Finite Group Databasebeta

← #37547811

Subgroups — #37547811

Order 23168.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042007C213yesmaximal, complemented
2#26042005C211yesmaximal, characteristic, complemented
3#26041997C213yesmaximal, complemented
4#26041970C213yesmaximal, complemented
5#26041968C213yesmaximal, complemented
6Q8×D724C211yesmaximal, characteristic, complemented
7Q8×C22⋊D181C211yesmaximal, characteristic, complemented
8Q8×C22⋊D181C211yesmaximal, characteristic, complemented
9#26041891C213yesmaximal, complemented
10#26041887C213yesmaximal, complemented
11#26041833C213yesmaximal, complemented
12#26041807C213yesmaximal
13#26041755C211yesmaximal, characteristic, complemented
14C4×Q8×D181C211yesmaximal, characteristic, complemented
15#519×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18#19153816C2213yes—
19#19153815C2213yes—
20#19153814C2213yes—
21#19153814C2213yes—
22#19153811C2213yes—
23#19153810C2216yes—
24#19153809C2213yes—
25#19153809C2216yes—
26#19153808C2216yes—
27#19153808C2213yes—
28#19153807C2213yes—
29#19153806C2216yes—
30#19153806C2216yes—
31#19153806C2213yes—
32#19153802C2213yes—
33#19153801C2213yes—
34#19153798C2213yes—
35#19153796C2213yes—
36#19153796C2213yes—
37#19153795C2213yes—
38#19153795C2213yes—
39#19153794C2213yes—
40#19153792C2213yes—
41#19153792C2213yes—
42#19153789C2211yescharacteristic
43Q8×Dic181C2211yescharacteristic
44Q8×Dic181C2211yescharacteristic
45C4×Dic362C2213yes—
46C4×Dic362C2213yes—
47C4×Dic362C2216yes—
48C4×Dic362C2213yes—
49#19153781C2213yes—
50#19153781C2213yes—
51C4×D724C2213yes—
52C4×C22⋊D181C2213yes—
53C4×C22⋊D181C2213yes—
54#19153774C2213yes—
55#19153773C2213yes—
56#19153773C2216yes—
57C4⋊C4×D181C2213yes—
58C42×D181C2213yes—
59C22×Q8×C181C2211yescharacteristic
60C4×Q8×C181C2213yes—
61C4×Q8×C181C2211yescharacteristic
62C4×Q8×C181C2213yes—
63C4×Q8×C181C2211yescharacteristic
64#117×C181C2213yes—
65#117×C181C2213yes—
66C2×Dic362C2313yes—
67C2×Dic362C2313yes—
68C2×Dic362C2313yes—
69Q8×D181—24no—
70Q8×D181—24no—
71C2×D724C2311yescharacteristic
72C2×C22⋊D181C2311yescharacteristic
73C2×C22⋊D181C2311yescharacteristic
74C2×C4×D181C2311yescharacteristic
75C2×C4×D181C2313yes—
76C2×C4×D181C2313yes—
77C2×C4×D181C2313yes—
78C4⋊Dic181C2313yes—
79C4⋊Dic181C2313yes—
80C4⋊Dic181C2313yes—
81C4⋊Dic181C2316yes—
82C4⋊Dic181C2313yes—
83#15668428C2316yes—
84#15668428C2313yes—
85#15668428C2313yes—
86#15668428C2316yes—
87#15668428C2316yes—
88#15668428C2313yes—
89#15668428C2313yes—
90#15668428C2311yescharacteristic
91#15668428C2316yes—
92#15668428C2313yes—
93C22⋊Dic181C2313yes—
94C22⋊Dic181C2313yes—
95#15668426C2311yescharacteristic
96#15668426C2313yes—
97#15668426C2313yes—
98#15668426C2313yes—
99#15668426C2313yes—
100#15668426C2313yes—
101C4×Dic181C2313yes—
102C4×Dic181C2313yes—
103C4×Dic181C2313yes—
104C4×Dic181C2316yes—
