Finite Group Databasebeta

← #37535373

Subgroups — #37535373

Order 23168.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041851C211yesmaximal, characteristic
2#26041260C211yesmaximal, characteristic
3#26041251C211yesmaximal, characteristic
4#26041070C211yesmaximal, characteristic, complemented
5#26041069C211yesmaximal, characteristic, complemented
6#26041068C211yesmaximal, characteristic, complemented
7#352×C181C211yesmaximal, characteristic, complemented
8C2×#15668448C2211yescharacteristic
9C22×Dic362C411yescharacteristic, complemented
10M16×D181—21no—
11#19153815—21no—
12#19153802—21no—
13#19153800C411yescharacteristic, complemented
14#19153799C411yescharacteristic, complemented
15#19153797C2211yescharacteristic
16#19153794—21no—
17#19153793—21no—
18#19153772—21no—
19#19153772C411yescharacteristic, complemented
20#19153734—21no—
21#19153733C2211yescharacteristic
22#19153731C2211yescharacteristic
23#19153730—21no—
24#123×C181C2211yescharacteristic
25#100×C181C2211yescharacteristic
26#99×C181C2211yescharacteristic
27#15668448—41no—
28#15668448—41no—
29C2×Dic362—21no—
30C2×Dic362—21no—
31C2×Dic362—41no—
32C2×Dic362—41no—
33C2×Dic362D411yescharacteristic
34C2×C22⋊D181C2×C411yescharacteristic
35C2×C22⋊D181D411yescharacteristic
36C22×Dic181C2×C411yescharacteristic
37C22×Dic181C2×C411yescharacteristic
38C22×Dic181D411yescharacteristic
39C2×C4×D181D411yescharacteristic
40#15668433—21no—
41#15668432—41no—
42C8×D181—41no—
43C4⋊Dic181—21no—
44#15668428—21no—
45#15668428—21no—
46#15668428—21no—
47#15668428—41no—
48C22⋊Dic181—21no—
49C22⋊Dic181—21no—
50C22⋊Dic181C2×C411yescharacteristic
51#15668426—21no—
52#15668426—21no—
53C4×Dic181—21no—
54C4×Dic181—41no—
55C2×D4×C181C2311yescharacteristic
56C22×C4×C181C2×C411yescharacteristic
57M16×C181—21no—
58C4⋊C4×C181—21no—
59C22⋊C4×C181—21no—
60C22⋊C4×C181—21no—
61C22⋊C4×C181C2×C411yescharacteristic
62C22×D181C22⋊C411yescharacteristic
63Dic362—41no—
64Dic362—42no—
65Dic362—41no—
66Dic362—41no—
67C22⋊D181—41no—
68C22⋊D181—41no—
69C2×Dic181—21no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—21no—
74C2×Dic181—21no—
75C2×Dic181—41no—
76C2×Dic181—41no—
77C2×Dic181C22⋊C411yescharacteristic
78C2×Dic181C22⋊C411yescharacteristic
79C2×Dic181C22⋊C411yescharacteristic
80C4×D181—21no—
81C4×D181—21no—
82#13089678—41no—
83C23×C181C22×C411yescharacteristic
84C23×C181C2×D411yescharacteristic
85D4×C181—41no—
86C2×C4×C181—21no—
87C2×C4×C181—21no—
88C2×C4×C181—41no—
89C2×C4×C181—41no—
90C2×C4×C181C2×D411yescharacteristic
91C8×C181—41no—
92C22×C181—21no—
93C22×C181—21no—
94C22×C181—41no—
95C22×C181—41no—
96C22×C181C2×C22⋊C411yescharacteristic
97C2×D181—21no—
98C2×D181—41no—
99C4×C181—21no—
100C4×C181—41no—
101C4×C181—41no—
102C4×C181—81no—
103Dic181—21no—
104Dic181—21no—
105Dic181—41no—
106Dic181—41no—
107Dic181—81no—
108Dic181#9911yescharacteristic
109Dic181#9911yescharacteristic
