Group #34175291 — subgroups

Subgroups — #34175291

Order 20224.

212 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 95 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	#23765199	C₂	1	1	yes	maximal, characteristic
2	C₈×#11929608	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#2150	C₂	1	1	yes	maximal, characteristic
4	#19098616	C₄	1	1	yes	characteristic
5	C₄×#11929608	C₄	1	1	yes	characteristic
6	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
7	C₂²⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
8	C₄×C₇₉⋊C₁₆	—	2	1	no	—
9	#19098285	—	2	1	no	—
10	C₈×C₇₉⋊C₈	—	2	1	no	—
11	C₂×M₃₂×C₇₉	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₉	—	2	1	no	—
14	C₄⋊C₁₆×C₇₉	—	2	1	no	—
15	C₂×#11929608	—	2	1	no	—
16	C₂×C₇₉⋊C₁₆	—	2	2	no	—
17	C₄⋊(C₇₉⋊C₈)	C₈	1	2	yes	—
18	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
19	#14579072	C₈	1	2	yes	—
20	C₄×C₇₉⋊C₈	—	2	1	no	—
21	C₄×C₇₉⋊C₈	—	2	1	no	—
22	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₇₉	—	4	2	no	—
26	C₂×C₁₆×C₇₉	—	2	2	no	—
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
31	#11929608	—	2	4	no	—
32	C₂×C₇₉⋊C₈	—	2	2	no	—
33	C₂×C₇₉⋊C₈	—	2	2	no	—
34	C₇₉⋊C₁₆	—	4	2	no	—
35	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₂×C₈×C₇₉	—	2	1	no	—
40	C₂×C₈×C₇₉	—	4	1	no	—
41	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
49	C₁₆×C₇₉	—	4	2	no	—
50	C₂³×C₇₉	#97	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	2	1	no	—
53	C₂×C₄×C₇₉	—	2	2	no	—
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	2	1	no	—
63	C₈×C₇₉	—	2	1	no	—
64	C₈×C₇₉	#100	1	1	yes	characteristic
65	C₈×C₇₉	#101	1	1	yes	characteristic
66	C₇₉⋊C₈	—	2	4	no	—
67	C₇₉⋊C₈	—	4	2	no	—
68	C₂²×C₇₉	—	2	1	no	—
69	C₂²×C₇₉	—	2	1	no	—
70	C₂²×C₇₉	—	2	1	no	—
71	C₂²×C₇₉	#336	1	1	yes	characteristic
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	#343	1	1	yes	characteristic
76	C₄×C₇₉	#344	1	1	yes	characteristic
77	C₄×C₇₉	#350	1	2	yes	—
78	C₄×C₇₉	#364	1	2	yes	—
79	#7212	—	79	1	no	maximal, Sylow
80	C₂×C₇₉	—	4	1	no	—
81	C₂×C₇₉	#1283	1	1	yes	characteristic
82	C₂×C₇₉	#1376	1	1	yes	characteristic
83	C₂×C₇₉	#1379	1	1	yes	characteristic
84	#2162	—	79	1	no	—
85	#2150	D₇₉	1	1	yes	characteristic
86	C₈×M₁₆	—	79	1	no	—
87	C₇₉	#7212	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₃₂	Dic₇₉	1	2	yes	—
89	#431	—	79	1	no	—
90	C₄×M₁₆	—	79	1	no	—
91	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
92	C₄⋊C₁₆	—	2	1	no	—
93	C₄⋊C₁₆	—	2	1	no	—
94	C₄⋊C₁₆	—	158	1	no	—
95	C₂²⋊C₁₆	—	79	2	no	—
96	C₄×C₁₆	—	158	1	no	—
97	#322	—	158	1	no	—
98	C₈²	—	158	1	no	—
99	C₂×M₁₆	—	158	1	no	—
100	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
101	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
102	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
103	M₃₂	—	4	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	158	2	no	—
106	C₄⋊C₈	—	79	2	no	—
107	C₂²⋊C₈	—	158	1	no	—
108	#97	—	79	2	no	—
109	C₄×C₈	—	158	1	no	—
110	C₄×C₈	—	158	1	no	—
111	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
112	C₄×C₈	D₃₁₆	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
115	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
116	C₂²×C₄	#11929607	1	1	yes	characteristic
117	C₂²×C₄	#11929607	1	1	yes	characteristic
118	M₁₆	—	158	4	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	158	2	no	—
123	C₂×C₈	—	158	2	no	—
124	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
126	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
128	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
129	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
130	C₄²	C₈×D₇₉	1	1	yes	characteristic
131	C₄²	C₈×D₇₉	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	C₁₆	—	316	2	no	—
134	C₂³	#14579071	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
140	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
141	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
142	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
143	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
144	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	316	2	no	—
149	C₈	—	158	4	no	—
150	C₈	#14579078	1	1	yes	characteristic
151	C₈	#14579080	1	1	yes	characteristic
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#19098317	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#19098332	1	1	yes	characteristic
160	C₄	#19098334	1	1	yes	characteristic
161	C₄	#19098348	1	2	yes	—
162	C₄	#19098383	1	2	yes	—
163	C₂	—	4	1	no	—
164	C₂	#23761250	1	1	yes	characteristic
165	C₂	#23761481	1	1	yes	characteristic
166	C₂	#23761490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765199

