Finite Group Databasebeta

← #37545141

Subgroups — #37545141

Order 23168.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041900C211yesmaximal, characteristic
2#26041583C211yesmaximal, characteristic, complemented
3#26041576C211yesmaximal, characteristic, complemented
4#26041574C211yesmaximal, characteristic, complemented
5#26041564C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041527C211yesmaximal, characteristic
8#26041518C211yesmaximal, characteristic, complemented
9#26041510C211yesmaximal, characteristic
10#26041501C211yesmaximal, characteristic, complemented
11#26041501C211yesmaximal, characteristic, complemented
12#26041486C211yesmaximal, characteristic, complemented
13#26041481C211yesmaximal, characteristic, complemented
14#26041306C211yesmaximal, characteristic, complemented
15#541×C181C211yesmaximal, characteristic, complemented
16C2×#15668439C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668433C2211yescharacteristic
21C22×#13089678C2211yescharacteristic
22#19153816C2211yescharacteristic
23#19153811—21no—
24#19153804C2211yescharacteristic
25#19153802—21no—
26#19153798—21no—
27#19153795C2211yescharacteristic
28#19153794—21no—
29#19153790C2211yescharacteristic
30C4×Dic362C2211yescharacteristic
31C4×D724C2211yescharacteristic
32#19153749C2211yescharacteristic
33#19153749C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36C4⋊#13089678C2211yescharacteristic
37C4⋊#13089678C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39C4⋊#13089678C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153739C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic
48C4⋊Q8×C181C2211yescharacteristic
49#126×C181—21no—
50#126×C181—21no—
51#125×C181—21no—
52#125×C181—21no—
53#125×C181C2211yescharacteristic
54C22⋊Q8×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×Q8×C181C2211yescharacteristic
58#117×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C4×D181—21no—
64#15668439—41no—
65#15668439—41no—
66#15668436—41no—
67#15668436—41no—
68#15668433—41no—
69C2×#13089678—41no—
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C4⋊Dic181—21no—
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79C22⋊Dic181—21no—
80#15668426—21no—
81#15668426—21no—
82#15668426—21no—
83#15668426—21no—
84C4×Dic181—21no—
85C2×Q8×C181C2311yescharacteristic
86C2×Q8×C181C2311yescharacteristic
87C22×C4×C181C2311yescharacteristic
88C4⋊C4×C181—21no—
89C4⋊C4×C181—21no—
90C4⋊C4×C181—21no—
91C4⋊C4×C181—21no—
92C4⋊C4×C181—21no—
93C4⋊C4×C181—21no—
94C4⋊C4×C181—21no—
95C4⋊C4×C181—21no—
96C4⋊C4×C181D411yescharacteristic
97C4⋊C4×C181D411yescharacteristic
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C22⋊C4×C181—21no—
101C22⋊C4×C181—21no—
102C22⋊C4×C181D411yescharacteristic
103C22⋊C4×C181D411yescharacteristic
104C42×C181—21no—
105C42×C181—21no—
106C42×C181C2311yescharacteristic
107C42×C181C2311yescharacteristic
108C22×D181—21no—
109Dic362—21no—
110Dic362—21no—
111Dic362—81no—
112D724—21no—
113D724—21no—
114C22⋊D181—81no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C4×D181—81no—
119#13089678—21no—
120#13089678—21no—
121#13089678—21no—
122#13089678—21no—
123C23×C181C2×D411yescharacteristic
124Q8×C181—21no—
125Q8×C181—21no—
126Q8×C181—21no—
127Q8×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2411yescharacteristic
140C22×C181—21no—
141C22×C181—21no—
142C22×C181—21no—
143C22×C181C22×D411yescharacteristic
144C2×D181—41no—
145C2×D181—41no—
146C2×D181—41no—
147C4×C181—21no—
148C4×C181—21no—
149C4×C181—21no—
150C4×C181—21no—
151C4×C181—21no—
152C4×C181—41no—
153C4×C181—41no—
154C4×C181—41no—
155C4×C181—41no—
156C4×C181H(5,𝔽2)11yescharacteristic
157C4×C1812−1+411yescharacteristic
158Dic181—41no—
159Dic181—41no—
160Dic181—41no—
161C2×C181—41no—
162C2×C181#53711yescharacteristic
163C2×C181#57711yescharacteristic
164C2×C181#57811yescharacteristic
