Finite Group Databasebeta

← #37545154

Subgroups — #37545154

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#26041903C211yesmaximal, characteristic
2#26041887C211yesmaximal, characteristic, complemented
3#26041589C211yesmaximal, characteristic, complemented
4#26041578C211yesmaximal, characteristic, complemented
5#26041571C211yesmaximal, characteristic, complemented
6#26041538C211yesmaximal, characteristic, complemented
7#26041528C211yesmaximal, characteristic
8#26041519C211yesmaximal, characteristic, complemented
9#26041509C211yesmaximal, characteristic
10#26041502C211yesmaximal, characteristic, complemented
11#26041501C211yesmaximal, characteristic, complemented
12#26041488C211yesmaximal, characteristic, complemented
13#26041483C211yesmaximal, characteristic, complemented
14#26041304C211yesmaximal, characteristic, complemented
15#475×C181C211yesmaximal, characteristic, complemented
16C2×Dic724C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668432C2211yescharacteristic
21C2×C8×D181C2211yescharacteristic
22#19153814C2211yescharacteristic
23#19153811—21no—
24#19153811—21no—
25#19153810—21no—
26#19153808—21no—
27#19153807C2211yescharacteristic
28#19153806—21no—
29#19153805—21no—
30#19153804C2211yescharacteristic
31#19153801—21no—
32#19153796C2211yescharacteristic
33#19153796C2211yescharacteristic
34#19153792—21no—
35#19153792C2211yescharacteristic
36#19153791C2211yescharacteristic
37C4×Dic362C2211yescharacteristic
38#19153781C2211yescharacteristic
39#19153778C2211yescharacteristic
40#19153773C2211yescharacteristic
41#19153749C2211yescharacteristic
42#19153749C2211yescharacteristic
43C4⋊#13089678C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153745C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153739C2211yescharacteristic
50#19153738C2211yescharacteristic
51#19153736C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54#106×C181C2211yescharacteristic
55#106×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59C2×Dic362C2311yescharacteristic
60C2×Dic362C2311yescharacteristic
61C2×D724—21no—
62C2×D724C2311yescharacteristic
63C2×C4×D181—21no—
64C2×C4×D181C2311yescharacteristic
65Dic724—41no—
66#15668439—41no—
67#15668438—41no—
68#15668436—41no—
69#15668432—41no—
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C8×D181—41no—
73C4⋊Dic181—21no—
74C4⋊Dic181—21no—
75C4⋊Dic181—21no—
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84#15668428D411yescharacteristic
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426D411yescharacteristic
93#15668426D411yescharacteristic
94C4×Dic181—21no—
95C4×Dic181—21no—
96C4×Dic181C2311yescharacteristic
97C2×Q8×C181—21no—
98C2×Q8×C181C2311yescharacteristic
99C2×C8×C181C2311yescharacteristic
100C2×C8×C181C2311yescharacteristic
101C4⋊C4×C181—21no—
102C4⋊C4×C181—21no—
103C4⋊C4×C181C2311yescharacteristic
104C4⋊C4×C181C2311yescharacteristic
105C4⋊C4×C181C2311yescharacteristic
106C42×C181—21no—
107C42×C181C2311yescharacteristic
108C22×D181—21no—
109C22×D181C2×D411yescharacteristic
110Dic362—21no—
111Dic362—21no—
112Dic362—21no—
113Dic362—21no—
114D724—21no—
115D724—21no—
116D724—81no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122C2×Dic181C2×D411yescharacteristic
123C4×D181—21no—
124C4×D181—21no—
125C4×D181—81no—
126#13089678—21no—
127#13089678—21no—
128Q8×C181—21no—
129Q8×C181—21no—
130Q8×C181—81no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2411yescharacteristic
138C8×C181—21no—
139C8×C181—21no—
140C22×C181C22×D411yescharacteristic
141C2×D181—21no—
142C2×D181—21no—
143C2×D181—21no—
144C2×D181—41no—
145C2×D181—41no—
146C2×D181—41no—
147C4×C181—21no—
148C4×C181—21no—
149C4×C181—41no—
150C4×C181—41no—
151C4×C181—41no—
152C4×C181—41no—
153C4×C181H(5,𝔽2)11yescharacteristic
154C4×C1812−1+411yescharacteristic
155Dic181—21no—
156Dic181—21no—
