Group #34203705 — subgroups

Subgroups — #34203705

Order 20224.

958 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 219 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×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	6	yes	maximal, complemented
2	C₂²×#131×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇₉⋊C₁₆	C₂²	1	4	yes	—
4	C₂²⋊(C₇₉⋊C₁₆)	—	2	12	no	—
5	C₂²×#41×C₇₉	C₄	1	1	yes	characteristic
6	C₂×#131×C₇₉	—	2	12	no	—
7	C₂²×M₁₆×C₇₉	C₄	1	3	yes	—
8	C₂³×C₈×C₇₉	C₂²	1	3	yes	—
9	C₂×C₇₉⋊C₁₆	—	2	12	no	—
10	C₂×#41×C₇₉	—	2	6	no	—
11	C₂×#41×C₇₉	—	2	6	no	—
12	C₂²×Q₈×C₇₉	C₈	1	1	yes	characteristic
13	C₂²×D₄×C₇₉	C₈	1	3	yes	—
14	C₂³×C₄×C₇₉	C₂×C₄	1	3	yes	—
15	#131×C₇₉	—	2	32	no	—
16	C₂×M₁₆×C₇₉	—	2	6	no	—
17	C₂×M₁₆×C₇₉	—	2	12	no	—
18	C₂²×C₈×C₇₉	—	2	12	no	—
19	C₂²×C₈×C₇₉	C₂×C₄	1	3	yes	—
20	C₂²×C₈×C₇₉	D₄	1	12	yes	—
21	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
22	C₇₉⋊C₁₆	—	4	4	no	—
23	C₂⁴×C₇₉	C₂×C₈	1	3	yes	—
24	#41×C₇₉	—	2	8	no	—
25	#41×C₇₉	—	2	24	no	—
26	C₂×Q₈×C₇₉	—	2	6	no	—
27	C₂×D₄×C₇₉	—	2	12	no	—
28	C₂×D₄×C₇₉	—	2	6	no	—
29	C₂²×C₄×C₇₉	—	2	6	no	—
30	C₂²×C₄×C₇₉	—	2	6	no	—
31	C₂²×C₄×C₇₉	C₂²⋊C₄	1	6	yes	—
32	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
33	C₂²×C₄×C₇₉	M₁₆	1	6	yes	—
34	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₇₉	—	2	24	no	—
36	C₂×C₈×C₇₉	—	2	24	no	—
37	C₂×C₈×C₇₉	—	2	6	no	—
38	C₂×C₈×C₇₉	C₂²⋊C₄	1	6	yes	—
39	C₂×C₈×C₇₉	C₂×D₄	1	6	yes	—
40	C₂³×C₇₉	—	2	6	no	—
41	C₂³×C₇₉	—	2	12	no	—
42	C₂³×C₇₉	C₂²⋊C₈	1	6	yes	—
43	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
44	Q₈×C₇₉	—	2	8	no	—
45	D₄×C₇₉	—	2	24	no	—
46	C₂×C₄×C₇₉	—	2	6	no	—
47	C₂×C₄×C₇₉	—	2	6	no	—
48	C₂×C₄×C₇₉	—	2	6	no	—
49	C₂×C₄×C₇₉	—	2	12	no	—
50	C₂×C₄×C₇₉	C₂²⋊C₈	1	6	yes	—
51	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	3	yes	—
52	C₂×C₄×C₇₉	C₂×M₁₆	1	3	yes	—
53	C₈×C₇₉	—	2	6	no	—
54	C₈×C₇₉	#120	1	4	yes	—
55	C₂²×C₇₉	—	2	12	no	—
56	C₂²×C₇₉	—	2	6	no	—
57	C₂²×C₇₉	—	4	12	no	—
58	C₂²×C₇₉	C₂×C₂²⋊C₈	1	3	yes	—
59	C₂²×C₇₉	#504	1	4	yes	—
60	C₄×C₇₉	—	2	6	no	—
61	C₄×C₇₉	#379	1	1	yes	characteristic
62	C₄×C₇₉	#407	1	3	yes	—
