Finite Group Databasebeta

← #37544902

Subgroups — #37544902

Order 23168.

332 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 121 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#26041836C211yesmaximal, characteristic, complemented
2#26041813C211yesmaximal, characteristic
3#26041681C212yesmaximal, complemented
4#26041634C212yesmaximal, complemented
5#26041621C212yesmaximal, complemented
6#26041604C212yesmaximal
7#26041575C211yesmaximal, characteristic, complemented
8#26041570C211yesmaximal, characteristic, complemented
9#26041565C211yesmaximal, characteristic, complemented
10#26041560C211yesmaximal, characteristic, complemented
11#431×C181C211yesmaximal, characteristic, complemented
12#19153816C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153810—21no—
15#19153808C2212yes—
16#19153808C2211yescharacteristic
17#19153804C2211yescharacteristic
18#19153799—21no—
19#19153795—21no—
20#19153795—21no—
21#19153795C2211yescharacteristic
22#19153790C2211yescharacteristic
23C4×Dic362—21no—
24C4×Dic362—21no—
25C4×Dic362C2211yescharacteristic
26C4×D724C2211yescharacteristic
27C4×C22⋊D181—21no—
28#19153775C2212yes—
29#19153773—21no—
30#19153750C2212yes—
31#19153750C2212yes—
32#19153748C2212yes—
33#19153748C2212yes—
34#19153741C2212yes—
35#19153741C2212yes—
36#19153738C2212yes—
37#19153738C2212yes—
38C2×C42×C181C2211yescharacteristic
39C4⋊C8×C181C2211yescharacteristic
40C4⋊C8×C181C2211yescharacteristic
41C22⋊C8×C181C2212yes—
42C4×C8×C181C2212yes—
43C2×Dic362—21no—
44C2×Dic362—21no—
45C2×Dic362C2311yescharacteristic
46C2×D724C2311yescharacteristic
47C2×C22⋊D181—21no—
48C2×C4×D181—21no—
49C4⋊Dic181—21no—
50C4⋊Dic181—21no—
51C4⋊Dic181C2312yes—
52C4⋊Dic181C2312yes—
53C4⋊Dic181C2311yescharacteristic
54C4⋊Dic181C2311yescharacteristic
55#15668428—21no—
56#15668428—21no—
57#15668428—21no—
58#15668428—22no—
59#15668428—22no—
60#15668428—21no—
61C22⋊Dic181—22no—
62C22⋊Dic181—21no—
63#15668426—21no—
64#15668426—21no—
65#15668426—21no—
66#15668426—21no—
67C4×Dic181—21no—
68C4×Dic181—22no—
69C22×C4×C181D411yescharacteristic
70C22×C4×C181D411yescharacteristic
71C22×C4×C181C2311yescharacteristic
72C2×C8×C181C2312yes—
73C2×C8×C181C2312yes—
74C42×C181D411yescharacteristic
75C42×C181D411yescharacteristic
76C42×C181C2311yescharacteristic
77C42×C181C2311yescharacteristic
78C22×D181—21no—
79Dic362—21no—
80Dic362—21no—
81Dic362—42no—
82Dic362—42no—
83D724—21no—
84D724—21no—
85C22⋊D181—42no—
86C2×Dic181—21no—
87C2×Dic181—21no—
88C2×Dic181—22no—
89C2×Dic181—22no—
90C2×Dic181—21no—
91C4×D181—42no—
92C23×C181C2×D411yescharacteristic
93C2×C4×C181—22no—
94C2×C4×C181—21no—
95C2×C4×C181—21no—
96C2×C4×C181—41no—
97C2×C4×C181C2×D411yescharacteristic
98C2×C4×C181C2×D411yescharacteristic
99C2×C4×C181C2×D411yescharacteristic
100C2×C4×C181C2×D411yescharacteristic
101C2×C4×C181C2×D411yescharacteristic
102C2×C4×C181C2411yescharacteristic
103C8×C181—22no—
104C8×C181—22no—
105C22×C181—21no—
106C22×C181—21no—
107C22×C181—21no—
108C22×C181C22×D411yescharacteristic
109C2×D181—41no—
110C2×D181—41no—
111C2×D181—41no—
112C4×C181—21no—
113C4×C181—21no—
114C4×C181—21no—
115C4×C181—21no—
116C4×C181#13512yes—
117C4×C1812−1+411yescharacteristic
118C4×C1812−1+411yescharacteristic
119Dic181—41no—
120Dic181—41no—
121Dic181—42no—
122Dic181—41no—
123Dic181—42no—
124C2×C181—41no—
125C2×C181#53611yescharacteristic
