Group #34202956 — subgroups

Subgroups — #34202956

Order 20224.

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

normal subgroups only

This group has 306 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	#23765625	C₂	1	4	yes	maximal, complemented
2	#23763015	C₂	1	1	yes	maximal, characteristic
3	C₈×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098940	—	2	1	no	—
6	#19098940	—	2	1	no	—
7	#19098940	C₂²	1	2	yes	—
8	#19098363	—	2	8	no	—
9	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	#500×C₇₉	—	2	1	no	—
11	#500×C₇₉	C₂²	1	2	yes	—
12	#494×C₇₉	—	2	1	no	—
13	C₈²×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×D₆₃₂	—	2	4	no	—
15	C₂×D₆₃₂	—	4	2	no	—
16	C₂×D₆₃₂	—	4	2	no	—
17	#14579158	—	2	2	no	—
18	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	—
19	C₂×Q₁₆×C₇₉	—	4	2	no	—
20	C₄⋊Q₈×C₇₉	—	2	2	no	—
21	#107×C₇₉	—	2	4	no	—
22	#107×C₇₉	—	4	2	no	—
23	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×D₃₁₆	—	4	2	no	—
27	C₂×D₃₁₆	—	4	2	no	—
28	C₂×D₃₁₆	—	4	2	no	—
29	D₆₃₂	—	4	8	no	—
30	D₆₃₂	—	8	4	no	—
31	D₆₃₂	—	8	4	no	—
32	C₇₉⋊C₁₆	—	2	4	no	—
33	C₂×Q₈×C₇₉	—	4	2	no	—
34	Q₁₆×C₇₉	—	8	4	no	—
35	C₂×C₈×C₇₉	D₈	1	2	yes	—
36	C₂×C₈×C₇₉	D₈	1	2	yes	—
37	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₇₉	—	4	2	no	—
39	C₄⋊C₄×C₇₉	—	4	2	no	—
40	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₇₉	—	8	2	no	—
42	Q₈×C₇₉	—	8	4	no	—
43	C₂×C₄×C₇₉	—	8	2	no	—
44	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
47	D₃₁₆	—	8	4	no	—
48	D₃₁₆	—	8	4	no	—
49	D₃₁₆	—	8	4	no	—
50	C₈×C₇₉	D₁₆	1	4	yes	—
51	C₈×C₇₉	SD₃₂	1	4	yes	—
52	C₈×C₇₉	#135	1	4	yes	—
53	C₂²×C₇₉	#493	1	1	yes	characteristic
54	C₂×D₇₉	—	16	2	no	—
55	C₂×D₇₉	—	16	2	no	—
56	C₂×D₇₉	—	16	2	no	—
57	C₄×C₇₉	—	16	2	no	—
58	C₄×C₇₉	#495	1	2	yes	—
59	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
60	C₄×C₇₉	C₂×SD₃₂	1	2	yes	—
61	#12509	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	#1709	1	1	yes	characteristic
63	C₂×C₇₉	#2246	1	1	yes	characteristic
64	C₂×C₇₉	#2248	1	1	yes	characteristic
65	D₇₉	—	32	2	no	—
66	#2240	—	79	4	no	—
67	#1713	D₇₉	1	1	yes	characteristic, complemented
68	#1712	—	79	1	no	—
69	C₈×C₁₆	—	79	1	no	—
70	C₇₉	#12509	1	1	yes	characteristic, complemented, Sylow
71	#500	—	2	1	no	—
72	#500	C₂×D₇₉	1	2	yes	—
73	#494	—	2	1	no	—
74	#493	—	79	2	no	—
75	#493	—	158	1	no	—
76	#493	—	158	1	no	—
77	#357	—	158	8	no	—
78	C₄×C₁₆	—	79	2	no	—
79	C₈²	C₂×D₇₉	1	1	yes	characteristic
80	C₂×Q₁₆	—	4	2	no	—
81	C₂×D₈	—	316	2	no	—
82	C₂×D₈	—	316	2	no	—
83	C₂×D₈	—	158	4	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	#127	—	158	2	no	—
86	C₂×C₁₆	—	79	4	no	—
87	#107	—	2	4	no	—
88	#107	—	4	2	no	—
89	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×D₄	—	316	2	no	—
94	C₂×D₄	—	316	2	no	—
95	C₂×D₄	—	316	2	no	—
96	Q₁₆	—	8	4	no	—
97	D₈	—	316	8	no	—
98	D₈	—	632	4	no	—
99	D₈	—	632	4	no	—
100	C₂×C₈	#11929609	1	2	yes	—
101	C₂×C₈	#11929609	1	2	yes	—
102	C₂×C₈	D₄×D₇₉	1	2	yes	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
106	C₁₆	—	158	4	no	—
107	C₂³	—	632	2	no	—
108	Q₈	—	8	4	no	—
109	D₄	—	632	4	no	—
110	D₄	—	632	4	no	—
111	D₄	—	632	4	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	#14579157	1	1	yes	characteristic
114	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
115	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
116	C₈	#14579102	1	4	yes	—
117	C₈	#14579104	1	4	yes	—
118	C₈	#14579184	1	4	yes	—
119	C₂²	—	1264	2	no	—
120	C₂²	—	1264	2	no	—
121	C₂²	—	1264	2	no	—
122	C₂²	#19098939	1	1	yes	characteristic
123	C₄	—	16	2	no	—
124	C₄	#19098948	1	2	yes	—
125	C₄	C₂×#14579102	1	2	yes	—
126	C₄	C₂×#14579104	1	2	yes	—
127	C₂	—	2528	2	no	—
128	C₂	#23763007	1	1	yes	characteristic
129	C₂	#23765649	1	1	yes	characteristic
130	C₂	#23765657	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765625