105C4×Dic181C2313yes—
106C2×Q8×C181—24no—
107C2×Q8×C181C2313yes—
108C2×Q8×C181C2311yescharacteristic
109C22×C4×C181C2313yes—
110C4⋊C4×C181C2313yes—
111C4⋊C4×C181C2313yes—
112C4⋊C4×C181C2313yes—
113C4⋊C4×C181C2313yes—
114C22⋊C4×C181C2311yescharacteristic
115C22⋊C4×C181C2313yes—
116C42×C181C2313yes—
117C42×C181C2316yes—
118C42×C181C2313yes—
119C22×D181C2411yescharacteristic
120C22×D181C2411yescharacteristic
121Dic362—26no—
122Dic362—26no—
123Dic362—26no—
124D724—22no—
125C22⋊D181—22no—
126C22⋊D181—22no—
127C2×Dic181C2413yes—
128C2×Dic181C2411yescharacteristic
129C2×Dic181C2413yes—
130C2×Dic181C2413yes—
131C2×Dic181C2413yes—
132C2×Dic181C2411yescharacteristic
133C4×D181—26no—
134C4×D181—26no—
135C4×D181—26no—
136C4×D181—22no—
137C23×C181C2411yescharacteristic
138Q8×C181—26no—
139Q8×C181#4114yes—
140C2×C4×C181—26no—
141C2×C4×C181C2411yescharacteristic
142C2×C4×C181C2413yes—
143C2×C4×C181C2413yes—
144C2×C4×C181C2411yescharacteristic
145C2×C4×C181C2413yes—
146C2×C4×C181C2413yes—
147C22×C181—21no—
148C22×C181—21no—
149C22×C181—21no—
150C22×C181C2511yescharacteristic
151C2×D181—21no—
152C2×D181—21no—
153C2×D181—21no—
154C2×D181—21no—
155C2×D181—21no—
156C2×D1812−1+412yes—
157C4×C181—23no—
158C4×C181—23no—
159C4×C181—21no—
160C4×C181—21no—
161C4×C181—23no—
162C4×C181C2×#4116yes—
163Dic181—23no—
164Dic181—23no—
165Dic181—23no—
166Dic181—21no—
167Dic181—23no—
168Dic1812−1+412yes—
169C2×C181—41no—
170C2×C181C22×#4111yescharacteristic
171C2×C181C2×2−1+411yescharacteristic
172C2×C181C2×2−1+411yescharacteristic
173D181—22no—
174D181—41no—
175C181#352311yescharacteristic, complemented, Sylow
176#3523—1811nomaximal, Sylow
177#557—1811no—
178#557—1813no—
179#556—1813no—
180#552—1813no—
181#552—1813no—
182D4×Q8—1811no—
183D4×Q8—1811no—
184D4×Q8—1811no—
185#541—1813no—
186#541—1813no—
187#533—1813no—
188#529—1813no—
189#519—1811no—
190#519D18111yescharacteristic, complemented
191C2×C4×Q8—1811no—
192C22×Q8—1811no—
193C22×Q8—1811no—
194C22×Q8C2×D18111yescharacteristic
195C4⋊Q8—1813no—
196C4⋊Q8—1813no—
197C4⋊Q8—1813no—
198C4⋊Q8—1813no—
199#126—1813no—
200#126—1813no—
201#126—1816no—
202#126—1816no—
203#125—1813no—
204#125—1816no—
205#125—1816no—
206#125—1813no—
207#125—1816no—
208#125—1813no—
209#124—1813no—
210#123—1813no—
211#123—1813no—
212C22⋊Q8—1813no—
213C22⋊Q8—1813no—
214C22⋊Q8—1813no—
215C22⋊Q8—1813no—
216C22⋊Q8—1813no—
217C22⋊Q8—1813no—
218C22⋊Q8—1813no—
219#121—1811no—
220C4×Q8—1811no—
221C4×Q8—1811no—
222C4×Q8—1813no—
223C4×Q8—1813no—
224C4×Q8—1813no—
225C4×Q8—1813no—
226C4×Q8—1813no—
227C4×Q8—1816no—
228C4×Q8C2×D18111yescharacteristic
229C4×Q8C2×D18113yes—
230C4×Q8C2×D18111yescharacteristic
231C4×Q8C2×D18113yes—
232C4×D4—1813no—
233C4×D4—1813no—
234C4×D4—1813no—
235#117—1813no—
236#117—1813no—
237#117—1816no—
238#117C2×D18113yes—
239#117C2×D18113yes—
240C2×C4⋊C4—1813no—
241C2×C42—1813no—
