Group #34011803 — subgroups

Subgroups — #34011803

Order 18688.

242 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 100 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	#22625489	C₂	1	2	yes	maximal
2	#22624963	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989591	—	2	2	no	—
5	#17989450	—	2	2	no	—
6	#17989450	C₂²	1	4	yes	—
7	C₂×C₁₆×D₇₃	C₄	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₄×#11924003	C₄	1	1	yes	characteristic
10	C₁₆×Dic₇₃	C₂²	1	2	yes	—
11	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924038	—	4	2	no	—
13	C₄⋊#10596833	—	2	4	no	—
14	C₄⋊#10596833	—	4	2	no	—
15	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
17	C₄²×D₇₃	D₄	1	1	yes	characteristic
18	C₂×#11924003	C₂×C₄	1	2	yes	—
19	C₁₆×D₇₃	—	2	4	no	—
20	C₄×#10596831	Q₈	1	1	yes	characteristic
21	C₈×Dic₇₃	D₄	1	1	yes	characteristic
22	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	#11924038	—	8	4	no	—
26	C₂×#10596833	—	4	4	no	—
27	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
29	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
30	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
31	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
32	C₈×D₇₃	—	2	2	no	—
33	C₈×D₇₃	—	2	2	no	—
34	C₄×Dic₇₃	D₈	1	1	yes	characteristic
35	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
36	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
37	#11924003	M₁₆	1	4	yes	complemented
38	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₁₆×C₇₃	—	2	2	no	—
42	C₂²×D₇₃	#107	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₇₃	#107	1	1	yes	characteristic
47	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×D₇₃	—	2	2	no	—
50	C₄×D₇₃	—	2	1	no	—
51	C₄×D₇₃	—	2	1	no	—
52	C₄×D₇₃	D₁₆	1	2	yes	—
53	C₄×D₇₃	Q₃₂	1	2	yes	—
54	#10596833	—	8	4	no	—
55	C₈×C₇₃	#108	1	2	yes	—
56	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
57	#10596831	—	2	1	no	—
58	#10596831	—	2	1	no	—
59	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
60	C₂×D₇₃	—	2	1	no	—
61	C₂×D₇₃	—	2	1	no	—
62	C₂×D₇₃	#366	1	2	yes	—
63	C₄×C₇₃	#366	1	2	yes	—
64	C₄×C₇₃	#368	1	2	yes	—
65	C₄×C₇₃	#423	1	1	yes	characteristic
66	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	—	2	1	no	—
69	Dic₇₃	C₂×D₁₆	1	1	yes	characteristic
70	Dic₇₃	C₂×Q₃₂	1	1	yes	characteristic
71	#12022	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	#1567	1	1	yes	characteristic
73	C₂×C₇₃	C₂×#366	1	1	yes	characteristic
74	C₂×C₇₃	C₂×#368	1	1	yes	characteristic
75	D₇₃	—	2	2	no	—
76	C₂×C₄×C₁₆	—	73	1	no	—
77	#1567	—	73	2	no	—
78	#1371	—	73	4	no	—
79	C₇₃	#12022	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₁₆	—	73	2	no	—
81	#423	—	146	2	no	—
82	C₂×C₄×C₈	—	73	1	no	—
83	C₄×C₁₆	—	73	2	no	—
84	C₄×C₁₆	—	73	1	no	—
85	C₄×C₁₆	#64408	1	1	yes	characteristic
86	#335	—	73	4	no	—
87	#335	—	146	2	no	—
88	C₂×M₁₆	—	292	2	no	—
89	C₂²×C₈	—	73	1	no	—
90	C₂²×C₈	—	73	1	no	—
91	C₂×C₄²	—	73	1	no	—
92	C₂×C₁₆	—	73	2	no	—
93	C₂×C₁₆	—	146	4	no	—
94	C₂×C₁₆	C₂×#64408	1	2	yes	—
95	C₄⋊C₈	—	292	2	no	—
96	C₄⋊C₈	—	146	4	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	C₂×#64408	1	1	yes	characteristic
101	C₂²×C₄	—	73	1	no	—
102	C₂²×C₄	—	73	1	no	—
103	C₂²×C₄	—	73	1	no	—
104	M₁₆	—	584	4	no	—
105	C₂×C₈	—	292	4	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	146	2	no	—
109	C₂×C₈	—	146	2	no	—
110	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
111	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
112	C₄²	—	73	1	no	—
113	C₄²	—	73	1	no	—
114	C₄²	—	73	1	no	—
115	C₄²	C₄⋊#64408	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₁₆	—	73	4	no	—
118	C₂³	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	2	no	—
121	C₂×C₄	—	73	2	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	—	146	2	no	—
125	C₂×C₄	—	146	1	no	—
126	C₂×C₄	—	146	1	no	—
127	C₂×C₄	#14556660	1	1	yes	characteristic
128	C₂×C₄	#14556660	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
130	C₈	—	584	4	no	—
131	C₈	—	146	1	no	—
132	C₈	—	146	1	no	—
133	C₈	#14556661	1	2	yes	—
134	C₈	D₇₃⋊M₁₆	1	2	yes	—
135	C₂²	—	73	2	no	—
136	C₂²	—	146	1	no	—
137	C₂²	—	146	1	no	—
138	C₂²	C₂×#14556660	1	1	yes	characteristic
139	C₄	—	73	1	no	—
140	C₄	—	73	1	no	—
141	C₄	—	146	1	no	—
142	C₄	—	146	1	no	—
143	C₄	#17989513	1	2	yes	—
144	C₄	#17989516	1	2	yes	—
145	C₄	#17989592	1	1	yes	characteristic
146	C₄	C₂×#14556661	1	1	yes	characteristic
147	C₂	—	146	2	no	—
148	C₂	#22625490	1	1	yes	characteristic
149	C₂	C₂×#17989513	1	1	yes	characteristic
150	C₂	C₂×#17989516	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625489