157Dic181—21no—
158Dic181—41no—
159Dic181—41no—
160Dic181—41no—
161C2×C181#53711yescharacteristic
162C2×C181#57711yescharacteristic
163C2×C181#57811yescharacteristic
164D181—41no—
165D181—81no—
166C181#325811yescharacteristic, complemented, Sylow
167#3258—1811nomaximal, Sylow
168#542—1811no—
169#541—1811no—
170#479—1811no—
171#478—1811no—
172#477—1811no—
173#475D18111yescharacteristic, complemented
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—3621no—
192#126—3621no—
193#126—3621no—
194#125—3621no—
195#125—3621no—
196#125—1811no—
197#124—3621no—
198#124—1811no—
199#123—3621no—
200C22⋊Q8—3621no—
201C22⋊Q8—1811no—
202C22⋊Q8—1811no—
203C22⋊Q8—1811no—
204#121—1811no—
205C4×Q8—1811no—
206C4×Q8—1811no—
207C4×Q8C2×D18111yescharacteristic
208C4×D4—1811no—
209#117—1811no—
210#107—1811no—
211#107—1811no—
212#106C2×D18111yescharacteristic
213#106C2×D18111yescharacteristic
214C4⋊C8—1811no—
215C4⋊C8—1811no—
216C4⋊C8—1811no—
217C4⋊C8C2×D18111yescharacteristic
218#103—1811no—
219#103—1811no—
220#103—1811no—
221#103—1811no—
222#103C2×D18111yescharacteristic
223#103C2×D18111yescharacteristic
224#102—1811no—
225#102—1811no—
226C2×Q8—21no—
227C2×Q8—1811no—
228C2×Q8—1811no—
229C2×Q8C22×D18111yescharacteristic
230C2×D4—3621no—
231C2×D4—1811no—
232C22×C4—3621no—
233C22×C4—1811no—
234Q16—7241no—
235Q16—7241no—
236SD16—7241no—
237SD16—7241no—
238M16—7241no—
239C2×C8—1811no—
240C2×C8—1811no—
241C2×C8—7241no—
242C2×C8C22×D18111yescharacteristic
243C2×C8C22×D18111yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—3621no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—1811no—
256C4⋊C4—1811no—
257C4⋊C4—1811no—
258C4⋊C4C22×D18111yescharacteristic
259C4⋊C4C22×D18111yescharacteristic
260C4⋊C4C22×D18111yescharacteristic
261C22⋊C4—3621no—
262C22⋊C4—3621no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C22⋊C4—3621no—
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C22⋊C4—1811no—
269C22⋊C4—1811no—
270C42—21no—
271C42—3621no—
272C42—3621no—
273C42—1811no—
274C42C22×D18111yescharacteristic
275C23—3621no—
276C23—1811no—
277Q8—21no—
278Q8—21no—
279Q8—81no—
280Q8—3621no—
281Q8—3621no—
282Q8—3621no—
283Q8—3621no—
284D4—3621no—
285D4—3621no—
286D4—14481no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3621no—
292C2×C4—3621no—
293C2×C4—3621no—
294C2×C4—3621no—
295C2×C4—3621no—
296C2×C4—14481no—
297C2×C4—1811no—
298C2×C4—1811no—
299C2×C4—1811no—
300C2×C4D4×D18111yescharacteristic
301C2×C4D4×D18111yescharacteristic
302C2×C4C23×D18111yescharacteristic
303C8—21no—
304C8—21no—
305C8—3621no—
306C8—3621no—
307C22—3621no—
308C22—3621no—
309C22—3621no—
310C22—7241no—
311C22—7241no—
312C22—7241no—
313C22C2×D4×D18111yescharacteristic
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—3621no—
321C4—3621no—
322C4—3621no—
323C4—7241no—
324C4—7241no—
325C4—7241no—
326C4#1915386211yescharacteristic
327C4#1915386411yescharacteristic
328C2—14481no—
329C2—7241no—
330C2#2604185911yescharacteristic
331C2#2604208511yescharacteristic
332C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#475×C181C2
socleC22×C181C22×D4
prime radical#19153791C22
hypercenter#475D181
hypocenterC181#3258
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #26041887C2semidirect
3. #26041589C2semidirect
4. #26041578C2semidirect
5. #26041571C2semidirect
6. #26041538C2semidirect
8. #26041519C2semidirect
10. #26041502C2semidirect
11. #26041501C2semidirect
12. #26041488C2semidirect
13. #26041483C2semidirect
14. #26041304C2semidirect
15. #475×C181C2semidirect
166. C181#3258semidirect
173. #475D181semidirect

Maximal subgroups

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

subgroupindex
#260419032
#260418872
#260415892
#260415782
#260415712
#260415382
#260415282
#260415192
#260415092
#260415022
#260415012
#260414882
#260414832
#260413042
#475×C1812
#3258181