Group #34175522 — subgroups

Subgroups — #34175522

Order 20224.

216 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₄×#14579097	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579097	—	2	1	no	—
5	C₄×#11929608	—	2	1	no	—
6	C₄⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₇₉⋊C₁₆)	—	2	1	no	—
8	C₄×C₇₉⋊C₁₆	—	2	1	no	—
9	#19098328	C₄	1	2	yes	—
10	#423×C₇₉	—	2	1	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#335×C₇₉	—	2	1	no	—
13	#335×C₇₉	C₄	1	2	yes	—
14	C₂×#11929608	—	2	2	no	—
15	C₂×C₇₉⋊C₁₆	—	2	2	no	—
16	#14579097	—	2	4	no	—
17	#14579072	—	4	1	no	—
18	C₄×C₇₉⋊C₈	—	4	1	no	—
19	C₂×M₁₆×C₇₉	—	4	1	no	—
20	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
23	C₄⋊C₈×C₇₉	—	2	2	no	—
24	C₄⋊C₈×C₇₉	—	4	1	no	—
25	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
29	#11929608	—	2	4	no	—
30	#11929608	—	4	2	no	—
31	C₂×C₇₉⋊C₈	—	4	2	no	—
32	C₇₉⋊C₁₆	—	4	2	no	—
33	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
36	M₁₆×C₇₉	—	8	2	no	—
37	C₂×C₈×C₇₉	—	2	2	no	—
38	C₂×C₈×C₇₉	—	4	1	no	—
39	C₂×C₈×C₇₉	—	4	2	no	—
40	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₉	D₈	1	1	yes	characteristic
47	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
48	C₂³×C₇₉	#103	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
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	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	2	1	no	—
63	C₈×C₇₉	—	8	2	no	—
64	C₈×C₇₉	#104	1	2	yes	—
65	C₇₉⋊C₈	—	4	4	no	—
66	C₂²×C₇₉	—	2	1	no	—
67	C₂²×C₇₉	—	2	1	no	—
68	C₂²×C₇₉	—	2	1	no	—
69	C₂²×C₇₉	#340	1	1	yes	characteristic
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	—	2	1	no	—
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	#339	1	1	yes	characteristic
74	C₄×C₇₉	#344	1	1	yes	characteristic
75	C₄×C₇₉	#361	1	1	yes	characteristic
76	C₄×C₇₉	#362	1	1	yes	characteristic
77	C₄×C₇₉	#368	1	2	yes	—
78	#7298	—	79	1	no	maximal, Sylow
79	C₂×C₇₉	—	4	1	no	—
80	C₂×C₇₉	#1303	1	1	yes	characteristic
81	C₂×C₇₉	#1381	1	1	yes	characteristic
82	C₂×C₇₉	#1387	1	1	yes	characteristic
83	#2162	—	79	1	no	—
84	C₄×#108	—	79	1	no	—
85	#1567	D₇₉	1	1	yes	characteristic
86	C₇₉	#7298	1	1	yes	characteristic, complemented, Sylow
87	C₂×#108	—	158	1	no	—
88	#423	—	2	1	no	—
89	C₄×M₁₆	—	158	1	no	—
90	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
91	C₄⋊C₁₆	—	79	2	no	—
92	C₂²⋊C₁₆	—	158	1	no	—
93	C₄×C₁₆	—	158	1	no	—
94	#341	—	79	2	no	—
95	#335	—	2	1	no	—
96	#335	Dic₇₉	1	2	yes	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂×M₁₆	—	158	2	no	—
99	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
100	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
102	C₂×C₁₆	—	158	2	no	—
103	#108	—	158	4	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	1	no	—
106	#97	—	316	1	no	—
107	C₄×C₈	—	316	1	no	—
108	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
109	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
113	C₂²×C₄	#11929612	1	1	yes	characteristic
114	C₂²×C₄	#11929614	1	1	yes	characteristic
115	M₁₆	—	8	2	no	—
116	M₁₆	—	316	2	no	—
117	M₁₆	—	158	4	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	316	2	no	—
122	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
123	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
124	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
126	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
127	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
128	C₄²	#11929609	1	1	yes	characteristic
129	C₄²	#11929611	1	1	yes	characteristic
130	C₁₆	—	316	2	no	—
131	C₂³	#14579084	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
138	C₂×C₄	#14579081	1	1	yes	characteristic
139	C₂×C₄	#14579082	1	1	yes	characteristic
140	C₂×C₄	#14579085	1	1	yes	characteristic
141	C₂×C₄	#14579092	1	1	yes	characteristic
142	C₂×C₄	#14579094	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	316	4	no	—
148	C₈	#14579087	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#19098325	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#19098324	1	1	yes	characteristic
157	C₄	#19098334	1	1	yes	characteristic
158	C₄	#19098376	1	1	yes	characteristic
159	C₄	#19098379	1	1	yes	characteristic
160	C₄	#19098392	1	2	yes	—
161	C₂	—	4	1	no	—
162	C₂	#23761301	1	1	yes	characteristic
163	C₂	#23761493	1	1	yes	characteristic
164	C₂	#23761508	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765199

