Group #34204666 — subgroups

Subgroups — #34204666

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	#23763524	C₂	1	2	yes	maximal
2	#23761504	C₂	1	1	yes	maximal, characteristic
3	#23761502	C₂	1	1	yes	maximal, characteristic
4	#23761496	C₂	1	2	yes	maximal
5	C₇₉×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	C₂²	1	2	yes	—
7	#19098601	C₂²	1	2	yes	—
8	#19098334	—	2	2	no	—
9	#19098319	—	2	2	no	—
10	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
12	#405×C₇₉	C₂²	1	1	yes	characteristic
13	#346×C₇₉	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₇₉	C₄	1	2	yes	complemented
15	C₂×#11929608	—	2	2	no	—
16	#14579120	—	2	2	no	—
17	#14579097	—	2	2	no	—
18	#14579097	—	2	2	no	—
19	#14579095	—	2	2	no	—
20	#14579093	—	2	2	no	—
21	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
23	#117×C₇₉	D₄	1	1	yes	characteristic
24	M₃₂×C₇₉	—	2	2	no	—
25	C₂×C₁₆×C₇₉	—	2	2	no	—
26	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
29	#97×C₇₉	Q₈	1	2	yes	—
30	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
31	#11929608	—	4	2	no	—
32	#11929608	—	4	2	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₂²⋊Dic₇₉	—	4	2	no	—
36	C₄×Dic₇₉	—	4	2	no	—
37	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₇₉	—	2	2	no	—
39	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₇₉	#41	1	2	yes	—
43	C₂×C₈×C₇₉	#41	1	2	yes	—
44	C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₂²⋊C₄×C₇₉	—	2	1	no	—
46	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
47	C₁₆×C₇₉	—	2	1	no	—
48	C₁₆×C₇₉	—	2	1	no	—
49	C₁₆×C₇₉	—	2	2	no	—
50	C₂³×C₇₉	#135	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#124	1	2	yes	—
54	C₂×C₄×C₇₉	#135	1	2	yes	—
55	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
56	C₂×Dic₇₉	—	8	2	no	—
57	C₈×C₇₉	—	2	2	no	—
58	C₈×C₇₉	#117	1	1	yes	characteristic
59	C₈×C₇₉	#117	1	1	yes	characteristic
60	C₈×C₇₉	#125	1	2	yes	—
61	C₇₉⋊C₈	—	8	2	no	—
62	C₂²×C₇₉	—	4	1	no	—
63	C₂²×C₇₉	#427	1	1	yes	characteristic
64	C₂²×C₇₉	#488	1	2	yes	—
65	C₄×C₇₉	—	4	2	no	—
66	C₄×C₇₉	#382	1	1	yes	characteristic
67	C₄×C₇₉	#427	1	1	yes	characteristic
68	C₄×C₇₉	#488	1	2	yes	—
69	Dic₇₉	—	16	2	no	—
70	#12893	—	79	1	no	maximal, Sylow
71	C₂×C₇₉	—	2	2	no	—
72	C₂×C₇₉	—	2	1	no	—
73	C₂×C₇₉	#1835	1	1	yes	characteristic
74	#2108	D₇₉	1	1	yes	characteristic
75	#1833	—	79	2	no	—
76	#1385	—	79	1	no	—
77	#1384	—	79	1	no	—
78	#1382	—	79	2	no	—
79	C₇₉	#12893	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
82	C₂×#108	—	79	2	no	—
83	#427	—	79	2	no	—
84	#405	C₂×D₇₉	1	1	yes	characteristic
85	#346	Dic₇₉	1	2	yes	complemented
86	C₄×C₁₆	Dic₇₉	1	2	yes	complemented
87	#344	—	158	2	no	—
88	H(3,ℤ/4)	—	158	2	no	—
89	C₂×M₁₆	—	158	2	no	—
90	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
92	#117	—	158	2	no	—
93	#117	D₃₁₆	1	1	yes	characteristic
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
98	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
99	#108	—	158	2	no	—
100	#108	—	158	2	no	—
101	#107	—	158	2	no	—
102	#106	—	158	2	no	—
103	#97	Dic₁₅₈	1	2	yes	—
104	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
105	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	316	2	no	—
108	M₁₆	—	316	2	no	—
109	C₂×C₈	—	316	2	no	—
110	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
111	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
113	C₂×C₈	#11929624	1	2	yes	—
114	C₂×C₈	#11929624	1	2	yes	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	316	2	no	—
117	C₂²⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	—	316	2	no	—
119	C₄²	—	316	2	no	—
120	C₄²	C₄⋊Dic₇₉	1	2	yes	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	2	no	—
124	C₂³	#14579183	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	632	2	no	—
127	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
128	C₂×C₄	#14579149	1	2	yes	—
129	C₂×C₄	#14579183	1	1	yes	characteristic
130	C₂×C₄	#14579183	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	632	2	no	—
133	C₈	#14579120	1	1	yes	characteristic
134	C₈	#14579120	1	1	yes	characteristic
135	C₈	#14579153	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#19098601	1	1	yes	characteristic
138	C₂²	#19098918	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	1264	2	no	—
141	C₄	#19098429	1	1	yes	characteristic
142	C₄	#19098601	1	1	yes	characteristic
143	C₄	#19098918	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#23763531	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763524