63	#12680	—	79	1	no	maximal, Sylow
64	C₂×C₇₉	—	4	6	no	—
65	C₂×C₇₉	#1779	1	1	yes	characteristic
66	C₂×C₇₉	#2113	1	6	yes	—
67	C₂²×#131	D₇₉	1	1	yes	characteristic
68	C₂×C₂²⋊C₁₆	—	79	6	no	—
69	C₇₉	#12680	1	1	yes	characteristic, complemented, Sylow
70	C₂²×#41	Dic₇₉	1	1	yes	characteristic
71	C₂×#131	—	2	12	no	—
72	C₂²×M₁₆	Dic₇₉	1	3	yes	—
73	C₂³×C₈	C₂×D₇₉	1	3	yes	—
74	C₂²×C₁₆	—	79	4	no	—
75	C₂²⋊C₁₆	—	158	12	no	—
76	C₂×#41	—	2	6	no	—
77	C₂×#41	—	2	6	no	—
78	C₂²×Q₈	C₇₉⋊C₈	1	1	yes	characteristic
79	C₂²×D₄	C₇₉⋊C₈	1	3	yes	—
80	C₂³×C₄	C₂×Dic₇₉	1	3	yes	—
81	#131	—	2	32	no	—
82	C₂×M₁₆	—	2	6	no	—
83	C₂×M₁₆	—	2	12	no	—
84	C₂²×C₈	—	2	12	no	—
85	C₂²×C₈	C₂×Dic₇₉	1	3	yes	—
86	C₂²×C₈	C₂²⋊D₇₉	1	12	yes	—
87	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
88	C₂×C₁₆	—	158	12	no	—
89	C₂⁴	C₂×C₇₉⋊C₈	1	3	yes	—
90	#41	—	2	8	no	—
91	#41	—	2	24	no	—
92	C₂×Q₈	—	2	6	no	—
93	C₂×D₄	—	2	12	no	—
94	C₂×D₄	—	2	6	no	—
95	C₂²×C₄	—	2	6	no	—
96	C₂²×C₄	—	2	6	no	—
97	C₂²×C₄	C₂²⋊Dic₇₉	1	6	yes	—
98	C₂²×C₄	C₂×C₇₉⋊C₈	1	3	yes	—
99	C₂²×C₄	#11929608	1	6	yes	—
100	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
101	M₁₆	—	2	24	no	—
102	C₂×C₈	—	2	6	no	—
103	C₂×C₈	—	2	24	no	—
104	C₂×C₈	C₂²⋊Dic₇₉	1	6	yes	—
105	C₂×C₈	C₂×C₂²⋊D₇₉	1	6	yes	—
106	C₁₆	—	316	4	no	—
107	C₂³	—	2	12	no	—
108	C₂³	—	2	6	no	—
109	C₂³	C₂²⋊(C₇₉⋊C₈)	1	6	yes	—
110	C₂³	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
111	Q₈	—	2	8	no	—
112	D₄	—	2	24	no	—
113	C₂×C₄	—	2	6	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	2	12	no	—
117	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	6	yes	—
118	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	3	yes	—
119	C₂×C₄	C₂×#11929608	1	3	yes	—
120	C₈	—	2	6	no	—
121	C₈	#14579130	1	4	yes	—
122	C₂²	—	2	12	no	—
123	C₂²	—	2	6	no	—
124	C₂²	—	4	12	no	—
125	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	3	yes	—
126	C₂²	#19098980	1	4	yes	—
127	C₄	—	2	6	no	—
128	C₄	#19098419	1	1	yes	characteristic
129	C₄	#19098527	1	3	yes	—
130	C₂	—	4	6	no	—
131	C₂	#23763310	1	1	yes	characteristic
132	C₂	#23765014	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₇₉⋊C₁₆)

