Finite Group Databasebeta

← #37544998

Subgroups — #37544998

Order 23168.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041907C211yesmaximal, characteristic, complemented
2#26041685C211yesmaximal, characteristic, complemented
3#26041682C211yesmaximal, characteristic, complemented
4#26041638C211yesmaximal, characteristic, complemented
5#26041633C211yesmaximal, characteristic, complemented
6#26041617C211yesmaximal, characteristic, complemented
7#26041612C211yesmaximal, characteristic, complemented
8#26041600C211yesmaximal, characteristic
9#26041590C211yesmaximal, characteristic
10#26041586C211yesmaximal, characteristic, complemented
11#26041579C211yesmaximal, characteristic, complemented
12#26041574C211yesmaximal, characteristic, complemented
13#26041559C211yesmaximal, characteristic, complemented
14#26041340C211yesmaximal, characteristic, complemented
15#542×C181C211yesmaximal, characteristic
16#19153816C2211yescharacteristic
17#19153816C2211yescharacteristic
18#19153815—21no—
19#19153808C2211yescharacteristic
20#19153808C2211yescharacteristic
21#19153807—21no—
22#19153793—21no—
23#19153793—21no—
24#19153775C2211yescharacteristic
25C2×C4⋊Dic181C2211yescharacteristic
26#19153749C2211yescharacteristic
27#19153749C2211yescharacteristic
28#19153749C2211yescharacteristic
29#19153749C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35C4⋊#13089678C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153737C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44C22⋊#13089678C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46#19153727C2211yescharacteristic
47C4×#13089678C2211yescharacteristic
48C4⋊Q8×C181C2211yescharacteristic
49#126×C181—21no—
50#124×C181—21no—
51#124×C181C2211yescharacteristic
52#123×C181—21no—
53C22⋊Q8×C181—21no—
54C22⋊Q8×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×D4×C181C2211yescharacteristic
58#117×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C22×Dic181—21no—
62C2×#13089678C2311yescharacteristic
63C2×#13089678C2311yescharacteristic
64C2×#13089678C2311yescharacteristic
65C2×#13089678C2311yescharacteristic
66C4⋊Dic181—21no—
67C4⋊Dic181—21no—
68C4⋊Dic181—21no—
69C4⋊Dic181C2311yescharacteristic
70C4⋊Dic181C2311yescharacteristic
71C4⋊Dic181C2311yescharacteristic
72C4⋊Dic181C2311yescharacteristic
73#15668428—21no—
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77C22⋊Dic181—21no—
78C4×Dic181—21no—
79C2×Q8×C181—21no—
80C2×Q8×C181C2311yescharacteristic
81C2×D4×C181—21no—
82C2×D4×C181C2311yescharacteristic
83C22×C4×C181—21no—
84C22×C4×C181C2311yescharacteristic
85C4⋊C4×C181—21no—
86C4⋊C4×C181—21no—
87C4⋊C4×C181—21no—
88C4⋊C4×C181D411yescharacteristic
89C4⋊C4×C181D411yescharacteristic
90C4⋊C4×C181C2311yescharacteristic
91C4⋊C4×C181C2311yescharacteristic
92C22⋊C4×C181—21no—
93C22⋊C4×C181—21no—
94C22⋊C4×C181—21no—
95C22⋊C4×C181—21no—
96C22⋊C4×C181—21no—
97C22⋊C4×C181D411yescharacteristic
98C22⋊C4×C181D411yescharacteristic
99C42×C181—21no—
100C42×C181C2311yescharacteristic
101C42×C181C2311yescharacteristic
102Dic362—81no—
103Dic362—81no—
104C2×Dic181—21no—
105C2×Dic181—21no—
106C2×Dic181—21no—
107C2×Dic181—21no—
108C2×Dic181—81no—
109#13089678—21no—
110#13089678—21no—
111#13089678—21no—
112#13089678—21no—
113C23×C181—21no—
114C23×C181C2×D411yescharacteristic
115Q8×C181—21no—
116Q8×C181—21no—
117Q8×C181—81no—
118D4×C181—21no—
119D4×C181—21no—
120D4×C181—81no—
121C2×C4×C181—21no—
122C2×C4×C181—21no—
123C2×C4×C181—21no—
124C2×C4×C181—21no—
125C2×C4×C181—21no—
126C2×C4×C181—81no—
127C2×C4×C181C2×D411yescharacteristic
128C2×C4×C181C2×D411yescharacteristic
129C2×C4×C181C2×D411yescharacteristic
130C2×C4×C181C2×D411yescharacteristic
131C2×C4×C181C2×D411yescharacteristic
132C2×C4×C181C2411yescharacteristic
133C22×C181—21no—
134C22×C181—21no—
135C22×C181—21no—
136C22×C181—41no—
137C22×C181—41no—
138C22×C181—41no—
139C22×C181C22×D411yescharacteristic
140C4×C181—21no—
141C4×C181—21no—
142C4×C181—21no—
143C4×C181—21no—
144C4×C181—21no—
145C4×C181—41no—
146C4×C181—41no—
147C4×C181—41no—
