Group #34199906 — subgroups

Subgroups — #34199906

Order 20224.

216 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23764993	C₂	1	1	yes	maximal, characteristic
2	#23762160	C₂	1	2	yes	maximal
3	#23761466	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761465	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579100	C₄	1	1	yes	characteristic
7	C₂×#14579100	C₄	1	1	yes	characteristic
8	#19098619	—	2	2	no	—
9	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
10	#19098339	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₇₉	C₄	1	1	yes	characteristic
12	C₁₆×Dic₇₉	C₄	1	1	yes	characteristic
13	#19098315	—	2	2	no	—
14	#19098314	C₂²	1	2	yes	—
15	#346×C₇₉	C₂²	1	1	yes	characteristic
16	#334×C₇₉	C₂²	1	2	yes	—
17	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₇₉	Q₈	1	1	yes	characteristic
19	C₄²×D₇₉	D₄	1	1	yes	characteristic
20	#14579100	—	2	2	no	—
21	#14579100	—	2	2	no	—
22	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	(C₂×D₇₉)⋊C₈	—	4	2	no	—
26	#14579071	—	4	2	no	—
27	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
28	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₉	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₇₉	—	2	2	no	—
33	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₇₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₇₉	SD₁₆	1	2	yes	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₂×C₇₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₇₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₈×D₇₉	—	2	1	no	—
40	C₈×D₇₉	—	2	1	no	—
41	C₈×D₇₉	—	2	2	no	—
42	C₄×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₇₉	SD₁₆	1	2	yes	—
44	C₇₉⋊C₁₆	—	2	1	no	—
45	C₇₉⋊C₁₆	M₁₆	1	2	yes	—
46	C₂×C₈×C₇₉	—	4	2	no	—
47	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₇₉	—	2	1	no	—
51	C₁₆×C₇₉	M₁₆	1	2	yes	—
52	C₂²×D₇₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
55	C₂×Dic₇₉	#106	1	2	yes	—
56	C₂×Dic₇₉	#106	1	1	yes	characteristic
57	C₄×D₇₉	—	2	1	no	—
58	C₄×D₇₉	—	2	1	no	—
59	C₄×D₇₉	—	4	2	no	—
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	—	8	2	no	—
62	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
64	C₇₉⋊C₈	—	2	1	no	—
65	C₇₉⋊C₈	—	8	2	no	—
66	C₇₉⋊C₈	#108	1	2	yes	—
67	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
68	C₂×D₇₉	—	2	1	no	—
69	C₂×D₇₉	—	2	2	no	—
70	C₄×C₇₉	—	2	2	no	—
71	C₄×C₇₉	#423	1	1	yes	characteristic
72	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
73	Dic₇₉	—	2	1	no	—
74	Dic₇₉	—	2	2	no	—
75	#12021	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	#1566	1	1	yes	characteristic
77	C₂×C₇₉	#2155	1	2	yes	—
78	D₇₉	—	4	1	no	—
79	C₄×M₃₂	—	79	1	no	—
80	#1568	—	79	2	no	—
81	#1368	—	79	1	no	—
82	#1368	—	79	2	no	—
83	#1368	D₇₉	1	1	yes	characteristic, complemented
84	C₇₉	#12021	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	79	1	no	—
86	C₂×M₃₂	—	79	1	no	—
87	#432	—	158	2	no	—
88	C₂×C₄×C₈	—	79	1	no	—
89	#346	—	79	1	no	—
90	#346	C₂×D₇₉	1	1	yes	characteristic
91	C₄×C₁₆	—	79	1	no	—
92	C₄×C₁₆	—	79	1	no	—
93	#335	—	158	2	no	—
94	#334	—	79	2	no	—
95	#334	C₂×D₇₉	1	2	yes	—
96	C₂²×C₈	—	79	1	no	—
97	C₂²×C₈	—	79	1	no	—
98	C₂×C₄²	—	79	1	no	—
99	M₃₂	—	158	2	no	—
100	M₃₂	—	158	2	no	—
101	C₂×C₁₆	—	79	1	no	—
102	C₂×C₁₆	—	79	1	no	—
103	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
104	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	158	2	no	—
107	C₂²⋊C₈	—	316	2	no	—
108	#97	—	316	2	no	—
109	C₄×C₈	—	79	1	no	—
110	C₄×C₈	—	79	1	no	—
111	C₄×C₈	—	79	1	no	—
112	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
113	C₂²×C₄	—	79	2	no	—
114	C₂²×C₄	—	79	1	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	316	2	no	—
117	C₂×C₈	—	79	1	no	—
118	C₂×C₈	—	79	1	no	—
119	C₂×C₈	—	158	2	no	—
120	C₂×C₈	—	158	1	no	—
121	C₂×C₈	—	158	1	no	—
122	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
123	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
124	C₄²	—	79	1	no	—
125	C₄²	—	79	2	no	—
126	C₄²	D₄×D₇₉	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	79	2	no	—
129	C₁₆	—	158	1	no	—
130	C₁₆	#11929607	1	2	yes	—
131	C₂³	—	79	1	no	—
132	C₂×C₄	—	316	2	no	—
133	C₂×C₄	—	79	2	no	—
134	C₂×C₄	—	79	1	no	—
135	C₂×C₄	—	158	1	no	—
136	C₂×C₄	—	158	1	no	—
137	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
138	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	—	79	2	no	—
142	C₈	—	632	2	no	—
143	C₈	—	158	1	no	—
144	C₈	M₁₆×D₇₉	1	1	yes	characteristic
145	C₈	C₂×#11929607	1	1	yes	characteristic
146	C₂²	—	158	2	no	—
147	C₂²	—	158	1	no	—
148	C₂²	#106×D₇₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	158	2	no	—
151	C₄	—	158	1	no	—
152	C₄	#19098585	1	1	yes	characteristic
153	C₄	#108×D₇₉	1	1	yes	characteristic
154	C₂	—	316	1	no	—
155	C₂	#23762154	1	1	yes	characteristic
156	C₂	#23765171	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764993

