Finite Group Databasebeta

← #37545816

Subgroups — #37545816

Order 23168.

382 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#26041967C211yesmaximal, characteristic, complemented
2#26041932C211yesmaximal, characteristic
3#26041619C211yesmaximal, characteristic, complemented
4#26041613C211yesmaximal, characteristic, complemented
5#26041576C211yesmaximal, characteristic, complemented
6#26041536C211yesmaximal, characteristic, complemented
7#26041519C211yesmaximal, characteristic, complemented
8#26041486C211yesmaximal, characteristic, complemented
9#26041444C211yesmaximal, characteristic, complemented
10#26041444C211yesmaximal, characteristic, complemented
11#26041381C211yesmaximal, characteristic, complemented
12#26041352C211yesmaximal, characteristic, complemented
13#26041279C211yesmaximal, characteristic, complemented
14#26041274C211yesmaximal, characteristic, complemented
15#491×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668450C2211yescharacteristic
18C2×#15668439C2211yescharacteristic
19C2×#15668439C2211yescharacteristic
20C2×#15668438C2211yescharacteristic
21C2×#15668432C2211yescharacteristic
22C2×C8×D181C2211yescharacteristic
23#19153811—21no—
24#19153807—21no—
25#19153801—21no—
26#19153801—21no—
27#19153801—21no—
28#19153796C2211yescharacteristic
29#19153796C2211yescharacteristic
30#19153792C2211yescharacteristic
31#19153792C2211yescharacteristic
32#19153791—21no—
33C4×Dic362C2211yescharacteristic
34C4×D724C2211yescharacteristic
35C4×D724C2211yescharacteristic
36#19153774—21no—
37#19153774C2211yescharacteristic
38C4⋊C4×D181—21no—
39C4⋊C4×D181C2211yescharacteristic
40#19153750C2211yescharacteristic
41#19153749C2211yescharacteristic
42#19153747C2211yescharacteristic
43C4⋊#13089678C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153728C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic
53#125×C181C2211yescharacteristic
54#103×C181C2211yescharacteristic
55#103×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59#15668450—22no—
60#15668450—22no—
61#15668450—41no—
62#15668450—41no—
63C2×Dic362C2311yescharacteristic
64C2×D724—21no—
65C2×D724—21no—
66C2×D724D411yescharacteristic
67C2×D724C2311yescharacteristic
68C2×C4×D181—21no—
69C2×C4×D181—21no—
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72C2×C4×D181C2311yescharacteristic
73C2×C4×D181C2311yescharacteristic
74#15668439—41no—
75#15668439—41no—
76#15668438—22no—
77#15668432—41no—
78C2×#13089678C2311yescharacteristic
79C2×#13089678C2311yescharacteristic
80C8×D181—41no—
81C4⋊Dic181—21no—
82C4⋊Dic181—21no—
83C4⋊Dic181D411yescharacteristic
84C4⋊Dic181C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86#15668428—21no—
87#15668428—21no—
88#15668428—21no—
89#15668428—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426D411yescharacteristic
97#15668426D411yescharacteristic
98C4×Dic181—21no—
99C4×Dic181—21no—
100C2×Q8×C181C2311yescharacteristic
101C2×Q8×C181C2311yescharacteristic
102C2×C8×C181C2311yescharacteristic
103C2×C8×C181C2311yescharacteristic
104C4⋊C4×C181—21no—
105C4⋊C4×C181—21no—
106C4⋊C4×C181—21no—
107C4⋊C4×C181—21no—
108C4⋊C4×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181C2×D411yescharacteristic
113C22×D181C2×D411yescharacteristic
114Dic362—21no—
115Dic362—21no—
116D724—21no—
117D724—41no—
118D724—42no—
119D724—42no—
120D724#4112yes—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181C2×D411yescharacteristic
125C2×Dic181C2×D411yescharacteristic
126C4×D181—21no—
127C4×D181—21no—
128C4×D181—21no—
129C4×D181—21no—
130C4×D181—42no—
131C4×D181—42no—
132C4×D181—42no—
133C4×D181—42no—
134#13089678—21no—
135#13089678—21no—
136Q8×C181—21no—
137Q8×C181—21no—
138Q8×C181—21no—
139Q8×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2411yescharacteristic
147C8×C181—21no—
148C8×C181—21no—
149C22×C181C22×D411yescharacteristic
150C2×D181—21no—
151C2×D181—21no—
152C2×D181—22no—
153C2×D181—21no—
154C2×D181—21no—
155C2×D181—21no—
