yes

characteristic

106

C₂×Q₈

—

268

—

107

C₂×Q₈

D₅₃₆

yes

characteristic

108

Q₁₆

—

536

—

109

C₂×C₈

—

110

C₂×C₈

—

268

—

111

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

112

C₂×C₈

#11761628

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

268

—

115

C₄⋊C₄

—

268

—

116

C₄⋊C₄

D₅₃₆

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₂₆₈

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#11761618

yes

complemented

122

Q₈

—

123

Q₈

—

536

—

124

Q₈

—

536

—

125

Q₈

D₁₀₇₂

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

536

—

129

C₂×C₄

#14524017

yes

characteristic

130

C₂×C₄

C₂×D₅₃₆

yes

characteristic

131

C₂×C₄

#14524065

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

536

—

135

C₈

C₂×#11761618

yes

characteristic

136

C₈

#14524073

yes

characteristic

137

C₂²

#16863632

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

1072

—

142

C₄

#16863606

yes

characteristic

143

C₄

#16863671

yes

characteristic

144

C₄

C₂×D₁₀₇₂

yes

characteristic

145

C₄

#16863828

yes

characteristic

146

C₂

#22559603

yes

characteristic

147

C₂

#22562295

yes

characteristic

148

C₂

#22562334

yes

characteristic

	subgroup	G/H
center	C₂²	#16863632
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	Q₈×C₁₆×C₆₇	C₂
socle, prime radical	C₂²×C₆₇	#466
hypercenter	Q₈×C₁₆	D₆₇
hypocenter	C₆₇	#12457
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#16863632

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

Q₈×C₁₆×C₆₇

C₂

socle, prime radical

C₂²×C₆₇

#466

hypercenter

Q₈×C₁₆

D₆₇

hypocenter

C₆₇

#12457

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₆₇	Q₁₆	semidirect
80. C₆₇	#12457	semidirect
121. C₁₆	#11761618	semidirect

H ≅ G/N

type

46. C₁₆×C₆₇

Q₁₆

semidirect

80. C₆₇

#12457

semidirect

121. C₁₆

#11761618

semidirect

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

subgroup	index
#22562484	2
#22559603	2
#22558281	2
#22558263	2
Q₈×C₁₆×C₆₇	2
#12457	67

subgroup

index

Q₈×C₁₆×C₆₇

Subgroups — #30770190

Conjugacy classes of subgroups

Complements

Maximal subgroups