Group #34202940 — subgroups

Subgroups — #34202940

Order 20224.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#23765650	C₂	1	1	yes	maximal, characteristic
2	#23765625	C₂	1	2	yes	maximal, complemented
3	#23762993	C₂	1	2	yes	maximal, complemented
4	#23761316	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2254	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₆₄	—	2	2	no	—
7	#19098940	C₂²	1	2	yes	—
8	#19098937	—	2	2	no	—
9	#19098911	—	2	2	no	—
10	#19098363	—	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	#367×C₇₉	—	2	2	no	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#11929611	—	4	4	no	—
17	C₂×D₆₃₂	—	2	2	no	—
18	C₂×D₆₃₂	—	2	2	no	—
19	#14579158	—	2	2	no	—
20	D₁₂₆₄	—	4	4	no	—
21	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
22	#14579081	—	4	4	no	—
23	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₉	—	2	2	no	—
25	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
26	#107×C₇₉	—	2	2	no	—
27	#107×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×D₃₁₆	—	4	2	no	—
30	C₂×D₃₁₆	—	4	2	no	—
31	C₂×D₃₁₆	—	4	2	no	—
32	#11929611	—	8	8	no	—
33	D₆₃₂	—	4	4	no	—
34	D₆₃₂	—	4	4	no	—
35	C₂×C₇₉⋊C₈	D₈	1	4	yes	—
36	C₇₉⋊C₁₆	—	2	2	no	—
37	C₂×Q₈×C₇₉	—	4	2	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	2	no	—
41	C₄⋊C₄×C₇₉	—	4	2	no	—
42	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₉	SD₁₆	1	4	yes	complemented
44	C₂²×D₇₉	—	8	2	no	—
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	2	yes	—
49	D₃₁₆	—	8	2	no	—
50	D₃₁₆	—	8	2	no	—
51	D₃₁₆	—	8	4	no	—
52	D₃₁₆	—	8	4	no	—
53	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
54	C₈×C₇₉	#135	1	2	yes	—
55	C₇₉⋊C₈	—	2	4	no	—
56	C₂²×C₇₉	#493	1	1	yes	characteristic
57	C₂×D₇₉	—	16	2	no	—
58	C₂×D₇₉	—	16	2	no	—
59	C₂×D₇₉	—	16	2	no	—
60	C₄×C₇₉	—	16	2	no	—
61	C₄×C₇₉	#492	1	1	yes	characteristic
62	C₄×C₇₉	#495	1	1	yes	characteristic
63	C₄×C₇₉	#509	1	4	yes	—
64	#12506	—	79	1	no	maximal, Sylow
65	C₂×C₇₉	#1706	1	1	yes	characteristic
66	C₂×C₇₉	#2250	1	2	yes	—
67	D₇₉	—	32	2	no	—
68	#2254	D₇₉	1	1	yes	characteristic, complemented
69	#2246	—	79	1	no	—
70	#2240	—	79	2	no	—
71	#1706	—	79	2	no	—
72	#1311	—	79	1	no	—
73	C₇₉	#12506	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₁₆	—	158	2	no	—
75	#500	C₂×D₇₉	1	2	yes	—
76	#493	—	79	2	no	—
77	#492	—	158	2	no	—
78	#486	—	158	2	no	—
79	#367	—	2	2	no	—
80	#357	—	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×SD₁₆	—	316	4	no	—
85	C₂×D₈	—	158	2	no	—
86	C₂×D₈	—	158	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	#127	—	158	2	no	—
89	D₁₆	—	316	4	no	—
90	C₂×C₁₆	—	79	2	no	—
91	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#102	—	316	4	no	—
95	C₄×C₈	—	79	2	no	—
96	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
97	C₂×Q₈	—	4	2	no	—
98	C₂×D₄	—	316	2	no	—
99	C₂×D₄	—	316	2	no	—
100	C₂×D₄	—	316	2	no	—
101	SD₁₆	—	632	8	no	—
102	D₈	—	316	4	no	—
103	D₈	—	316	4	no	—
104	C₂×C₈	—	79	4	no	—
105	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	D₄×D₇₉	1	1	yes	characteristic
110	C₁₆	—	158	2	no	—
111	C₁₆	#11929611	1	4	yes	complemented
112	C₂³	—	632	2	no	—
113	Q₈	—	8	4	no	—
114	D₄	—	632	2	no	—
115	D₄	—	632	2	no	—
116	D₄	—	632	4	no	—
117	D₄	—	632	4	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	#14579157	1	1	yes	characteristic
120	C₂×C₄	D₈×D₇₉	1	2	yes	—
121	C₈	—	158	4	no	—
122	C₈	C₂×#11929611	1	2	yes	—
123	C₈	#14579184	1	2	yes	—
124	C₂²	—	1264	2	no	—
125	C₂²	—	1264	2	no	—
126	C₂²	—	1264	2	no	—
127	C₂²	#19098938	1	1	yes	characteristic
128	C₄	—	16	2	no	—
129	C₄	#19098937	1	1	yes	characteristic
130	C₄	#19098948	1	1	yes	characteristic
131	C₄	#19099000	1	4	yes	—
132	C₂	—	2528	2	no	—
133	C₂	#23762993	1	1	yes	characteristic
134	C₂	#23765665	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765650

