Group #34204636 — subgroups

Subgroups — #34204636

Order 20224.

234 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 111 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	#23765166	C₂	1	1	yes	maximal, characteristic
2	#23765159	C₂	1	1	yes	maximal, characteristic
3	#23761489	C₂	1	2	yes	maximal
4	#23761486	C₂	1	2	yes	maximal
5	C₇₉×#1747	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	1	yes	characteristic
7	C₂×#14579101	C₂²	1	2	yes	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	#19098384	—	2	2	no	—
10	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
11	#405×C₇₉	C₂²	1	2	yes	—
12	#405×C₇₉	C₂²	1	1	yes	characteristic
13	#338×C₇₉	C₄	1	1	yes	characteristic
14	H(3,ℤ/4)×C₇₉	C₄	1	1	yes	characteristic
15	H(3,ℤ/4)×C₇₉	C₄	1	2	yes	—
16	#14579101	—	2	2	no	—
17	#14579101	—	2	2	no	—
18	#14579101	—	2	2	no	—
19	C₂×C₇₉⋊C₁₆	—	2	1	no	—
20	C₂×C₇₉⋊C₁₆	—	2	1	no	—
21	C₂×C₇₉⋊C₁₆	—	2	2	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	#117×C₇₉	C₂×C₄	1	2	yes	—
27	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
28	#97×C₇₉	—	2	2	no	—
29	#97×C₇₉	Q₈	1	2	yes	—
30	C₄×C₈×C₇₉	—	2	2	no	—
31	C₄×C₈×C₇₉	D₄	1	2	yes	—
32	C₇₉⋊C₁₆	—	4	1	no	—
33	C₇₉⋊C₁₆	—	4	1	no	—
34	C₇₉⋊C₁₆	—	4	2	no	—
35	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
36	M₁₆×C₇₉	—	2	2	no	—
37	M₁₆×C₇₉	—	2	2	no	—
38	M₁₆×C₇₉	—	2	2	no	—
39	C₂×C₈×C₇₉	—	2	2	no	—
40	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₇₉	#41	1	2	yes	—
44	C₂×C₈×C₇₉	#41	1	2	yes	—
45	C₄⋊C₄×C₇₉	—	2	1	no	—
46	C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₂²⋊C₄×C₇₉	—	2	1	no	—
48	C₂²⋊C₄×C₇₉	—	2	2	no	—
49	C₄²×C₇₉	—	2	2	no	—
50	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
51	C₂³×C₇₉	#131	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	—	4	2	no	—
54	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#117	1	2	yes	—
56	C₂×C₄×C₇₉	#131	1	2	yes	—
57	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
58	C₈×C₇₉	—	2	2	no	—
59	C₈×C₇₉	—	4	2	no	—
60	C₈×C₇₉	#124	1	1	yes	characteristic
61	C₈×C₇₉	#124	1	1	yes	characteristic
62	C₈×C₇₉	#125	1	2	yes	—
63	C₂²×C₇₉	—	4	1	no	—
64	C₂²×C₇₉	#424	1	1	yes	characteristic
65	C₂²×C₇₉	#433	1	2	yes	—
66	C₄×C₇₉	—	4	2	no	—
67	C₄×C₇₉	—	8	2	no	—
68	C₄×C₇₉	#382	1	1	yes	characteristic
69	C₄×C₇₉	#424	1	1	yes	characteristic
70	C₄×C₇₉	#433	1	2	yes	—
71	#12885	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	—	2	1	no	—
73	C₂×C₇₉	—	2	2	no	—
74	C₂×C₇₉	#1832	1	1	yes	characteristic
75	#2153	—	79	1	no	—
76	#2151	—	79	1	no	—
77	#1747	D₇₉	1	1	yes	characteristic
78	#1379	—	79	2	no	—
79	#1378	—	79	2	no	—
80	C₇₉	#12885	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	79	1	no	—
82	C₂×M₃₂	—	79	2	no	—
83	C₂²×C₁₆	—	79	1	no	—
84	#405	C₂×D₇₉	1	2	yes	—
85	#405	C₂×D₇₉	1	1	yes	characteristic
86	#364	—	158	2	no	—
87	C₄⋊C₁₆	—	158	2	no	—
88	#338	Dic₇₉	1	1	yes	characteristic
89	H(3,ℤ/4)	Dic₇₉	1	1	yes	characteristic
90	H(3,ℤ/4)	Dic₇₉	1	2	yes	—
91	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
92	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
94	#117	C₂×Dic₇₉	1	2	yes	—
95	#117	C₂×Dic₇₉	1	1	yes	characteristic
96	M₃₂	—	158	2	no	—
97	M₃₂	—	158	2	no	—
98	M₃₂	—	158	2	no	—
99	C₂×C₁₆	—	158	1	no	—
100	C₂×C₁₆	—	158	1	no	—
101	C₂×C₁₆	—	158	2	no	—
102	C₂×C₁₆	—	158	2	no	—
103	#97	—	2	2	no	—
104	#97	Dic₁₅₈	1	2	yes	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	D₃₁₆	1	2	yes	—
107	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
113	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
114	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
115	C₂×C₈	#11929624	1	2	yes	—
116	C₂×C₈	#11929624	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₄⋊Dic₇₉	1	2	yes	—
123	C₁₆	—	316	1	no	—
124	C₁₆	—	316	1	no	—
125	C₁₆	—	316	2	no	—
126	C₂³	#14579168	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
130	C₂×C₄	#14579120	1	2	yes	—
131	C₂×C₄	#14579168	1	1	yes	characteristic
132	C₂×C₄	#14579168	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	#14579149	1	1	yes	characteristic
136	C₈	#14579149	1	1	yes	characteristic
137	C₈	#14579153	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#19098590	1	1	yes	characteristic
140	C₂²	#19098624	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	#19098429	1	1	yes	characteristic
144	C₄	#19098590	1	1	yes	characteristic
145	C₄	#19098624	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#23763520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765166