C₂

yes

maximal, characteristic

C₄×#14579097

C₂

yes

maximal, characteristic

C₇₉×#1567

C₂

yes

maximal, characteristic

C₂×#14579097

—

C₄×#11929608

—

C₄⋊(C₇₉⋊C₁₆)

C₄

yes

—

C₂²⋊(C₇₉⋊C₁₆)

—

C₄×C₇₉⋊C₁₆

—

#19098328

C₄

yes

—

#423×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#335×C₇₉

—

#335×C₇₉

C₄

yes

—

C₂×#11929608

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

Q₁₆

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

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

D₈

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₂³×C₇₉

#103

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#104

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#340

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#7298

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

C₇₉

#7298

yes

characteristic, complemented, Sylow

C₂×#108

—

158

—

#423

—

C₄×M₁₆

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

158

—

C₄×C₁₆

—

158

—

#341

—

#335

—

#335

Dic₇₉

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

158

—

C₂²×C₈

C₄×D₇₉

yes

characteristic

100

C₂²×C₈

D₃₁₆

yes

characteristic

101

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

102

C₂×C₁₆

—

158

—

103

#108

—

158

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#97

—

316

—

107

C₄×C₈

—

316

—

108

C₄×C₈

Dic₁₅₈

yes

characteristic

109

C₄×C₈

C₄×D₇₉

yes

characteristic

110

C₄×C₈

C₂×Dic₇₉

yes

characteristic

111

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

112

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

113

C₂²×C₄

#11929612

yes

characteristic

114

C₂²×C₄

#11929614

yes

characteristic

115

M₁₆

—

116

M₁₆

—

316

—

117

M₁₆

—

158

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

316

—

122

C₂×C₈

C₄×Dic₇₉

yes

characteristic

123

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

characteristic

124

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

125

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

126

C₄²

C₂²⋊Dic₇₉

yes

characteristic

127

C₄²

Dic₇₉⋊C₄

yes

characteristic

128

C₄²

#11929609

yes

characteristic

129

C₄²

#11929611

yes

characteristic

130

C₁₆

—

316

—

131

C₂³

#14579084

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₇₉⋊#95

yes

characteristic

138

C₂×C₄

#14579081

yes

characteristic

139

C₂×C₄

#14579082

yes

characteristic

140

C₂×C₄

#14579085

yes

characteristic

141

C₂×C₄

#14579092

yes

characteristic

142

C₂×C₄

#14579094

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

316

—

148

C₈

#14579087

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#19098325

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#19098324

yes

characteristic

157

C₄

#19098334

yes

characteristic

158

C₄

#19098376

yes

characteristic

159

C₄

#19098379

yes

characteristic

160

C₄

#19098392

yes

—

161

C₂

—

162

C₂

#23761301

yes

characteristic

163

C₂

#23761493

yes

characteristic

164

C₂

#23761508

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579082
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₇₉×#1567	C₂
socle	C₂²×C₇₉	#340
prime radical	C₂³×C₇₉	#103
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#1567	D₇₉
hypocenter	C₇₉	#7298
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

subgroup

G/H

center

C₂×C₄

#14579082

commutator

C₂×C₈×C₇₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₇₉×#1567

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

N	H ≅ G/N	type
86. C₇₉	#7298	semidirect

H ≅ G/N

type

86. C₇₉

#7298

semidirect

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

subgroup	index
#23765199	2
C₄×#14579097	2
C₇₉×#1567	2
#7298	79

subgroup

index

#23765199

C₄×#14579097

C₇₉×#1567

#7298

Subgroups — #34175522

Conjugacy classes of subgroups

Complements

Maximal subgroups