Finite Group Databasebeta

← #37546753

Subgroups — #37546753

Order 23168.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042007C211yesmaximal, characteristic, complemented
2#26042001C211yesmaximal, characteristic
3#26041673C211yesmaximal, characteristic
4#26041672C211yesmaximal, characteristic
5#26041657C211yesmaximal, characteristic, complemented
6#26041504C211yesmaximal, characteristic
7#26041501C211yesmaximal, characteristic
8#26041489C211yesmaximal, characteristic, complemented
9#26041488C211yesmaximal, characteristic, complemented
10#26041488C211yesmaximal, characteristic, complemented
11#26041483C211yesmaximal, characteristic
12#26041407C211yesmaximal, characteristic, complemented
13#26041385C211yesmaximal, characteristic, complemented
14#26041366C211yesmaximal, characteristic
15#441×C181C211yesmaximal, characteristic, complemented
16C2×Dic724C2211yescharacteristic
17C2×Dic724C2211yescharacteristic
18C2×#15668439C2211yescharacteristic
19C2×#15668439C2211yescharacteristic
20C2×#15668439C2211yescharacteristic
21C2×#15668438C2211yescharacteristic
22C2×#15668438C2211yescharacteristic
23C2×#15668436C2211yescharacteristic
24#19153816C2211yescharacteristic
25#19153814C2211yescharacteristic
26#19153814C2211yescharacteristic
27#19153814C2211yescharacteristic
28#19153811—21no—
29#19153810—21no—
30#19153808—21no—
31#19153806—21no—
32#19153806—21no—
33#19153805C2211yescharacteristic
34#19153804C2211yescharacteristic
35#19153794—21no—
36C4×Dic362—21no—
37C4×Dic362C2211yescharacteristic
38#19153781C2211yescharacteristic
39#19153781C2211yescharacteristic
40#19153781C2211yescharacteristic
41#19153778C2211yescharacteristic
42#19153773—21no—
43C4⋊#13089678C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153745C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×Q16×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54C4×Q8×C181C2211yescharacteristic
55#106×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59C2×Dic362D411yescharacteristic
60C2×Dic362C2311yescharacteristic
61C2×Dic362C2311yescharacteristic
62C2×Dic362C2311yescharacteristic
63C2×D724C2311yescharacteristic
64C2×C4×D181—21no—
65Dic724—22no—
66Dic724—41no—
67#15668439—41no—
68#15668439—41no—
69#15668439—41no—
70#15668438—22no—
71#15668438—41no—
72#15668436—41no—
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C4⋊Dic181—21no—
76C4⋊Dic181—21no—
77C4⋊Dic181D411yescharacteristic
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428—21no—
87#15668428D411yescharacteristic
88#15668428D411yescharacteristic
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94C4×Dic181—21no—
95C4×Dic181—21no—
96C4×Dic181—21no—
97C4×Dic181C2311yescharacteristic
98C4×Dic181C2311yescharacteristic
99C2×Q8×C181C2311yescharacteristic
100C2×Q8×C181C2311yescharacteristic
101Q16×C181—22no—
102C2×C8×C181C2311yescharacteristic
103C2×C8×C181C2311yescharacteristic
104C4⋊C4×C181—21no—
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C42×C181—21no—
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112Dic362—22no—
113Dic362—21no—
114Dic362—21no—
115Dic362—21no—
116Dic362—21no—
117Dic362—21no—
118Dic362—21no—
119D724—21no—
120D724—21no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181C2×D411yescharacteristic
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C4×D181—42no—
129#13089678—21no—
130#13089678—21no—
131Q8×C181—21no—
132Q8×C181—21no—
133Q8×C181—21no—
134Q8×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181C2×D411yescharacteristic
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2411yescharacteristic
142C8×C181—21no—
143C8×C181#4112yes—
144C22×C181C22×D411yescharacteristic
145C2×D181—41no—
146C2×D181—41no—
147C2×D181—41no—
148C4×C181—21no—
149C4×C181—21no—
150C4×C181—41no—
151C4×C181—41no—
152C4×C181—41no—
153C4×C181—41no—
154C4×C181C2×#4111yescharacteristic
155C4×C181H(5,𝔽2)11yescharacteristic
156Dic181—21no—
157Dic181—21no—
158Dic181—21no—
159Dic181—41no—
160Dic181—41no—
