Finite Group Databasebeta

← #37544627

Subgroups — #37544627

Order 23168.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041857C211yesmaximal, characteristic, complemented
2#26041609C211yesmaximal, characteristic, complemented
3#26041607C211yesmaximal, characteristic, complemented
4#26041597C211yesmaximal, characteristic, complemented
5#26041595C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041530C211yesmaximal, characteristic, complemented
8#26041510C211yesmaximal, characteristic, complemented
9#26041506C211yesmaximal, characteristic, complemented
10#26041441C211yesmaximal, characteristic, complemented
11#26041439C211yesmaximal, characteristic, complemented
12#26041424C211yesmaximal, characteristic, complemented
13#26041422C211yesmaximal, characteristic, complemented
14#26041246C211yesmaximal, characteristic, complemented
15#537×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668434C2211yescharacteristic
21C2×#15668433C2211yescharacteristic
22C22×#13089678C2211yescharacteristic
23#19153795C2211yescharacteristic
24#19153795C2211yescharacteristic
25#19153792—21no—
26#19153790C2211yescharacteristic
27#19153790C2211yescharacteristic
28#19153788—21no—
29#19153788—21no—
30#19153787—21no—
31C2×C4⋊Dic181C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153739C2211yescharacteristic
39#19153739C2211yescharacteristic
40#19153737C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44C22⋊#13089678C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47C22⋊#13089678C2211yescharacteristic
48C2×#41×C181C2211yescharacteristic
49C2×#41×C181C2211yescharacteristic
50C22⋊Q8×C181—21no—
51C22⋊Q8×C181C2211yescharacteristic
52C22⋊Q8×C181C2211yescharacteristic
53#121×C181—21no—
54#121×C181—21no—
55#121×C181—21no—
56#121×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic
58C2×C4⋊C4×C181C2211yescharacteristic
59#15668449—41no—
60#15668449—41no—
61C2×Dic362C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C22×Dic181—21no—
67C2×C4×D181—21no—
68#15668439—41no—
69#15668437—41no—
70#15668436—41no—
71#15668434—41no—
72#15668433—41no—
73C2×#13089678—41no—
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C2×#13089678C2311yescharacteristic
77C2×#13089678C2311yescharacteristic
78C4⋊Dic181—21no—
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81#15668428—21no—
82#15668428—21no—
83C22⋊Dic181—21no—
84C22⋊Dic181—21no—
85#15668426—21no—
86#15668426—21no—
87#41×C181—41no—
88#41×C181—41no—
89#41×C181—41no—
90#41×C181—41no—
91C2×Q8×C181D411yescharacteristic
92C2×Q8×C181C2311yescharacteristic
93C2×D4×C181—21no—
94C2×D4×C181—21no—
95C2×D4×C181—21no—
96C2×D4×C181D411yescharacteristic
97C2×D4×C181D411yescharacteristic
98C2×D4×C181D411yescharacteristic
99C2×D4×C181C2311yescharacteristic
100C22×C4×C181—21no—
101C22×C4×C181—21no—
102C22×C4×C181C2311yescharacteristic
103C22×C4×C181C2311yescharacteristic
104C22×C4×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181—21no—
107C4⋊C4×C181—21no—
108C4⋊C4×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22⋊C4×C181—21no—
112C22⋊C4×C181—21no—
113C22⋊C4×C181—21no—
114C22×D181—21no—
115Dic362—21no—
116Dic362—21no—
117D724—21no—
118D724—21no—
119C22⋊D181—42no—
120C22⋊D181—81no—
121C22⋊D181—81no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—21no—
125C2×Dic181—81no—
126C4×D181—42no—
127#13089678—21no—
128#13089678—21no—
129#13089678—21no—
130#13089678—21no—
131C23×C181—21no—
132C23×C181C2×D411yescharacteristic
133C23×C181C2×D411yescharacteristic
134C23×C181C2×D411yescharacteristic
135Q8×C181—21no—
136Q8×C181—21no—
137Q8×C181—41no—
138D4×C181—21no—
139D4×C181—21no—
140D4×C181—41no—
141D4×C181—41no—
142D4×C181—41no—
143D4×C181—42no—
144D4×C181—81no—
145D4×C181—81no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—42no—
156C2×C4×C181—81no—
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2×D411yescharacteristic
160C2×C4×C181C2411yescharacteristic
161C22×C181—21no—
162C22×C181—21no—
163C22×C181—21no—
164C22×C181—21no—
165C22×C181—21no—
166C22×C181—21no—
167C22×C181—21no—
168C22×C181—21no—
169C22×C181—21no—
170C22×C181—41no—
171C22×C181—41no—
172C22×C181—41no—
173C22×C181C22×D411yescharacteristic
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C4×C181—21no—
178C4×C181—21no—
179C4×C181—21no—
180C4×C181—41no—
181C4×C181—41no—
182C4×C181—41no—
183C4×C181H(5,𝔽2)11yescharacteristic
184C4×C181H(5,𝔽2)11yescharacteristic
185Dic181—41no—
186Dic181—41no—
