Group #34012043 — subgroups

Subgroups — #34012043

Order 18688.

210 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 68 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	#22625571	C₂	1	2	yes	maximal
2	#22624962	C₂	1	1	yes	maximal, characteristic
3	#22624887	C₂	1	2	yes	maximal
4	C₁₆×#10596833	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989638	—	2	2	no	—
7	#17989473	C₂²	1	1	yes	characteristic
8	#17989450	C₂²	1	2	yes	—
9	Q₈⋊#10596833	—	2	2	no	—
10	C₈×#10596833	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
12	#17988413	C₄	1	1	yes	characteristic
13	#17988391	C₄	1	2	yes	—
14	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924031	—	2	1	no	—
17	#14556658	—	4	2	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	#14556646	—	4	2	no	—
20	C₄×#10596833	D₄	1	1	yes	characteristic
21	C₄×#10596833	D₄	1	1	yes	characteristic
22	Q₈×Dic₇₃	—	2	2	no	—
23	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
24	#14556525	C₂×C₄	1	2	yes	—
25	#14556515	—	2	2	no	—
26	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
28	Q₃₂×C₇₃	—	2	2	no	—
29	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#10596833	—	2	1	no	—
31	C₂×#10596833	—	4	2	no	—
32	C₂×#10596833	D₈	1	1	yes	characteristic
33	C₂×#10596833	D₈	1	1	yes	characteristic
34	#11924031	—	2	2	no	—
35	C₂×#10596831	#41	1	1	yes	characteristic
36	C₄⋊Dic₇₃	—	2	2	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	C₄×Dic₇₃	—	4	2	no	—
39	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
40	#11924003	—	2	1	no	—
41	C₂×Q₈×C₇₃	—	2	2	no	—
42	Q₁₆×C₇₃	—	2	2	no	—
43	Q₁₆×C₇₃	—	2	2	no	—
44	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₇₃	M₁₆	1	2	yes	—
46	Q₈×C₇₃	—	4	2	no	—
47	Q₈×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
50	C₂×Dic₇₃	—	4	2	no	—
51	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂×Dic₇₃	#135	1	1	yes	characteristic
53	#10596833	—	2	2	no	—
54	#10596833	—	2	1	no	—
55	#10596833	—	8	2	no	—
56	#10596833	Q₃₂	1	2	yes	complemented
57	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₇₃	#131	1	1	yes	characteristic
59	#10596831	—	2	1	no	—
60	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
61	C₄×C₇₃	—	8	2	no	—
62	C₄×C₇₃	#436	1	1	yes	characteristic
63	C₄×C₇₃	#443	1	1	yes	characteristic
64	Dic₇₃	—	2	1	no	—
65	Dic₇₃	—	8	2	no	—
66	Dic₇₃	C₂×Q₃₂	1	1	yes	characteristic
67	Dic₇₃	#508	1	1	yes	characteristic
68	#12083	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	#1589	1	1	yes	characteristic
70	C₂×C₇₃	C₄×Q₃₂	1	1	yes	characteristic
71	C₂×C₇₃	#2178	1	1	yes	characteristic
72	C₄×Q₃₂	—	73	1	no	—
73	#1591	—	73	2	no	—
74	#1371	—	73	1	no	—
75	#1332	—	73	2	no	—
76	C₈×C₁₆	—	73	1	no	—
77	C₇₃	#12083	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₃₂	#64408	1	1	yes	characteristic
79	#446	—	146	2	no	—
80	C₄×Q₁₆	—	73	2	no	—
81	#366	—	73	1	no	—
82	#358	—	73	2	no	—
83	C₄×C₁₆	—	73	1	no	—
84	C₄×C₁₆	—	73	1	no	—
85	#335	—	73	2	no	—
86	Q₈⋊C₈	—	146	2	no	—
87	C₈²	—	73	1	no	—
88	C₂×Q₁₆	C₂×#64408	1	2	yes	—
89	C₄×Q₈	—	146	2	no	—
90	Q₃₂	—	2	2	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	146	1	no	—
93	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
94	#107	—	73	2	no	—
95	C₄⋊C₈	—	292	2	no	—
96	C₄⋊C₈	—	146	2	no	—
97	#103	—	146	2	no	—
98	#97	—	292	2	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	—	73	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	Q₁₆	—	2	2	no	—
104	Q₁₆	—	2	2	no	—
105	C₂×C₈	—	292	2	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	146	1	no	—
110	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
111	C₄⋊C₄	—	292	2	no	—
112	C₄⋊C₄	—	146	2	no	—
113	C₄²	—	292	2	no	—
114	C₄²	—	73	1	no	—
115	C₁₆	—	146	2	no	—
116	C₁₆	—	146	1	no	—
117	C₁₆	#11924037	1	2	yes	—
118	Q₈	—	4	2	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	292	2	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	D₄×#64408	1	1	yes	characteristic
125	C₈	—	584	2	no	—
126	C₈	—	73	2	no	—
127	C₈	—	146	1	no	—
128	C₈	—	146	2	no	—
129	C₈	—	146	1	no	—
130	C₈	C₂×#11924037	1	1	yes	characteristic
131	C₈	#14556678	1	1	yes	characteristic
132	C₂²	D₈×#64408	1	1	yes	characteristic
133	C₄	—	8	2	no	—
134	C₄	—	584	2	no	—
135	C₄	—	73	1	no	—
136	C₄	—	73	1	no	—
137	C₄	—	146	1	no	—
138	C₄	#17989622	1	1	yes	characteristic
139	C₄	#17989629	1	1	yes	characteristic
140	C₂	#22625564	1	1	yes	characteristic
141	C₂	Q₃₂×#64408	1	1	yes	characteristic
142	C₂	#22626332	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625571