148C4×C1812−1+411yescharacteristic
149C4×C1812−1+411yescharacteristic
150Dic181—41no—
151Dic181—41no—
152Dic181—41no—
153Dic181—41no—
154C2×C181—41no—
155C2×C181—81no—
156C2×C181#53611yescharacteristic
157C2×C181#57611yescharacteristic
158C2×C181#57811yescharacteristic
159C181#324311yescharacteristic, complemented, Sylow
160#3243—1811nomaximal, Sylow
161#544—1811no—
162#542D18111yescharacteristic
163#501—1811no—
164#500—1811no—
165#489—1811no—
166#488—1811no—
167#485—1811no—
168#484—1811no—
169#482—1811no—
170#480—1811no—
171#479—1811no—
172#478—1811no—
173#477—1811no—
174#474—1811no—
175#433—1811no—
176C4⋊Q8—3621no—
177C4⋊Q8—1811no—
178C4⋊Q8—1811no—
179C4⋊Q8C2×D18111yescharacteristic
180#126—21no—
181#125—3621no—
182#125—1811no—
183#125—1811no—
184#124—21no—
185#124C2×D18111yescharacteristic
186#123—21no—
187C22⋊Q8—21no—
188C22⋊Q8—3621no—
189C22⋊Q8—3621no—
190C22⋊Q8C2×D18111yescharacteristic
191#121C2×D18111yescharacteristic
192C4×Q8C2×D18111yescharacteristic
193C4×D4C2×D18111yescharacteristic
194#117—1811no—
195#117C2×D18111yescharacteristic
196C2×C4⋊C4—1811no—
197#107—1811no—
198#107—1811no—
199#107—1811no—
200#107—1811no—
201#107—1811no—
202#107—1811no—
203#106—1811no—
204#106—1811no—
205C4⋊C8—1811no—
206C4⋊C8—1811no—
207#103—1811no—
208#103—1811no—
209#103—1811no—
210#103—1811no—
211#102—1811no—
212#102—1811no—
213#102—1811no—
214#102—1811no—
215C22⋊C8—1811no—
216C22⋊C8—1811no—
217#97—1811no—
218C4×C8—1811no—
219C2×Q8—21no—
220C2×Q8—3621no—
221C2×Q8—3621no—
222C2×Q8C22×D18111yescharacteristic
223C2×D4—21no—
224C2×D4C22×D18111yescharacteristic
225C22×C4—21no—
226C22×C4—3621no—
227C22×C4C22×D18111yescharacteristic
228C2×C8—1811no—
229C2×C8—1811no—
230C2×C8—1811no—
231C2×C8—1811no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—3621no—
236C4⋊C4—3621no—
237C4⋊C4—3621no—
238C4⋊C4—3621no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—1811no—
243C4⋊C4—1811no—
244C4⋊C4—1811no—
245C4⋊C4—1811no—
246C4⋊C4C22⋊D18111yescharacteristic
247C4⋊C4C22⋊D18111yescharacteristic
248C4⋊C4C22×D18111yescharacteristic
249C4⋊C4C22×D18111yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—3621no—
256C22⋊C4C22⋊D18111yescharacteristic
257C22⋊C4C22⋊D18111yescharacteristic
258C42—21no—
259C42—3621no—
260C42C22×D18111yescharacteristic
261C42C22×D18111yescharacteristic
262C23—21no—
263C23C2×C22⋊D18111yescharacteristic
264Q8—21no—
265Q8—21no—
266Q8—81no—
267Q8—14481no—
268Q8—14481no—
269D4—21no—
270D4—21no—
271D4—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—81no—
278C2×C4—3621no—
279C2×C4—3621no—
280C2×C4—3621no—
281C2×C4—3621no—
282C2×C4—14481no—
283C2×C4C2×C22⋊D18111yescharacteristic
284C2×C4C2×C22⋊D18111yescharacteristic
285C2×C4C2×C22⋊D18111yescharacteristic
286C2×C4C2×C22⋊D18111yescharacteristic
287C2×C4C2×C22⋊D18111yescharacteristic
288C2×C4C23×D18111yescharacteristic
289C8—3621no—
290C8—3621no—
291C8—3621no—
292C8—3621no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22C22×C22⋊D18111yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—7241no—
309C4—7241no—
310C4—7241no—
311C4—7241no—
312C4#1915386311yescharacteristic
313C4#1915386311yescharacteristic
314C2—41no—
315C2—81no—
316C2#2604185011yescharacteristic
317C2#2604208211yescharacteristic
318C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#542×C181C2
socleC22×C181C22×D4
prime radical#121×C181C22
hypercenter#542D181
hypocenterC181#3243
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #26041907C2semidirect
2. #26041685C2semidirect
3. #26041682C2semidirect
4. #26041638C2semidirect
5. #26041633C2semidirect
6. #26041617C2semidirect
7. #26041612C2semidirect
10. #26041586C2semidirect
11. #26041579C2semidirect
12. #26041574C2semidirect
13. #26041559C2semidirect
14. #26041340C2semidirect
159. C181#3243semidirect

Maximal subgroups

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

subgroupindex
#260419072
#260416852
#260416822
#260416382
#260416332
#260416172
#260416122
#260416002
#260415902
#260415862
#260415792
#260415742
#260415592
#260413402
#542×C1812
#3243181