Finite Group Databasebeta

← #37545826

Subgroups — #37545826

Order 23168.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041938C211yesmaximal, characteristic
2#26041641C211yesmaximal, characteristic, complemented
3#26041609C211yesmaximal, characteristic, complemented
4#26041597C211yesmaximal, characteristic, complemented
5#26041561C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041501C211yesmaximal, characteristic, complemented
9#26041439C211yesmaximal, characteristic, complemented
10#26041434C211yesmaximal, characteristic, complemented
11#26041384C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041278C211yesmaximal, characteristic, complemented
14C2×#19153739C211yesmaximal, characteristic, complemented
15#554×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18C2×#15668439C2211yescharacteristic
19C2×#15668439C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
21C2×#15668433C2211yescharacteristic
22C22×#13089678C2211yescharacteristic
23#19153804C2211yescharacteristic
24#19153802—21no—
25#19153795C2211yescharacteristic
26#19153794—21no—
27#19153790C2211yescharacteristic
28#19153788—21no—
29C4×Dic362C2211yescharacteristic
30#19153776—21no—
31#19153747C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35#19153740C2211yescharacteristic
36#19153739C2211yescharacteristic
37#19153739C2211yescharacteristic
38#19153739C2211yescharacteristic
39#19153739C2211yescharacteristic
40#19153739C2211yescharacteristic
41#19153739C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44#19153736C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48C4⋊Q8×C181C2211yescharacteristic
49#125×C181—21no—
50C22⋊Q8×C181—21no—
51C22⋊Q8×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic
53C22⋊Q8×C181C2211yescharacteristic
54C4×Q8×C181C2211yescharacteristic
55C4×D4×C181—21no—
56C4×D4×C181C2211yescharacteristic
57C2×C4⋊C4×C181C2211yescharacteristic
58C2×C4⋊C4×C181C2211yescharacteristic
59#15668449—22no—
60#15668449—41no—
61C2×Dic362—22no—
62C2×Dic362—21no—
63C2×Dic362—41no—
64C2×Dic362C2311yescharacteristic
65C2×Dic362C2311yescharacteristic
66C2×D724C2311yescharacteristic
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C22×Dic181—21no—
70C2×C4×D181—21no—
71#15668439—22no—
72#15668439—41no—
73#15668436—41no—
74#15668433—41no—
75C2×#13089678—41no—
76C2×#13089678C2311yescharacteristic
77C2×#13089678C2311yescharacteristic
78C2×#13089678C2311yescharacteristic
79C2×#13089678C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81#15668428—21no—
82#15668428—21no—
83C22⋊Dic181—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87C4×Dic181—21no—
88C2×Q8×C181—21no—
89C2×Q8×C181C2311yescharacteristic
90C2×D4×C181D411yescharacteristic
91C22×C4×C181—21no—
92C22×C4×C181—21no—
93C22×C4×C181C2311yescharacteristic
94C22×C4×C181C2311yescharacteristic
95C4⋊C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181—21no—
98C4⋊C4×C181—21no—
99C4⋊C4×C181—21no—
100C4⋊C4×C181—21no—
101C4⋊C4×C181D411yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C4⋊C4×C181C2311yescharacteristic
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181D411yescharacteristic
108C22⋊C4×C181D411yescharacteristic
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112Dic362—21no—
113Dic362—21no—
114Dic362—21no—
115Dic362—42no—
116Dic362#4112yes—
117D724—21no—
118D724—21no—
119C22⋊D181—42no—
120C22⋊D181—42no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—42no—
125C4×D181—42no—
126#13089678—21no—
127#13089678—21no—
128#13089678—21no—
129#13089678—21no—
130C23×C181C2×D411yescharacteristic
131C23×C181C2×D411yescharacteristic
132Q8×C181—21no—
133Q8×C181—21no—
134Q8×C181—81no—
135D4×C181—22no—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—81no—
145C2×C4×C181—81no—
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2×D411yescharacteristic
150C2×C4×C181C2411yescharacteristic
151C22×C181—21no—
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181#13712yes—
157C22×C181C22×D411yescharacteristic
158C2×D181—41no—
159C2×D181—41no—
160C2×D181—41no—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—21no—
164C4×C181—21no—
165C4×C181—41no—
166C4×C181—41no—
167C4×C181—41no—
168C4×C181—41no—
169C4×C181C2×#4111yescharacteristic
170C4×C181H(5,𝔽2)11yescharacteristic
171Dic181—22no—