C₂

yes

maximal

#22624962

C₂

yes

maximal, characteristic

#22624887

C₂

yes

maximal

C₁₆×#10596833

C₂

yes

maximal, characteristic

Q₃₂×Dic₇₃

C₂

yes

maximal, characteristic

#17989638

—

#17989473

C₂²

yes

characteristic

#17989450

C₂²

yes

—

Q₈⋊#10596833

—

C₈×#10596833

C₂²

yes

characteristic

Q₁₆×Dic₇₃

C₂²

yes

—

#17988413

C₄

yes

characteristic

#17988391

C₄

yes

—

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

Q₈×Dic₇₃

—

C₂×#11924003

C₂×C₄

yes

characteristic

#14556525

C₂×C₄

yes

—

#14556515

—

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×Q₁₆×C₇₃

C₂×C₄

yes

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

—

D₈

yes

characteristic

C₂×#10596833

D₈

yes

—

yes

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

M₁₆

yes

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₈

yes

C₂×Dic₇₃

yes

—

—

—

Q₃₂

yes

complemented

C₈×C₇₃

C₂×M₁₆

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂×Q₃₂

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

C₄×Q₃₂

yes

characteristic

C₂×C₇₃

#2178

yes

characteristic

C₄×Q₃₂

—

#1591

—

#1371

—

#1332

—

C₈×C₁₆

—

C₇₃

#12083

yes

characteristic, complemented, Sylow

C₂×Q₃₂

#64408

yes

characteristic

#446

—

146

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

146

—

C₈²

—

C₂×Q₁₆

C₂×#64408

yes

—

C₄×Q₈

—

146

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×#64408

yes

characteristic

#107

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

#103

—

146

—

#97

—

292

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

Q₁₆

—

104

Q₁₆

—

105

C₂×C₈

—

292

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

146

—

110

C₂×C₈

C₂²×#64408

yes

characteristic

111

C₄⋊C₄

—

292

—

112

C₄⋊C₄

—

146

—

113

C₄²

—

292

—

114

C₄²

—

115

C₁₆

—

146

—

116

C₁₆

—

146

—

117

C₁₆

#11924037

yes

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

292

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

D₄×#64408

yes

characteristic

125

C₈

—

584

—

126

C₈

—

127

C₈

—

146

—

128

C₈

—

146

—

129

C₈

—

146

—

130

C₈

C₂×#11924037

yes

characteristic

131

C₈

#14556678

yes

characteristic

132

C₂²

D₈×#64408

yes

characteristic

133

C₄

—

134

C₄

—

584

—

135

C₄

—

136

C₄

—

137

C₄

—

146

—

138

C₄

#17989622

yes

characteristic

139

C₄

#17989629

yes

characteristic

140

C₂

#22625564

yes

characteristic

141

C₂

Q₃₂×#64408

yes

characteristic

142

C₂

#22626332

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#64408
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×Q₃₂×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₄×D₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂×Q₃₂	#64408
hypocenter	C₇₃	#12083
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×#64408

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₂×Q₃₂×C₇₃

C₄

socle, prime radical

C₂²×C₇₃

C₄×D₈

cosocle

C₈×Dic₇₃

C₂³

C₂×Q₃₂

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
56. #10596833	Q₃₂	semidirect
77. C₇₃	#12083	semidirect

H ≅ G/N

type

56. #10596833

Q₃₂

semidirect

77. C₇₃

#12083

semidirect

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

subgroup	index
#22625571	2
#22624962	2
#22624887	2
C₁₆×#10596833	2
Q₃₂×Dic₇₃	2
#12083	73

subgroup

index

#22625571

#22624962

#22624887

C₁₆×#10596833

Q₃₂×Dic₇₃

#12083

Subgroups — #34012043

Conjugacy classes of subgroups

Complements

Maximal subgroups