Finite Group Databasebeta

← #37544603

Subgroups — #37544603

Order 23168.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041857C211yesmaximal, characteristic
2#26041620C211yesmaximal, characteristic, complemented
3#26041612C211yesmaximal, characteristic, complemented
4#26041607C211yesmaximal, characteristic, complemented
5#26041603C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041527C211yesmaximal, characteristic, complemented
8#26041518C211yesmaximal, characteristic, complemented
9#26041510C211yesmaximal, characteristic, complemented
10#26041439C211yesmaximal, characteristic, complemented
11#26041439C211yesmaximal, characteristic, complemented
12#26041435C211yesmaximal, characteristic, complemented
13#26041418C211yesmaximal, characteristic, complemented
14#26041246C211yesmaximal, characteristic, complemented
15#536×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668439C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
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#19153749C2211yescharacteristic
33#19153749C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153739C2211yescharacteristic
41#19153739C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44C22⋊#13089678C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47C22⋊#13089678C2211yescharacteristic
48C2×#41×C181C2211yescharacteristic
49C22×Q8×C181C2211yescharacteristic
50C4⋊Q8×C181—21no—
51#124×C181—21no—
52#123×C181—21no—
53#123×C181—21no—
54C22⋊Q8×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56C22⋊Q8×C181C2211yescharacteristic
57C22⋊Q8×C181C2211yescharacteristic
58#117×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#15668439—41no—
70#15668436—41no—
71#15668436—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—
89C2×Q8×C181—21no—
90C2×Q8×C181—41no—
91C2×Q8×C181—41no—
92C2×Q8×C181D411yescharacteristic
93C2×Q8×C181C2311yescharacteristic
94C2×Q8×C181C2311yescharacteristic
95C2×D4×C181D411yescharacteristic
96C2×D4×C181D411yescharacteristic
97C2×D4×C181D411yescharacteristic
98C22×C4×C181—21no—
99C22×C4×C181C2311yescharacteristic
100C22×C4×C181C2311yescharacteristic
101C22×C4×C181C2311yescharacteristic
102C4⋊C4×C181—21no—
103C4⋊C4×C181—21no—
104C4⋊C4×C181—21no—
105C4⋊C4×C181—21no—
106C4⋊C4×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C22⋊C4×C181—21no—
109C22⋊C4×C181—21no—
110C22⋊C4×C181—21no—
111C22⋊C4×C181—21no—
112C22⋊C4×C181—21no—
113C42×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×C181C2×D411yescharacteristic
132C23×C181C2×D411yescharacteristic
133C23×C181C2×D411yescharacteristic
134Q8×C181—21no—
135Q8×C181—21no—
136Q8×C181—21no—
137Q8×C181—21no—
138Q8×C181—41no—
139Q8×C181—42no—
140D4×C181—41no—
141D4×C181—41no—
142D4×C181—41no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
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—42no—
154C2×C4×C181C2×D411yescharacteristic
155C2×C4×C181C2×D411yescharacteristic
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2411yescharacteristic
158C22×C181—21no—
159C22×C181—21no—
160C22×C181—21no—
161C22×C181—21no—
162C22×C181—21no—
163C22×C181—21no—
164C22×C181—21no—
165C22×C181—21no—
166C22×C181—21no—
167C22×C181C22×D411yescharacteristic
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C4×C181—21no—
172C4×C181—21no—
173C4×C181—21no—
174C4×C181—41no—
175C4×C181—41no—
176C4×C181—41no—
177C4×C181—41no—
178C4×C181H(5,𝔽2)11yescharacteristic
179C4×C181H(5,𝔽2)11yescharacteristic
180Dic181—41no—
181Dic181—41no—
182Dic181—41no—
183C2×C181—41no—
184C2×C181—41no—
185C2×C181—41no—
186C2×C181#53411yescharacteristic
187C2×C181#57711yescharacteristic
188C2×C181#57811yescharacteristic
189D181—81no—
190C181#320111yescharacteristic, complemented, Sylow
191#3201—1811nomaximal, Sylow
192#537—1811no—
193#536D18111yescharacteristic, complemented
194#485—1811no—
195#484—1811no—
196#483—1811no—
197#482—1811no—
198#470—1811no—
199#468—1811no—