172Dic181—41no—
173Dic181—41no—
174C2×C181—22no—
175C2×C181—41no—
176C2×C181#54611yescharacteristic
177C2×C181C2×#13711yescharacteristic
178C2×C181#57711yescharacteristic
179D181—81no—
180C181#331611yescharacteristic, complemented, Sylow
181#3316—1811nomaximal, Sylow
182#554D18111yescharacteristic, complemented
183#548—1811no—
184#489—1811no—
185#483—1811no—
186#481—1811no—
187#475—1811no—
188#470—1811no—
189#465—1811no—
190#464—1811no—
191#452—1811no—
192#451—1811no—
193#441—1811no—
194#435—1811no—
195#417—1811no—
196C2×#103—1811no—
197C2×#41—1811no—
198C22×Q8—1811no—
199C2×Q16—1811no—
200C2×Q16—1811no—
201C2×SD16—1811no—
202C2×M16—1811no—
203C22×C8—1811no—
204C4⋊Q8C2×D18111yescharacteristic
205#125—21no—
206#124—3621no—
207#124—1811no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—3621no—
211C22⋊Q8—1811no—
212C22⋊Q8C2×D18111yescharacteristic
213C22⋊Q8C2×D18111yescharacteristic
214#121—3621no—
215#121—1811no—
216C4×Q8—1811no—
217C4×Q8C2×D18111yescharacteristic
218C4×D4—21no—
219C4×D4—3621no—
220C4×D4C2×D18111yescharacteristic
221C2×C4⋊C4C2×D18111yescharacteristic
222C2×C4⋊C4C2×D18111yescharacteristic
223#106—1811no—
224#106—1811no—
225#106—1811no—
226C4⋊C8—1811no—
227#103—1811no—
228#103—1811no—
229#103—1811no—
230#103—1811no—
231#103—1811no—
232#103—1811no—
233#103—1811no—
234#102—1811no—
235#102—1811no—
236#102—1811no—
237C22⋊C8—1811no—
238C22⋊C8—1811no—
239#97—1811no—
240#41—3622no—
241#41—7241no—
242C2×Q8—21no—
243C2×Q8—3622no—
244C2×Q8—3621no—
245C2×Q8—1811no—
246C2×Q8—1811no—
247C2×Q8—7241no—
248C2×Q8C22×D18111yescharacteristic
249C2×D4—3621no—
250C2×D4—3621no—
251C2×D4—1811no—
252C2×D4C22⋊D18111yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3621no—
256C22×C4—3621no—
257C22×C4C22×D18111yescharacteristic
258C22×C4C22×D18111yescharacteristic
259Q16—3622no—
260Q16—7241no—
261SD16—7241no—
262M16—7241no—
263C2×C8—1811no—
264C2×C8—1811no—
265C2×C8—1811no—
266C2×C8—1811no—
267C2×C8—7241no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—1811no—
277C4⋊C4C22⋊D18111yescharacteristic
278C4⋊C4C22×D18111yescharacteristic
279C4⋊C4C22×D18111yescharacteristic
280C4⋊C4C22×D18111yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4C22⋊D18111yescharacteristic
288C22⋊C4C22⋊D18111yescharacteristic
289C42—21no—
290C42—3621no—
291C42C22×D18111yescharacteristic
292C23—3621no—
293C23C2×C22⋊D18111yescharacteristic
294C23C2×C22⋊D18111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—81no—
298Q8—3621no—
299Q8—3621no—
300Q8—3621no—
301Q8—1812no—
302Q8—7242no—
303D4—22no—
304D4—3621no—
305D4—3621no—
306D4—7242no—
307D4—7242no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3621no—
319C2×C4—3621no—
320C2×C4—3621no—
321C2×C4—7242no—
322C2×C4—7242no—
323C2×C4C2×C22⋊D18111yescharacteristic
324C2×C4C2×C22⋊D18111yescharacteristic
325C2×C4C2×C22⋊D18111yescharacteristic
326C2×C4C2×C22⋊D18111yescharacteristic
327C2×C4C23×D18111yescharacteristic
328C8—3621no—
329C8—3621no—
330C8—3621no—
331C8—3621no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—7241no—
338C22—7241no—
339C22—7241no—
340C22#1915384712yes—
341C22C22×C22⋊D18111yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3622no—
351C4—7241no—
352C4—7241no—
353C4#41×D18111yescharacteristic
354C4#1915386211yescharacteristic
355C2—22no—
356C2—41no—
357C2—14481no—
358C2#2604191711yescharacteristic
359C2C2×#1915384711yescharacteristic
360C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#554×C181C2
socleC22×C181C22×D4
prime radical#26041938C2
hypercenter#554D181
hypocenterC181#3316
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #26041641C2semidirect
3. #26041609C2semidirect
4. #26041597C2semidirect
5. #26041561C2semidirect
6. #26041535C2semidirect
7. #26041510C2semidirect
8. #26041501C2semidirect
9. #26041439C2semidirect
10. #26041434C2semidirect
11. #26041384C2semidirect
12. #26041351C2semidirect
13. #26041278C2semidirect
14. C2×#19153739C2semidirect
15. #554×C181C2semidirect
180. C181#3316semidirect
182. #554D181semidirect

Maximal subgroups

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

subgroupindex
#260419382
#260416412
#260416092
#260415972
#260415612
#260415352
#260415102
#260415012
#260414392
#260414342
#260413842
#260413512
#260412782
C2×#191537392
#554×C1812
#3316181