Group #34032826 — subgroups

Subgroups — #34032826

Order 18688.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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	#22627003	C₂	1	2	yes	maximal
2	#22626062	C₂	1	1	yes	maximal, characteristic
3	#17989706	—	2	2	no	—
4	C₂×#14556654	—	2	2	no	—
5	#17989546	C₂²	1	1	yes	characteristic
6	#17989531	C₄	1	1	yes	characteristic
7	C₄⋊Q₈×D₇₃	C₄	1	1	yes	characteristic
8	#14556688	—	4	2	no	—
9	#14556687	—	4	2	no	—
10	#14556686	—	2	4	no	—
11	C₂×#11924036	D₄	1	2	yes	—
12	C₂×C₄⋊#64408	—	2	1	no	—
13	#14556654	—	4	2	no	—
14	#14556654	—	4	2	no	—
15	#14556646	—	4	1	no	—
16	(C₂×#64408)⋊C₄	—	4	1	no	—
17	C₂×Q₈×D₇₃	—	2	1	no	—
18	#14556591	C₈	1	1	yes	characteristic
19	#14556590	—	4	1	no	—
20	#14556589	—	2	1	no	—
21	C₄⋊C₄×D₇₃	—	4	1	no	—
22	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
24	#11924040	—	4	4	no	—
25	C₂²×#64408	—	4	1	no	—
26	#11924036	—	2	2	no	—
27	#11924036	D₈	1	2	yes	—
28	#11924036	SD₁₆	1	2	yes	—
29	C₂×#10596833	—	2	2	no	—
30	C₄⋊#64408	—	4	2	no	—
31	C₂×Dic₁₄₆	—	2	1	no	—
32	C₂×Dic₁₄₆	—	4	1	no	—
33	Q₈×D₇₃	—	4	2	no	—
34	Q₈×D₇₃	—	4	2	no	—
35	C₂×C₄×D₇₃	—	2	1	no	—
36	C₂×C₄×D₇₃	—	4	1	no	—
37	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄⋊Dic₇₃	—	4	1	no	—
39	#11924007	—	4	2	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
42	C₂×Q₈×C₇₃	—	2	1	no	—
43	C₄⋊C₄×C₇₃	—	4	1	no	—
44	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₇₃	#100	1	1	yes	characteristic
46	C₂×#64408	—	8	2	no	—
47	C₂×#64408	—	8	1	no	—
48	Q₈×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×Dic₇₃	—	2	1	no	—
53	C₂×Dic₇₃	—	4	1	no	—
54	C₂×Dic₇₃	#99	1	1	yes	characteristic
55	C₄×D₇₃	—	4	2	no	—
56	C₄×D₇₃	—	8	1	no	—
57	C₄×D₇₃	—	8	1	no	—
58	C₄×D₇₃	#102	1	2	yes	—
59	C₄×D₇₃	#104	1	2	yes	—
60	Dic₁₄₆	—	4	2	no	—
61	Dic₁₄₆	—	8	2	no	—
62	#10596833	—	2	2	no	—
63	#10596833	—	2	2	no	—
64	C₂²×C₇₃	#323	1	1	yes	characteristic
65	C₂×D₇₃	—	2	2	no	—
66	C₂×D₇₃	#329	1	1	yes	characteristic
67	C₂×D₇₃	#331	1	1	yes	characteristic
68	#64408	—	16	1	no	—
69	C₄×C₇₃	—	4	1	no	—
70	C₄×C₇₃	—	8	1	no	—
71	C₄×C₇₃	D₄⋊C₈	1	2	yes	—
72	Dic₇₃	—	4	1	no	—
73	Dic₇₃	—	8	1	no	—
74	Dic₇₃	#327	1	2	yes	—
75	#7117	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	2	1	no	—
77	C₂×C₇₃	#1270	1	1	yes	characteristic
78	D₇₃	—	2	1	no	—
79	D₇₃	#1356	1	2	yes	—
80	#1960	—	73	1	no	—
81	#1375	—	73	2	no	—
82	C₇₃	#7117	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊Q₈	—	73	1	no	—
84	C₂×M₃₂	—	146	2	no	—
85	C₂×#103	—	146	2	no	—
86	C₂×#97	—	73	1	no	—
87	#382	—	73	1	no	—
88	C₂²×Q₈	—	146	1	no	—
89	C₂²×C₈	—	73	2	no	—
90	C₄⋊Q₈	—	292	1	no	—
91	C₄⋊Q₈	—	73	1	no	—
92	C₄⋊Q₈	—	146	1	no	—
93	C₄⋊Q₈	#10596842	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	292	1	no	—
95	C₂×C₄⋊C₄	—	146	1	no	—
96	C₂×C₄²	—	73	1	no	—
97	M₃₂	—	292	2	no	—
98	M₃₂	—	292	2	no	—
99	C₂×C₁₆	—	146	4	no	—
100	#103	—	292	2	no	—
101	#103	—	292	2	no	—
102	#97	—	292	1	no	—
103	#95	—	292	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	292	1	no	—
106	C₂×Q₈	—	292	2	no	—
107	C₂×Q₈	—	292	2	no	—
108	C₂×Q₈	—	146	1	no	—
109	C₂²×C₄	—	292	1	no	—
110	C₂²×C₄	—	292	1	no	—
111	C₂²×C₄	—	73	1	no	—
112	C₂²×C₄	—	146	1	no	—
113	C₂×C₈	—	73	2	no	—
114	C₂×C₈	—	73	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₂×C₈	—	146	2	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	292	1	no	—
119	C₄⋊C₄	—	292	2	no	—
120	C₄⋊C₄	—	292	2	no	—
121	C₄²	—	73	1	no	—
122	C₄²	—	146	1	no	—
123	C₄²	C₂×#10596842	1	1	yes	characteristic
124	C₁₆	—	292	4	no	—
125	C₂³	—	73	1	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	292	2	no	—
128	Q₈	—	584	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	292	2	no	—
132	C₂×C₄	—	292	1	no	—
133	C₂×C₄	—	584	1	no	—
134	C₂×C₄	—	584	1	no	—
135	C₂×C₄	—	584	1	no	—
136	C₂×C₄	—	584	2	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	73	2	no	—
139	C₂×C₄	—	73	2	no	—
140	C₂×C₄	—	146	1	no	—
141	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
142	C₈	—	146	2	no	—
143	C₈	—	146	2	no	—
144	C₂²	—	73	1	no	—
145	C₂²	—	73	1	no	—
146	C₂²	—	146	2	no	—
147	C₂²	#17989672	1	1	yes	characteristic
148	C₄	—	4	1	no	—
149	C₄	—	8	1	no	—
150	C₄	—	292	1	no	—
151	C₄	—	584	1	no	—
152	C₄	—	73	2	no	—
153	C₄	—	1168	1	no	—
154	C₄	D₄⋊#10596842	1	2	yes	—
155	C₂	—	2	1	no	—
156	C₂	—	73	2	no	—
157	C₂	—	146	1	no	—
158	C₂	#22626930	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627003