242C2×Q8—24no—
243C2×Q8—3624no—
244C2×Q8—3624no—
245C2×Q8—1813no—
246C2×Q8—1813no—
247C2×Q8—1813no—
248C2×Q8C22×D18113yes—
249C2×Q8C22×D18111yescharacteristic
250C2×D4—1811no—
251C2×D4—1811no—
252C2×D4—1811no—
253C22×C4—1811no—
254C22×C4—1813no—
255C22×C4—1813no—
256C22×C4—1813no—
257C22×C4C22×D18113yes—
258C4⋊C4—1813no—
259C4⋊C4—1816no—
260C4⋊C4—1813no—
261C4⋊C4—1813no—
262C4⋊C4—1813no—
263C4⋊C4—1816no—
264C4⋊C4—1816no—
265C4⋊C4—1813no—
266C4⋊C4—1813no—
267C4⋊C4—1813no—
268C4⋊C4—1811no—
269C4⋊C4—1816no—
270C4⋊C4—1816no—
271C4⋊C4—1813no—
272C4⋊C4—1813no—
273C4⋊C4C22×D18113yes—
274C4⋊C4C22×D18113yes—
275C4⋊C4C22×D18113yes—
276C4⋊C4C22×D18113yes—
277C22⋊C4—1813no—
278C22⋊C4—1813no—
279C22⋊C4—1811no—
280C22⋊C4—1813no—
281C22⋊C4—1813no—
282C22⋊C4—1813no—
283C22⋊C4—1813no—
284C22⋊C4—1813no—
285C22⋊C4C22×D18111yescharacteristic
286C22⋊C4C22×D18113yes—
287C42—1813no—
288C42—1813no—
289C42—1813no—
290C42—1816no—
291C42—1813no—
292C42C22×D18113yes—
293C42C22×D18116yes—
294C42C22×D18113yes—
295C23—1811no—
296C23—1811no—
297C23C23×D18111yescharacteristic
298Q8—26no—
299Q8—3626no—
300Q8—3626no—
301Q8—3626no—
302Q8#1566844914yes—
303D4—3622no—
304D4—3622no—
305D4—3622no—
306C2×C4—26no—
307C2×C4—3626no—
308C2×C4—3622no—
309C2×C4—3626no—
310C2×C4—3626no—
311C2×C4—1813no—
312C2×C4—1811no—
313C2×C4—1813no—
314C2×C4—1813no—
315C2×C4—1813no—
316C2×C4—1811no—
317C2×C4C23×D18113yes—
318C2×C4C23×D18111yescharacteristic
319C2×C4C23×D18113yes—
320C2×C4C23×D18111yescharacteristic
321C2×C4C23×D18113yes—
322C2×C4C23×D18113yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—3621no—
327C22—3621no—
328C22—3621no—
329C22—3621no—
330C22—3621no—
331C22—1812no—
332C22C24×D18111yescharacteristic
333C4—23no—
334C4—23no—
335C4—21no—
336C4—21no—
337C4—23no—
338C4—3623no—
339C4—3623no—
340C4—3623no—
341C4—3621no—
342C4—3623no—
343C4—1812no—
344C4C2×#1566844916yes—
345C2—41no—
346C2—3622no—
347C2—7241no—
348C2C22×#1566844911yescharacteristic
349C22−1+4×D18111yescharacteristic
350C22−1+4×D18111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#519×C181C2
prime radical#19153789C22
hypercenter#519D181
hypocenterC181#3523
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #26042007C2semidirect
2. #26042005C2semidirect
3. #26041997C2semidirect
4. #26041970C2semidirect
5. #26041968C2semidirect
6. Q8×D724C2semidirect
7. Q8×C22⋊D181C2semidirect
8. Q8×C22⋊D181C2semidirect
9. #26041891C2semidirect
10. #26041887C2semidirect
11. #26041833C2semidirect
13. #26041755C2semidirect
14. C4×Q8×D181C2semidirect
15. #519×C181C2semidirect
175. C181#3523semidirect
190. #519D181semidirect

Maximal subgroups

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

subgroupindex
#260420072
#260420052
#260419972
#260419702
#260419682
Q8×D7242
Q8×C22⋊D1812
#260418912
#260418872
#260418332
#260418072
#260417552
C4×Q8×D1812
#519×C1812
#3523181