Group #34202928 — subgroups

Subgroups — #34202928

Order 20224.

262 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23765653	C₂	1	1	yes	maximal, characteristic
2	#23765629	C₂	1	2	yes	maximal
3	#23762998	C₂	1	2	yes	maximal
4	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#2252	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	—	2	2	no	—
7	#19098959	—	2	2	no	—
8	#19098943	C₂²	1	2	yes	—
9	#19098913	—	2	2	no	—
10	#19098366	—	2	4	no	—
11	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₇₉	C₂²	1	2	yes	—
14	#366×C₇₉	—	2	2	no	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₁₆	—	2	2	no	—
17	C₂×Dic₃₁₆	—	2	2	no	—
18	#14579161	—	2	2	no	—
19	Dic₆₃₂	—	4	4	no	—
20	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
21	#14579092	—	4	4	no	—
22	#14579085	—	4	4	no	—
23	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
24	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₇₉	—	2	2	no	—
26	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
27	#107×C₇₉	—	2	2	no	—
28	#107×C₇₉	—	2	2	no	—
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₅₈	—	4	2	no	—
31	Dic₃₁₆	—	4	4	no	—
32	Dic₃₁₆	—	4	4	no	—
33	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
34	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₄⋊Dic₇₉	—	4	2	no	—
37	C₇₉⋊C₁₆	—	2	2	no	—
38	C₂×Q₈×C₇₉	—	4	2	no	—
39	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₇₉	—	4	2	no	—
42	C₄⋊C₄×C₇₉	—	4	2	no	—
43	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₉	SD₁₆	1	4	yes	—
45	Q₈×C₇₉	—	8	4	no	—
46	C₂×C₄×C₇₉	—	8	2	no	—
47	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₇₉	—	8	2	no	—
51	Dic₁₅₈	—	8	2	no	—
52	Dic₁₅₈	—	8	2	no	—
53	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
54	C₈×C₇₉	#135	1	2	yes	—
55	C₇₉⋊C₈	—	2	2	no	—
56	C₇₉⋊C₈	Q₃₂	1	4	yes	complemented
57	C₂²×C₇₉	#493	1	1	yes	characteristic
58	C₄×C₇₉	—	16	2	no	—
59	C₄×C₇₉	#492	1	1	yes	characteristic
60	C₄×C₇₉	#495	1	1	yes	characteristic
61	C₄×C₇₉	C₂×Q₃₂	1	2	yes	—
62	C₄×C₇₉	#508	1	2	yes	—
63	Dic₇₉	—	16	2	no	—
64	#12504	—	79	1	no	maximal, Sylow
65	C₂×C₇₉	#1706	1	1	yes	characteristic
66	C₂×C₇₉	#2247	1	1	yes	characteristic
67	C₂×C₇₉	#2249	1	1	yes	characteristic
68	#2252	D₇₉	1	1	yes	characteristic
69	#2247	—	79	1	no	—
70	#2241	—	79	2	no	—
71	#1707	—	79	2	no	—
72	C₈×C₁₆	—	79	1	no	—
73	C₇₉	#12504	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	158	2	no	—
75	#500	C₂×D₇₉	1	2	yes	—
76	#498	—	158	2	no	—
77	#494	—	79	2	no	—
78	#487	—	158	2	no	—
79	#366	—	2	2	no	—
80	#358	—	158	4	no	—
81	C₄×C₁₆	—	79	1	no	—
82	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
83	C₈²	—	79	1	no	—
84	C₂×Q₁₆	—	158	2	no	—
85	C₂×Q₁₆	—	158	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	158	2	no	—
88	Q₃₂	—	316	4	no	—
89	C₂×C₁₆	—	79	2	no	—
90	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
91	#107	—	2	2	no	—
92	#107	—	2	2	no	—
93	#106	—	316	4	no	—
94	#103	—	316	4	no	—
95	C₄×C₈	—	79	1	no	—
96	C₄×C₈	—	79	1	no	—
97	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	316	2	no	—
100	Q₁₆	—	316	4	no	—
101	Q₁₆	—	316	4	no	—
102	C₂×C₈	—	79	2	no	—
103	C₂×C₈	—	79	2	no	—
104	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	316	2	no	—
109	C₄⋊C₄	—	316	2	no	—
110	C₄²	D₄×D₇₉	1	1	yes	characteristic
111	C₁₆	—	158	2	no	—
112	C₁₆	#11929611	1	4	yes	—
113	Q₈	—	8	4	no	—
114	Q₈	—	632	2	no	—
115	Q₈	—	632	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	632	2	no	—
118	C₂×C₄	#14579157	1	1	yes	characteristic
119	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
120	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
121	C₈	—	79	4	no	—
122	C₈	—	158	2	no	—
123	C₈	C₂×#11929611	1	2	yes	—
124	C₈	#14579184	1	2	yes	—
125	C₂²	#19098938	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	1264	2	no	—
128	C₄	#19098937	1	1	yes	characteristic
129	C₄	#19098948	1	1	yes	characteristic
130	C₄	Q₃₂×D₇₉	1	2	yes	—
131	C₄	#19098992	1	2	yes	—
132	C₂	#23762993	1	1	yes	characteristic
133	C₂	#23765651	1	1	yes	characteristic
134	C₂	#23765658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765653