126C2×C181C2×#13511yescharacteristic
127C2×C181#57511yescharacteristic
128D181—81no—
129C181#323311yescharacteristic, complemented, Sylow
130#3233—1811nomaximal, Sylow
131#533—1811no—
132#529—1811no—
133#499—1812no—
134#488—1812no—
135#485—1812no—
136#482—1812no—
137#477—1811no—
138#476—1811no—
139#475—1811no—
140#474—1811no—
141#431D18111yescharacteristic, complemented
142C4⋊Q8—1811no—
143C4⋊Q8—1811no—
144#126—3621no—
145#125—1811no—
146#125—1812no—
147#124—1811no—
148#123—3621no—
149C22⋊Q8—3621no—
150C22⋊Q8—3621no—
151C22⋊Q8—1811no—
152#121—1811no—
153C4×Q8—3621no—
154C4×Q8—3621no—
155C4×Q8—1811no—
156C4×D4—3621no—
157C4×D4—1811no—
158#117—3621no—
159#117—1812no—
160C2×C42C2×D18111yescharacteristic
161#107—1812no—
162#107—1812no—
163#106—1812no—
164#106—1812no—
165C4⋊C8C2×D18111yescharacteristic
166C4⋊C8C2×D18111yescharacteristic
167#103—1812no—
168#103—1812no—
169#102—1812no—
170#102—1812no—
171C22⋊C8C2×D18112yes—
172C4×C8C2×D18112yes—
173C2×Q8—3621no—
174C2×Q8—3621no—
175C2×Q8—1811no—
176C2×D4—3621no—
177C2×D4—1811no—
178C22×C4—3621no—
179C22×C4D72411yescharacteristic
180C22×C4D72411yescharacteristic
181C22×C4C22×D18111yescharacteristic
182C2×C8C22×D18112yes—
183C2×C8C22×D18112yes—
184C4⋊C4—3621no—
185C4⋊C4—3621no—
186C4⋊C4—3621no—
187C4⋊C4—3622no—
188C4⋊C4—3622no—
189C4⋊C4—3621no—
190C4⋊C4—3621no—
191C4⋊C4—3621no—
192C4⋊C4—1812no—
193C4⋊C4—1812no—
194C4⋊C4—1811no—
195C4⋊C4—1811no—
196C22⋊C4—3622no—
197C22⋊C4—3621no—
198C22⋊C4—3621no—
199C22⋊C4—3621no—
200C22⋊C4—3621no—
201C22⋊C4—3621no—
202C42—3621no—
203C42—3622no—
204C42D72411yescharacteristic
205C42D72411yescharacteristic
206C42C22×D18111yescharacteristic
207C42C22×D18111yescharacteristic
208C23—3621no—
209C23C2×D72411yescharacteristic
210Q8—3621no—
211Q8—3621no—
212Q8—7242no—
213Q8—7242no—
214D4—3621no—
215D4—3621no—
216D4—7242no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—3621no—
222C2×C4—3621no—
223C2×C4—3622no—
224C2×C4—3622no—
225C2×C4—3621no—
226C2×C4—7242no—
227C2×C4C2×D72411yescharacteristic
228C2×C4C2×D72411yescharacteristic
229C2×C4C2×D72411yescharacteristic
230C2×C4C2×D72411yescharacteristic
231C2×C4C2×D72411yescharacteristic
232C2×C4C23×D18111yescharacteristic
233C8—22no—
234C8—22no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—7241no—
239C22—7241no—
240C22—7241no—
241C22C22×D72411yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—7241no—
247C4—7242no—
248C4—7241no—
249C4—7242no—
250C4—7241no—
251C4#1915383812yes—
252C4#1915386311yescharacteristic
253C4#1915386311yescharacteristic
254C2—41no—
255C2—14481no—
256C2#2604185511yescharacteristic
257C2C2×#1915383811yescharacteristic
258C2#2604207911yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#431×C181C2
socleC22×C181C22×D4
prime radical#19153790C22
hypercenter#431D181
hypocenterC181#3233
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #26041836C2semidirect
3. #26041681C2semidirect
4. #26041634C2semidirect
5. #26041621C2semidirect
7. #26041575C2semidirect
8. #26041570C2semidirect
9. #26041565C2semidirect
10. #26041560C2semidirect
11. #431×C181C2semidirect
129. C181#3233semidirect
141. #431D181semidirect

Maximal subgroups

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

subgroupindex
#260418362
#260418132
#260416812
#260416342
#260416212
#260416042
#260415752
#260415702
#260415652
#260415602
#431×C1812
#3233181