Group #30772250 — subgroups

Subgroups — #30772250

Order 17152.

342 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 159 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×#16863216	C₂	1	4	yes	maximal
2	#22561965	C₂	1	1	yes	maximal, characteristic
3	#22559895	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523954	C₄	1	1	yes	characteristic
7	C₂×#14523953	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
9	#16863242	C₄	1	1	yes	characteristic
10	#16863216	—	2	8	no	—
11	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₆₇	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
17	#14523983	—	2	4	no	—
18	C₂×C₆₇⋊C₁₆	—	2	2	no	—
19	#14523954	C₈	1	4	yes	—
20	#14523953	—	2	4	no	—
21	C₈×Dic₆₇	D₄	1	4	yes	—
22	C₈×Dic₆₇	Q₈	1	4	yes	—
23	C₆₇⋊#95	—	2	2	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₆₇	—	2	4	no	—
28	C₂×C₁₆×C₆₇	—	2	2	no	—
29	C₄×C₈×C₆₇	D₄	1	2	yes	—
30	C₄×C₈×C₆₇	Q₈	1	2	yes	—
31	C₂²×Dic₆₇	—	2	2	no	—
32	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
33	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
34	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
35	C₄×Dic₆₇	—	2	2	no	—
36	C₄×Dic₆₇	C₄⋊C₄	1	4	yes	—
37	C₆₇⋊C₁₆	—	4	2	no	—
38	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₄²×C₆₇	C₂×C₈	1	2	yes	—
48	C₁₆×C₆₇	—	4	2	no	—
49	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
56	C₂×Dic₆₇	—	2	4	no	—
57	C₂×Dic₆₇	—	4	4	no	—
58	C₈×C₆₇	—	2	2	no	—
59	C₈×C₆₇	#127	1	1	yes	characteristic
60	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
62	C₆₇⋊C₈	C₄⋊C₈	1	8	yes	—
63	C₂²×C₆₇	—	2	2	no	—
64	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂²×C₆₇	#423	1	2	yes	—
66	C₄×C₆₇	—	2	4	no	—
67	C₄×C₆₇	#384	1	1	yes	characteristic
68	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
69	C₄×C₆₇	#423	1	1	yes	characteristic
70	Dic₆₇	—	4	4	no	—
71	#12901	—	67	1	no	maximal, Sylow
72	C₂×C₆₇	—	2	2	no	—
73	C₂×C₆₇	#1842	1	1	yes	characteristic
74	C₂×C₆₇	#2153	1	2	yes	—
75	C₂×#364	—	67	4	no	—
76	#2150	—	67	1	no	—
77	#2150	D₆₇	1	1	yes	characteristic
78	#1726	—	67	1	no	—
79	C₆₇	#12901	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	67	2	no	—
81	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
82	C₂×#97	—	67	1	no	—
83	C₂×#97	—	67	2	no	—
84	C₂×C₄×C₈	—	67	2	no	—
85	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
86	#376	—	67	1	no	—
87	#364	—	134	8	no	—
88	C₄⋊C₁₆	—	2	2	no	—
89	C₄⋊C₁₆	—	134	2	no	—
90	C₂²×C₈	—	67	2	no	—
91	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
93	C₂×C₄²	—	67	2	no	—
94	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	M₃₂	—	134	4	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	134	2	no	—
99	#97	—	67	4	no	—
100	#97	—	134	4	no	—
101	C₄×C₈	—	67	4	no	—
102	C₄×C₈	—	67	4	no	—
103	C₄×C₈	Dic₁₃₄	1	2	yes	—
104	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
105	#95	—	134	2	no	—
106	C₂²×C₄	—	134	2	no	—
107	C₂²×C₄	#11761611	1	2	yes	—
108	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	67	4	no	—
111	C₂×C₈	—	67	4	no	—
112	C₂×C₈	—	67	4	no	—
113	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
114	C₂×C₈	D₄×D₆₇	1	2	yes	—
115	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂×C₈	Q₈×D₆₇	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
118	C₄²	—	67	4	no	—
119	C₄²	—	134	2	no	—
120	C₄²	Dic₆₇⋊C₄	1	2	yes	—
121	C₄²	C₈×D₆₇	1	2	yes	—
122	C₁₆	—	4	2	no	—
123	C₁₆	—	268	2	no	—
124	C₂³	C₂×#11761611	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	268	4	no	—
127	C₂×C₄	—	134	4	no	—
128	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
129	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
130	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
132	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	67	8	no	—
135	C₈	#14524039	1	1	yes	characteristic
136	C₈	#14524041	1	1	yes	characteristic
137	C₈	#14524042	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
140	C₂²	#16863418	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	268	4	no	—
143	C₄	#16863265	1	1	yes	characteristic
144	C₄	C₄⋊C₈×D₆₇	1	2	yes	—
145	C₄	#16863419	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#22560360	1	1	yes	characteristic
148	C₂	#22561974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863216