C₂

yes

maximal

#23761504

C₂

yes

maximal, characteristic

#23761502

C₂

yes

maximal, characteristic

#23761496

C₂

yes

maximal

C₇₉×#2108

C₂

yes

maximal, characteristic

C₂×#14579097

C₂²

yes

—

#19098601

C₂²

yes

—

#19098334

—

#19098319

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

#405×C₇₉

C₂²

yes

characteristic

#346×C₇₉

C₄

yes

complemented

C₄×C₁₆×C₇₉

C₄

yes

—

—

—

—

—

—

C₂×M₁₆×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

#97×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#124

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#427

yes

characteristic

C₂²×C₇₉

#488

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#12893

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1835

yes

characteristic

#2108

D₇₉

yes

—

—

—

—

C₇₉

#12893

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂²×C₁₆

C₂×D₇₉

yes

—

—

C₂×D₇₉

yes

characteristic

#346

Dic₇₉

yes

complemented

C₄×C₁₆

Dic₇₉

yes

complemented

#344

—

158

—

H(3,ℤ/4)

—

158

—

C₂×M₁₆

—

158

—

C₂×M₁₆

D₃₁₆

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

#117

—

158

—

#117

D₃₁₆

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

C₂×Dic₇₉

yes

—

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

#108

—

158

—

100

#108

—

158

—

101

#107

—

158

—

102

#106

—

158

—

103

#97

Dic₁₅₈

yes

—

104

C₄×C₈

C₂×Dic₇₉

yes

—

105

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

106

M₁₆

—

107

M₁₆

—

316

—

108

M₁₆

—

316

—

109

C₂×C₈

—

316

—

110

C₂×C₈

C₄⋊Dic₇₉

yes

—

111

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

112

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

113

C₂×C₈

#11929624

yes

—

114

C₂×C₈

#11929624

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

316

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

316

—

119

C₄²

—

316

—

120

C₄²

C₄⋊Dic₇₉

yes

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#14579183

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

632

—

127

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

128

C₂×C₄

#14579149

yes

—

129

C₂×C₄

#14579183

yes

characteristic

130

C₂×C₄

#14579183

yes

—

131

C₈

—

132

C₈

—

632

—

133

C₈

#14579120

yes

characteristic

134

C₈

#14579120

yes

characteristic

135

C₈

#14579153

yes

—

136

C₂²

—

137

C₂²

#19098601

yes

characteristic

138

C₂²

#19098918

yes

—

139

C₄

—

140

C₄

—

1264

—

141

C₄

#19098429

yes

characteristic

142

C₄

#19098601

yes

characteristic

143

C₄

#19098918

yes

—

144

C₂

—

145

C₂

—

146

C₂

#23763531

yes

characteristic

	subgroup	G/H
center	C₄	#19098601
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2108	C₂
socle	C₂×C₇₉	#1835
prime radical	C₂³×C₇₉	#135
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2108	D₇₉
hypocenter	C₇₉	#12893
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

subgroup

G/H

center

C₄

#19098601

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2108

C₂

C₂×C₇₉

C₂³×C₇₉

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

N	H ≅ G/N	type
13. #346×C₇₉	C₄	semidirect
14. C₄×C₁₆×C₇₉	C₄	semidirect
79. C₇₉	#12893	semidirect
85. #346	Dic₇₉	semidirect
86. C₄×C₁₆	Dic₇₉	semidirect

H ≅ G/N

type

13. #346×C₇₉

C₄

semidirect

14. C₄×C₁₆×C₇₉

C₄

semidirect

79. C₇₉

#12893

semidirect

85. #346

Dic₇₉

semidirect

86. C₄×C₁₆

Dic₇₉

semidirect

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

subgroup	index
#23763524	2
#23761504	2
#23761502	2
#23761496	2
C₇₉×#2108	2
#12893	79

subgroup

index

C₇₉×#2108

Subgroups — #34204666

Conjugacy classes of subgroups

Complements

Maximal subgroups