C₂

yes

maximal, complemented

#23763015

C₂

yes

maximal, characteristic

C₈×C₇₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₇₉×#1713

C₂

yes

maximal, characteristic, complemented

#19098940

—

#19098940

—

#19098940

C₂²

yes

—

#19098363

—

C₄×C₇₉⋊C₁₆

C₂²

yes

—

#500×C₇₉

—

#500×C₇₉

C₂²

yes

—

#494×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

#14579158

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

C₂×Q₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

—

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

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

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

D₁₆

yes

—

C₈×C₇₉

SD₃₂

yes

—

C₈×C₇₉

#135

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#495

yes

—

C₄×C₇₉

C₂×D₁₆

yes

—

C₄×C₇₉

C₂×SD₃₂

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#2240

—

#1713

D₇₉

yes

characteristic, complemented

#1712

—

C₈×C₁₆

—

C₇₉

#12509

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₇₉

yes

—

#494

—

#493

—

#493

—

158

—

#493

—

158

—

#357

—

158

—

C₄×C₁₆

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

316

—

C₂×D₈

—

316

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

#127

—

158

—

C₂×C₁₆

—

#107

—

#107

—

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

Q₁₆

—

D₈

—

316

—

D₈

—

632

—

D₈

—

632

—

100

C₂×C₈

#11929609

yes

—

101

C₂×C₈

#11929609

yes

—

102

C₂×C₈

D₄×D₇₉

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

106

C₁₆

—

158

—

107

C₂³

—

632

—

108

Q₈

—

109

D₄

—

632

—

110

D₄

—

632

—

111

D₄

—

632

—

112

C₂×C₄

—

113

C₂×C₄

#14579157

yes

characteristic

114

C₂×C₄

C₂×#11929609

yes

characteristic

115

C₂×C₄

C₂×#11929609

yes

characteristic

116

C₈

#14579102

yes

—

117

C₈

#14579104

yes

—

118

C₈

#14579184

yes

—

119

C₂²

—

1264

—

120

C₂²

—

1264

—

121

C₂²

—

1264

—

122

C₂²

#19098939

yes

characteristic

123

C₄

—

124

C₄

#19098948

yes

—

125

C₄

C₂×#14579102

yes

—

126

C₄

C₂×#14579104

yes

—

127

C₂

—

2528

—

128

C₂

#23763007

yes

characteristic

129

C₂

#23765649

yes

characteristic

130

C₂

#23765657

yes

characteristic

	subgroup	G/H
center	C₂²	#19098939
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1713	C₂
socle	C₂²×C₇₉	#493
prime radical	#23763015	C₂
hypercenter	#1713	D₇₉
hypocenter	C₇₉	#12509
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#19098939

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1713

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. #23765625	C₂	semidirect
3. C₈×C₇₉⋊C₁₆	C₂	semidirect
4. C₇₉×#1713	C₂	semidirect
67. #1713	D₇₉	semidirect
70. C₇₉	#12509	semidirect

H ≅ G/N

type

1. #23765625

C₂

semidirect

3. C₈×C₇₉⋊C₁₆

C₂

semidirect

4. C₇₉×#1713

C₂

semidirect

67. #1713

D₇₉

semidirect

70. C₇₉

#12509

semidirect

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

subgroup	index
#23765625	2
#23763015	2
C₈×C₇₉⋊C₁₆	2
C₇₉×#1713	2
#12509	79

subgroup

index

#23765625

#23763015

C₈×C₇₉⋊C₁₆

C₇₉×#1713

#12509

Subgroups — #34202956

Conjugacy classes of subgroups

Complements

Maximal subgroups