200#467—1811no—
201#465—1811no—
202#452—1811no—
203#452—1811no—
204#451—1811no—
205#448—1811no—
206#408—1811no—
207C2×#41—1811no—
208C2×#41C2×D18111yescharacteristic
209C22×Q8C2×D18111yescharacteristic
210C2×Q16—1811no—
211C2×Q16—1811no—
212C2×SD16—1811no—
213C2×SD16—1811no—
214C2×M16—1811no—
215C22×C8—1811no—
216C4⋊Q8—21no—
217#124—21no—
218#123—21no—
219#123—21no—
220C22⋊Q8—3621no—
221C22⋊Q8—1811no—
222C22⋊Q8—1811no—
223C22⋊Q8C2×D18111yescharacteristic
224C22⋊Q8C2×D18111yescharacteristic
225C22⋊Q8C2×D18111yescharacteristic
226C22⋊Q8C2×D18111yescharacteristic
227#121—3621no—
228#121—3621no—
229#121—3621no—
230#121—1811no—
231#121—1811no—
232#117C2×D18111yescharacteristic
233C2×C4⋊C4—1811no—
234#107—1811no—
235#107—1811no—
236#106—1811no—
237#106—1811no—
238#103—1811no—
239#103—1811no—
240#103—1811no—
241#103—1811no—
242#103—1811no—
243#103—1811no—
244#102—1811no—
245#102—1811no—
246C22⋊C8—1811no—
247C22⋊C8—1811no—
248C22⋊C8—1811no—
249C22⋊C8—1811no—
250#41—41no—
251#41—41no—
252#41—7241no—
253#41—7241no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—1811no—
258C2×Q8C22⋊D18111yescharacteristic
259C2×Q8C22×D18111yescharacteristic
260C2×Q8C22×D18111yescharacteristic
261C2×D4—3621no—
262C2×D4—3621no—
263C2×D4—3621no—
264C2×D4—1811no—
265C2×D4C22⋊D18111yescharacteristic
266C2×D4C22⋊D18111yescharacteristic
267C2×D4C22⋊D18111yescharacteristic
268C22×C4—21no—
269C22×C4—3621no—
270C22×C4—3621no—
271C22×C4C22×D18111yescharacteristic
272C22×C4C22×D18111yescharacteristic
273C22×C4C22×D18111yescharacteristic
274Q16—7241no—
275Q16—7241no—
276SD16—7241no—
277SD16—7241no—
278M16—7241no—
279C2×C8—1811no—
280C2×C8—1811no—
281C2×C8—1811no—
282C2×C8—1811no—
283C2×C8—7241no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—3621no—
289C4⋊C4—3621no—
290C4⋊C4—3621no—
291C4⋊C4—1811no—
292C4⋊C4—1811no—
293C4⋊C4C22×D18111yescharacteristic
294C4⋊C4C22×D18111yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C42—21no—
305C23—3621no—
306C23C2×C22⋊D18111yescharacteristic
307C23C2×C22⋊D18111yescharacteristic
308C23C2×C22⋊D18111yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—41no—
314Q8—42no—
315Q8—3621no—
316Q8—3621no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—3621no—
321D4—3621no—
322D4—14481no—
323D4—14481no—
324D4—7242no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—3621no—
337C2×C4—3621no—
338C2×C4—3621no—
339C2×C4—14481no—
340C2×C4—7242no—
341C2×C4C2×C22⋊D18111yescharacteristic
342C2×C4C2×C22⋊D18111yescharacteristic
343C2×C4C2×C22⋊D18111yescharacteristic
344C2×C4C23×D18111yescharacteristic
345C8—3621no—
346C8—3621no—
347C8—3621no—
348C8—3621no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—7241no—
359C22—7241no—
360C22—7241no—
361C22C22×C22⋊D18111yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—7241no—
370C4—7241no—
371C4—7241no—
372C4#1915386111yescharacteristic
373C4#1915386111yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—14481no—
378C2#2604183811yescharacteristic
379C2#2604208711yescharacteristic
380C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#536×C181C2
socleC22×C181C22×D4
prime radical#26041857C2
hypercenter#536D181
hypocenterC181#3201
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #26041620C2semidirect
3. #26041612C2semidirect
4. #26041607C2semidirect
5. #26041603C2semidirect
6. #26041535C2semidirect
7. #26041527C2semidirect
8. #26041518C2semidirect
9. #26041510C2semidirect
10. #26041439C2semidirect
11. #26041439C2semidirect
12. #26041435C2semidirect
13. #26041418C2semidirect
14. #26041246C2semidirect
15. #536×C181C2semidirect
190. C181#3201semidirect
193. #536D181semidirect

Maximal subgroups

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

subgroupindex
#260418572
#260416202
#260416122
#260416072
#260416032
#260415352
#260415272
#260415182
#260415102
#260414392
#260414352
#260414182
#260412462
#536×C1812
#3201181