187Dic181—41no—
188C2×C181—41no—
189C2×C181—41no—
190C2×C181—41no—
191C2×C181—81no—
192C2×C181#53411yescharacteristic
193C2×C181#57711yescharacteristic
194C2×C181#57711yescharacteristic
195D181—81no—
196C181#320411yescharacteristic, complemented, Sylow
197#3204—1811nomaximal, Sylow
198#537—1811no—
199#537D18111yescharacteristic, complemented
200#483—1811no—
201#483—1811no—
202#481—1811no—
203#481—1811no—
204#470—1811no—
205#469—1811no—
206#465—1811no—
207#465—1811no—
208#452—1811no—
209#452—1811no—
210#449—1811no—
211#449—1811no—
212#408—1811no—
213C2×#41—1811no—
214C2×#41C2×D18111yescharacteristic
215C2×#41C2×D18111yescharacteristic
216C2×Q16—1811no—
217C2×SD16—1811no—
218C2×SD16—1811no—
219C2×D8—1811no—
220C2×M16—1811no—
221C22×C8—1811no—
222C22⋊Q8—21no—
223C22⋊Q8—3621no—
224C22⋊Q8—1811no—
225C22⋊Q8—1811no—
226C22⋊Q8C2×D18111yescharacteristic
227C22⋊Q8C2×D18111yescharacteristic
228#121—21no—
229#121—21no—
230#121—21no—
231#121—3621no—
232#121—3621no—
233#121—3621no—
234#121—1811no—
235#121—1811no—
236#121C2×D18111yescharacteristic
237#121C2×D18111yescharacteristic
238C2×C4⋊C4—1811no—
239C2×C4⋊C4C2×D18111yescharacteristic
240#106—1811no—
241#106—1811no—
242#106—1811no—
243#106—1811no—
244#103—1811no—
245#103—1811no—
246#103—1811no—
247#103—1811no—
248#102—1811no—
249#102—1811no—
250#102—1811no—
251#102—1811no—
252C22⋊C8—1811no—
253C22⋊C8—1811no—
254C22⋊C8—1811no—
255C22⋊C8—1811no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—7241no—
261#41—7241no—
262C2×Q8—1811no—
263C2×Q8C22⋊D18111yescharacteristic
264C2×Q8C22×D18111yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—3621no—
269C2×D4—3621no—
270C2×D4—3621no—
271C2×D4—1811no—
272C2×D4C22⋊D18111yescharacteristic
273C2×D4C22⋊D18111yescharacteristic
274C2×D4C22⋊D18111yescharacteristic
275C2×D4C22×D18111yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—3621no—
279C22×C4—3621no—
280C22×C4C22×D18111yescharacteristic
281C22×C4C22×D18111yescharacteristic
282C22×C4C22×D18111yescharacteristic
283Q16—7241no—
284SD16—7241no—
285SD16—7241no—
286D8—7241no—
287M16—7241no—
288C2×C8—1811no—
289C2×C8—1811no—
290C2×C8—1811no—
291C2×C8—1811no—
292C2×C8—7241no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—3621no—
297C4⋊C4—3621no—
298C4⋊C4—3621no—
299C4⋊C4—1811no—
300C4⋊C4—1811no—
301C4⋊C4C22×D18111yescharacteristic
302C4⋊C4C22×D18111yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—3621no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C22⋊C4—3621no—
311C23—21no—
312C23—3621no—
313C23C2×C22⋊D18111yescharacteristic
314C23C2×C22⋊D18111yescharacteristic
315C23C2×C22⋊D18111yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—41no—
319Q8—3621no—
320Q8—3621no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—3621no—
330D4—3621no—
331D4—14481no—
332D4—14481no—
333D4—7242no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—3621no—
346C2×C4—3621no—
347C2×C4—3621no—
348C2×C4—14481no—
349C2×C4—7242no—
350C2×C4C2×C22⋊D18111yescharacteristic
351C2×C4C2×C22⋊D18111yescharacteristic
352C2×C4C2×C22⋊D18111yescharacteristic
353C2×C4C23×D18111yescharacteristic
354C8—3621no—
355C8—3621no—
356C8—3621no—
357C8—3621no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—7241no—
371C22—7241no—
372C22—7241no—
373C22C22×C22⋊D18111yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—7241no—
381C4—7241no—
382C4—7241no—
383C4#1915386111yescharacteristic
384C4#1915386111yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—81no—
389C2—14481no—
390C2#2604183811yescharacteristic
391C2#2604208711yescharacteristic
392C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#537×C181C2
socleC22×C181C22×D4
hypercenter#537D181
hypocenterC181#3204
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #26041857C2semidirect
2. #26041609C2semidirect
3. #26041607C2semidirect
4. #26041597C2semidirect
5. #26041595C2semidirect
6. #26041535C2semidirect
7. #26041530C2semidirect
8. #26041510C2semidirect
9. #26041506C2semidirect
10. #26041441C2semidirect
11. #26041439C2semidirect
12. #26041424C2semidirect
13. #26041422C2semidirect
14. #26041246C2semidirect
15. #537×C181C2semidirect
196. C181#3204semidirect
199. #537D181semidirect

Maximal subgroups

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

subgroupindex
#260418572
#260416092
#260416072
#260415972
#260415952
#260415352
#260415302
#260415102
#260415062
#260414412
#260414392
#260414242
#260414222
#260412462
#537×C1812
#3204181