Group #34204664 — subgroups

Subgroups — #34204664

Order 20224.

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

normal subgroups only

This group has 234 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	#23764999	C₂	1	1	yes	maximal, characteristic
2	#23763523	C₂	1	2	yes	maximal
3	#23761504	C₂	1	1	yes	maximal, characteristic
4	#23761495	C₂	1	2	yes	maximal
5	C₇₉×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14579097	C₂²	1	2	yes	—
8	#19098518	C₂²	1	1	yes	characteristic
9	#19098338	C₄	1	2	yes	complemented
10	C₁₆×Dic₇₉	C₄	1	2	yes	complemented
11	#19098333	—	2	2	no	—
12	#19098319	—	2	2	no	—
13	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
14	#427×C₇₉	C₂²	1	2	yes	—
15	C₂×#11929608	—	2	2	no	—
16	C₂×#11929608	D₄	1	1	yes	characteristic
17	#14579120	D₄	1	1	yes	characteristic
18	C₂×C₇₉⋊C₁₆	—	2	2	no	—
19	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	#14579097	—	2	2	no	—
22	#14579097	—	2	2	no	—
23	#14579071	Q₈	1	2	yes	—
24	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	#117×C₇₉	—	2	2	no	—
27	M₃₂×C₇₉	—	2	2	no	—
28	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
29	#107×C₇₉	—	2	2	no	—
30	#106×C₇₉	—	2	2	no	—
31	#11929608	—	2	2	no	—
32	#11929608	—	4	2	no	—
33	#11929608	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
36	C₄⋊Dic₇₉	—	2	1	no	—
37	C₂²⋊Dic₇₉	—	2	1	no	—
38	C₄×Dic₇₉	C₄⋊C₄	1	2	yes	—
39	C₇₉⋊C₁₆	—	2	1	no	—
40	C₇₉⋊C₁₆	—	2	1	no	—
41	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₇₉	#41	1	2	yes	—
45	C₂×C₈×C₇₉	#41	1	2	yes	—
46	C₄⋊C₄×C₇₉	—	4	2	no	—
47	C₂²⋊C₄×C₇₉	—	4	2	no	—
48	C₄²×C₇₉	—	4	2	no	—
49	C₁₆×C₇₉	—	2	2	no	—
50	C₂³×C₇₉	#135	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	8	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	—	2	2	no	—
57	C₈×C₇₉	#117	1	1	yes	characteristic
58	C₈×C₇₉	#117	1	1	yes	characteristic
59	C₈×C₇₉	#125	1	2	yes	—
60	C₇₉⋊C₈	—	2	2	no	—
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	—	16	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	—	4	2	no	—
70	#12893	—	79	1	no	maximal, Sylow
71	C₂×C₇₉	—	2	1	no	—
72	C₂×C₇₉	—	2	2	no	—
73	C₂×C₇₉	#1835	1	1	yes	characteristic
74	#2108	—	79	1	no	—
75	#1833	—	79	2	no	—
76	#1385	—	79	1	no	—
77	#1384	D₇₉	1	1	yes	characteristic
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	—	79	1	no	—
82	C₂×#108	—	79	2	no	—
83	#427	C₂×D₇₉	1	2	yes	—
84	#405	—	79	1	no	—
85	#346	—	79	2	no	—
86	C₄×C₁₆	—	79	2	no	—
87	#344	—	158	2	no	—
88	H(3,ℤ/4)	—	158	2	no	—
89	C₂×M₁₆	—	79	1	no	—
90	C₂×M₁₆	—	158	2	no	—
91	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
92	#117	—	2	2	no	—
93	#117	—	79	1	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	79	1	no	—
96	C₂×C₁₆	—	79	1	no	—
97	C₂×C₁₆	—	158	2	no	—
98	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
99	#108	—	158	2	no	—
100	#108	—	158	2	no	—
101	#107	—	2	2	no	—
102	#106	—	2	2	no	—
103	#97	—	79	2	no	—
104	C₄×C₈	—	79	2	no	—
105	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
106	M₁₆	—	316	2	no	—
107	M₁₆	—	316	2	no	—
108	M₁₆	—	158	2	no	—
109	C₂×C₈	—	316	2	no	—
110	C₂×C₈	—	79	2	no	—
111	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
112	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
113	C₂×C₈	#11929623	1	2	yes	—
114	C₂×C₈	#11929624	1	2	yes	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	158	1	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	158	1	no	—
119	C₄²	—	4	2	no	—
120	C₄²	—	79	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	158	1	no	—
123	C₁₆	—	158	1	no	—
124	C₂³	#14579180	1	1	yes	characteristic
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	—	158	2	no	—
127	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
128	C₂×C₄	#14579147	1	2	yes	—
129	C₂×C₄	#14579182	1	2	yes	—
130	C₂×C₄	#14579184	1	1	yes	characteristic
131	C₈	—	632	2	no	—
132	C₈	—	158	2	no	—
133	C₈	#14579117	1	1	yes	characteristic
134	C₈	#14579118	1	1	yes	characteristic
135	C₈	#14579151	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#19098600	1	1	yes	characteristic
138	C₂²	#19098916	1	2	yes	—
139	C₄	—	16	2	no	—
140	C₄	—	316	2	no	—
141	C₄	#19098427	1	1	yes	characteristic
142	C₄	#19098598	1	1	yes	characteristic
143	C₄	#19098919	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	2	1	no	—
146	C₂	#23763529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764999

