Finite Group Databasebeta

← #37545602

Subgroups — #37545602

Order 23168.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041938C211yesmaximal, characteristic
2#26041636C211yesmaximal, characteristic, complemented
3#26041620C211yesmaximal, characteristic, complemented
4#26041592C211yesmaximal, characteristic, complemented
5#26041583C211yesmaximal, characteristic, complemented
6#26041518C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041501C211yesmaximal, characteristic, complemented
9#26041439C211yesmaximal, characteristic, complemented
10#26041434C211yesmaximal, characteristic, complemented
11C4×#15668439C211yesmaximal, characteristic, complemented
12D4×#13089678C211yesmaximal, characteristic, complemented
13#26041273C211yesmaximal, characteristic, complemented
14C2×#19153739C211yesmaximal, characteristic, complemented
15#552×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18C2×#15668439C2211yescharacteristic
19C2×#15668439C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
21C22×#13089678C2211yescharacteristic
22C22×#13089678C2211yescharacteristic
23#19153804C2211yescharacteristic
24#19153802—21no—
25#19153795C2211yescharacteristic
26#19153794—21no—
27#19153790C2211yescharacteristic
28#19153788—21no—
29C4×Dic362C2211yescharacteristic
30#19153776—21no—
31#19153749C2211yescharacteristic
32#19153749C2211yescharacteristic
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
47C4×#13089678C2211yescharacteristic
48#126×C181—21no—
49#125×C181—21no—
50#125×C181C2211yescharacteristic
51#123×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic
53C22⋊Q8×C181C2211yescharacteristic
54C4×Q8×C181C2211yescharacteristic
55C4×D4×C181C2211yescharacteristic
56#117×C181—21no—
57#117×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—
74C2×#13089678—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×C181C2311yescharacteristic
89C2×D4×C181D411yescharacteristic
90C22×C4×C181—21no—
91C22×C4×C181C2311yescharacteristic
92C22×C4×C181C2311yescharacteristic
93C4⋊C4×C181—21no—
94C4⋊C4×C181—21no—
95C4⋊C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181—21no—
98C4⋊C4×C181—21no—
99C4⋊C4×C181D411yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C22⋊C4×C181—21no—
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181D411yescharacteristic
107C22⋊C4×C181D411yescharacteristic
108C42×C181—21no—
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—
134D4×C181—22no—
135C2×C4×C181—21no—
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—42no—
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2411yescharacteristic
149C22×C181—21no—
150C22×C181—21no—
151C22×C181—21no—
152C22×C181—21no—
153C22×C181—21no—
154C22×C181#13512yes—
155C22×C181C22×D411yescharacteristic
156C2×D181—41no—
157C2×D181—41no—
158C2×D181—41no—
159C4×C181—21no—
160C4×C181—21no—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—41no—
164C4×C181—41no—
165C4×C181—41no—
166C4×C181—41no—
167C4×C181C2×#4111yescharacteristic
168C4×C181H(5,𝔽2)11yescharacteristic
169Dic181—22no—
170Dic181—41no—
171Dic181—41no—
172C2×C181—22no—
173C2×C181—41no—
174C2×C181#54611yescharacteristic
175C2×C181C2×#13511yescharacteristic
176C2×C181#57811yescharacteristic
177D181—81no—
178C181#330011yescharacteristic, complemented, Sylow
179#3300—1811nomaximal, Sylow
180#552D18111yescharacteristic, complemented
181#548—1811no—
182#488—1811no—
183#485—1811no—
184#480—1811no—
185#479—1811no—
186#467—1811no—
187#465—1811no—
188#464—1811no—
189#452—1811no—
190#451—1811no—
191C4×Q16—1811no—
192C8×D4—1811no—
193#416—1811no—
194C2×#103—1811no—
195C2×#41—1811no—
196C22×Q8—1811no—
197C2×Q16—1811no—