C₂

yes

maximal, complemented

C₂²×#131×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

—

C₂²⋊(C₇₉⋊C₁₆)

—

C₂²×#41×C₇₉

C₄

yes

characteristic

C₂×#131×C₇₉

—

C₂²×M₁₆×C₇₉

C₄

yes

—

C₂³×C₈×C₇₉

C₂²

yes

—

C₂×C₇₉⋊C₁₆

—

C₂×#41×C₇₉

—

C₂×#41×C₇₉

—

C₂²×Q₈×C₇₉

C₈

yes

characteristic

C₂²×D₄×C₇₉

C₈

yes

—

C₂³×C₄×C₇₉

C₂×C₄

yes

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₇₉⋊C₁₆

—

C₂⁴×C₇₉

C₂×C₈

yes

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₈

yes

—

C₂²×C₄×C₇₉

M₁₆

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂²⋊C₈

yes

—

C₂³×C₇₉

C₂²×C₈

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

#120

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₈

yes

—

C₂²×C₇₉

#504

yes

—

C₄×C₇₉

—

C₄×C₇₉

#379

yes

characteristic

C₄×C₇₉

#407

yes

—

#12680

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1779

yes

characteristic

C₂×C₇₉

#2113

yes

—

C₂²×#131

D₇₉

yes

characteristic

C₂×C₂²⋊C₁₆

—

C₇₉

#12680

yes

characteristic, complemented, Sylow

C₂²×#41

Dic₇₉

yes

characteristic

C₂×#131

—

C₂²×M₁₆

Dic₇₉

yes

—

C₂³×C₈

C₂×D₇₉

yes

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

158

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

C₇₉⋊C₈

yes

characteristic

C₂²×D₄

C₇₉⋊C₈

yes

—

C₂³×C₄

C₂×Dic₇₉

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₇₉

yes

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₂×C₁₆

—

158

—

C₂⁴

C₂×C₇₉⋊C₈

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₉

yes

—

C₂²×C₄

C₂×C₇₉⋊C₈

yes

—

C₂²×C₄

#11929608

yes

—

100

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²⋊Dic₇₉

yes

—

105

C₂×C₈

C₂×C₂²⋊D₇₉

yes

—

106

C₁₆

—

316

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²⋊(C₇₉⋊C₈)

yes

—

110

C₂³

C₂²×C₇₉⋊C₈

yes

characteristic

111

Q₈

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

—

118

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

—

119

C₂×C₄

C₂×#11929608

yes

—

120

C₈

—

121

C₈

#14579130

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×C₂²⋊(C₇₉⋊C₈)

yes

—

126

C₂²

#19098980

yes

—

127

C₄

—

128

C₄

#19098419

yes

characteristic

129

C₄

#19098527

yes

—

130

C₂

—

131

C₂

#23763310

yes

characteristic

132

C₂

#23765014

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator, socle	C₂³×C₇₉	C₂²×C₈
Fitting	C₂²×#131×C₇₉	C₂
prime radical	C₂²×#41×C₇₉	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂²×#131	D₇₉
hypocenter	C₇₉	#12680
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×#131

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₉

commutator, socle

C₂³×C₇₉

C₂²×C₈

Fitting

C₂²×#131×C₇₉

C₂

prime radical

C₂²×#41×C₇₉

C₄

cosocle

C₂²×C₈×C₇₉

C₂³

hypercenter

C₂²×#131

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂²×C₈ ◁ C₂²×#131

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	semidirect
69. C₇₉	#12680	semidirect

H ≅ G/N

type

1. C₂×C₂²⋊(C₇₉⋊C₁₆)

C₂

semidirect

69. C₇₉

#12680

semidirect

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

subgroup	index
C₂×C₂²⋊(C₇₉⋊C₁₆)	2
C₂²×#131×C₇₉	2
#12680	79

subgroup

index

C₂×C₂²⋊(C₇₉⋊C₁₆)

C₂²×#131×C₇₉

#12680

Subgroups — #34203705

Conjugacy classes of subgroups

Complements

Maximal subgroups