165D181—81no—
166C181#325811yescharacteristic, complemented, Sylow
167#3258—1811nomaximal, Sylow
168#542—1811no—
169#541D18111yescharacteristic, complemented
170#479—1811no—
171#478—1811no—
172#477—1811no—
173#475—1811no—
174#470—1811no—
175#468—1811no—
176#467—1811no—
177#465—1811no—
178#464—1811no—
179#464—1811no—
180#462—1811no—
181#461—1811no—
182#424—1811no—
183C2×Q16—1811no—
184C2×Q16—1811no—
185C2×SD16—1811no—
186C2×SD16—1811no—
187C2×M16—1811no—
188C22×C8—1811no—
189C4⋊Q8—1811no—
190C4⋊Q8C2×D18111yescharacteristic
191#126—21no—
192#126—21no—
193#126—3621no—
194#125—21no—
195#125—21no—
196#125C2×D18111yescharacteristic
197#124—3621no—
198#124—1811no—
199#123—3621no—
200C22⋊Q8—3621no—
201C22⋊Q8—1811no—
202C22⋊Q8C2×D18111yescharacteristic
203C22⋊Q8C2×D18111yescharacteristic
204#121—1811no—
205C4×Q8—1811no—
206C4×Q8C2×D18111yescharacteristic
207C4×Q8C2×D18111yescharacteristic
208C4×D4—1811no—
209#117C2×D18111yescharacteristic
210#107—1811no—
211#107—1811no—
212#106—1811no—
213#106—1811no—
214C4⋊C8—1811no—
215C4⋊C8—1811no—
216C4⋊C8—1811no—
217C4⋊C8—1811no—
218#103—1811no—
219#103—1811no—
220#103—1811no—
221#103—1811no—
222#103—1811no—
223#103—1811no—
224#102—1811no—
225#102—1811no—
226C2×Q8—3621no—
227C2×Q8—1811no—
228C2×Q8C22×D18111yescharacteristic
229C2×Q8C22×D18111yescharacteristic
230C2×D4—3621no—
231C2×D4—1811no—
232C22×C4—3621no—
233C22×C4C22×D18111yescharacteristic
234Q16—7241no—
235Q16—7241no—
236SD16—7241no—
237SD16—7241no—
238M16—7241no—
239C2×C8—1811no—
240C2×C8—1811no—
241C2×C8—1811no—
242C2×C8—1811no—
243C2×C8—7241no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—1811no—
256C4⋊C4—1811no—
257C4⋊C4C22⋊D18111yescharacteristic
258C4⋊C4C22⋊D18111yescharacteristic
259C4⋊C4C22×D18111yescharacteristic
260C4⋊C4C22×D18111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C22⋊C4—3621no—
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C22⋊C4C22⋊D18111yescharacteristic
269C22⋊C4C22⋊D18111yescharacteristic
270C42—21no—
271C42—21no—
272C42—3621no—
273C42C22×D18111yescharacteristic
274C42C22×D18111yescharacteristic
275C23—3621no—
276C23C2×C22⋊D18111yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—3621no—
282Q8—3621no—
283Q8—14481no—
284D4—3621no—
285D4—3621no—
286D4—14481no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—3621no—
294C2×C4—3621no—
295C2×C4—3621no—
296C2×C4—14481no—
297C2×C4C2×C22⋊D18111yescharacteristic
298C2×C4C2×C22⋊D18111yescharacteristic
299C2×C4C2×C22⋊D18111yescharacteristic
300C2×C4C2×C22⋊D18111yescharacteristic
301C2×C4C2×C22⋊D18111yescharacteristic
302C2×C4C23×D18111yescharacteristic
303C8—3621no—
304C8—3621no—
305C8—3621no—
306C8—3621no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—7241no—
311C22—7241no—
312C22—7241no—
313C22C22×C22⋊D18111yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—7241no—
324C4—7241no—
325C4—7241no—
326C4#1915386211yescharacteristic
327C4#1915386311yescharacteristic
328C2—41no—
329C2—14481no—
330C2#2604185711yescharacteristic
331C2#2604208711yescharacteristic
332C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#541×C181C2
socleC22×C181C22×D4
prime radical#19153790C22
hypercenter#541D181
hypocenterC181#3258
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #26041583C2semidirect
3. #26041576C2semidirect
4. #26041574C2semidirect
5. #26041564C2semidirect
6. #26041535C2semidirect
8. #26041518C2semidirect
10. #26041501C2semidirect
11. #26041501C2semidirect
12. #26041486C2semidirect
13. #26041481C2semidirect
14. #26041306C2semidirect
15. #541×C181C2semidirect
166. C181#3258semidirect
169. #541D181semidirect

Maximal subgroups

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

subgroupindex
#260419002
#260415832
#260415762
#260415742
#260415642
#260415352
#260415272
#260415182
#260415102
#260415012
#260414862
#260414812
#260413062
#541×C1812
#3258181