Group #30686385 — subgroups

Subgroups — C₆₇×#7134 #30686385

Order 17152.

230 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 70 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	C₆₇×#1994	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1276	C₂	1	2	yes	maximal
3	#533×C₆₇	C₄	1	1	yes	characteristic
4	#431×C₆₇	—	2	2	no	—
5	#423×C₆₇	C₄	1	1	yes	characteristic
6	C₄³×C₆₇	C₂²	1	1	yes	characteristic
7	#339×C₆₇	—	2	2	no	—
8	C₂×M₁₆×C₆₇	—	2	1	no	—
9	C₄⋊Q₈×C₆₇	—	2	1	no	—
10	#125×C₆₇	C₈	1	2	yes	complemented
11	C₂²⋊Q₈×C₆₇	—	4	1	no	—
12	C₄×Q₈×C₆₇	—	4	1	no	—
13	#117×C₆₇	—	2	1	no	—
14	C₂×C₄²×C₆₇	—	4	1	no	—
15	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	D₄	1	2	yes	—
17	C₄⋊C₈×C₆₇	—	2	4	no	—
18	C₄⋊C₈×C₆₇	—	4	1	no	—
19	C₂²⋊C₈×C₆₇	—	4	2	no	—
20	C₄×C₈×C₆₇	—	4	2	no	—
21	C₂×Q₈×C₆₇	—	4	1	no	—
22	C₂²×C₄×C₆₇	—	2	1	no	—
23	C₂²×C₄×C₆₇	—	2	2	no	—
24	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
25	M₁₆×C₆₇	—	4	2	no	—
26	C₂×C₈×C₆₇	—	4	4	no	—
27	C₂×C₈×C₆₇	—	4	1	no	—
28	C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₄⋊C₄×C₆₇	—	4	2	no	—
30	C₄⋊C₄×C₆₇	—	4	1	no	—
31	C₂²⋊C₄×C₆₇	—	4	1	no	—
32	C₄²×C₆₇	—	2	1	no	—
33	C₄²×C₆₇	—	2	2	no	—
34	C₄²×C₆₇	—	4	4	no	—
35	C₄²×C₆₇	—	4	1	no	—
36	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
38	C₄²×C₆₇	SD₁₆	1	2	yes	—
39	C₄²×C₆₇	Q₁₆	1	2	yes	—
40	C₂³×C₆₇	#99	1	1	yes	characteristic
41	Q₈×C₆₇	—	8	2	no	—
42	C₂×C₄×C₆₇	—	2	1	no	—
43	C₂×C₄×C₆₇	—	2	2	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	4	4	no	—
47	C₂×C₄×C₆₇	—	4	2	no	—
48	C₂×C₄×C₆₇	—	4	2	no	—
49	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#100	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#103	1	2	yes	—
52	C₂×C₄×C₆₇	#104	1	2	yes	—
53	C₈×C₆₇	—	8	4	no	—
54	C₈×C₆₇	—	8	1	no	—
55	C₂²×C₆₇	—	4	1	no	—
56	C₂²×C₆₇	#323	1	1	yes	characteristic
57	C₂²×C₆₇	#328	1	2	yes	—
58	C₄×C₆₇	—	2	4	no	—
59	C₄×C₆₇	—	2	2	no	—
60	C₄×C₆₇	—	4	1	no	—
61	C₄×C₆₇	—	4	2	no	—
62	C₄×C₆₇	—	8	2	no	—
63	C₄×C₆₇	Q₈⋊C₈	1	2	yes	—
64	C₄×C₆₇	#329	1	1	yes	characteristic
65	C₄×C₆₇	#333	1	1	yes	characteristic
66	#7134	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂×C₆₇	—	2	1	no	—
68	C₂×C₆₇	—	2	2	no	—
69	C₂×C₆₇	#1272	1	1	yes	characteristic
70	#1994	C₂×C₆₇	1	1	yes	characteristic
71	#1276	C₂×C₆₇	1	2	yes	—
72	C₆₇	#7134	1	1	yes	characteristic, direct product factor, Sylow
73	#533	C₄×C₆₇	1	1	yes	characteristic
74	#431	—	2	2	no	—
75	#423	C₄×C₆₇	1	1	yes	characteristic
76	C₄³	C₂²×C₆₇	1	1	yes	characteristic
77	#339	—	2	2	no	—
78	C₂×M₁₆	—	2	1	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	#125	C₈×C₆₇	1	2	yes	complemented
81	C₂²⋊Q₈	—	4	1	no	—
82	C₄×Q₈	—	4	1	no	—
83	#117	—	2	1	no	—
84	C₂×C₄²	—	4	1	no	—
85	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
86	C₂×C₄²	D₄×C₆₇	1	2	yes	—
87	C₄⋊C₈	—	2	4	no	—
88	C₄⋊C₈	—	4	1	no	—
89	C₂²⋊C₈	—	4	2	no	—
90	C₄×C₈	—	4	2	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
95	M₁₆	—	4	2	no	—
96	C₂×C₈	—	4	4	no	—
97	C₂×C₈	—	4	1	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₂²⋊C₄	—	4	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	4	4	no	—
105	C₄²	—	4	1	no	—
106	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
107	C₄²	M₁₆×C₆₇	1	1	yes	characteristic
108	C₄²	SD₁₆×C₆₇	1	2	yes	—
109	C₄²	Q₁₆×C₆₇	1	2	yes	—
110	C₂³	#99×C₆₇	1	1	yes	characteristic
111	Q₈	—	8	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₂²⋊C₈×C₆₇	1	1	yes	characteristic
120	C₂×C₄	#100×C₆₇	1	1	yes	characteristic
121	C₂×C₄	#103×C₆₇	1	2	yes	—
122	C₂×C₄	#104×C₆₇	1	2	yes	—
123	C₈	—	8	4	no	—
124	C₈	—	8	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	#323×C₆₇	1	1	yes	characteristic
127	C₂²	#328×C₆₇	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	8	2	no	—
133	C₄	Q₈⋊C₈×C₆₇	1	2	yes	—
134	C₄	#329×C₆₇	1	1	yes	characteristic
135	C₄	#333×C₆₇	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	2	2	no	—
138	C₂	C₆₇×#1272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#1994

