Group #34202993 — subgroups

Subgroups — #34202993

Order 20224.

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

normal subgroups only

This group has 330 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	#23765657	C₂	1	2	yes	maximal, complemented
2	#23765650	C₂	1	1	yes	maximal, characteristic
3	#23763007	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579104	—	2	4	no	—
7	C₂×D₁₂₆₄	—	2	2	no	—
8	#19098940	C₂²	1	2	yes	—
9	#19098911	—	2	4	no	—
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×D₆₃₂	—	2	2	no	—
16	C₂×D₆₃₂	—	2	2	no	—
17	#14579158	—	2	2	no	—
18	#14579104	—	4	8	no	—
19	D₁₂₆₄	—	4	4	no	—
20	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
21	#14579081	—	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×D₃₁₆	—	4	2	no	—
30	C₂×D₃₁₆	—	4	4	no	—
31	D₆₃₂	—	4	4	no	—
32	D₆₃₂	—	4	4	no	—
33	C₂×C₇₉⋊C₈	D₈	1	4	yes	—
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	C₂²×D₇₉	—	8	2	no	—
44	Q₈×C₇₉	—	8	2	no	—
45	Q₈×C₇₉	—	8	2	no	—
46	C₂×C₄×C₇₉	—	8	2	no	—
47	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
49	D₃₁₆	—	8	2	no	—
50	D₃₁₆	—	8	2	no	—
51	D₃₁₆	—	8	8	no	—
52	C₈×C₇₉	C₂×D₈	1	2	yes	—
53	C₈×C₇₉	C₂×D₈	1	2	yes	—
54	C₇₉⋊C₈	—	2	4	no	—
55	C₂²×C₇₉	#493	1	1	yes	characteristic
56	C₂×D₇₉	—	16	2	no	—
57	C₂×D₇₉	—	16	4	no	—
58	C₄×C₇₉	—	16	2	no	—
59	C₄×C₇₉	#493	1	1	yes	characteristic
60	C₄×C₇₉	#493	1	1	yes	characteristic
61	C₄×C₇₉	#508	1	4	yes	—
62	#12518	—	79	1	no	maximal, Sylow
63	C₂×C₇₉	#1712	1	1	yes	characteristic
64	C₂×C₇₉	#2249	1	2	yes	—
65	D₇₉	—	32	2	no	—
66	#2248	—	79	2	no	—
67	#2247	D₇₉	1	1	yes	characteristic, complemented
68	#2246	—	79	1	no	—
69	#1709	—	79	2	no	—
70	C₈×C₁₆	—	79	1	no	—
71	C₇₉	#12518	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	2	2	no	—
73	C₂×SD₃₂	—	158	4	no	—
74	C₂×D₁₆	—	158	2	no	—
75	#494	C₂×D₇₉	1	2	yes	—
76	#493	—	79	2	no	—
77	#486	—	158	4	no	—
78	C₄×C₁₆	—	79	1	no	—
79	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
80	C₈²	—	79	1	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×D₈	—	158	2	no	—
84	C₂×D₈	—	158	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	#127	—	158	2	no	—
87	Q₃₂	—	4	4	no	—
88	SD₃₂	—	316	8	no	—
89	D₁₆	—	316	4	no	—
90	C₂×C₁₆	—	79	2	no	—
91	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
92	#102	—	316	8	no	—
93	C₄×C₈	—	79	2	no	—
94	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	316	2	no	—
97	C₂×D₄	—	316	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	316	4	no	—
101	D₈	—	316	4	no	—
102	C₂×C₈	—	79	4	no	—
103	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	D₄×D₇₉	1	1	yes	characteristic
107	C₁₆	—	79	4	no	—
108	C₁₆	#11929609	1	4	yes	—
109	C₂³	—	632	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	632	2	no	—
113	D₄	—	632	2	no	—
114	D₄	—	632	8	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#14579157	1	1	yes	characteristic
117	C₂×C₄	D₈×D₇₉	1	2	yes	—
118	C₈	—	158	4	no	—
119	C₈	D₈×D₇₉	1	2	yes	—
120	C₈	C₂×#11929609	1	2	yes	—
121	C₂²	—	1264	2	no	—
122	C₂²	—	1264	4	no	—
123	C₂²	#19098938	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#19098938	1	1	yes	characteristic
126	C₄	#19098939	1	1	yes	characteristic
127	C₄	#19098996	1	4	yes	—
128	C₂	—	2528	2	no	—
129	C₂	#23763014	1	1	yes	characteristic
130	C₂	#23765662	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765657

