Group #30767591 — subgroups

Subgroups — #30767591

Order 17152.

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

normal subgroups only

This group has 212 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	#22562091	C₂	1	1	yes	maximal, characteristic
2	#22562045	C₂	1	2	yes	maximal, complemented
3	#22558181	C₂	1	2	yes	maximal, complemented
4	#22558125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863828	C₄	1	1	yes	characteristic
7	#16863814	—	2	2	no	—
8	#16863514	C₂²	1	2	yes	—
9	#16863226	C₄	1	1	yes	characteristic
10	#16863204	C₄	1	2	yes	—
11	#16863181	—	2	2	no	—
12	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	#364×C₆₇	C₂²	1	2	yes	—
15	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#14524065	C₂×C₄	1	2	yes	—
17	#14524049	—	2	2	no	—
18	Dic₅₃₆	C₈	1	1	yes	characteristic, complemented
19	#14523988	C₈	1	2	yes	complemented
20	D₁₀₇₂	C₈	1	1	yes	characteristic, complemented
21	#14523982	—	4	2	no	—
22	C₂×C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₆₇	—	4	2	no	—
25	#14523979	C₂×C₄	1	2	yes	—
26	#14523971	—	2	2	no	—
27	C₄×C₆₇⋊C₈	—	2	1	no	—
28	M₃₂×C₆₇	—	2	2	no	—
29	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	#11761628	—	2	2	no	—
33	Dic₂₆₈	C₂×C₈	1	2	yes	—
34	#11761617	—	2	2	no	—
35	D₅₃₆	C₂×C₈	1	2	yes	—
36	#11761612	—	2	2	no	—
37	#11761611	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	2	2	no	—
39	C₈×D₆₇	—	4	2	no	—
40	C₆₇⋊C₁₆	—	2	1	no	—
41	C₆₇⋊C₁₆	D₈	1	2	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₆₇	#41	1	1	yes	characteristic
45	C₁₆×C₆₇	—	2	1	no	—
46	C₁₆×C₆₇	—	4	2	no	—
47	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	1	no	—
50	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
51	C₂²⋊D₆₇	—	4	2	no	—
52	D₂₆₈	—	2	2	no	—
53	C₄×D₆₇	—	4	2	no	—
54	Dic₁₃₄	—	2	2	no	—
55	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₆₇	#131	1	1	yes	characteristic
57	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
58	C₈×C₆₇	#135	1	1	yes	characteristic
59	C₆₇⋊C₈	—	4	2	no	—
60	C₆₇⋊C₈	—	4	2	no	—
61	C₂²×C₆₇	#443	1	1	yes	characteristic
62	C₂×D₆₇	—	4	2	no	—
63	C₄×C₆₇	—	2	2	no	—
64	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
65	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
66	Dic₆₇	—	4	2	no	—
67	#12047	—	67	1	no	maximal, Sylow
68	C₂×C₆₇	—	2	1	no	—
69	C₂×C₆₇	C₈×D₈	1	1	yes	characteristic
70	D₆₇	—	8	2	no	—
71	#2178	—	67	1	no	—
72	#2170	—	67	2	no	—
73	#1373	D₆₇	1	1	yes	characteristic, complemented
74	#1335	—	67	2	no	—
75	#1311	—	67	1	no	—
76	C₆₇	#12047	1	1	yes	characteristic, complemented, Sylow
77	#508	—	67	1	no	—
78	#504	—	134	2	no	—
79	#443	—	67	2	no	—
80	#368	—	67	1	no	—
81	#364	C₂×D₆₇	1	2	yes	—
82	#359	—	67	2	no	—
83	#350	—	134	2	no	—
84	C₄×C₁₆	—	67	1	no	—
85	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
86	C₈²	—	67	1	no	—
87	#135	—	67	2	no	—
88	#131	—	134	2	no	—
89	Q₃₂	—	67	1	no	—
90	SD₃₂	—	67	2	no	—
91	D₁₆	—	67	1	no	—
92	M₃₂	—	2	2	no	—
93	M₃₂	—	268	2	no	—
94	C₂×C₁₆	—	268	2	no	—
95	C₂×C₁₆	—	67	1	no	—
96	C₂×C₁₆	—	67	1	no	—
97	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
98	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
99	#108	—	67	2	no	—
100	#104	—	134	2	no	—
101	C₄×C₈	—	134	1	no	—
102	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
103	#41	—	134	2	no	—
104	Q₁₆	—	67	2	no	—
105	SD₁₆	—	134	2	no	—
106	D₈	—	67	2	no	—
107	M₁₆	—	268	2	no	—
108	M₁₆	—	134	2	no	—
109	C₂×C₈	—	268	2	no	—
110	C₂×C₈	—	134	2	no	—
111	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
112	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
113	C₄²	#11761629	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	2	no	—
116	C₁₆	—	67	2	no	—
117	C₁₆	—	134	1	no	—
118	C₁₆	C₈×D₆₇	1	1	yes	characteristic
119	C₁₆	C₈×D₆₇	1	1	yes	characteristic
120	Q₈	—	134	2	no	—
121	D₄	—	268	2	no	—
122	D₄	—	134	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	268	2	no	—
125	C₂×C₄	#14524005	1	1	yes	characteristic
126	C₈	—	268	2	no	—
127	C₈	—	268	2	no	—
128	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
129	C₈	#14524051	1	1	yes	characteristic
130	C₈	D₈×D₆₇	1	1	yes	characteristic
131	C₈	#14524066	1	1	yes	characteristic
132	C₂²	—	268	2	no	—
133	C₂²	#16863512	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	268	2	no	—
136	C₄	#16863461	1	1	yes	characteristic
137	C₄	#16863478	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	536	2	no	—
140	C₂	#22559003	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562091