C₂

yes

maximal, characteristic

C₆₇×#1276

C₂

yes

maximal

#533×C₆₇

C₄

yes

characteristic

#431×C₆₇

—

#423×C₆₇

C₄

yes

characteristic

C₄³×C₆₇

C₂²

yes

characteristic

#339×C₆₇

—

C₂×M₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

—

#125×C₆₇

C₈

yes

complemented

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₄²×C₆₇

M₁₆

yes

characteristic

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₂³×C₆₇

#99

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#104

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#323

yes

characteristic

C₂²×C₆₇

#328

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

Q₈⋊C₈

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1272

yes

characteristic

#1994

C₂×C₆₇

yes

characteristic

#1276

C₂×C₆₇

yes

—

C₆₇

#7134

yes

characteristic, direct product factor, Sylow

#533

C₄×C₆₇

yes

characteristic

#431

—

#423

C₄×C₆₇

yes

characteristic

C₄³

C₂²×C₆₇

yes

characteristic

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

C₈×C₆₇

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₆₇

yes

characteristic

C₂×C₄²

D₄×C₆₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₆₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×C₈×C₆₇

yes

characteristic

107

C₄²

M₁₆×C₆₇

yes

characteristic

108

C₄²

SD₁₆×C₆₇

yes

—

109

C₄²

Q₁₆×C₆₇

yes

—

110

C₂³

#99×C₆₇

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊C₈×C₆₇

yes

characteristic

120

C₂×C₄

#100×C₆₇

yes

characteristic

121

C₂×C₄

#103×C₆₇

yes

—

122

C₂×C₄

#104×C₆₇

yes

—

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

#323×C₆₇

yes

characteristic

127

C₂²

#328×C₆₇

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

Q₈⋊C₈×C₆₇

yes

—

134

C₄

#329×C₆₇

yes

characteristic

135

C₄

#333×C₆₇

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

C₆₇×#1272

yes

characteristic

	subgroup	G/H
center	C₄×C₆₇	#333
commutator	C₄²	C₂×C₈×C₆₇
Frattini, cosocle	C₄³	C₂²×C₆₇
socle	C₂×C₆₇	#1272
prime radical	C₂³×C₆₇	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₄³×C₆₇ ◁ G

subgroup

G/H

center

C₄×C₆₇

#333

commutator

C₄²

C₂×C₈×C₆₇

Frattini, cosocle

C₄³

C₂²×C₆₇

C₂×C₆₇

C₂³×C₆₇

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₄³×C₆₇ ◁ G

N	H ≅ G/N	type
10. #125×C₆₇	C₈	semidirect
66. #7134	C₆₇	direct (N × H)
72. C₆₇	#7134	direct (N × H)
80. #125	C₈×C₆₇	semidirect

H ≅ G/N

type

10. #125×C₆₇

C₈

semidirect

66. #7134

C₆₇

direct (N × H)

72. C₆₇

#7134

direct (N × H)

80. #125

C₈×C₆₇

semidirect

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

subgroup	index
C₆₇×#1994	2
C₆₇×#1276	2
#7134	67

subgroup

index

C₆₇×#1994

C₆₇×#1276

#7134

Subgroups — C₆₇×#7134 #30686385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#7134 #30686385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#7134 #30686385