C₂

yes

maximal, complemented

#23765650

C₂

yes

maximal, characteristic

#23763007

C₂

yes

maximal, complemented

C₁₆×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₇₉×#2247

C₂

yes

maximal, characteristic, complemented

C₂×#14579104

—

C₂×D₁₂₆₄

—

#19098940

C₂²

yes

—

#19098911

—

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×D₆₃₂

—

C₂×D₆₃₂

—

#14579158

—

#14579104

—

D₁₂₆₄

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579081

—

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×D₃₁₆

—

C₂×D₃₁₆

—

D₆₃₂

—

D₆₃₂

—

C₂×C₇₉⋊C₈

D₈

yes

—

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

—

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

C₂×D₈

yes

—

C₈×C₇₉

C₂×D₈

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

#2248

—

#2247

D₇₉

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₇₉

#12518

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

158

—

C₂×D₁₆

—

158

—

#494

C₂×D₇₉

yes

—

#493

—

#486

—

158

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

#127

—

158

—

Q₃₂

—

SD₃₂

—

316

—

D₁₆

—

316

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

—

#102

—

316

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

Q₁₆

—

Q₁₆

—

100

D₈

—

316

—

101

D₈

—

316

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₇₉

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

D₄×D₇₉

yes

characteristic

107

C₁₆

—

108

C₁₆

#11929609

yes

—

109

C₂³

—

632

—

110

Q₈

—

111

Q₈

—

112

D₄

—

632

—

113

D₄

—

632

—

114

D₄

—

632

—

115

C₂×C₄

—

116

C₂×C₄

#14579157

yes

characteristic

117

C₂×C₄

D₈×D₇₉

yes

—

118

C₈

—

158

—

119

C₈

D₈×D₇₉

yes

—

120

C₈

C₂×#11929609

yes

—

121

C₂²

—

1264

—

122

C₂²

—

1264

—

123

C₂²

#19098938

yes

characteristic

124

C₄

—

125

C₄

#19098938

yes

characteristic

126

C₄

#19098939

yes

characteristic

127

C₄

#19098996

yes

—

128

C₂

—

2528

—

129

C₂

#23763014

yes

characteristic

130

C₂

#23765662

yes

—

	subgroup	G/H
center	C₂²	#19098938
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2247	C₂
socle	C₂²×C₇₉	#493
prime radical	#23765650	C₂
hypercenter	#2247	D₇₉
hypocenter	C₇₉	#12518
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

C₂

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #23765657	C₂	semidirect
3. #23763007	C₂	semidirect
4. C₁₆×C₇₉⋊C₈	C₂	semidirect
5. C₇₉×#2247	C₂	semidirect
67. #2247	D₇₉	semidirect
71. C₇₉	#12518	semidirect

H ≅ G/N

type

1. #23765657

C₂

semidirect

3. #23763007

C₂

semidirect

4. C₁₆×C₇₉⋊C₈

C₂

semidirect

5. C₇₉×#2247

C₂

semidirect

67. #2247

D₇₉

semidirect

71. C₇₉

#12518

semidirect

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

subgroup	index
#23765657	2
#23765650	2
#23763007	2
C₁₆×C₇₉⋊C₈	2
C₇₉×#2247	2
#12518	79

subgroup

index

#23765657

#23765650

#23763007

C₁₆×C₇₉⋊C₈

C₇₉×#2247

#12518

Subgroups — #34202993

Conjugacy classes of subgroups

Complements

Maximal subgroups