C₂

yes

maximal, characteristic

#23765625

C₂

yes

maximal, complemented

#23762993

C₂

yes

maximal, complemented

#23761316

C₂

yes

maximal, characteristic, complemented

C₇₉×#2254

C₂

yes

maximal, characteristic, complemented

C₂×D₁₂₆₄

—

#19098940

C₂²

yes

—

#19098937

—

#19098911

—

#19098363

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

#500×C₇₉

C₂²

yes

—

#367×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#11929611

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

#14579158

—

D₁₂₆₄

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579081

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₄⋊Q₈×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929611

—

D₆₃₂

—

D₆₃₂

—

C₂×C₇₉⋊C₈

D₈

yes

—

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

complemented

C₂²×D₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#135

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

#2254

D₇₉

yes

characteristic, complemented

—

—

—

—

C₇₉

#12506

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

158

—

#500

C₂×D₇₉

yes

—

#493

—

#492

—

158

—

#486

—

158

—

#367

—

#357

—

158

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

—

C₂×SD₁₆

—

316

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₄⋊Q₈

—

#127

—

158

—

D₁₆

—

316

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

—

#107

—

#107

—

#102

—

316

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

100

C₂×D₄

—

316

—

101

SD₁₆

—

632

—

102

D₈

—

316

—

103

D₈

—

316

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₇₉

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

D₄×D₇₉

yes

characteristic

110

C₁₆

—

158

—

111

C₁₆

#11929611

yes

complemented

112

C₂³

—

632

—

113

Q₈

—

114

D₄

—

632

—

115

D₄

—

632

—

116

D₄

—

632

—

117

D₄

—

632

—

118

C₂×C₄

—

119

C₂×C₄

#14579157

yes

characteristic

120

C₂×C₄

D₈×D₇₉

yes

—

121

C₈

—

158

—

122

C₈

C₂×#11929611

yes

—

123

C₈

#14579184

yes

—

124

C₂²

—

1264

—

125

C₂²

—

1264

—

126

C₂²

—

1264

—

127

C₂²

#19098938

yes

characteristic

128

C₄

—

129

C₄

#19098937

yes

characteristic

130

C₄

#19098948

yes

characteristic

131

C₄

#19099000

yes

—

132

C₂

—

2528

—

133

C₂

#23762993

yes

characteristic

134

C₂

#23765665

yes

—

	subgroup	G/H
center	C₂²	#19098938
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2254	C₂
socle	C₂²×C₇₉	#493
prime radical	#23765650	C₂
hypercenter	#2254	D₇₉
hypocenter	C₇₉	#12506
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#19098938

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#2254

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
2. #23765625	C₂	semidirect
3. #23762993	C₂	semidirect
4. #23761316	C₂	semidirect
5. C₇₉×#2254	C₂	semidirect
43. C₁₆×C₇₉	SD₁₆	semidirect
68. #2254	D₇₉	semidirect
73. C₇₉	#12506	semidirect
111. C₁₆	#11929611	semidirect

H ≅ G/N

type

2. #23765625

C₂

semidirect

3. #23762993

C₂

semidirect

4. #23761316

C₂

semidirect

5. C₇₉×#2254

C₂

semidirect

43. C₁₆×C₇₉

SD₁₆

semidirect

68. #2254

D₇₉

semidirect

73. C₇₉

#12506

semidirect

111. C₁₆

#11929611

semidirect

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

subgroup	index
#23765650	2
#23765625	2
#23762993	2
#23761316	2
C₇₉×#2254	2
#12506	79

subgroup

index

C₇₉×#2254

Subgroups — #34202940

Conjugacy classes of subgroups

Complements

Maximal subgroups