C₂

yes

maximal, characteristic

#23762160

C₂

yes

maximal

#23761466

C₂

yes

maximal, characteristic, complemented

#23761465

C₂

yes

maximal, complemented

C₇₉×#1368

C₂

yes

maximal, characteristic, complemented

C₂×#14579100

C₄

yes

characteristic

C₂×#14579100

C₄

yes

characteristic

#19098619

—

C₄×C₈×D₇₉

C₂²

yes

characteristic

#19098339

C₂²

yes

characteristic

C₁₆×Dic₇₉

C₄

yes

characteristic

C₁₆×Dic₇₉

C₄

yes

characteristic

#19098315

—

#19098314

C₂²

yes

—

#346×C₇₉

C₂²

yes

characteristic

#334×C₇₉

C₂²

yes

—

C₂×C₈×D₇₉

C₂×C₄

yes

characteristic

C₂×C₈×D₇₉

Q₈

yes

characteristic

C₄²×D₇₉

D₄

yes

characteristic

#14579100

—

#14579100

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

(C₂×D₇₉)⋊C₈

—

#14579071

—

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₈×Dic₇₉

C₂×C₄

yes

characteristic

C₈×Dic₇₉

Q₈

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄×D₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₉

SD₁₆

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₇₉

—

C₈×D₇₉

—

C₈×D₇₉

—

C₄×Dic₇₉

C₄⋊C₄

yes

characteristic

C₄×Dic₇₉

SD₁₆

yes

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

M₁₆

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

M₁₆

yes

—

C₂²×D₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×Dic₇₉

#106

yes

—

C₂×Dic₇₉

#106

yes

characteristic

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×M₁₆

yes

characteristic

C₈×C₇₉

C₂×M₁₆

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

#108

yes

—

C₂²×C₇₉

C₂×#106

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

C₄×M₃₂

—

—

—

—

D₇₉

yes

characteristic, complemented

C₇₉

#12021

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

#432

—

158

—

C₂×C₄×C₈

—

#346

—

#346

C₂×D₇₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

158

—

#334

—

#334

C₂×D₇₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

158

—

100

M₃₂

—

158

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₄×D₇₉

yes

characteristic

104

C₂×C₁₆

C₄×D₇₉

yes

characteristic

105

C₄⋊C₈

—

106

C₄⋊C₈

—

158

—

107

C₂²⋊C₈

—

316

—

108

#97

—

316

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₂²×D₇₉

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂×C₈

—

116

C₂×C₈

—

316

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

158

—

120

C₂×C₈

—

158

—

121

C₂×C₈

—

158

—

122

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

123

C₂×C₈

Q₈×D₇₉

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

D₄×D₇₉

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

158

—

130

C₁₆

#11929607

yes

—

131

C₂³

—

132

C₂×C₄

—

316

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

158

—

136

C₂×C₄

—

158

—

137

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

138

C₂×C₄

SD₁₆×D₇₉

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

632

—

143

C₈

—

158

—

144

C₈

M₁₆×D₇₉

yes

characteristic

145

C₈

C₂×#11929607

yes

characteristic

146

C₂²

—

158

—

147

C₂²

—

158

—

148

C₂²

#106×D₇₉

yes

characteristic

149

C₄

—

150

C₄

—

158

—

151

C₄

—

158

—

152

C₄

#19098585

yes

characteristic

153

C₄

#108×D₇₉

yes

characteristic

154

C₂

—

316

—

155

C₂

#23762154

yes

characteristic

156

C₂

#23765171

yes

—

	subgroup	G/H
center	C₂²	#106×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1368	C₂
socle	C₂²×C₇₉	C₂×#106
prime radical	C₂²×D₇₉	#106
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1368	D₇₉
hypocenter	C₇₉	#12021
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#106×D₇₉

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1368

C₂

C₂²×C₇₉

C₂²×D₇₉

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
3. #23761466	C₂	semidirect
4. #23761465	C₂	semidirect
5. C₇₉×#1368	C₂	semidirect
83. #1368	D₇₉	semidirect
84. C₇₉	#12021	semidirect

H ≅ G/N

type

3. #23761466

C₂

semidirect

4. #23761465

C₂

semidirect

5. C₇₉×#1368

C₂

semidirect

83. #1368

D₇₉

semidirect

84. C₇₉

#12021

semidirect

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

subgroup	index
#23764993	2
#23762160	2
#23761466	2
#23761465	2
C₇₉×#1368	2
#12021	79

subgroup

index

C₇₉×#1368

Subgroups — #34199906

Conjugacy classes of subgroups

Complements

Maximal subgroups