Group #30742930 — subgroups

Subgroups — #30742930

Order 17152.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22558968	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863420	—	2	2	no	—
4	#16863149	—	2	2	no	—
5	#16863149	C₄	1	4	yes	—
6	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₆₇	—	2	2	no	—
9	C₂×#11761612	—	4	2	no	—
10	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
11	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₆₇	—	2	2	no	—
16	C₂×C₁₆×C₆₇	—	4	2	no	—
17	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₆₇	D₄	1	2	yes	—
19	#11761612	—	8	2	no	—
20	#11761612	—	8	2	no	—
21	C₂×C₆₇⋊C₈	—	4	4	no	—
22	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
25	C₂×C₈×C₆₇	—	2	1	no	—
26	C₂×C₈×C₆₇	—	2	1	no	—
27	C₂×C₈×C₆₇	—	4	1	no	—
28	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
31	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
32	C₄²×C₆₇	SD₁₆	1	2	yes	—
33	C₁₆×C₆₇	—	4	2	no	—
34	C₂³×C₆₇	#107	1	1	yes	characteristic
35	C₂×C₄×C₆₇	—	2	1	no	—
36	C₂×C₄×C₆₇	—	2	1	no	—
37	C₂×C₄×C₆₇	—	4	1	no	—
38	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
39	C₂×C₄×C₆₇	#102	1	2	yes	—
40	C₂×C₄×C₆₇	#103	1	2	yes	—
41	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
42	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
44	C₂×C₄×C₆₇	Q₃₂	1	1	yes	characteristic
45	C₈×C₆₇	—	2	2	no	—
46	C₈×C₆₇	—	2	1	no	—
47	C₈×C₆₇	#100	1	1	yes	characteristic
48	C₈×C₆₇	#101	1	1	yes	characteristic
49	C₆₇⋊C₈	—	8	4	no	—
50	C₂²×C₆₇	—	2	1	no	—
51	C₂²×C₆₇	—	2	1	no	—
52	C₂²×C₆₇	#340	1	1	yes	characteristic
53	C₂²×C₆₇	#366	1	1	yes	characteristic
54	C₂²×C₆₇	#367	1	1	yes	characteristic
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	#338	1	1	yes	characteristic
58	C₄×C₆₇	#343	1	1	yes	characteristic
59	C₄×C₆₇	#357	1	2	yes	—
60	C₄×C₆₇	#358	1	2	yes	—
61	#7241	—	67	1	no	maximal, Sylow
62	C₂×C₆₇	—	2	1	no	—
63	C₂×C₆₇	—	2	1	no	—
64	C₂×C₆₇	#1292	1	1	yes	characteristic
65	C₂×C₆₇	#1380	1	1	yes	characteristic
66	C₂×C₆₇	#1385	1	1	yes	characteristic
67	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
68	#1567	—	67	2	no	—
69	C₆₇	#7241	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₁₆	Dic₆₇	1	2	yes	—
71	#423	—	134	2	no	—
72	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
73	C₄⋊C₁₆	—	2	2	no	—
74	#335	—	67	4	no	—
75	#335	—	134	2	no	—
76	C₂×M₁₆	—	268	2	no	—
77	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
79	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₄⋊C₈	—	268	2	no	—
83	C₄⋊C₈	—	134	4	no	—
84	C₄×C₈	C₄×D₆₇	1	2	yes	—
85	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
86	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
87	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
88	C₂²×C₄	Dic₂₆₈	1	1	yes	characteristic
89	M₁₆	—	536	2	no	—
90	M₁₆	—	536	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	268	4	no	—
95	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
97	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
98	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
99	C₄²	#11761617	1	2	yes	—
100	C₁₆	—	4	2	no	—
101	C₂³	#14523977	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
106	C₂×C₄	#14523965	1	2	yes	—
107	C₂×C₄	#14523968	1	2	yes	—
108	C₂×C₄	#14523975	1	1	yes	characteristic
109	C₂×C₄	D₁₀₇₂	1	1	yes	characteristic
110	C₂×C₄	#14523988	1	2	yes	—
111	C₂×C₄	Dic₅₃₆	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	536	4	no	—
115	C₈	#14523960	1	1	yes	characteristic
116	C₈	#14523962	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#16863159	1	1	yes	characteristic
120	C₂²	#16863222	1	1	yes	characteristic
121	C₂²	#16863224	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	C₆₇⋊#338	1	1	yes	characteristic
125	C₄	#16863165	1	1	yes	characteristic
126	C₄	#16863198	1	2	yes	—
127	C₄	#16863201	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#22558084	1	1	yes	characteristic
131	C₂	#22558301	1	1	yes	characteristic
132	C₂	#22558313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22558968

