Group #34032852 — subgroups

Subgroups — #34032852

Order 18688.

230 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 43 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	#22626936	C₂	1	2	yes	maximal
2	#22626117	C₂	1	1	yes	maximal, characteristic
3	#17989700	—	2	2	no	—
4	#17989592	C₄	1	1	yes	characteristic
5	C₄²×#64408	C₂²	1	1	yes	characteristic
6	#17989457	—	2	2	no	—
7	#17989143	C₄	1	1	yes	characteristic
8	C₄⋊#10596842	—	2	4	no	—
9	C₂²⋊#10596842	—	4	2	no	—
10	C₄×#10596842	—	4	2	no	—
11	C₂×#11924038	—	2	1	no	—
12	C₂×C₄×#64408	—	4	1	no	—
13	C₂×C₄×#64408	D₄	1	2	yes	—
14	#14556658	—	4	1	no	—
15	#14556590	—	2	1	no	—
16	#14556583	C₈	1	1	yes	characteristic, complemented
17	#14556569	—	4	1	no	—
18	#14556556	—	4	1	no	—
19	#14556549	—	2	1	no	—
20	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
21	#125×C₇₃	C₈	1	1	yes	characteristic, complemented
22	C₂×#10596842	—	4	4	no	—
23	C₂²×#64408	—	2	2	no	—
24	#11924038	—	4	2	no	—
25	C₂×#10596833	—	4	1	no	—
26	C₄×#64408	—	2	2	no	—
27	C₄×#64408	—	4	4	no	—
28	C₄×#64408	SD₁₆	1	2	yes	—
29	C₄×#64408	Q₁₆	1	2	yes	—
30	C₂×Dic₁₄₆	—	4	1	no	—
31	C₂×C₄×D₇₃	—	2	1	no	—
32	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₄⋊Dic₇₃	—	2	1	no	—
34	#11924007	—	4	1	no	—
35	#11924007	—	4	1	no	—
36	#11924005	—	4	1	no	—
37	C₄×Dic₇₃	—	2	1	no	—
38	C₄×Dic₇₃	—	4	1	no	—
39	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	—	4	1	no	—
42	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×D₇₃	#99	1	1	yes	characteristic
44	C₂×#64408	—	2	2	no	—
45	C₂×#64408	—	2	2	no	—
46	C₂×#64408	—	4	4	no	—
47	#10596842	—	8	4	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	—	4	1	no	—
53	C₂×Dic₇₃	#100	1	1	yes	characteristic
54	C₄×D₇₃	—	4	2	no	—
55	C₄×D₇₃	#103	1	2	yes	—
56	C₄×D₇₃	#104	1	2	yes	—
57	Dic₁₄₆	—	8	2	no	—
58	#10596833	—	8	1	no	—
59	C₂²×C₇₃	#323	1	1	yes	characteristic
60	C₂×D₇₃	—	4	1	no	—
61	C₂×D₇₃	#328	1	2	yes	—
62	#64408	—	2	4	no	—
63	#64408	—	4	2	no	—
64	C₄×C₇₃	—	4	1	no	—
65	C₄×C₇₃	—	8	1	no	—
66	C₄×C₇₃	Q₈⋊C₈	1	2	yes	—
67	Dic₇₃	—	2	2	no	—
68	Dic₇₃	—	8	1	no	—
69	Dic₇₃	#329	1	1	yes	characteristic
70	Dic₇₃	#333	1	1	yes	characteristic
71	#7134	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	—	2	1	no	—
73	C₂×C₇₃	#1272	1	1	yes	characteristic
74	D₇₃	—	2	2	no	—
75	#1994	—	73	1	no	—
76	#1276	—	73	2	no	—
77	C₇₃	#7134	1	1	yes	characteristic, complemented, Sylow
78	#533	—	73	1	no	—
79	#431	—	146	2	no	—
80	#423	—	73	1	no	—
81	C₄³	—	73	1	no	—
82	#339	—	146	2	no	—
83	C₂×M₁₆	—	146	1	no	—
84	C₄⋊Q₈	—	146	1	no	—
85	#125	—	73	1	no	—
86	#125	#10596842	1	1	yes	characteristic, complemented
87	C₂²⋊Q₈	—	292	1	no	—
88	C₄×Q₈	—	292	1	no	—
89	#117	—	146	1	no	—
90	C₂×C₄²	—	292	1	no	—
91	C₂×C₄²	—	73	2	no	—
92	C₂×C₄²	—	73	1	no	—
93	C₄⋊C₈	—	292	1	no	—
94	C₄⋊C₈	—	146	4	no	—
95	C₂²⋊C₈	—	292	2	no	—
96	C₄×C₈	—	292	2	no	—
97	C₂×Q₈	—	292	1	no	—
98	C₂²×C₄	—	73	1	no	—
99	C₂²×C₄	—	146	1	no	—
100	C₂²×C₄	—	146	2	no	—
101	M₁₆	—	292	2	no	—
102	C₂×C₈	—	292	1	no	—
103	C₂×C₈	—	292	4	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	292	1	no	—
107	C₄⋊C₄	—	292	1	no	—
108	C₄⋊C₄	—	146	1	no	—
109	C₂²⋊C₄	—	292	1	no	—
110	C₄²	—	292	1	no	—
111	C₄²	—	292	4	no	—
112	C₄²	—	73	2	no	—
113	C₄²	—	73	2	no	—
114	C₄²	—	73	1	no	—
115	C₄²	—	146	1	no	—
116	C₄²	—	146	2	no	—
117	C₄²	C₂×#10596842	1	1	yes	characteristic
118	C₂³	—	73	1	no	—
119	Q₈	—	584	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	292	2	no	—
123	C₂×C₄	—	292	4	no	—
124	C₂×C₄	—	292	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	73	2	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	—	146	2	no	—
130	C₂×C₄	—	146	1	no	—
131	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
132	C₈	—	584	1	no	—
133	C₈	—	584	4	no	—
134	C₂²	—	292	1	no	—
135	C₂²	—	73	2	no	—
136	C₂²	#17989672	1	1	yes	characteristic
137	C₄	—	4	1	no	—
138	C₄	—	8	1	no	—
139	C₄	—	292	2	no	—
140	C₄	—	584	1	no	—
141	C₄	—	73	1	no	—
142	C₄	—	73	1	no	—
143	C₄	—	146	2	no	—
144	C₄	—	146	4	no	—
145	C₄	Q₈⋊#10596842	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	146	2	no	—
148	C₂	#22626932	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626936