C₂

yes

maximal

#22626062

C₂

yes

maximal, characteristic

#17989706

—

C₂×#14556654

—

#17989546

C₂²

yes

characteristic

#17989531

C₄

yes

characteristic

C₄⋊Q₈×D₇₃

C₄

yes

—

—

—

D₄

yes

—

C₂×C₄⋊#64408

—

—

—

—

—

C₂×Q₈×D₇₃

—

#14556591

C₈

yes

characteristic

#14556590

—

#14556589

—

C₄⋊C₄×D₇₃

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₇₃

C₈

yes

characteristic

#11924040

—

C₂²×#64408

—

#11924036

—

#11924036

D₈

yes

—

#11924036

SD₁₆

yes

—

C₂×#10596833

—

C₄⋊#64408

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂×Q₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

C₂²×D₇₃

yes

—

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#99

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

#102

yes

—

C₄×D₇₃

#104

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

#323

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

D₄⋊C₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#327

yes

—

#7117

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1270

yes

characteristic

D₇₃

—

D₇₃

#1356

yes

—

#1960

—

#1375

—

C₇₃

#7117

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

146

—

C₂×#103

—

146

—

C₂×#97

—

#382

—

C₂²×Q₈

—

146

—

C₂²×C₈

—

C₄⋊Q₈

—

292

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

#10596842

yes

characteristic

C₂×C₄⋊C₄

—

292

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

M₃₂

—

292

—

M₃₂

—

292

—

C₂×C₁₆

—

146

—

100

#103

—

292

—

101

#103

—

292

—

102

#97

—

292

—

103

#95

—

292

—

104

C₂×Q₈

—

105

C₂×Q₈

—

292

—

106

C₂×Q₈

—

292

—

107

C₂×Q₈

—

292

—

108

C₂×Q₈

—

146

—

109

C₂²×C₄

—

292

—

110

C₂²×C₄

—

292

—

111

C₂²×C₄

—

112

C₂²×C₄

—

146

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

146

—

116

C₂×C₈

—

146

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

292

—

119

C₄⋊C₄

—

292

—

120

C₄⋊C₄

—

292

—

121

C₄²

—

122

C₄²

—

146

—

123

C₄²

C₂×#10596842

yes

characteristic

124

C₁₆

—

292

—

125

C₂³

—

126

Q₈

—

127

Q₈

—

292

—

128

Q₈

—

584

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

292

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

584

—

134

C₂×C₄

—

584

—

135

C₂×C₄

—

584

—

136

C₂×C₄

—

584

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

146

—

141

C₂×C₄

C₂²⋊#10596842

yes

characteristic

142

C₈

—

146

—

143

C₈

—

146

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

146

—

147

C₂²

#17989672

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

292

—

151

C₄

—

584

—

152

C₄

—

153

C₄

—

1168

—

154

C₄

D₄⋊#10596842

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

146

—

158

C₂

#22626930

yes

characteristic

	subgroup	G/H
center	C₂	#22626930
commutator	C₄²×C₇₃	C₂×C₈
Frattini	C₄²	C₂×#10596842
Fitting	C₄⋊Q₈×C₇₃	C₈
socle	C₂×C₇₃	#1270
prime radical	C₂²×D₇₃	#100
cosocle	#17989546	C₂²
hypercenter	C₄⋊Q₈	#10596842
hypocenter	C₇₃	#7117
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂

#22626930

commutator

C₄²×C₇₃

C₂×C₈

Frattini

C₄²

C₂×#10596842

Fitting

C₄⋊Q₈×C₇₃

C₈

C₂×C₇₃

C₂²×D₇₃

C₂²

C₄⋊Q₈

C₇₃

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
82. C₇₃	#7117	semidirect

H ≅ G/N

type

82. C₇₃

#7117

semidirect

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

subgroup	index
#22627003	2
#22626062	2
#7117	73

subgroup

index

#22627003

#22626062

#7117

Subgroups — #34032826

Conjugacy classes of subgroups

Complements

Maximal subgroups