Group #34209166 — subgroups

Subgroups — #34209166

Order 20224.

164 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 67 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	#23765675	C₂	1	1	yes	maximal, characteristic
2	#23761599	C₂	1	2	yes	maximal
3	C₄⋊(C₇₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#23761581	C₂	1	2	yes	maximal
5	C₄×Q₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098968	—	2	1	no	—
7	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
8	#19098389	—	2	1	no	—
9	#19098389	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
11	C₄×Q₁₆×C₇₉	—	2	1	no	—
12	#366×C₇₉	C₂²	1	1	yes	characteristic
13	#358×C₇₉	—	2	1	no	—
14	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	#14579161	—	4	1	no	—
16	#14579095	—	2	2	no	—
17	#14579095	—	4	1	no	—
18	C₇₉⋊C₃₂	—	2	2	no	—
19	C₂×Q₁₆×C₇₉	—	2	1	no	—
20	C₄×Q₈×C₇₉	—	4	1	no	—
21	Q₃₂×C₇₉	—	2	1	no	—
22	Q₃₂×C₇₉	Q₈	1	2	yes	complemented
23	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
25	#107×C₇₉	—	2	1	no	—
26	#103×C₇₉	—	4	1	no	—
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₂×Dic₁₅₈	—	8	1	no	—
29	C₄⋊Dic₇₉	—	4	2	no	—
30	C₄⋊Dic₇₉	—	8	1	no	—
31	C₂×Q₈×C₇₉	—	4	1	no	—
32	Q₁₆×C₇₉	—	2	2	no	—
33	Q₁₆×C₇₉	—	4	1	no	—
34	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₇₉	—	4	1	no	—
37	C₄⋊C₄×C₇₉	—	8	1	no	—
38	C₄²×C₇₉	—	8	1	no	—
39	C₄²×C₇₉	D₈	1	1	yes	characteristic
40	C₁₆×C₇₉	—	2	1	no	—
41	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₁₆×C₇₉	#41	1	1	yes	characteristic
43	Q₈×C₇₉	—	4	2	no	—
44	Q₈×C₇₉	—	8	1	no	—
45	C₂×C₄×C₇₉	—	8	1	no	—
46	C₂×C₄×C₇₉	—	8	1	no	—
47	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
48	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₇₉	—	8	2	no	—
51	Dic₁₅₈	—	16	2	no	—
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₈×C₇₉	#137	1	1	yes	characteristic
55	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₇₉	—	2	1	no	—
57	C₄×C₇₉	—	8	2	no	—
58	C₄×C₇₉	—	16	1	no	—
59	C₄×C₇₉	Q₆₄	1	2	yes	—
60	C₄×C₇₉	#474	1	1	yes	characteristic
61	C₄×C₇₉	#510	1	1	yes	characteristic
62	Dic₇₉	—	16	2	no	—
63	#13706	—	79	1	no	maximal, Sylow
64	C₂×C₇₉	#2224	1	1	yes	characteristic
65	C₂×C₇₉	C₂×Q₆₄	1	1	yes	characteristic
66	C₂×C₇₉	#2263	1	1	yes	characteristic
67	#2252	—	79	1	no	—
68	C₄×Q₃₂	D₇₉	1	1	yes	characteristic
69	#1423	—	79	2	no	—
70	C₄⋊C₃₂	—	79	1	no	—
71	#1416	—	79	2	no	—
72	C₇₉	#13706	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
74	#500	—	158	1	no	—
75	C₄×Q₁₆	—	2	1	no	—
76	C₂×C₃₂	—	79	2	no	—
77	#366	—	79	2	no	—
78	#366	—	158	1	no	—
79	#366	C₂×D₇₉	1	1	yes	characteristic
80	#358	—	2	1	no	—
81	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
82	C₂×Q₁₆	—	2	1	no	—
83	C₄⋊Q₈	—	316	1	no	—
84	C₄×Q₈	—	4	1	no	—
85	Q₃₂	—	2	1	no	—
86	Q₃₂	Dic₁₅₈	1	2	yes	complemented
87	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
88	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
89	#107	—	2	1	no	—
90	#107	—	316	1	no	—
91	#107	—	158	2	no	—
92	#103	—	4	1	no	—
93	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
94	C₃₂	—	158	2	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	632	1	no	—
97	Q₁₆	—	2	2	no	—
98	Q₁₆	—	4	1	no	—
99	C₂×C₈	#11929609	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	8	1	no	—
103	C₄⋊C₄	—	316	2	no	—
104	C₄⋊C₄	—	632	1	no	—
105	C₄²	—	8	1	no	—
106	C₄²	#11929609	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	C₂×Dic₁₅₈	1	1	yes	characteristic
109	C₁₆	#11929624	1	1	yes	characteristic
110	Q₈	—	4	2	no	—
111	Q₈	—	8	1	no	—
112	Q₈	—	1264	2	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	632	2	no	—
116	C₂×C₄	#14579102	1	1	yes	characteristic
117	C₂×C₄	#14579102	1	1	yes	characteristic
118	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	#14579140	1	1	yes	characteristic
121	C₈	#14579191	1	1	yes	characteristic
122	C₂²	C₂×#14579102	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	8	2	no	—
125	C₄	—	16	1	no	—
126	C₄	—	1264	2	no	—
127	C₄	#19098403	1	2	yes	—
128	C₄	#19098843	1	1	yes	characteristic
129	C₄	#19099003	1	1	yes	characteristic
130	C₂	#23765537	1	1	yes	characteristic
131	C₂	C₂×#19098403	1	1	yes	characteristic
132	C₂	#23765712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic

