Group #30689642 — subgroups

Subgroups — C₆₇×#10391 #30689642

Order 17152.

542 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2840	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1819	C₂	1	3	yes	maximal
3	C₆₇×#1304	C₂	1	3	yes	maximal
4	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
5	#393×C₆₇	—	2	6	no	—
6	#379×C₆₇	C₂²	1	3	yes	—
7	#379×C₆₇	C₂²	1	3	yes	—
8	#342×C₆₇	—	2	6	no	—
9	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
10	C₂²×Q₈×C₆₇	—	2	2	no	—
11	C₂³×C₄×C₆₇	Q₈	1	2	yes	complemented
12	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
13	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
14	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
15	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
16	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
17	C₂×C₂²⋊C₄×C₆₇	D₄	1	6	yes	—
18	#95×C₆₇	—	2	6	no	—
19	C₂⁴×C₆₇	—	4	2	no	—
20	C₂⁴×C₆₇	—	8	2	no	—
21	C₂⁴×C₆₇	C₂×D₄	1	3	yes	—
22	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
23	C₂⁴×C₆₇	#41	1	3	yes	—
24	C₂×Q₈×C₆₇	—	8	4	no	—
25	C₂×Q₈×C₆₇	—	8	2	no	—
26	C₂²×C₄×C₆₇	—	2	6	no	—
27	C₂²×C₄×C₆₇	—	2	6	no	—
28	C₂²×C₄×C₆₇	—	4	3	no	—
29	C₂²×C₄×C₆₇	—	8	2	no	—
30	C₂²⋊C₄×C₆₇	—	4	1	no	—
31	C₂²⋊C₄×C₆₇	—	4	3	no	—
32	C₂²⋊C₄×C₆₇	—	4	6	no	—
33	C₂²⋊C₄×C₆₇	—	4	6	no	—
34	C₂²⋊C₄×C₆₇	—	8	6	no	—
35	C₂²⋊C₄×C₆₇	—	8	6	no	—
36	C₂³×C₆₇	—	2	6	no	—
37	C₂³×C₆₇	—	4	2	no	—
38	C₂³×C₆₇	—	4	1	no	—
39	C₂³×C₆₇	—	4	6	no	—
40	C₂³×C₆₇	—	4	6	no	—
41	C₂³×C₆₇	—	4	3	no	—
42	C₂³×C₆₇	—	8	8	no	—
43	C₂³×C₆₇	#120	1	1	yes	characteristic
44	C₂³×C₆₇	C₂²⋊Q₈	1	3	yes	—
45	C₂³×C₆₇	#124	1	3	yes	—
46	Q₈×C₆₇	—	8	8	no	—
47	C₂×C₄×C₆₇	—	4	12	no	—
48	C₂×C₄×C₆₇	—	4	2	no	—
49	C₂×C₄×C₆₇	—	4	6	no	—
50	C₂×C₄×C₆₇	—	8	6	no	—
51	C₂×C₄×C₆₇	—	8	3	no	—
52	C₂×C₄×C₆₇	—	8	6	no	—
53	C₂²×C₆₇	—	2	2	no	—
54	C₂²×C₆₇	—	2	3	no	—
55	C₂²×C₆₇	—	2	3	no	—
56	C₂²×C₆₇	—	2	6	no	—
57	C₂²×C₆₇	—	2	6	no	—
58	C₂²×C₆₇	—	4	4	no	—
59	C₂²×C₆₇	—	8	6	no	—
60	C₂²×C₆₇	—	8	6	no	—
61	C₂²×C₆₇	#393	1	1	yes	characteristic
62	C₂²×C₆₇	#458	1	2	yes	—
63	C₄×C₆₇	—	8	6	no	—
64	C₄×C₆₇	—	8	2	no	—
65	C₄×C₆₇	—	16	3	no	—
66	#10391	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂×C₆₇	—	2	2	no	—
68	C₂×C₆₇	—	4	3	no	—
69	C₂×C₆₇	—	4	3	no	—
70	C₂×C₆₇	#2030	1	1	yes	characteristic
71	C₂×C₆₇	#2032	1	2	yes	—
72	#2840	C₂×C₆₇	1	1	yes	characteristic
73	#1819	C₂×C₆₇	1	3	yes	—
74	#1304	C₂×C₆₇	1	3	yes	—
75	C₆₇	#10391	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
77	#393	—	2	6	no	—
78	#379	C₂²×C₆₇	1	3	yes	—
79	#379	C₂²×C₆₇	1	3	yes	—
80	#342	—	2	6	no	—
81	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
82	C₂²×Q₈	—	2	2	no	—
83	C₂³×C₄	Q₈×C₆₇	1	2	yes	complemented
84	C₂×C₂²⋊C₄	—	2	3	no	—
85	C₂×C₂²⋊C₄	—	2	3	no	—
86	C₂×C₂²⋊C₄	—	2	3	no	—
87	C₂×C₂²⋊C₄	—	2	6	no	—
88	C₂×C₂²⋊C₄	—	2	6	no	—
89	C₂×C₂²⋊C₄	D₄×C₆₇	1	6	yes	—
90	#95	—	2	6	no	—
91	C₂⁴	—	4	2	no	—
92	C₂⁴	—	8	2	no	—
93	C₂⁴	C₂×D₄×C₆₇	1	3	yes	—
94	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
95	C₂⁴	#41×C₆₇	1	3	yes	—
96	C₂×Q₈	—	8	4	no	—
97	C₂×Q₈	—	8	2	no	—
98	C₂²×C₄	—	2	6	no	—
99	C₂²×C₄	—	2	6	no	—
100	C₂²×C₄	—	4	3	no	—
101	C₂²×C₄	—	8	2	no	—
102	C₂²⋊C₄	—	4	1	no	—
103	C₂²⋊C₄	—	4	3	no	—
104	C₂²⋊C₄	—	4	6	no	—
105	C₂²⋊C₄	—	4	6	no	—
106	C₂²⋊C₄	—	8	6	no	—
107	C₂²⋊C₄	—	8	6	no	—
108	C₂³	—	2	6	no	—
109	C₂³	—	4	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	6	no	—
112	C₂³	—	4	6	no	—
113	C₂³	—	4	3	no	—
114	C₂³	—	8	8	no	—
115	C₂³	#120×C₆₇	1	1	yes	characteristic
116	C₂³	C₂²⋊Q₈×C₆₇	1	3	yes	—
117	C₂³	#124×C₆₇	1	3	yes	—
118	Q₈	—	8	8	no	—
119	C₂×C₄	—	4	12	no	—
120	C₂×C₄	—	4	6	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	8	6	no	—
123	C₂×C₄	—	8	3	no	—
124	C₂×C₄	—	8	6	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	3	no	—
127	C₂²	—	2	3	no	—
128	C₂²	—	2	6	no	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	8	6	no	—
132	C₂²	—	8	6	no	—
133	C₂²	#393×C₆₇	1	1	yes	characteristic
134	C₂²	#458×C₆₇	1	2	yes	—
135	C₄	—	8	6	no	—
136	C₄	—	8	2	no	—
137	C₄	—	16	3	no	—
138	C₂	—	2	2	no	—
139	C₂	—	4	3	no	—
140	C₂	—	4	3	no	—
141	C₂	C₆₇×#2030	1	1	yes	characteristic
142	C₂	C₆₇×#2032	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2840

