Group #34178689 — subgroups

Subgroups — #34178689

Order 20224.

520 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 151 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	#23772032	C₂	1	3	yes	maximal
2	C₇₉⋊#1309	C₂	1	3	yes	maximal, complemented
3	C₇₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
5	#19098619	—	2	6	no	—
6	#19098615	—	2	6	no	—
7	#19098587	C₂²	1	3	yes	—
8	#19098526	C₄	1	6	yes	—
9	M₁₆×Dic₇₉	C₄	1	6	yes	—
10	#423×C₇₉	C₂²	1	3	yes	—
11	C₂²×C₄×D₇₉	C₂×C₄	1	3	yes	—
12	C₂×#11929608	C₂×C₄	1	6	yes	—
13	C₂×C₄×Dic₇₉	C₂×C₄	1	3	yes	—
14	C₄²×D₇₉	D₄	1	6	yes	—
15	C₄²×D₇₉	Q₈	1	2	yes	—
16	C₄⋊(C₇₉⋊C₈)	—	2	6	no	—
17	(C₂×D₇₉)⋊C₈	—	2	12	no	—
18	#14579071	—	2	6	no	—
19	C₈×Dic₇₉	—	2	6	no	—
20	C₂×M₁₆×C₇₉	C₂×C₄	1	6	yes	—
21	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₇₉	—	2	6	no	—
23	C₂³×D₇₉	C₄²	1	1	yes	characteristic
24	C₂²×Dic₇₉	C₄²	1	3	yes	—
25	C₂×C₄×D₇₉	—	4	3	no	—
26	C₂×C₄×D₇₉	—	4	3	no	—
27	C₂×C₄×D₇₉	C₂²⋊C₄	1	6	yes	—
28	C₂×C₄×D₇₉	C₄⋊C₄	1	6	yes	—
29	#11929608	—	4	6	no	—
30	C₂×C₇₉⋊C₈	—	2	6	no	—
31	C₄×Dic₇₉	C₂²⋊C₄	1	3	yes	—
32	C₄×Dic₇₉	C₂²⋊C₄	1	3	yes	—
33	C₄×Dic₇₉	C₄⋊C₄	1	6	yes	—
34	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
35	M₁₆×C₇₉	—	4	6	no	—
36	C₂×C₈×C₇₉	—	2	6	no	—
37	C₄²×C₇₉	C₂×D₄	1	3	yes	—
38	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
39	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
40	C₂²×D₇₉	—	4	3	no	—
41	C₂²×D₇₉	#95	1	2	yes	—
42	C₂×C₄×C₇₉	—	2	3	no	—
43	C₂×C₄×C₇₉	—	2	3	no	—
44	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	3	yes	—
45	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	3	yes	—
46	C₂×Dic₇₉	—	4	3	no	—
47	C₂×Dic₇₉	#95	1	6	yes	—
48	C₄×D₇₉	—	2	6	no	—
49	C₄×D₇₉	—	2	6	no	—
50	C₄×D₇₉	—	4	6	no	—
51	C₈×C₇₉	—	4	6	no	—
52	C₇₉⋊C₈	—	4	6	no	—
53	C₂²×C₇₉	—	2	3	no	—
54	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
55	C₂×D₇₉	—	2	6	no	—
56	C₂×D₇₉	—	4	4	no	—
57	C₄×C₇₉	—	2	3	no	—
58	C₄×C₇₉	#413	1	3	yes	—
59	C₄×C₇₉	#413	1	3	yes	—
60	Dic₇₉	—	2	6	no	—
61	#8130	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	—	4	1	no	—
63	C₂×C₇₉	#1742	1	3	yes	—
64	D₇₉	—	4	2	no	—
65	#2920	—	79	3	no	—
66	#1309	—	79	3	no	—
67	#1309	D₇₉	1	1	yes	characteristic, complemented
68	C₇₉	#8130	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₄²	—	79	1	no	—
70	#432	—	158	6	no	—
71	#431	—	158	6	no	—
72	#423	—	79	3	no	—
73	#423	C₂×D₇₉	1	3	yes	—
74	#407	—	79	6	no	—
75	C₄×M₁₆	—	79	6	no	—
76	C₂³×C₄	—	79	3	no	—
77	C₂×M₁₆	—	79	6	no	—
78	C₂×M₁₆	C₄×D₇₉	1	6	yes	—
79	C₂×C₄²	—	79	3	no	—
80	C₂×C₄²	—	79	6	no	—
81	C₂×C₄²	—	79	2	no	—
82	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
83	C₄⋊C₈	—	2	6	no	—
84	C₄⋊C₈	—	158	6	no	—
85	C₂²⋊C₈	—	158	12	no	—
86	#97	—	158	6	no	—
87	C₄×C₈	—	158	6	no	—
88	C₂⁴	—	79	1	no	—
89	C₂²×C₄	—	316	3	no	—
90	C₂²×C₄	—	316	3	no	—
91	C₂²×C₄	—	79	3	no	—
92	C₂²×C₄	—	79	6	no	—
93	C₂²×C₄	—	79	6	no	—
94	C₂²×C₄	C₂×C₄×D₇₉	1	3	yes	—
95	M₁₆	—	4	6	no	—
96	M₁₆	—	316	6	no	—
97	C₂×C₈	—	2	6	no	—
98	C₂×C₈	—	158	6	no	—
99	C₄²	—	79	3	no	—
100	C₄²	—	79	3	no	—
101	C₄²	—	79	6	no	—
102	C₄²	D₄×D₇₉	1	3	yes	—
103	C₄²	Q₈×D₇₉	1	1	yes	characteristic
104	C₂³	—	316	3	no	—
105	C₂³	—	79	2	no	—
106	C₂³	C₄²×D₇₉	1	1	yes	characteristic
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	316	6	no	—
110	C₂×C₄	—	316	3	no	—
111	C₂×C₄	—	79	6	no	—
112	C₂×C₄	—	158	6	no	—
113	C₂×C₄	—	158	6	no	—
114	C₂×C₄	C₂²⋊C₄×D₇₉	1	3	yes	—
115	C₂×C₄	C₄⋊C₄×D₇₉	1	3	yes	—
116	C₈	—	4	6	no	—
117	C₈	—	316	6	no	—
118	C₂²	—	2	3	no	—
119	C₂²	—	316	4	no	—
120	C₂²	—	158	6	no	—
121	C₂²	#95×D₇₉	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	158	6	no	—
124	C₄	#19098544	1	3	yes	—
125	C₄	#19098546	1	3	yes	—
126	C₂	—	4	1	no	—
127	C₂	—	316	2	no	—
128	C₂	#23763137	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23772032