110C2×C181—21no—
111C2×C181—41no—
112C2×C181—41no—
113C2×C181C2×#9911yescharacteristic
114D181—41no—
115C181#212311yescharacteristic, complemented, Sylow
116#2123—1811nomaximal, Sylow
117#536—1811no—
118#413—1811no—
119#410—1811no—
120#352—1811no—
121#352—1811no—
122#352—1811no—
123#352D18111yescharacteristic, complemented
124C2×#41—1811no—
125C22×Q8—1811no—
126C2×M16—3621no—
127C4⋊Q8—3621no—
128#124—3621no—
129#123—1811no—
130#123—1811no—
131#123—1811no—
132#123C2×D18111yescharacteristic
133C22⋊Q8—3621no—
134C22⋊Q8—3621no—
135#117—3621no—
136#117—1811no—
137#101—3621no—
138#100—1811no—
139#100C2×D18111yescharacteristic
140#99—3621no—
141#99—1811no—
142#99C2×D18111yescharacteristic
143#41—7241no—
144#41—7241no—
145C2×Q8—3621no—
146C2×Q8—3621no—
147C2×Q8—1811no—
148C2×Q8—7241no—
149C2×Q8—7241no—
150C2×D4—1811no—
151C2×D4—1811no—
152C2×D4C22×D18111yescharacteristic
153C22×C4—1811no—
154C22×C4—1811no—
155C22×C4—1811no—
156C22×C4—1811no—
157C22×C4C4×D18111yescharacteristic
158M16—21no—
159M16—3621no—
160M16—7241no—
161C2×C8—7241no—
162C4⋊C4—21no—
163C4⋊C4—3621no—
164C4⋊C4—3621no—
165C4⋊C4—3621no—
166C4⋊C4—3621no—
167C4⋊C4—7241no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—3621no—
171C22⋊C4—3621no—
172C22⋊C4—3621no—
173C22⋊C4—3621no—
174C22⋊C4—1811no—
175C22⋊C4C4×D18111yescharacteristic
176C42—3621no—
177C42—7241no—
178C23—1811no—
179C23C2×C4×D18111yescharacteristic
180C23D4×D18111yescharacteristic
181Q8—7241no—
182Q8—7242no—
183Q8—7241no—
184Q8—7241no—
185D4—41no—
186D4—7241no—
187D4—7241no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—3621no—
193C2×C4—3621no—
194C2×C4—3621no—
195C2×C4—3621no—
196C2×C4—3621no—
197C2×C4—3621no—
198C2×C4—3621no—
199C2×C4—3621no—
200C2×C4—1811no—
201C2×C4—1811no—
202C2×C4—1811no—
203C2×C4—7241no—
204C2×C4—7241no—
205C2×C4D4×D18111yescharacteristic
206C8—41no—
207C8—7241no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—3621no—
213C22—7241no—
214C22C22⋊C4×D18111yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—3621no—
220C4—3621no—
221C4—14481no—
222C4—1811no—
223C4—1811no—
224C4—7241no—
225C4—7241no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—7241no—
230C2#99×D18111yescharacteristic
subgroupG/H
centerC2#99×D181
commutatorC23×C181C22×C4
FrattiniC2×D4C22×D181
Fitting#352×C181C2
socleC2×C181C2×#99
prime radicalC2×#15668448C22
cosocleC2×D4×C181C23
hypercenter#352D181
hypocenterC181#2123
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #26041070C2semidirect
5. #26041069C2semidirect
6. #26041068C2semidirect
7. #352×C181C2semidirect
9. C22×Dic362C4semidirect
13. #19153800C4semidirect
14. #19153799C4semidirect
19. #19153772C4semidirect
115. C181#2123semidirect
123. #352D181semidirect

Maximal subgroups

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

subgroupindex
#260418512
#260412602
#260412512
#260410702
#260410692
#260410682
#352×C1812
#2123181