C₂

yes

maximal, characteristic

C₈×#11929608

C₂

yes

maximal, characteristic

C₇₉×#2150

C₂

yes

maximal, characteristic

#19098616

C₄

yes

characteristic

C₄×#11929608

C₄

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

—

C₂²⋊(C₇₉⋊C₁₆)

C₄

yes

—

C₄×C₇₉⋊C₁₆

—

#19098285

—

C₈×C₇₉⋊C₈

—

C₂×M₃₂×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₄⋊C₁₆×C₇₉

—

C₂×#11929608

—

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

C₈

yes

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

C₈

yes

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

M₁₆

yes

characteristic

C₂²×C₄×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

#97

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

C₄×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#336

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₄×C₇₉

#364

yes

—

#7212

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

C₈×M₁₆

—

C₇₉

#7212

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₇₉

yes

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

158

—

#322

—

158

—

C₈²

—

158

—

C₂×M₁₆

—

158

—

100

C₂²×C₈

C₄×D₇₉

yes

characteristic

101

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

102

C₂×C₄²

C₄×D₇₉

yes

characteristic

103

M₃₂

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

158

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

158

—

108

#97

—

109

C₄×C₈

—

158

—

110

C₄×C₈

—

158

—

111

C₄×C₈

Dic₁₅₈

yes

characteristic

112

C₄×C₈

D₃₁₆

yes

characteristic

113

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

114

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

115

C₂²×C₄

C₄×Dic₇₉

yes

characteristic

116

C₂²×C₄

#11929607

yes

characteristic

117

C₂²×C₄

#11929607

yes

characteristic

118

M₁₆

—

158

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

158

—

123

C₂×C₈

—

158

—

124

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

characteristic

125

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

126

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

127

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

128

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

129

C₄²

Dic₇₉⋊C₄

yes

characteristic

130

C₄²

C₈×D₇₉

yes

characteristic

131

C₄²

C₈×D₇₉

yes

characteristic

132

C₁₆

—

133

C₁₆

—

316

—

134

C₂³

#14579071

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₇₉⋊#95

yes

characteristic

140

C₂×C₄

C₈×Dic₇₉

yes

characteristic

141

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

characteristic

142

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

characteristic

143

C₂×C₄

Dic₇₉⋊C₈

yes

characteristic

144

C₂×C₄

Dic₇₉⋊C₈

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

316

—

149

C₈

—

158

—

150

C₈

#14579078

yes

characteristic

151

C₈

#14579080

yes

characteristic

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#19098317

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#19098332

yes

characteristic

160

C₄

#19098334

yes

characteristic

161

C₄

#19098348

yes

—

162

C₄

#19098383

yes

—

163

C₂

—

164

C₂

#23761250

yes

characteristic

165

C₂

#23761481

yes

characteristic

166

C₂

#23761490

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₇₉×#2150	C₂
socle	C₂²×C₇₉	#336
prime radical	C₂³×C₇₉	#97
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#7212
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₇₉

commutator

C₂×C₄×C₇₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₇₉×#2150

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
87. C₇₉	#7212	semidirect

H ≅ G/N

type

87. C₇₉

#7212

semidirect

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

subgroup	index
#23765199	2
C₈×#11929608	2
C₇₉×#2150	2
#7212	79

subgroup

index

#23765199

C₈×#11929608

C₇₉×#2150

#7212

Subgroups — #34175291

Conjugacy classes of subgroups

Complements

Maximal subgroups