Group #34204721 — subgroups

Subgroups — #34204721

Order 20224.

306 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₂×#19098393	C₂	1	4	yes	maximal
2	#23763572	C₂	1	2	yes	maximal
3	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579097	—	2	2	no	—
5	C₂×#14579097	C₂²	1	4	yes	—
6	#19098587	—	2	2	no	—
7	#19098393	—	2	8	no	—
8	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	#346×C₇₉	C₄	1	4	yes	—
11	C₂×#11929608	—	2	4	no	—
12	#14579097	—	2	8	no	—
13	#14579097	—	4	4	no	—
14	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
15	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
20	C₂×C₁₆×C₇₉	Q₈	1	4	yes	—
21	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₉	Q₈	1	2	yes	—
23	#11929608	—	4	4	no	—
24	#11929608	—	4	4	no	—
25	C₂×C₇₉⋊C₈	—	4	4	no	—
26	C₂²×C₄×C₇₉	Q₁₆	1	2	yes	—
27	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₇₉	D₈	1	2	yes	—
30	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
31	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
33	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
35	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
36	C₄²×C₇₉	D₈	1	2	yes	—
37	C₁₆×C₇₉	C₄⋊C₄	1	8	yes	—
38	C₂³×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	4	2	no	—
40	C₂×C₄×C₇₉	#107	1	4	yes	—
41	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₈×C₇₉	#107	1	4	yes	—
48	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
49	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
50	C₇₉⋊C₈	—	8	4	no	—
51	C₂²×C₇₉	—	2	2	no	—
52	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
53	C₂²×C₇₉	#494	1	1	yes	characteristic
54	C₂²×C₇₉	#500	1	1	yes	characteristic
55	C₄×C₇₉	—	2	4	no	—
56	C₄×C₇₉	#384	1	1	yes	characteristic
57	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
58	C₄×C₇₉	#493	1	1	yes	characteristic
59	C₄×C₇₉	#500	1	1	yes	characteristic
60	#12910	—	79	1	no	maximal, Sylow
61	C₂×C₇₉	—	2	1	no	—
62	C₂×C₇₉	—	2	1	no	—
63	C₂×C₇₉	#1845	1	1	yes	characteristic
64	C₂×C₇₉	#2161	1	2	yes	—
65	C₂×#368	—	79	4	no	—
66	C₂×#346	D₇₉	1	1	yes	characteristic
67	#1848	—	79	2	no	—
68	C₇₉	#12910	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
70	C₂×#108	—	79	4	no	—
71	C₂×#108	—	158	2	no	—
72	#423	—	158	2	no	—
73	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
74	#368	—	158	8	no	—
75	#346	Dic₇₉	1	4	yes	—
76	C₂×M₁₆	—	158	4	no	—
77	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
79	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
80	C₂×C₁₆	Dic₁₅₈	1	4	yes	—
81	C₂×C₁₆	D₃₁₆	1	4	yes	—
82	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
83	#108	—	316	4	no	—
84	#108	—	158	8	no	—
85	C₄⋊C₈	—	316	4	no	—
86	C₄×C₈	Dic₁₅₈	1	2	yes	—
87	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
88	C₂²×C₄	Dic₃₁₆	1	2	yes	—
89	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
90	M₁₆	—	316	4	no	—
91	M₁₆	—	316	4	no	—
92	C₂×C₈	—	316	4	no	—
93	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
94	C₂×C₈	D₆₃₂	1	2	yes	—
95	C₂×C₈	Dic₃₁₆	1	2	yes	—
96	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
97	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
98	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
99	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
100	C₄²	C₄⋊Dic₇₉	1	2	yes	—
101	C₄²	D₆₃₂	1	2	yes	—
102	C₁₆	C₄⋊Dic₇₉	1	8	yes	—
103	C₂³	C₂×Dic₃₁₆	1	1	yes	characteristic
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	#14579095	1	4	yes	—
106	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
107	C₂×C₄	#14579158	1	1	yes	characteristic
108	C₂×C₄	#14579161	1	1	yes	characteristic
109	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
110	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
112	C₈	—	632	4	no	—
113	C₈	#14579095	1	4	yes	—
114	C₈	C₂×C₄⋊Dic₇₉	1	2	yes	—
115	C₈	#14579161	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#14579095	1	1	yes	characteristic
118	C₂²	#19098943	1	1	yes	characteristic
119	C₂²	#19098968	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	#19098433	1	1	yes	characteristic
122	C₄	C₂×#14579095	1	1	yes	characteristic
123	C₄	#19098940	1	1	yes	characteristic
124	C₄	#19098968	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	#23763560	1	1	yes	characteristic
128	C₂	#23765195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098393