198C2×Q16—1811no—
199C2×SD16—1811no—
200C22×C8—1811no—
201C22×C8—1811no—
202#126—21no—
203#125—21no—
204#125C2×D18111yescharacteristic
205#124—3621no—
206#124—1811no—
207#123—21no—
208C22⋊Q8—3621no—
209C22⋊Q8—1811no—
210C22⋊Q8C2×D18111yescharacteristic
211C22⋊Q8C2×D18111yescharacteristic
212#121—3621no—
213#121—1811no—
214C4×Q8—1811no—
215C4×Q8C2×D18111yescharacteristic
216C4×D4—3621no—
217C4×D4C2×D18111yescharacteristic
218#117—21no—
219#117C2×D18111yescharacteristic
220C2×C4⋊C4C2×D18111yescharacteristic
221#107—1811no—
222#107—1811no—
223#106—1811no—
224C4⋊C8—1811no—
225#103—1811no—
226#103—1811no—
227#103—1811no—
228#103—1811no—
229#103—1811no—
230#103—1811no—
231#103—1811no—
232#102—1811no—
233#102—1811no—
234#102—1811no—
235C22⋊C8—1811no—
236C22⋊C8—1811no—
237C4×C8—1811no—
238#41—3622no—
239#41—7241no—
240C2×Q8—3622no—
241C2×Q8—3621no—
242C2×Q8—1811no—
243C2×Q8—1811no—
244C2×Q8—7241no—
245C2×Q8C22×D18111yescharacteristic
246C2×D4—3621no—
247C2×D4—3621no—
248C2×D4—1811no—
249C2×D4C22⋊D18111yescharacteristic
250C22×C4—21no—
251C22×C4—3621no—
252C22×C4—3621no—
253C22×C4C22×D18111yescharacteristic
254C22×C4C22×D18111yescharacteristic
255Q16—3622no—
256Q16—7241no—
257SD16—7241no—
258C2×C8—1811no—
259C2×C8—1811no—
260C2×C8—1811no—
261C2×C8—1811no—
262C2×C8—7241no—
263C2×C8—7241no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—3621no—
271C4⋊C4—3621no—
272C4⋊C4—1811no—
273C4⋊C4C22⋊D18111yescharacteristic
274C4⋊C4C22×D18111yescharacteristic
275C4⋊C4C22×D18111yescharacteristic
276C4⋊C4C22×D18111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C22⋊C4C22⋊D18111yescharacteristic
285C22⋊C4C22⋊D18111yescharacteristic
286C42—21no—
287C42—21no—
288C42—3621no—
289C42C22×D18111yescharacteristic
290C23—3621no—
291C23C2×C22⋊D18111yescharacteristic
292C23C2×C22⋊D18111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3621no—
296Q8—3621no—
297Q8—3621no—
298Q8—1812no—
299Q8—7242no—
300D4—22no—
301D4—3621no—
302D4—3621no—
303D4—7242no—
304D4—7242no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—7242no—
318C2×C4—7242no—
319C2×C4C2×C22⋊D18111yescharacteristic
320C2×C4C2×C22⋊D18111yescharacteristic
321C2×C4C2×C22⋊D18111yescharacteristic
322C2×C4C2×C22⋊D18111yescharacteristic
323C2×C4C23×D18111yescharacteristic
324C8—3621no—
325C8—3621no—
326C8—3621no—
327C8—3621no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—7241no—
334C22—7241no—
335C22—7241no—
336C22#1915383612yes—
337C22C22×C22⋊D18111yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3622no—
347C4—7241no—
348C4—7241no—
349C4#41×D18111yescharacteristic
350C4#1915386211yescharacteristic
351C2—22no—
352C2—41no—
353C2—14481no—
354C2#2604191711yescharacteristic
355C2C2×#1915383611yescharacteristic
356C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#552×C181C2
socleC22×C181C22×D4
prime radical#26041938C2
hypercenter#552D181
hypocenterC181#3300
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #26041636C2semidirect
3. #26041620C2semidirect
4. #26041592C2semidirect
5. #26041583C2semidirect
6. #26041518C2semidirect
7. #26041510C2semidirect
8. #26041501C2semidirect
9. #26041439C2semidirect
10. #26041434C2semidirect
11. C4×#15668439C2semidirect
12. D4×#13089678C2semidirect
13. #26041273C2semidirect
14. C2×#19153739C2semidirect
15. #552×C181C2semidirect
178. C181#3300semidirect
180. #552D181semidirect

Maximal subgroups

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

subgroupindex
#260419382
#260416362
#260416202
#260415922
#260415832
#260415182
#260415102
#260415012
#260414392
#260414342
C4×#156684392
D4×#130896782
#260412732
C2×#191537392
#552×C1812
#3300181