Group #34202982 — subgroups

Subgroups — #34202982

Order 20224.

310 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 95 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	#23765652	C₂	1	2	yes	maximal
3	#23763016	C₂	1	2	yes	maximal
4	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	—	2	2	no	—
7	C₂×#14579107	—	2	4	no	—
8	#19098966	—	2	4	no	—
9	#19098943	C₂²	1	2	yes	—
10	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₇₉	—	2	2	no	—
13	#494×C₇₉	C₂²	1	2	yes	—
14	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₆	—	2	2	no	—
16	C₂×Dic₃₁₆	—	2	2	no	—
17	#14579161	—	2	2	no	—
18	Dic₆₃₂	—	4	4	no	—
19	#14579107	—	4	8	no	—
20	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
21	#14579094	—	4	8	no	—
22	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₇₉	—	2	2	no	—
24	C₂×Q₁₆×C₇₉	—	2	2	no	—
25	C₄⋊Q₈×C₇₉	—	2	2	no	—
26	Q₃₂×C₇₉	—	4	4	no	—
27	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₅₈	—	4	2	no	—
30	Dic₃₁₆	—	4	4	no	—
31	Dic₃₁₆	—	4	4	no	—
32	C₂×C₇₉⋊C₈	D₈	1	4	yes	—
33	C₄⋊Dic₇₉	—	4	4	no	—
34	C₇₉⋊C₁₆	D₈	1	4	yes	—
35	C₂×Q₈×C₇₉	—	4	2	no	—
36	Q₁₆×C₇₉	—	4	4	no	—
37	Q₁₆×C₇₉	—	4	4	no	—
38	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₉	—	4	4	no	—
41	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₉	D₈	1	4	yes	—
43	Q₈×C₇₉	—	8	2	no	—
44	Q₈×C₇₉	—	8	2	no	—
45	C₂×C₄×C₇₉	—	8	2	no	—
46	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
48	C₂×Dic₇₉	—	8	2	no	—
49	Dic₁₅₈	—	8	2	no	—
50	Dic₁₅₈	—	8	2	no	—
51	C₈×C₇₉	C₂×D₈	1	2	yes	—
52	C₈×C₇₉	C₂×D₈	1	2	yes	—
53	C₇₉⋊C₈	Q₃₂	1	8	yes	complemented
54	C₂²×C₇₉	#493	1	1	yes	characteristic
55	C₄×C₇₉	—	16	2	no	—
56	C₄×C₇₉	#493	1	1	yes	characteristic
57	C₄×C₇₉	#493	1	1	yes	characteristic
58	C₄×C₇₉	C₂×Q₃₂	1	4	yes	—
59	Dic₇₉	—	16	2	no	—
60	#12516	—	79	1	no	maximal, Sylow
61	C₂×C₇₉	#1712	1	1	yes	characteristic
62	C₂×C₇₉	#2247	1	2	yes	—
63	#2247	—	79	2	no	—
64	#2247	—	79	1	no	—
65	#2247	D₇₉	1	1	yes	characteristic
66	#1713	—	79	2	no	—
67	C₈×C₁₆	—	79	1	no	—
68	C₇₉	#12516	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	2	2	no	—
70	C₂×Q₃₂	—	158	4	no	—
71	C₂×Q₃₂	—	158	2	no	—
72	#500	—	158	4	no	—
73	#494	—	79	2	no	—
74	#494	C₂×D₇₉	1	2	yes	—
75	C₄×C₁₆	—	79	1	no	—
76	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
77	C₈²	—	79	1	no	—
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₂×Q₁₆	—	158	2	no	—
81	C₂×Q₁₆	—	158	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	158	2	no	—
84	Q₃₂	—	4	4	no	—
85	Q₃₂	—	316	8	no	—
86	Q₃₂	—	316	4	no	—
87	C₂×C₁₆	—	79	2	no	—
88	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
89	#107	—	316	8	no	—
90	C₄×C₈	—	79	2	no	—
91	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	316	2	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	316	4	no	—
97	Q₁₆	—	316	4	no	—
98	C₂×C₈	—	79	4	no	—
99	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	316	4	no	—
103	C₄²	D₄×D₇₉	1	1	yes	characteristic
104	C₁₆	—	79	4	no	—
105	C₁₆	#11929609	1	4	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	632	2	no	—
109	Q₈	—	632	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	632	2	no	—
112	C₂×C₄	#14579157	1	1	yes	characteristic
113	C₂×C₄	D₈×D₇₉	1	2	yes	—
114	C₈	—	79	8	no	—
115	C₈	D₈×D₇₉	1	2	yes	—
116	C₈	C₂×#11929609	1	2	yes	—
117	C₂²	#19098938	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	1264	2	no	—
120	C₄	#19098938	1	1	yes	characteristic
121	C₄	#19098939	1	1	yes	characteristic
122	C₄	Q₃₂×D₇₉	1	4	yes	—
123	C₂	#23763014	1	1	yes	characteristic
124	C₂	#23765651	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765653

