Group #34202923 — subgroups

Subgroups — #34202923

Order 20224.

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

normal subgroups only

This group has 306 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	#23765650	C₂	1	1	yes	maximal, characteristic
2	#23765625	C₂	1	2	yes	maximal, complemented
3	#23762993	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2252	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	#366×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	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×D₃₁₆	—	4	2	no	—
31	C₂×D₃₁₆	—	4	2	no	—
32	C₂×D₃₁₆	—	4	2	no	—
33	#11929611	—	8	8	no	—
34	D₆₃₂	—	4	4	no	—
35	D₆₃₂	—	4	4	no	—
36	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
37	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
38	C₇₉⋊C₁₆	—	2	2	no	—
39	C₂×Q₈×C₇₉	—	4	2	no	—
40	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₇₉	—	4	2	no	—
43	C₄⋊C₄×C₇₉	—	4	2	no	—
44	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₇₉	SD₁₆	1	4	yes	complemented
46	C₂²×D₇₉	—	8	2	no	—
47	Q₈×C₇₉	—	8	4	no	—
48	C₂×C₄×C₇₉	—	8	2	no	—
49	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
52	D₃₁₆	—	8	2	no	—
53	D₃₁₆	—	8	2	no	—
54	D₃₁₆	—	8	4	no	—
55	D₃₁₆	—	8	4	no	—
56	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
57	C₈×C₇₉	#135	1	2	yes	—
58	C₇₉⋊C₈	—	2	2	no	—
59	C₇₉⋊C₈	D₁₆	1	4	yes	complemented
60	C₂²×C₇₉	#493	1	1	yes	characteristic
61	C₂×D₇₉	—	16	2	no	—
62	C₂×D₇₉	—	16	2	no	—
63	C₂×D₇₉	—	16	2	no	—
64	C₄×C₇₉	—	16	2	no	—
65	C₄×C₇₉	#492	1	1	yes	characteristic
66	C₄×C₇₉	#495	1	1	yes	characteristic
67	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
68	C₄×C₇₉	#508	1	2	yes	—
69	#12503	—	79	1	no	maximal, Sylow
70	C₂×C₇₉	#1706	1	1	yes	characteristic
71	C₂×C₇₉	#2246	1	1	yes	characteristic
72	C₂×C₇₉	#2249	1	1	yes	characteristic
73	D₇₉	—	32	2	no	—
74	#2252	D₇₉	1	1	yes	characteristic, complemented
75	#2246	—	79	1	no	—
76	#2240	—	79	2	no	—
77	#1706	—	79	2	no	—
78	C₈×C₁₆	—	79	1	no	—
79	C₇₉	#12503	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₁₆	—	158	2	no	—
81	#500	C₂×D₇₉	1	2	yes	—
82	#493	—	79	2	no	—
83	#492	—	158	2	no	—
84	#486	—	158	2	no	—
85	#366	—	2	2	no	—
86	#357	—	158	4	no	—
87	C₄×C₁₆	—	79	1	no	—
88	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
89	C₈²	—	79	1	no	—
90	C₂×SD₁₆	—	316	4	no	—
91	C₂×D₈	—	158	2	no	—
92	C₂×D₈	—	158	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	#127	—	158	2	no	—
95	D₁₆	—	316	4	no	—
96	C₂×C₁₆	—	79	2	no	—
97	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	#102	—	316	4	no	—
101	C₄×C₈	—	79	1	no	—
102	C₄×C₈	—	79	1	no	—
103	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×D₄	—	316	2	no	—
106	C₂×D₄	—	316	2	no	—
107	C₂×D₄	—	316	2	no	—
108	SD₁₆	—	632	8	no	—
109	D₈	—	316	4	no	—
110	D₈	—	316	4	no	—
111	C₂×C₈	—	79	2	no	—
112	C₂×C₈	—	79	2	no	—
113	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
114	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄²	D₄×D₇₉	1	1	yes	characteristic
118	C₁₆	—	158	2	no	—
119	C₁₆	#11929611	1	4	yes	complemented
120	C₂³	—	632	2	no	—
121	Q₈	—	8	4	no	—
122	D₄	—	632	2	no	—
123	D₄	—	632	2	no	—
124	D₄	—	632	4	no	—
125	D₄	—	632	4	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	#14579157	1	1	yes	characteristic
128	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
129	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
130	C₈	—	79	4	no	—
131	C₈	—	158	2	no	—
132	C₈	C₂×#11929611	1	2	yes	—
133	C₈	#14579184	1	2	yes	—
134	C₂²	—	1264	2	no	—
135	C₂²	—	1264	2	no	—
136	C₂²	—	1264	2	no	—
137	C₂²	#19098938	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	#19098937	1	1	yes	characteristic
140	C₄	#19098948	1	1	yes	characteristic
141	C₄	D₁₆×D₇₉	1	2	yes	—
142	C₄	#19098996	1	2	yes	—
143	C₂	—	2528	2	no	—
144	C₂	#23762993	1	1	yes	characteristic
145	C₂	#23765648	1	1	yes	characteristic
146	C₂	#23765662	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765650