C₂

yes

maximal, characteristic

#23763523

C₂

yes

maximal

#23761504

C₂

yes

maximal, characteristic

#23761495

C₂

yes

maximal

C₇₉×#1384

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14579097

C₂²

yes

—

#19098518

C₂²

yes

characteristic

#19098338

C₄

yes

complemented

C₁₆×Dic₇₉

C₄

yes

complemented

#19098333

—

#19098319

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

#427×C₇₉

C₂²

yes

—

C₂×#11929608

—

C₂×#11929608

D₄

yes

characteristic

#14579120

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

—

Q₈

yes

—

C₈×Dic₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

#107×C₇₉

—

#106×C₇₉

—

#11929608

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

C₄⋊C₄

yes

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

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

—

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

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

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

yes

—

—

—

D₇₉

yes

characteristic

#1382

—

C₇₉

#12893

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

#427

C₂×D₇₉

yes

—

#405

—

#346

—

C₄×C₁₆

—

#344

—

158

—

H(3,ℤ/4)

—

158

—

C₂×M₁₆

—

C₂×M₁₆

—

158

—

C₂²×C₈

C₂²×D₇₉

yes

characteristic

#117

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

158

—

C₂×C₁₆

C₄×D₇₉

yes

—

#108

—

158

—

100

#108

—

158

—

101

#107

—

102

#106

—

103

#97

—

104

C₄×C₈

—

105

C₂²×C₄

D₄×D₇₉

yes

characteristic

106

M₁₆

—

316

—

107

M₁₆

—

316

—

108

M₁₆

—

158

—

109

C₂×C₈

—

316

—

110

C₂×C₈

—

111

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

112

C₂×C₈

Q₈×D₇₉

yes

characteristic

113

C₂×C₈

#11929623

yes

—

114

C₂×C₈

#11929624

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

158

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

158

—

119

C₄²

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

158

—

123

C₁₆

—

158

—

124

C₂³

#14579180

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

158

—

127

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

128

C₂×C₄

#14579147

yes

—

129

C₂×C₄

#14579182

yes

—

130

C₂×C₄

#14579184

yes

characteristic

131

C₈

—

632

—

132

C₈

—

158

—

133

C₈

#14579117

yes

characteristic

134

C₈

#14579118

yes

characteristic

135

C₈

#14579151

yes

—

136

C₂²

—

137

C₂²

#19098600

yes

characteristic

138

C₂²

#19098916

yes

—

139

C₄

—

140

C₄

—

316

—

141

C₄

#19098427

yes

characteristic

142

C₄

#19098598

yes

characteristic

143

C₄

#19098919

yes

—

144

C₂

—

145

C₂

—

146

C₂

#23763529

yes

characteristic

	subgroup	G/H
center	C₄	#19098598
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1384	C₂
socle	C₂×C₇₉	#1835
prime radical	C₂³×C₇₉	#135
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1384	D₇₉
hypocenter	C₇₉	#12893
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#19098598

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1384

C₂

C₂×C₇₉

C₂³×C₇₉

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
9. #19098338	C₄	semidirect
10. C₁₆×Dic₇₉	C₄	semidirect
79. C₇₉	#12893	semidirect

H ≅ G/N

type

9. #19098338

C₄

semidirect

10. C₁₆×Dic₇₉

C₄

semidirect

79. C₇₉

#12893

semidirect

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

subgroup	index
#23764999	2
#23763523	2
#23761504	2
#23761495	2
C₇₉×#1384	2
#12893	79

subgroup

index

C₇₉×#1384

Subgroups — #34204664

Conjugacy classes of subgroups

Complements

Maximal subgroups