C₂

yes

maximal, characteristic

#23765652

C₂

yes

maximal

#23763016

C₂

yes

maximal

C₁₆×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₇₉×#2247

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

—

C₂×#14579107

—

#19098966

—

#19098943

C₂²

yes

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₇₉

—

#494×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

Dic₆₃₂

—

#14579107

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579094

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

—

Q₃₂×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

D₈

yes

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

D₈

yes

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂×D₈

yes

—

C₈×C₇₉

C₂×D₈

yes

—

C₇₉⋊C₈

Q₃₂

yes

complemented

C₂²×C₇₉

#493

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×Q₃₂

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2247

—

#2247

—

#2247

D₇₉

yes

characteristic

#1713

—

C₈×C₁₆

—

C₇₉

#12516

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

158

—

C₂×Q₃₂

—

158

—

#500

—

158

—

#494

—

#494

C₂×D₇₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

Q₃₂

—

Q₃₂

—

316

—

Q₃₂

—

316

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

—

#107

—

316

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

316

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

316

—

Q₁₆

—

316

—

C₂×C₈

—

C₂×C₈

D₄×D₇₉

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

316

—

103

C₄²

D₄×D₇₉

yes

characteristic

104

C₁₆

—

105

C₁₆

#11929609

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

632

—

109

Q₈

—

632

—

110

C₂×C₄

—

111

C₂×C₄

—

632

—

112

C₂×C₄

#14579157

yes

characteristic

113

C₂×C₄

D₈×D₇₉

yes

—

114

C₈

—

115

C₈

D₈×D₇₉

yes

—

116

C₈

C₂×#11929609

yes

—

117

C₂²

#19098938

yes

characteristic

118

C₄

—

119

C₄

—

1264

—

120

C₄

#19098938

yes

characteristic

121

C₄

#19098939

yes

characteristic

122

C₄

Q₃₂×D₇₉

yes

—

123

C₂

#23763014

yes

characteristic

124

C₂

#23765651

yes

—

	subgroup	G/H
center	C₂²	#19098938
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2247	C₂
socle, prime radical	C₂²×C₇₉	#493
hypercenter	#2247	D₇₉
hypocenter	C₇₉	#12516
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#19098938

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#2247

C₂

C₂²×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
53. C₇₉⋊C₈	Q₃₂	semidirect
68. C₇₉	#12516	semidirect

H ≅ G/N

type

53. C₇₉⋊C₈

Q₃₂

semidirect

68. C₇₉

#12516

semidirect

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

subgroup	index
#23765653	2
#23765652	2
#23763016	2
C₁₆×C₇₉⋊C₈	2
C₇₉×#2247	2
#12516	79

subgroup

index

#23765653

#23765652

#23763016

C₁₆×C₇₉⋊C₈

C₇₉×#2247

#12516

Subgroups — #34202982

Conjugacy classes of subgroups

Complements

Maximal subgroups