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic

#16863420

—

#16863149

—

#16863149

C₄

yes

—

C₂²×C₁₆×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

D₈

yes

characteristic

C₂²×C₄×C₆₇

Q₁₆

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

SD₁₆

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

#107

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#95

yes

characteristic

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

#106

yes

characteristic

C₂×C₄×C₆₇

D₁₆

yes

characteristic

C₂×C₄×C₆₇

SD₃₂

yes

—

C₂×C₄×C₆₇

Q₃₂

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#358

yes

—

#7241

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₄⋊C₁₆

D₆₇

yes

characteristic

#1567

—

C₆₇

#7241

yes

characteristic, complemented, Sylow

C₂²×C₁₆

Dic₆₇

yes

—

#423

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

—

134

—

C₂×M₁₆

—

268

—

C₂²×C₈

Dic₁₃₄

yes

characteristic

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₄²

D₂₆₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

268

—

C₄⋊C₈

—

134

—

C₄×C₈

C₄×D₆₇

yes

—

C₄×C₈

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂²×C₄

D₅₃₆

yes

characteristic

C₂²×C₄

Dic₂₆₈

yes

characteristic

M₁₆

—

536

—

M₁₆

—

536

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

C₄×Dic₆₇

yes

characteristic

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

C₂×C₈

Dic₆₇⋊C₄

yes

—

C₄²

(C₂×D₆₇)⋊C₄

yes

—

C₄²

#11761617

yes

—

100

C₁₆

—

101

C₂³

#14523977

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₆₇⋊#95

yes

characteristic

106

C₂×C₄

#14523965

yes

—

107

C₂×C₄

#14523968

yes

—

108

C₂×C₄

#14523975

yes

characteristic

109

C₂×C₄

D₁₀₇₂

yes

characteristic

110

C₂×C₄

#14523988

yes

—

111

C₂×C₄

Dic₅₃₆

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

536

—

115

C₈

#14523960

yes

characteristic

116

C₈

#14523962

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

#16863159

yes

characteristic

120

C₂²

#16863222

yes

characteristic

121

C₂²

#16863224

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₆₇⋊#338

yes

characteristic

125

C₄

#16863165

yes

characteristic

126

C₄

#16863198

yes

—

127

C₄

#16863201

yes

—

128

C₂

—

129

C₂

—

130

C₂

#22558084

yes

characteristic

131

C₂

#22558301

yes

characteristic

132

C₂

#22558313

yes

characteristic

	subgroup	G/H
center	C₂²	#16863159
commutator	C₂×C₈×C₆₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#340
prime radical	C₂³×C₆₇	#107
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#7241
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#16863159

commutator

C₂×C₈×C₆₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₇

Fitting

C₂×C₄⋊C₁₆×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄×C₈×C₆₇

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₆₇

hypocenter

C₆₇

#7241

LCS

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
69. C₆₇	#7241	semidirect

H ≅ G/N

type

69. C₆₇

#7241

semidirect

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

subgroup	index
#22558968	2
C₂×C₄⋊C₁₆×C₆₇	2
#7241	67

subgroup

index

#22558968

C₂×C₄⋊C₁₆×C₆₇

#7241

Subgroups — #30742930

Conjugacy classes of subgroups

Complements

Maximal subgroups