Group #30712487 — subgroups

Subgroups — C₆₇×#33236 #30712487

Order 17152.

910 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 318 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	C₂×#487×C₆₇	C₂	1	4	yes	maximal
2	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1960	C₂	1	8	yes	maximal
5	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
9	#487×C₆₇	—	2	16	no	—
10	C₂×#103×C₆₇	C₂²	1	16	yes	—
11	C₂×#97×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
13	#391×C₆₇	—	2	4	no	—
14	#384×C₆₇	—	2	4	no	—
15	#382×C₆₇	C₂²	1	4	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₆₇	—	2	2	no	—
18	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
19	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
20	C₄⋊Q₈×C₆₇	—	2	8	no	—
21	C₄⋊Q₈×C₆₇	—	2	8	no	—
22	C₄⋊Q₈×C₆₇	—	2	16	no	—
23	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
27	C₂×C₄²×C₆₇	D₄	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
30	#103×C₆₇	—	2	32	no	—
31	#97×C₆₇	—	2	8	no	—
32	C₄×C₈×C₆₇	D₄	1	8	yes	—
33	#95×C₆₇	—	2	8	no	—
34	C₂×Q₈×C₆₇	—	2	8	no	—
35	C₂×Q₈×C₆₇	—	2	16	no	—
36	C₂×Q₈×C₆₇	—	4	4	no	—
37	C₂×Q₈×C₆₇	—	4	8	no	—
38	C₂²×C₄×C₆₇	—	2	4	no	—
39	C₂²×C₄×C₆₇	—	2	4	no	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
43	C₂×C₈×C₆₇	—	2	8	no	—
44	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
45	C₄⋊C₄×C₆₇	—	2	8	no	—
46	C₄⋊C₄×C₆₇	—	4	16	no	—
47	C₄⋊C₄×C₆₇	—	4	4	no	—
48	C₄⋊C₄×C₆₇	—	4	8	no	—
49	C₄²×C₆₇	—	2	8	no	—
50	C₄²×C₆₇	C₂×D₄	1	4	yes	—
51	C₄²×C₆₇	#41	1	8	yes	—
52	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
53	Q₈×C₆₇	—	4	16	no	—
54	Q₈×C₆₇	—	8	8	no	—
55	C₂×C₄×C₆₇	—	2	8	no	—
56	C₂×C₄×C₆₇	—	4	8	no	—
57	C₂×C₄×C₆₇	—	4	4	no	—
58	C₂×C₄×C₆₇	—	4	4	no	—
59	C₂×C₄×C₆₇	—	4	8	no	—
60	C₂×C₄×C₆₇	#124	1	8	yes	—
61	C₂×C₄×C₆₇	#127	1	4	yes	—
62	C₂×C₄×C₆₇	#137	1	8	yes	—
63	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
65	C₈×C₆₇	—	2	8	no	—
66	C₂²×C₆₇	C₂×#124	1	2	yes	—
67	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
68	C₂²×C₆₇	C₂×#137	1	4	yes	—
69	C₄×C₆₇	—	2	8	no	—
70	C₄×C₆₇	—	8	4	no	—
71	C₄×C₆₇	—	8	4	no	—
72	C₄×C₆₇	#497	1	8	yes	—
73	C₄×C₆₇	#532	1	4	yes	—
74	#33236	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₆₇	#2858	1	1	yes	characteristic
76	C₂×C₆₇	#3139	1	4	yes	—
77	C₂×C₆₇	C₂×#497	1	2	yes	—
78	C₂×#487	C₂×C₆₇	1	4	yes	—
79	#2868	C₂×C₆₇	1	1	yes	characteristic
80	#2862	C₂×C₆₇	1	1	yes	characteristic
81	#1960	C₂×C₆₇	1	8	yes	—
82	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
83	C₆₇	#33236	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
85	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
86	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
87	#487	—	2	16	no	—
88	C₂×#103	C₂²×C₆₇	1	16	yes	—
89	C₂×#97	C₂²×C₆₇	1	4	yes	—
90	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
91	#391	—	2	4	no	—
92	#384	—	2	4	no	—
93	#382	C₂²×C₆₇	1	4	yes	—
94	C₄³	C₂²×C₆₇	1	1	yes	characteristic
95	C₂²×Q₈	—	2	2	no	—
96	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
97	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
98	C₄⋊Q₈	—	2	8	no	—
99	C₄⋊Q₈	—	2	8	no	—
100	C₄⋊Q₈	—	2	16	no	—
101	C₂×C₄⋊C₄	—	2	8	no	—
102	C₂×C₄⋊C₄	—	2	4	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
105	C₂×C₄²	D₄×C₆₇	1	4	yes	—
106	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
107	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
108	#103	—	2	32	no	—
109	#97	—	2	8	no	—
110	C₄×C₈	D₄×C₆₇	1	8	yes	—
111	#95	—	2	8	no	—
112	C₂×Q₈	—	2	8	no	—
113	C₂×Q₈	—	2	16	no	—
114	C₂×Q₈	—	4	4	no	—
115	C₂×Q₈	—	4	8	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
119	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
120	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
121	C₂×C₈	—	2	8	no	—
122	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
123	C₄⋊C₄	—	2	8	no	—
124	C₄⋊C₄	—	4	16	no	—
125	C₄⋊C₄	—	4	4	no	—
126	C₄⋊C₄	—	4	8	no	—
127	C₄²	—	2	8	no	—
128	C₄²	C₂×D₄×C₆₇	1	4	yes	—
129	C₄²	#41×C₆₇	1	8	yes	—
130	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
131	Q₈	—	4	16	no	—
132	Q₈	—	8	8	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	4	8	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	4	8	no	—
138	C₂×C₄	#124×C₆₇	1	8	yes	—
139	C₂×C₄	#127×C₆₇	1	4	yes	—
140	C₂×C₄	#137×C₆₇	1	8	yes	—
141	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
142	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
143	C₈	—	2	8	no	—
144	C₂²	C₂×#124×C₆₇	1	2	yes	—
145	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
146	C₂²	C₂×#137×C₆₇	1	4	yes	—
147	C₄	—	2	8	no	—
148	C₄	—	8	4	no	—
149	C₄	—	8	4	no	—
150	C₄	#497×C₆₇	1	8	yes	—
151	C₄	#532×C₆₇	1	4	yes	—
152	C₂	C₆₇×#2858	1	1	yes	characteristic
153	C₂	C₆₇×#3139	1	4	yes	—
154	C₂	C₂×#497×C₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#487×C₆₇

