Finite Group Databasebeta

← #37545019

Subgroups — #37545019

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#26041900C211yesmaximal, characteristic
2#26041689C211yesmaximal, characteristic, complemented
3#26041675C211yesmaximal, characteristic, complemented
4#26041641C211yesmaximal, characteristic, complemented
5#26041636C211yesmaximal, characteristic, complemented
6#26041620C211yesmaximal, characteristic, complemented
7#26041609C211yesmaximal, characteristic, complemented
8#26041603C211yesmaximal, characteristic
9#26041597C211yesmaximal, characteristic
10#26041583C211yesmaximal, characteristic, complemented
11#26041576C211yesmaximal, characteristic, complemented
12#26041566C211yesmaximal, characteristic, complemented
13#26041561C211yesmaximal, characteristic, complemented
14#26041340C211yesmaximal, characteristic, complemented
15#544×C181C211yesmaximal, characteristic, complemented
16#19153816C2211yescharacteristic
17#19153811—21no—
18#19153804C2211yescharacteristic
19#19153802—21no—
20#19153798—21no—
21#19153795C2211yescharacteristic
22#19153794—21no—
23#19153790C2211yescharacteristic
24C4×Dic362C2211yescharacteristic
25C4×D724C2211yescharacteristic
26#19153749C2211yescharacteristic
27#19153749C2211yescharacteristic
28#19153747C2211yescharacteristic
29#19153747C2211yescharacteristic
30#19153747C2211yescharacteristic
31#19153747C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35C4⋊#13089678C2211yescharacteristic
36#19153739C2211yescharacteristic
37#19153739C2211yescharacteristic
38#19153739C2211yescharacteristic
39#19153739C2211yescharacteristic
40#19153736C2211yescharacteristic
41#19153736C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44C22⋊#13089678C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46#19153727C2211yescharacteristic
47C4×#13089678C2211yescharacteristic
48C4⋊Q8×C181—21no—
49C4⋊Q8×C181C2211yescharacteristic
50C4⋊Q8×C181C2211yescharacteristic
51#125×C181—21no—
52#125×C181C2211yescharacteristic
53#125×C181C2211yescharacteristic
54C22⋊Q8×C181—21no—
55C22⋊Q8×C181—21no—
56#117×C181C2211yescharacteristic
57#117×C181C2211yescharacteristic
58C2×C4⋊C4×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C4×D181—21no—
64C2×#13089678C2311yescharacteristic
65C2×#13089678C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C4⋊Dic181—21no—
69C4⋊Dic181C2311yescharacteristic
70C4⋊Dic181C2311yescharacteristic
71#15668428—21no—
72#15668428—21no—
73C22⋊Dic181—21no—
74#15668426—21no—
75#15668426—21no—
76#15668426—21no—
77#15668426—21no—
78C4×Dic181—21no—
79C2×Q8×C181—21no—
80C2×Q8×C181—21no—
81C22×C4×C181—21no—
82C22×C4×C181C2311yescharacteristic
83C4⋊C4×C181—21no—
84C4⋊C4×C181—21no—
85C4⋊C4×C181—21no—
86C4⋊C4×C181—21no—
87C4⋊C4×C181—21no—
88C4⋊C4×C181—21no—
89C4⋊C4×C181—21no—
90C4⋊C4×C181D411yescharacteristic
91C4⋊C4×C181D411yescharacteristic
92C4⋊C4×C181C2311yescharacteristic
93C4⋊C4×C181C2311yescharacteristic
94C4⋊C4×C181C2311yescharacteristic
95C4⋊C4×C181C2311yescharacteristic
96C22⋊C4×C181—21no—
97C22⋊C4×C181D411yescharacteristic
98C22⋊C4×C181D411yescharacteristic
99C42×C181—21no—
100C42×C181C2311yescharacteristic
101C42×C181C2311yescharacteristic
102C22×D181—21no—
103Dic362—21no—
104Dic362—21no—
105Dic362—81no—
106D724—21no—
107D724—21no—
108C22⋊D181—81no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C4×D181—81no—
113#13089678—21no—
114#13089678—21no—
115#13089678—21no—
116#13089678—21no—
117C23×C181C2×D411yescharacteristic
118Q8×C181—81no—
119Q8×C181—81no—
120C2×C4×C181—21no—
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×C181C22×D411yescharacteristic