C₂

yes

maximal, characteristic

C₆₇×#1819

C₂

yes

maximal

C₆₇×#1304

C₂

yes

maximal

C₂²×C₂²⋊C₄×C₆₇

C₂²

yes

characteristic

#393×C₆₇

—

#379×C₆₇

C₂²

yes

—

#379×C₆₇

C₂²

yes

—

#342×C₆₇

—

C₂⁵×C₆₇

C₂³

yes

characteristic

C₂²×Q₈×C₆₇

—

C₂³×C₄×C₆₇

Q₈

yes

complemented

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

D₄

yes

—

#95×C₆₇

—

C₂⁴×C₆₇

—

C₂⁴×C₆₇

—

C₂⁴×C₆₇

C₂×D₄

yes

—

C₂⁴×C₆₇

C₂×Q₈

yes

characteristic

C₂⁴×C₆₇

#41

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#120

yes

characteristic

C₂³×C₆₇

C₂²⋊Q₈

yes

—

C₂³×C₆₇

#124

yes

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#393

yes

characteristic

C₂²×C₆₇

#458

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

#10391

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2030

yes

characteristic

C₂×C₆₇

#2032

yes

—

#2840

C₂×C₆₇

yes

characteristic

#1819

C₂×C₆₇

yes

—

#1304

C₂×C₆₇

yes

—

C₆₇

#10391

yes

characteristic, direct product factor, Sylow

C₂²×C₂²⋊C₄

C₂²×C₆₇

yes

characteristic

#393

—

#379

C₂²×C₆₇

yes

—

#379

C₂²×C₆₇

yes

—

#342

—

C₂⁵

C₂³×C₆₇

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

Q₈×C₆₇

yes

complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

D₄×C₆₇

yes

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×D₄×C₆₇

yes

—

C₂⁴

C₂×Q₈×C₆₇

yes

characteristic

C₂⁴

#41×C₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

#120×C₆₇

yes

characteristic

116

C₂³

C₂²⋊Q₈×C₆₇

yes

—

117

C₂³

#124×C₆₇

yes

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#393×C₆₇

yes

characteristic

134

C₂²

#458×C₆₇

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

C₆₇×#2030

yes

characteristic

142

C₂

C₆₇×#2032

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₇	#393
commutator, Frattini, cosocle	C₂⁵	C₂³×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂⁵×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

#393

commutator, Frattini, cosocle

C₂⁵

C₂³×C₆₇

prime radical

C₂⁵×C₆₇

C₂³

LCS

G ▷ C₂⁵ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₂⁵×C₆₇ ◁ G

N	H ≅ G/N	type
11. C₂³×C₄×C₆₇	Q₈	semidirect
66. #10391	C₆₇	direct (N × H)
75. C₆₇	#10391	direct (N × H)
83. C₂³×C₄	Q₈×C₆₇	semidirect

H ≅ G/N

type

11. C₂³×C₄×C₆₇

Q₈

semidirect

66. #10391

C₆₇

direct (N × H)

75. C₆₇

#10391

direct (N × H)

83. C₂³×C₄

Q₈×C₆₇

semidirect

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

subgroup	index
C₆₇×#2840	2
C₆₇×#1819	2
C₆₇×#1304	2
#10391	67

subgroup

index

C₆₇×#2840

C₆₇×#1819

C₆₇×#1304

#10391

Subgroups — C₆₇×#10391 #30689642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#10391 #30689642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10391 #30689642