C₂

yes

maximal, characteristic

#23765159

C₂

yes

maximal, characteristic

#23761489

C₂

yes

maximal

#23761486

C₂

yes

maximal

C₇₉×#1747

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

characteristic

C₂×#14579101

C₂²

yes

—

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098384

—

C₄⋊(C₇₉⋊C₁₆)

—

#405×C₇₉

C₂²

yes

—

#405×C₇₉

C₂²

yes

characteristic

#338×C₇₉

C₄

yes

characteristic

H(3,ℤ/4)×C₇₉

C₄

yes

characteristic

H(3,ℤ/4)×C₇₉

C₄

yes

—

#14579101

—

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

C₂×C₄

yes

—

#117×C₇₉

C₂×C₄

yes

characteristic

#97×C₇₉

—

#97×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₄⋊C₄

yes

—

C₂³×C₇₉

#131

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#131

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

#424

yes

characteristic

C₂²×C₇₉

#433

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#12885

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

#1379

—

#1378

—

C₇₉

#12885

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

C₂×D₇₉

yes

—

#405

C₂×D₇₉

yes

characteristic

#364

—

158

—

C₄⋊C₁₆

—

158

—

#338

Dic₇₉

yes

characteristic

H(3,ℤ/4)

Dic₇₉

yes

characteristic

H(3,ℤ/4)

Dic₇₉

yes

—

C₂×M₁₆

C₂×Dic₇₉

yes

—

C₂×M₁₆

C₂×Dic₇₉

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

#117

C₂×Dic₇₉

yes

—

#117

C₂×Dic₇₉

yes

characteristic

M₃₂

—

158

—

M₃₂

—

158

—

M₃₂

—

158

—

C₂×C₁₆

—

158

—

100

C₂×C₁₆

—

158

—

101

C₂×C₁₆

—

158

—

102

C₂×C₁₆

—

158

—

103

#97

—

104

#97

Dic₁₅₈

yes

—

105

C₄×C₈

—

106

C₄×C₈

D₃₁₆

yes

—

107

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₄⋊Dic₇₉

yes

—

113

C₂×C₈

C₂×D₃₁₆

yes

characteristic

114

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

115

C₂×C₈

#11929624

yes

—

116

C₂×C₈

#11929624

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₄⋊Dic₇₉

yes

—

123

C₁₆

—

316

—

124

C₁₆

—

316

—

125

C₁₆

—

316

—

126

C₂³

#14579168

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

130

C₂×C₄

#14579120

yes

—

131

C₂×C₄

#14579168

yes

characteristic

132

C₂×C₄

#14579168

yes

—

133

C₈

—

134

C₈

—

135

C₈

#14579149

yes

characteristic

136

C₈

#14579149

yes

characteristic

137

C₈

#14579153

yes

—

138

C₂²

—

139

C₂²

#19098590

yes

characteristic

140

C₂²

#19098624

yes

—

141

C₄

—

142

C₄

—

143

C₄

#19098429

yes

characteristic

144

C₄

#19098590

yes

characteristic

145

C₄

#19098624

yes

—

146

C₂

—

147

C₂

—

148

C₂

#23763520

yes

characteristic

	subgroup	G/H
center	C₈	#14579149
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1747	C₂
socle	C₂×C₇₉	#1832
prime radical	C₂³×C₇₉	#131
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1747	D₇₉
hypocenter	C₇₉	#12885
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

subgroup

G/H

center

C₈

#14579149

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1747

C₂

C₂×C₇₉

C₂³×C₇₉

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

N	H ≅ G/N	type
80. C₇₉	#12885	semidirect

H ≅ G/N

type

80. C₇₉

#12885

semidirect

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

subgroup	index
#23765166	2
#23765159	2
#23761489	2
#23761486	2
C₇₉×#1747	2
#12885	79

subgroup

index

C₇₉×#1747

Subgroups — #34204636

Conjugacy classes of subgroups

Complements

Maximal subgroups