C₂

yes

maximal

C₇₉⋊#1309

C₂

yes

maximal, complemented

C₇₉×#1309

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₇₉

C₂²

yes

—

—

C₂²

yes

—

#19098526

C₄

yes

—

M₁₆×Dic₇₉

C₄

yes

—

#423×C₇₉

C₂²

yes

—

C₂²×C₄×D₇₉

C₂×C₄

yes

—

C₂×#11929608

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

C₄²×D₇₉

D₄

yes

—

C₄²×D₇₉

Q₈

yes

—

C₄⋊(C₇₉⋊C₈)

—

(C₂×D₇₉)⋊C₈

—

#14579071

—

C₈×Dic₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂³×D₇₉

C₄²

yes

characteristic

C₂²×Dic₇₉

C₄²

yes

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

C₂²⋊C₄

yes

—

C₂×C₄×D₇₉

C₄⋊C₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄×Dic₇₉

C₂²⋊C₄

yes

—

C₄×Dic₇₉

C₂²⋊C₄

yes

—

C₄×Dic₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

C₂×C₄²

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

#95

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

#95

yes

—

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#95

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#413

yes

—

C₄×C₇₉

#413

yes

—

Dic₇₉

—

#8130

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1742

yes

—

D₇₉

—

#2920

—

#1309

—

#1309

D₇₉

yes

characteristic, complemented

C₇₉

#8130

yes

characteristic, complemented, Sylow

C₂²×C₄²

—

#432

—

158

—

#431

—

158

—

#423

—

#423

C₂×D₇₉

yes

—

#407

—

C₄×M₁₆

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₂²⋊C₈

—

158

—

#97

—

158

—

C₄×C₈

—

158

—

C₂⁴

—

C₂²×C₄

—

316

—

C₂²×C₄

—

316

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₇₉

yes

—

M₁₆

—

M₁₆

—

316

—

C₂×C₈

—

C₂×C₈

—

158

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

D₄×D₇₉

yes

—

103

C₄²

Q₈×D₇₉

yes

characteristic

104

C₂³

—

316

—

105

C₂³

—

106

C₂³

C₄²×D₇₉

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

316

—

110

C₂×C₄

—

316

—

111

C₂×C₄

—

112

C₂×C₄

—

158

—

113

C₂×C₄

—

158

—

114

C₂×C₄

C₂²⋊C₄×D₇₉

yes

—

115

C₂×C₄

C₄⋊C₄×D₇₉

yes

—

116

C₈

—

117

C₈

—

316

—

118

C₂²

—

119

C₂²

—

316

—

120

C₂²

—

158

—

121

C₂²

#95×D₇₉

yes

characteristic

122

C₄

—

123

C₄

—

158

—

124

C₄

#19098544

yes

—

125

C₄

#19098546

yes

—

126

C₂

—

127

C₂

—

316

—

128

C₂

#23763137

yes

—

	subgroup	G/H
center	C₂²	#95×D₇₉
commutator	C₂³×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1309	C₂
socle	C₂²×C₇₉	C₂×#95
prime radical	C₂³×D₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1309	D₇₉
hypocenter	C₇₉	#8130
LCS	G ▷ C₂³×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#95×D₇₉

commutator

C₂³×C₇₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1309

C₂

socle

C₂²×C₇₉

C₂×#95

prime radical

C₂³×D₇₉

C₄²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
2. C₇₉⋊#1309	C₂	semidirect
3. C₇₉×#1309	C₂	semidirect
67. #1309	D₇₉	semidirect
68. C₇₉	#8130	semidirect

H ≅ G/N

type

2. C₇₉⋊#1309

C₂

semidirect

3. C₇₉×#1309

C₂

semidirect

67. #1309

D₇₉

semidirect

68. C₇₉

#8130

semidirect

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

subgroup	index
#23772032	2
C₇₉⋊#1309	2
C₇₉×#1309	2
#8130	79

subgroup

index

#23772032

C₇₉⋊#1309

C₇₉×#1309

#8130

Subgroups — #34178689

Conjugacy classes of subgroups

Complements

Maximal subgroups