C₂

yes

maximal

C₆₇×#2868

C₂

yes

maximal, characteristic

C₆₇×#2862

C₂

yes

maximal, characteristic

C₆₇×#1960

C₂

yes

maximal

C₄×#97×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#487×C₆₇

—

C₂×#103×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

#391×C₆₇

—

#384×C₆₇

—

#382×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

—

#103×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

#95×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂²×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

#127

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₂×C₄×C₆₇

C₂×#41

yes

—

C₈×C₆₇

—

C₂²×C₆₇

C₂×#124

yes

—

C₂²×C₆₇

C₂×#127

yes

characteristic

C₂²×C₆₇

C₂×#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#497

yes

—

C₄×C₆₇

#532

yes

—

#33236

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

#2858

yes

characteristic

C₂×C₆₇

#3139

yes

—

C₂×C₆₇

C₂×#497

yes

—

C₂×#487

C₂×C₆₇

yes

—

#2868

C₂×C₆₇

yes

characteristic

#2862

C₂×C₆₇

yes

characteristic

#1960

C₂×C₆₇

yes

—

C₄×#97

C₂×C₆₇

yes

characteristic

C₆₇

#33236

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

#487

—

C₂×#103

C₂²×C₆₇

yes

—

C₂×#97

C₂²×C₆₇

yes

—

C₂×C₄×C₈

C₂²×C₆₇

yes

—

#391

—

#384

—

#382

C₂²×C₆₇

yes

—

C₄³

C₂²×C₆₇

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

C₂³×C₆₇

yes

—

C₂²×C₈

C₂³×C₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂³×C₆₇

yes

—

105

C₂×C₄²

D₄×C₆₇

yes

—

106

C₂×C₄²

C₂³×C₆₇

yes

characteristic

107

C₂×C₄²

C₂³×C₆₇

yes

—

108

#103

—

109

#97

—

110

C₄×C₈

D₄×C₆₇

yes

—

111

#95

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂×D₄×C₆₇

yes

—

119

C₂²×C₄

C₂×D₄×C₆₇

yes

—

120

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

121

C₂×C₈

—

122

C₂×C₈

C₂×D₄×C₆₇

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄²

—

128

C₄²

C₂×D₄×C₆₇

yes

—

129

C₄²

#41×C₆₇

yes

—

130

C₂³

C₂²×D₄×C₆₇

yes

characteristic

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#124×C₆₇

yes

—

139

C₂×C₄

#127×C₆₇

yes

—

140

C₂×C₄

#137×C₆₇

yes

—

141

C₂×C₄

C₂²×D₄×C₆₇

yes

—

142

C₂×C₄

C₂×#41×C₆₇

yes

—

143

C₈

—

144

C₂²

C₂×#124×C₆₇

yes

—

145

C₂²

C₂×#127×C₆₇

yes

characteristic

146

C₂²

C₂×#137×C₆₇

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#497×C₆₇

yes

—

151

C₄

#532×C₆₇

yes

—

152

C₂

C₆₇×#2858

yes

characteristic

153

C₂

C₆₇×#3139

yes

—

154

C₂

C₂×#497×C₆₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₆₇

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

N	H ≅ G/N	type
74. #33236	C₆₇	direct (N × H)
83. C₆₇	#33236	direct (N × H)

H ≅ G/N

type

74. #33236

C₆₇

direct (N × H)

83. C₆₇

#33236

direct (N × H)

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

subgroup	index
C₂×#487×C₆₇	2
C₆₇×#2868	2
C₆₇×#2862	2
C₆₇×#1960	2
C₄×#97×C₆₇	2
#33236	67

subgroup

index

C₂×#487×C₆₇

C₆₇×#2868

C₆₇×#2862

C₆₇×#1960

C₄×#97×C₆₇

#33236

Subgroups — C₆₇×#33236 #30712487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#33236 #30712487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33236 #30712487