C₂

yes

maximal

#22626117

C₂

yes

maximal, characteristic

#17989700

—

#17989592

C₄

yes

characteristic

C₄²×#64408

C₂²

yes

characteristic

#17989457

—

#17989143

C₄

yes

—

—

—

—

C₂×C₄×#64408

—

C₂×C₄×#64408

D₄

yes

—

#14556658

—

#14556590

—

#14556583

C₈

yes

characteristic, complemented

#14556569

—

#14556556

—

#14556549

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

#125×C₇₃

C₈

yes

characteristic, complemented

C₂×#10596842

—

C₂²×#64408

—

—

—

—

—

SD₁₆

yes

—

C₄×#64408

Q₁₆

yes

—

C₂×Dic₁₄₆

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

M₁₆

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

C₂²×D₇₃

yes

—

—

—

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#100

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#103

yes

—

C₄×D₇₃

#104

yes

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

#323

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#328

yes

—

#64408

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

Q₈⋊C₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1272

yes

characteristic

D₇₃

—

#1994

—

#1276

—

C₇₃

#7134

yes

characteristic, complemented, Sylow

#533

—

#431

—

146

—

#423

—

C₄³

—

#339

—

146

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

146

—

#125

—

#125

#10596842

yes

characteristic, complemented

C₂²⋊Q₈

—

292

—

C₄×Q₈

—

292

—

#117

—

146

—

C₂×C₄²

—

292

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

292

—

C₂×Q₈

—

292

—

C₂²×C₄

—

C₂²×C₄

—

146

—

100

C₂²×C₄

—

146

—

101

M₁₆

—

292

—

102

C₂×C₈

—

292

—

103

C₂×C₈

—

292

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

292

—

107

C₄⋊C₄

—

292

—

108

C₄⋊C₄

—

146

—

109

C₂²⋊C₄

—

292

—

110

C₄²

—

292

—

111

C₄²

—

292

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

146

—

116

C₄²

—

146

—

117

C₄²

C₂×#10596842

yes

characteristic

118

C₂³

—

119

Q₈

—

584

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

292

—

123

C₂×C₄

—

292

—

124

C₂×C₄

—

292

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

146

—

129

C₂×C₄

—

146

—

130

C₂×C₄

—

146

—

131

C₂×C₄

C₂²⋊#10596842

yes

characteristic

132

C₈

—

584

—

133

C₈

—

584

—

134

C₂²

—

292

—

135

C₂²

—

136

C₂²

#17989672

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

292

—

140

C₄

—

584

—

141

C₄

—

142

C₄

—

143

C₄

—

146

—

144

C₄

—

146

—

145

C₄

Q₈⋊#10596842

yes

—

146

C₂

—

147

C₂

—

146

—

148

C₂

#22626932

yes

characteristic

	subgroup	G/H
center	C₂	#22626932
commutator	C₄²×C₇₃	C₂×C₈
Frattini	C₄²	C₂×#10596842
Fitting	#125×C₇₃	C₈
socle	C₂×C₇₃	#1272
prime radical	C₂²×D₇₃	#99
cosocle	C₄²×#64408	C₂²
hypercenter	#125	#10596842
hypocenter	C₇₃	#7134
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

subgroup

G/H

center

C₂

#22626932

commutator

C₄²×C₇₃

C₂×C₈

Frattini

C₄²

C₂×#10596842

Fitting

#125×C₇₃

C₈

C₂×C₇₃

C₂²×D₇₃

C₄²×#64408

C₂²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

N	H ≅ G/N	type
16. #14556583	C₈	semidirect
21. #125×C₇₃	C₈	semidirect
77. C₇₃	#7134	semidirect
86. #125	#10596842	semidirect

H ≅ G/N

type

16. #14556583

C₈

semidirect

21. #125×C₇₃

C₈

semidirect

77. C₇₃

#7134

semidirect

86. #125

#10596842

semidirect

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

subgroup	index
#22626936	2
#22626117	2
#7134	73

subgroup

index

#22626936

#22626117

#7134

Subgroups — #34032852

Conjugacy classes of subgroups

Complements

Maximal subgroups