#23761599

C₂

yes

maximal

C₄⋊(C₇₉⋊C₃₂)

C₂

yes

maximal, characteristic

#23761581

C₂

yes

maximal

C₄×Q₃₂×C₇₉

C₂

yes

maximal, characteristic

#19098968

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

#19098389

—

#19098389

C₂²

yes

—

C₂×Q₃₂×C₇₉

C₂²

yes

characteristic

C₄×Q₁₆×C₇₉

—

#366×C₇₉

C₂²

yes

characteristic

#358×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

—

—

—

C₇₉⋊C₃₂

—

C₂×Q₁₆×C₇₉

—

C₄×Q₈×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

Q₈

yes

complemented

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#103×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×Dic₁₅₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×Q₈

yes

characteristic

C₁₆×C₇₉

#41

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

characteristic

C₈×C₇₉

#137

yes

characteristic

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

Q₆₄

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

C₂×Q₆₄

yes

C₂×C₇₉

yes

—

C₄×Q₃₂

D₇₉

yes

characteristic

#1423

—

C₄⋊C₃₂

—

#1416

—

C₇₉

#13706

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₉

yes

characteristic

#500

—

158

—

C₄×Q₁₆

—

C₂×C₃₂

—

#366

—

#366

—

158

—

#366

C₂×D₇₉

yes

characteristic

#358

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

316

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

Dic₁₅₈

yes

complemented

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#107

—

#107

—

316

—

#107

—

158

—

#103

—

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₃₂

—

158

—

C₂×Q₈

—

C₂×Q₈

—

632

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#11929609

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

316

—

104

C₄⋊C₄

—

632

—

105

C₄²

—

106

C₄²

#11929609

yes

characteristic

107

C₁₆

—

108

C₁₆

C₂×Dic₁₅₈

yes

characteristic

109

C₁₆

#11929624

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

1264

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

632

—

116

C₂×C₄

#14579102

yes

characteristic

117

C₂×C₄

#14579102

yes

characteristic

118

C₂×C₄

C₂×#11929609

yes

characteristic

119

C₈

—

120

C₈

#14579140

yes

characteristic

121

C₈

#14579191

yes

characteristic

122

C₂²

C₂×#14579102

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

1264

—

127

C₄

#19098403

yes

—

128

C₄

#19098843

yes

characteristic

129

C₄

#19099003

yes

characteristic

130

C₂

#23765537

yes

characteristic

131

C₂

C₂×#19098403

yes

characteristic

132

C₂

#23765712

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14579102
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₄×Q₃₂×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
hypercenter	C₄×Q₃₂	D₇₉
hypocenter	C₇₉	#13706
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂²

C₂×#14579102

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₄×Q₃₂×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

C₂×D₁₆

hypercenter

C₄×Q₃₂

D₇₉

hypocenter

C₇₉

#13706

LCS

G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

N	H ≅ G/N	type
22. Q₃₂×C₇₉	Q₈	semidirect
72. C₇₉	#13706	semidirect
86. Q₃₂	Dic₁₅₈	semidirect

H ≅ G/N

type

22. Q₃₂×C₇₉

Q₈

semidirect

72. C₇₉

#13706

semidirect

86. Q₃₂

Dic₁₅₈

semidirect

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

subgroup	index
#23765675	2
#23761599	2
C₄⋊(C₇₉⋊C₃₂)	2
#23761581	2
C₄×Q₃₂×C₇₉	2
#13706	79

subgroup

index

C₄⋊(C₇₉⋊C₃₂)

C₄×Q₃₂×C₇₉

Subgroups — #34209166

Conjugacy classes of subgroups

Complements

Maximal subgroups