137C2×D181—41no—
138C2×D181—41no—
139C2×D181—41no—
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×C181—41no—
149C4×C1812−1+411yescharacteristic
150C4×C1812−1+411yescharacteristic
151Dic181—41no—
152Dic181—41no—
153Dic181—41no—
154C2×C181—41no—
155C2×C181#53611yescharacteristic
156C2×C181#57711yescharacteristic
157C2×C181#57711yescharacteristic
158D181—81no—
159C181#324411yescharacteristic, complemented, Sylow
160#3244—1811nomaximal, Sylow
161#544D18111yescharacteristic, complemented
162#542—1811no—
163#501—1811no—
164#498—1811no—
165#489—1811no—
166#488—1811no—
167#485—1811no—
168#483—1811no—
169#482—1811no—
170#481—1811no—
171#479—1811no—
172#478—1811no—
173#476—1811no—
174#475—1811no—
175#433—1811no—
176C4⋊Q8—21no—
177C4⋊Q8—1811no—
178C4⋊Q8C2×D18111yescharacteristic
179C4⋊Q8C2×D18111yescharacteristic
180#126—3621no—
181#125—21no—
182#125C2×D18111yescharacteristic
183#125C2×D18111yescharacteristic
184#124—3621no—
185#124—1811no—
186#123—3621no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189C22⋊Q8—3621no—
190C22⋊Q8—1811no—
191#121—1811no—
192C4×Q8—1811no—
193C4×D4—1811no—
194#117C2×D18111yescharacteristic
195#117C2×D18111yescharacteristic
196C2×C4⋊C4C2×D18111yescharacteristic
197#107—1811no—
198#107—1811no—
199#106—1811no—
200#106—1811no—
201#106—1811no—
202#106—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—21no—
221C2×Q8—3621no—
222C2×Q8—1811no—
223C2×D4—3621no—
224C2×D4—1811no—
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—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—1811no—
243C4⋊C4—1811no—
244C4⋊C4C22⋊D18111yescharacteristic
245C4⋊C4C22⋊D18111yescharacteristic
246C4⋊C4C22×D18111yescharacteristic
247C4⋊C4C22×D18111yescharacteristic
248C4⋊C4C22×D18111yescharacteristic
249C4⋊C4C22×D18111yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—3621no—
252C22⋊C4—3621no—
253C22⋊C4—3621no—
254C22⋊C4—3621no—
255C22⋊C4—3621no—
256C22⋊C4C22⋊D18111yescharacteristic
257C22⋊C4C22⋊D18111yescharacteristic
258C42—21no—
259C42—3621no—
260C42C22×D18111yescharacteristic
261C42C22×D18111yescharacteristic
262C23—3621no—
263C23C2×C22⋊D18111yescharacteristic
264Q8—81no—
265Q8—81no—
266Q8—3621no—
267Q8—3621no—
268Q8—14481no—
269D4—3621no—
270D4—3621no—
271D4—14481no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—81no—
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—7241no—
297C22—7241no—
298C22—7241no—
299C22C22×C22⋊D18111yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—7241no—
310C4—7241no—
311C4—7241no—
312C4#1915386411yescharacteristic
313C4#1915386411yescharacteristic
314C2—41no—
315C2—14481no—
316C2#2604185311yescharacteristic
317C2#2604208711yescharacteristic
318C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#544×C181C2
socleC22×C181C22×D4
prime radical#19153790C22
hypercenter#544D181
hypocenterC181#3244
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
2. #26041689C2semidirect
3. #26041675C2semidirect
4. #26041641C2semidirect
5. #26041636C2semidirect
6. #26041620C2semidirect
7. #26041609C2semidirect
10. #26041583C2semidirect
11. #26041576C2semidirect
12. #26041566C2semidirect
13. #26041561C2semidirect
14. #26041340C2semidirect
15. #544×C181C2semidirect
159. C181#3244semidirect
161. #544D181semidirect

Maximal subgroups

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

subgroupindex
#260419002
#260416892
#260416752
#260416412
#260416362
#260416202
#260416092
#260416032
#260415972
#260415832
#260415762
#260415662
#260415612
#260413402
#544×C1812
#3244181