C₂

yes

maximal

#22624963

C₂

yes

maximal, complemented

C₄×C₁₆×D₇₃

C₂

yes

maximal, characteristic

#17989591

—

#17989450

—

#17989450

C₂²

yes

—

C₂×C₁₆×D₇₃

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×#11924003

C₄

yes

characteristic

C₁₆×Dic₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₄

yes

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

C₁₆×D₇₃

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924038

—

C₂×#10596833

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

Q₁₆

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

M₁₆

yes

complemented

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#107

yes

characteristic

C₂×Dic₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

C₂×Q₁₆

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

D₁₆

yes

—

C₄×D₇₃

Q₃₂

yes

—

#10596833

—

C₈×C₇₃

#108

yes

—

C₈×C₇₃

C₂×M₁₆

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#366

yes

—

C₄×C₇₃

#366

yes

—

C₄×C₇₃

#368

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂×D₁₆

yes

characteristic

Dic₇₃

C₂×Q₃₂

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

C₂×C₄×C₁₆

—

#1567

—

#1371

—

C₇₃

#12022

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#423

—

146

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

146

—

C₂×M₁₆

—

292

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×#64408

yes

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×#64408

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

M₁₆

—

584

—

105

C₂×C₈

—

292

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₂×C₈

C₄⋊#64408

yes

characteristic

111

C₂×C₈

C₂²×#64408

yes

characteristic

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₄⋊#64408

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₂³

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

146

—

125

C₂×C₄

—

146

—

126

C₂×C₄

—

146

—

127

C₂×C₄

#14556660

yes

characteristic

128

C₂×C₄

#14556660

yes

characteristic

129

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

130

C₈

—

584

—

131

C₈

—

146

—

132

C₈

—

146

—

133

C₈

#14556661

yes

—

134

C₈

D₇₃⋊M₁₆

yes

—

135

C₂²

—

136

C₂²

—

146

—

137

C₂²

—

146

—

138

C₂²

C₂×#14556660

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

146

—

142

C₄

—

146

—

143

C₄

#17989513

yes

—

144

C₄

#17989516

yes

—

145

C₄

#17989592

yes

characteristic

146

C₄

C₂×#14556661

yes

characteristic

147

C₂

—

146

—

148

C₂

#22625490

yes

characteristic

149

C₂

C₂×#17989513

yes

characteristic

150

C₂

C₂×#17989516

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556660
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂²×D₇₃	#107
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#12022
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#14556660

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₄×C₁₆×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₈×Dic₇₃

C₂³

C₄×C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #22624963	C₂	semidirect
37. #11924003	M₁₆	semidirect
79. C₇₃	#12022	semidirect

H ≅ G/N

type

2. #22624963

C₂

semidirect

37. #11924003

M₁₆

semidirect

79. C₇₃

#12022

semidirect

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

subgroup	index
#22625489	2
#22624963	2
C₄×C₁₆×D₇₃	2
#12022	73

subgroup

index

#22625489

#22624963

C₄×C₁₆×D₇₃

#12022

Subgroups — #34011803

Conjugacy classes of subgroups

Complements

Maximal subgroups