C₂

yes

maximal, characteristic

#23765629

C₂

yes

maximal

#23762998

C₂

yes

maximal

C₁₆×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₇₉×#2252

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

—

#19098959

—

#19098943

C₂²

yes

—

#19098913

—

#19098366

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

#500×C₇₉

C₂²

yes

—

#366×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

Dic₆₃₂

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579092

—

#14579085

—

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

SD₁₆

yes

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#135

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

Q₃₂

yes

complemented

C₂²×C₇₉

#493

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×Q₃₂

yes

—

C₄×C₇₉

#508

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

—

—

—

C₈×C₁₆

—

C₇₉

#12504

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

158

—

#500

C₂×D₇₉

yes

—

#498

—

158

—

#494

—

#487

—

158

—

#366

—

#358

—

158

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

Q₃₂

—

316

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

—

#107

—

#107

—

#106

—

316

—

#103

—

316

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

316

—

100

Q₁₆

—

316

—

101

Q₁₆

—

316

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₇₉

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

316

—

109

C₄⋊C₄

—

316

—

110

C₄²

D₄×D₇₉

yes

characteristic

111

C₁₆

—

158

—

112

C₁₆

#11929611

yes

—

113

Q₈

—

114

Q₈

—

632

—

115

Q₈

—

632

—

116

C₂×C₄

—

117

C₂×C₄

—

632

—

118

C₂×C₄

#14579157

yes

characteristic

119

C₂×C₄

D₈×D₇₉

yes

characteristic

120

C₂×C₄

D₈×D₇₉

yes

characteristic

121

C₈

—

122

C₈

—

158

—

123

C₈

C₂×#11929611

yes

—

124

C₈

#14579184

yes

—

125

C₂²

#19098938

yes

characteristic

126

C₄

—

127

C₄

—

1264

—

128

C₄

#19098937

yes

characteristic

129

C₄

#19098948

yes

characteristic

130

C₄

Q₃₂×D₇₉

yes

—

131

C₄

#19098992

yes

—

132

C₂

#23762993

yes

characteristic

133

C₂

#23765651

yes

characteristic

134

C₂

#23765658

yes

characteristic

	subgroup	G/H
center	C₂²	#19098938
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2252	C₂
socle, prime radical	C₂²×C₇₉	#493
hypercenter	#2252	D₇₉
hypocenter	C₇₉	#12504
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#19098938

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#2252

C₂

C₂²×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
56. C₇₉⋊C₈	Q₃₂	semidirect
73. C₇₉	#12504	semidirect

H ≅ G/N

type

56. C₇₉⋊C₈

Q₃₂

semidirect

73. C₇₉

#12504

semidirect

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

subgroup	index
#23765653	2
#23765629	2
#23762998	2
C₁₆×C₇₉⋊C₈	2
C₇₉×#2252	2
#12504	79

subgroup

index

#23765653

#23765629

#23762998

C₁₆×C₇₉⋊C₈

C₇₉×#2252

#12504

Subgroups — #34202928

Conjugacy classes of subgroups

Complements

Maximal subgroups