C₂

yes

maximal, characteristic

#23765625

C₂

yes

maximal, complemented

#23762993

C₂

yes

maximal, complemented

C₁₆×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₇₉×#2252

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

—

#366×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

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×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929611

—

D₆₃₂

—

D₆₃₂

—

C₂×C₇₉⋊C₈

D₈

yes

—

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

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#135

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

D₁₆

yes

complemented

C₂²×C₇₉

#493

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×D₁₆

yes

—

C₄×C₇₉

#508

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#2252

D₇₉

yes

characteristic, complemented

#2246

—

#2240

—

#1706

—

C₈×C₁₆

—

C₇₉

#12503

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

158

—

#500

C₂×D₇₉

yes

—

#493

—

#492

—

158

—

#486

—

158

—

#366

—

#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

—

100

#102

—

316

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₇₉

yes

characteristic

104

C₂×Q₈

—

105

C₂×D₄

—

316

—

106

C₂×D₄

—

316

—

107

C₂×D₄

—

316

—

108

SD₁₆

—

632

—

109

D₈

—

316

—

110

D₈

—

316

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₇₉

yes

characteristic

114

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

D₄×D₇₉

yes

characteristic

118

C₁₆

—

158

—

119

C₁₆

#11929611

yes

complemented

120

C₂³

—

632

—

121

Q₈

—

122

D₄

—

632

—

123

D₄

—

632

—

124

D₄

—

632

—

125

D₄

—

632

—

126

C₂×C₄

—

127

C₂×C₄

#14579157

yes

characteristic

128

C₂×C₄

D₈×D₇₉

yes

characteristic

129

C₂×C₄

D₈×D₇₉

yes

characteristic

130

C₈

—

131

C₈

—

158

—

132

C₈

C₂×#11929611

yes

—

133

C₈

#14579184

yes

—

134

C₂²

—

1264

—

135

C₂²

—

1264

—

136

C₂²

—

1264

—

137

C₂²

#19098938

yes

characteristic

138

C₄

—

139

C₄

#19098937

yes

characteristic

140

C₄

#19098948

yes

characteristic

141

C₄

D₁₆×D₇₉

yes

—

142

C₄

#19098996

yes

—

143

C₂

—

2528

—

144

C₂

#23762993

yes

characteristic

145

C₂

#23765648

yes

characteristic

146

C₂

#23765662

yes

characteristic

	subgroup	G/H
center	C₂²	#19098938
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2252	C₂
socle	C₂²×C₇₉	#493
prime radical	#23765650	C₂
hypercenter	#2252	D₇₉
hypocenter	C₇₉	#12503
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

C₂

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
2. #23765625	C₂	semidirect
3. #23762993	C₂	semidirect
4. C₁₆×C₇₉⋊C₈	C₂	semidirect
5. C₇₉×#2252	C₂	semidirect
45. C₁₆×C₇₉	SD₁₆	semidirect
59. C₇₉⋊C₈	D₁₆	semidirect
74. #2252	D₇₉	semidirect
79. C₇₉	#12503	semidirect
119. C₁₆	#11929611	semidirect

H ≅ G/N

type

2. #23765625

C₂

semidirect

3. #23762993

C₂

semidirect

4. C₁₆×C₇₉⋊C₈

C₂

semidirect

5. C₇₉×#2252

C₂

semidirect

45. C₁₆×C₇₉

SD₁₆

semidirect

59. C₇₉⋊C₈

D₁₆

semidirect

74. #2252

D₇₉

semidirect

79. C₇₉

#12503

semidirect

119. 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
C₁₆×C₇₉⋊C₈	2
C₇₉×#2252	2
#12503	79

subgroup

index

#23765650

#23765625

#23762993

C₁₆×C₇₉⋊C₈

C₇₉×#2252

#12503

Subgroups — #34202923

Conjugacy classes of subgroups

Complements

Maximal subgroups