C₂

yes

maximal

#22561965

C₂

yes

maximal, characteristic

#22559895

C₂

yes

maximal, characteristic

C₆₇×#2150

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

C₂×#14523954

C₄

yes

characteristic

C₂×#14523953

C₄

yes

—

C₂×C₈×Dic₆₇

C₂²

yes

—

#16863242

C₄

yes

characteristic

#16863216

—

C₄⋊(C₆₇⋊C₁₆)

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

#14523954

C₈

yes

—

#14523953

—

C₈×Dic₆₇

D₄

yes

—

C₈×Dic₆₇

Q₈

yes

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

M₁₆

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₄⋊C₄

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂×C₈

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#127

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

—

C₆₇⋊C₈

C₄⋊C₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#423

yes

—

C₄×C₆₇

—

C₄×C₆₇

#384

yes

characteristic

C₄×C₆₇

C₂×C₄⋊C₈

yes

—

C₄×C₆₇

#423

yes

characteristic

Dic₆₇

—

#12901

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1842

yes

characteristic

C₂×C₆₇

#2153

yes

—

C₂×#364

—

#2150

—

#2150

D₆₇

yes

characteristic

#1726

—

C₆₇

#12901

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#376

—

#364

—

134

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₆₇

yes

characteristic

M₃₂

—

M₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

#97

—

100

#97

—

134

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

Dic₁₃₄

yes

—

104

C₄×C₈

C₂²⋊D₆₇

yes

—

105

#95

—

134

—

106

C₂²×C₄

—

134

—

107

C₂²×C₄

#11761611

yes

—

108

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

Dic₆₇⋊C₄

yes

—

114

C₂×C₈

D₄×D₆₇

yes

—

115

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

116

C₂×C₈

Q₈×D₆₇

yes

—

117

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

118

C₄²

—

119

C₄²

—

134

—

120

C₄²

Dic₆₇⋊C₄

yes

—

121

C₄²

C₈×D₆₇

yes

—

122

C₁₆

—

123

C₁₆

—

268

—

124

C₂³

C₂×#11761611

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

268

—

127

C₂×C₄

—

134

—

128

C₂×C₄

Dic₆₇⋊C₈

yes

—

129

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

130

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

131

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

132

C₂×C₄

M₁₆×D₆₇

yes

—

133

C₈

—

134

C₈

—

135

C₈

#14524039

yes

characteristic

136

C₈

#14524041

yes

characteristic

137

C₈

#14524042

yes

—

138

C₂²

—

139

C₂²

C₂×Dic₆₇⋊C₈

yes

characteristic

140

C₂²

#16863418

yes

—

141

C₄

—

142

C₄

—

268

—

143

C₄

#16863265

yes

characteristic

144

C₄

C₄⋊C₈×D₆₇

yes

—

145

C₄

#16863419

yes

characteristic

146

C₂

—

147

C₂

#22560360

yes

characteristic

148

C₂

#22561974

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#12901
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2150

C₂

socle

C₂²×C₆₇

C₂×C₄⋊C₈

prime radical

C₂³×C₆₇

C₂×M₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
79. C₆₇	#12901	semidirect

H ≅ G/N

type

79. C₆₇

#12901

semidirect

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

subgroup	index
C₂×#16863216	2
#22561965	2
#22559895	2
C₆₇×#2150	2
#12901	67

subgroup

index

C₆₇×#2150

Subgroups — #30772250

Conjugacy classes of subgroups

Complements

Maximal subgroups