C₂

yes

maximal, characteristic

#22562045

C₂

yes

maximal, complemented

#22558181

C₂

yes

maximal, complemented

#22558125

C₂

yes

maximal, characteristic, complemented

C₆₇×#1373

C₂

yes

maximal, characteristic, complemented

#16863828

C₄

yes

characteristic

#16863814

—

#16863514

C₂²

yes

—

#16863226

C₄

yes

characteristic

#16863204

C₄

yes

—

#16863181

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

#364×C₆₇

C₂²

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

#14524065

C₂×C₄

yes

—

#14524049

—

Dic₅₃₆

C₈

yes

characteristic, complemented

#14523988

C₈

yes

complemented

D₁₀₇₂

C₈

yes

characteristic, complemented

#14523982

—

C₂×C₆₇⋊C₁₆

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

D₄

yes

characteristic

C₁₆×D₆₇

—

#14523979

C₂×C₄

yes

—

#14523971

—

C₄×C₆₇⋊C₈

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

#11761628

—

Dic₂₆₈

C₂×C₈

yes

—

#11761617

—

D₅₃₆

C₂×C₈

yes

—

#11761612

—

#11761611

—

C₂×C₆₇⋊C₈

—

C₈×D₆₇

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

D₈

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂²⋊D₆₇

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₂²×C₈

yes

characteristic

C₈×C₆₇

#131

yes

characteristic

C₈×C₆₇

C₂×D₈

yes

characteristic

C₈×C₆₇

#135

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#443

yes

characteristic

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₈×D₄

yes

characteristic

C₄×C₆₇

C₄×D₈

yes

characteristic

Dic₆₇

—

#12047

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₈×D₈

yes

characteristic

D₆₇

—

#2178

—

#2170

—

#1373

D₆₇

yes

characteristic, complemented

#1335

—

#1311

—

C₆₇

#12047

yes

characteristic, complemented, Sylow

#508

—

#504

—

134

—

#443

—

#368

—

#364

C₂×D₆₇

yes

—

#359

—

#350

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

#135

—

#131

—

134

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

268

—

C₂×C₁₆

—

268

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₆₇

yes

characteristic

C₂×C₁₆

C₄×D₆₇

yes

characteristic

#108

—

100

#104

—

134

—

101

C₄×C₈

—

134

—

102

C₄×C₈

C₂²×D₆₇

yes

characteristic

103

#41

—

134

—

104

Q₁₆

—

105

SD₁₆

—

134

—

106

D₈

—

107

M₁₆

—

268

—

108

M₁₆

—

134

—

109

C₂×C₈

—

268

—

110

C₂×C₈

—

134

—

111

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

112

C₂×C₈

D₄×D₆₇

yes

characteristic

113

C₄²

#11761629

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

134

—

118

C₁₆

C₈×D₆₇

yes

characteristic

119

C₁₆

C₈×D₆₇

yes

characteristic

120

Q₈

—

134

—

121

D₄

—

268

—

122

D₄

—

134

—

123

C₂×C₄

—

124

C₂×C₄

—

268

—

125

C₂×C₄

#14524005

yes

characteristic

126

C₈

—

268

—

127

C₈

—

268

—

128

C₈

C₂×C₈×D₆₇

yes

characteristic

129

C₈

#14524051

yes

characteristic

130

C₈

D₈×D₆₇

yes

characteristic

131

C₈

#14524066

yes

characteristic

132

C₂²

—

268

—

133

C₂²

#16863512

yes

characteristic

134

C₄

—

135

C₄

—

268

—

136

C₄

#16863461

yes

characteristic

137

C₄

#16863478

yes

characteristic

138

C₂

—

139

C₂

—

536

—

140

C₂

#22559003

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₆₇
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1373	C₂
socle	C₂×C₆₇	C₈×D₈
prime radical	#16863828	C₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1373	D₆₇
hypocenter	C₆₇	#12047
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₆₇

commutator

C₈×C₆₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1373

C₂

socle

C₂×C₆₇

C₈×D₈

prime radical

#16863828

C₄

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #22562045	C₂	semidirect
3. #22558181	C₂	semidirect
4. #22558125	C₂	semidirect
5. C₆₇×#1373	C₂	semidirect
18. Dic₅₃₆	C₈	semidirect
19. #14523988	C₈	semidirect
20. D₁₀₇₂	C₈	semidirect
73. #1373	D₆₇	semidirect
76. C₆₇	#12047	semidirect

H ≅ G/N

type

2. #22562045

C₂

semidirect

3. #22558181

C₂

semidirect

4. #22558125

C₂

semidirect

5. C₆₇×#1373

C₂

semidirect

18. Dic₅₃₆

C₈

semidirect

19. #14523988

C₈

semidirect

20. D₁₀₇₂

C₈

semidirect

73. #1373

D₆₇

semidirect

76. C₆₇

#12047

semidirect

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

subgroup	index
#22562091	2
#22562045	2
#22558181	2
#22558125	2
C₆₇×#1373	2
#12047	67

subgroup

index

C₆₇×#1373

Subgroups — #30767591

Conjugacy classes of subgroups

Complements

Maximal subgroups