C₂

yes

maximal

#23763572

C₂

yes

maximal

C₂×#346×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579097

—

C₂×#14579097

C₂²

yes

—

#19098587

—

#19098393

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#346×C₇₉

C₄

yes

—

C₂×#11929608

—

#14579097

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₂×C₁₆×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

Q₁₆

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₄²×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

D₈

yes

—

C₁₆×C₇₉

C₄⋊C₄

yes

—

C₂³×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#107

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₈×C₇₉

#107

yes

—

C₈×C₇₉

C₂×C₄⋊C₄

yes

—

C₈×C₇₉

C₄⋊Q₈

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1845

yes

characteristic

C₂×C₇₉

#2161

yes

—

C₂×#368

—

C₂×#346

D₇₉

yes

characteristic

#1848

—

C₇₉

#12910

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₉

yes

—

C₂×#108

—

C₂×#108

—

158

—

#423

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#368

—

158

—

#346

Dic₇₉

yes

—

C₂×M₁₆

—

158

—

C₂²×C₈

D₃₁₆

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₁₆

Dic₁₅₈

yes

—

C₂×C₁₆

D₃₁₆

yes

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

#108

—

316

—

#108

—

158

—

C₄⋊C₈

—

316

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₂²×C₄

Dic₃₁₆

yes

—

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

M₁₆

—

316

—

M₁₆

—

316

—

C₂×C₈

—

316

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

D₆₃₂

yes

—

C₂×C₈

Dic₃₁₆

yes

—

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

C₂×C₈

C₂×D₃₁₆

yes

—

C₂×C₈

C₂×Dic₁₅₈

yes

—

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

100

C₄²

C₄⋊Dic₇₉

yes

—

101

C₄²

D₆₃₂

yes

—

102

C₁₆

C₄⋊Dic₇₉

yes

—

103

C₂³

C₂×Dic₃₁₆

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#14579095

yes

—

106

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

107

C₂×C₄

#14579158

yes

characteristic

108

C₂×C₄

#14579161

yes

characteristic

109

C₂×C₄

C₂×D₆₃₂

yes

characteristic

110

C₂×C₄

C₂×D₆₃₂

yes

characteristic

111

C₂×C₄

C₂×Dic₃₁₆

yes

characteristic

112

C₈

—

632

—

113

C₈

#14579095

yes

—

114

C₈

C₂×C₄⋊Dic₇₉

yes

—

115

C₈

#14579161

yes

—

116

C₂²

—

117

C₂²

C₂×#14579095

yes

characteristic

118

C₂²

#19098943

yes

characteristic

119

C₂²

#19098968

yes

characteristic

120

C₄

—

121

C₄

#19098433

yes

characteristic

122

C₄

C₂×#14579095

yes

characteristic

123

C₄

#19098940

yes

characteristic

124

C₄

#19098968

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

#23763560

yes

characteristic

128

C₂

#23765195

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#107
prime radical	C₂³×C₇₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#12910
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₆₃₂

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×#346×C₇₉

C₂

socle

C₂²×C₇₉

C₂×#107

prime radical

C₂³×C₇₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
68. C₇₉	#12910	semidirect

H ≅ G/N

type

68. C₇₉

#12910

semidirect

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

subgroup	index
C₂×#19098393	2
#23763572	2
C₂×#346×C₇₉	2
#12910	79

subgroup

index

C₂×#19098393

#23763572

C₂×#346×C₇₉

#12910

Subgroups — #34204721

Conjugacy classes of subgroups

Complements

Maximal subgroups