161Dic181—41no—
162Dic181#13712yes—
163C2×C181#55011yescharacteristic
164C2×C181C2×#13711yescharacteristic
165C2×C181#57811yescharacteristic
166D181—81no—
167C181#339811yescharacteristic, complemented, Sylow
168#3398—1811nomaximal, Sylow
169#557—1811no—
170#556—1811no—
171#497—1811no—
172#497—1811no—
173#494—1811no—
174#464—1811no—
175#464—1811no—
176#462—1811no—
177#462—1811no—
178#462—1811no—
179#461—1811no—
180#446—1811no—
181#441—1811no—
182#441D18111yescharacteristic, complemented
183C4×SD16—1811no—
184C2×Q16—1811no—
185C2×Q16—1811no—
186C2×Q16—1811no—
187C2×Q16—1811no—
188C2×Q16—1811no—
189C2×Q16C2×D18111yescharacteristic
190C2×SD16—1811no—
191C2×SD16—1811no—
192C2×SD16—1811no—
193C4⋊Q8—1811no—
194C4⋊Q8—1811no—
195C4⋊Q8—1811no—
196C4⋊Q8—1811no—
197#126—3621no—
198#126—3621no—
199#125—3621no—
200#125—3621no—
201#125—3621no—
202#124—1811no—
203#124—1811no—
204C22⋊Q8—3621no—
205C4×Q8—3621no—
206C4×Q8—1811no—
207C4×Q8—1811no—
208C4×Q8—1811no—
209C4×Q8—1811no—
210C4×Q8C2×D18111yescharacteristic
211C4×Q8C2×D18111yescharacteristic
212C4×D4—1811no—
213#117—3621no—
214#106C2×D18111yescharacteristic
215C4⋊C8—1811no—
216C4⋊C8—1811no—
217C4⋊C8—1811no—
218#103—1811no—
219#103—1811no—
220#103—1811no—
221#103C2×D18111yescharacteristic
222#103C2×D18111yescharacteristic
223#102—1811no—
224#97—1811no—
225#97C2×D18111yescharacteristic
226C4×C8—1811no—
227C2×Q8—1811no—
228C2×Q8—1811no—
229C2×Q8—1811no—
230C2×Q8—1811no—
231C2×Q8C22×D18111yescharacteristic
232C2×Q8C22×D18111yescharacteristic
233C2×D4—1811no—
234C22×C4—3621no—
235Q16—22no—
236Q16—3622no—
237Q16—7241no—
238Q16—7241no—
239Q16—7241no—
240Q16—7241no—
241SD16—3622no—
242SD16—7241no—
243SD16—7241no—
244C2×C8—1811no—
245C2×C8—1811no—
246C2×C8C22×D18111yescharacteristic
247C2×C8C22×D18111yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—3621no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—3621no—
256C4⋊C4—3621no—
257C4⋊C4—3621no—
258C4⋊C4—3621no—
259C4⋊C4—3621no—
260C4⋊C4—3621no—
261C4⋊C4—1811no—
262C4⋊C4—1811no—
263C4⋊C4—1811no—
264C4⋊C4C22×D18111yescharacteristic
265C4⋊C4C22×D18111yescharacteristic
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C22⋊C4—3621no—
269C22⋊C4—3621no—
270C22⋊C4—3621no—
271C42—21no—
272C42—21no—
273C42—3621no—
274C42—3621no—
275C42—3621no—
276C42—1811no—
277C42—1811no—
278C42C22×D18111yescharacteristic
279C23—3621no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—3622no—
285Q8—3621no—
286Q8—3621no—
287Q8—3621no—
288Q8—3621no—
289Q8—3621no—
290Q8—3621no—
291D4—3621no—
292D4—3621no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—3621no—
300C2×C4—1811no—
301C2×C4—1811no—
302C2×C4—1811no—
303C2×C4—1811no—
304C2×C4—7242no—
305C2×C4D4×D18111yescharacteristic
306C2×C4D4×D18111yescharacteristic
307C2×C4C23×D18111yescharacteristic
308C8—21no—
309C8—3621no—
310C8—3621no—
311C8#1566844912yes—
312C22—7241no—
313C22—7241no—
314C22—7241no—
315C22C2×D4×D18111yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—3621no—
323C4—3621no—
324C4—3621no—
325C4—1812no—
326C4—7241no—
327C4—7241no—
328C4—7241no—
329C4C2×#1566844911yescharacteristic
330C4#1915386211yescharacteristic
331C2—14481no—
332C2#2604195011yescharacteristic
333C2#137×D18111yescharacteristic
334C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#441×C181C2
socleC22×C181C22×D4
prime radicalC2×D724C23
hypercenter#441D181
hypocenterC181#3398
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #26042007C2semidirect
5. #26041657C2semidirect
8. #26041489C2semidirect
9. #26041488C2semidirect
10. #26041488C2semidirect
12. #26041407C2semidirect
13. #26041385C2semidirect
15. #441×C181C2semidirect
167. C181#3398semidirect
182. #441D181semidirect

Maximal subgroups

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

subgroupindex
#260420072
#260420012
#260416732
#260416722
#260416572
#260415042
#260415012
#260414892
#260414882
#260414832
#260414072
#260413852
#260413662
#441×C1812
#3398181