156C2×D181—21no—
157C2×D181—41no—
158C2×D181—41no—
159C4×C181—21no—
160C4×C181—21no—
161C4×C181—41no—
162C4×C181—41no—
163C4×C181—41no—
164C4×C181—41no—
165C4×C181C2×#4111yescharacteristic
166C4×C181H(5,𝔽2)11yescharacteristic
167Dic181—21no—
168Dic181—21no—
169Dic181—41no—
170Dic181—41no—
171Dic181—41no—
172C2×C181#54611yescharacteristic
173C2×C181#57511yescharacteristic
174C2×C181#57811yescharacteristic
175D181—41no—
176D181—41no—
177D181—42no—
178C181#331511yescharacteristic, complemented, Sylow
179#3315—1811nomaximal, Sylow
180#552—1811no—
181#547—1811no—
182#491D18111yescharacteristic, complemented
183#485—1811no—
184#484—1811no—
185#478—1811no—
186#470—1811no—
187#467—1811no—
188#462—1811no—
189#452—1811no—
190#452—1811no—
191#440—1811no—
192#435—1811no—
193#417—1811no—
194#416—1811no—
195C2×#41—1811no—
196C2×#41—1811no—
197C2×Q16—1811no—
198C2×Q16—1811no—
199C2×SD16—1811no—
200C2×M16—1811no—
201C22×C8—1811no—
202C4⋊Q8C2×D18111yescharacteristic
203#126—3621no—
204#125—3621no—
205#125C2×D18111yescharacteristic
206#123—3621no—
207#123—3621no—
208#123—3621no—
209C22⋊Q8—1811no—
210C22⋊Q8—1811no—
211C22⋊Q8—1811no—
212C22⋊Q8—1811no—
213#121—3621no—
214C4×Q8—1811no—
215C4×D4—1811no—
216C4×D4—1811no—
217#117—3621no—
218#117—1811no—
219C2×C4⋊C4—3621no—
220C2×C4⋊C4—1811no—
221#107—1811no—
222#107—1811no—
223#106—1811no—
224C4⋊C8—1811no—
225#103—1811no—
226#103—1811no—
227#103—1811no—
228#103C2×D18111yescharacteristic
229#103C2×D18111yescharacteristic
230#103C2×D18111yescharacteristic
231#103C2×D18111yescharacteristic
232#102—1811no—
233#102—1811no—
234#102—1811no—
235C22⋊C8—1811no—
236C22⋊C8—1811no—
237#97C2×D18111yescharacteristic
238#41—3622no—
239#41—3622no—
240#41—7241no—
241#41—7241no—
242C2×Q8—1811no—
243C2×Q8C22×D18111yescharacteristic
244C2×Q8C22×D18111yescharacteristic
245C2×D4—3621no—
246C2×D4—3621no—
247C2×D4—1811no—
248C2×D4—1811no—
249C22×C4—3621no—
250C22×C4—3621no—
251C22×C4—3621no—
252C22×C4—3621no—
253C22×C4—1811no—
254C22×C4—1811no—
255Q16—7241no—
256Q16—7241no—
257SD16—3622no—
258M16—7241no—
259C2×C8—1811no—
260C2×C8—1811no—
261C2×C8—7241no—
262C2×C8C22×D18111yescharacteristic
263C2×C8C22×D18111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3621no—
269C4⋊C4—3621no—
270C4⋊C4—3621no—
271C4⋊C4—3621no—
272C4⋊C4—3621no—
273C4⋊C4—3621no—
274C4⋊C4—1811no—
275C4⋊C4—1811no—
276C4⋊C4—1811no—
277C4⋊C4C22×D18111yescharacteristic
278C4⋊C4C22×D18111yescharacteristic
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—1811no—
286C22⋊C4—1811no—
287C42—3621no—
288C42—3621no—
289C42C22×D18111yescharacteristic
290C23—3621no—
291C23—1811no—
292C23—1811no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—3621no—
298Q8—3621no—
299D4—3621no—
300D4—1812no—
301D4—7241no—
302D4—7242no—
303D4—7242no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—3621no—
309C2×C4—3621no—
310C2×C4—3621no—
311C2×C4—3621no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—3621no—
315C2×C4—1811no—
316C2×C4—1811no—
317C2×C4—7242no—
318C2×C4—7242no—
319C2×C4—7242no—
320C2×C4—7242no—
321C2×C4D4×D18111yescharacteristic
322C2×C4D4×D18111yescharacteristic
323C2×C4C23×D18111yescharacteristic
324C8—21no—
325C8—21no—
326C8—3621no—
327C8—3621no—
328C22—3621no—
329C22—3621no—
330C22—3622no—
331C22—3621no—
332C22—3621no—
333C22—3621no—
334C22—3621no—
335C22—7241no—
336C22—7241no—
337C22C2×D4×D18111yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3621no—
345C4—3621no—
346C4—7241no—
347C4—7241no—
348C4—7241no—
349C4#41×D18111yescharacteristic
350C4#1915386211yescharacteristic
351C2—7241no—
352C2—7241no—
353C2—7242no—
354C2#2604192211yescharacteristic
355C2#2604207811yescharacteristic
356C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#491×C181C2
socleC22×C181C22×D4
prime radical#26041932C2
hypercenter#491D181
hypocenterC181#3315
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #26041967C2semidirect
3. #26041619C2semidirect
4. #26041613C2semidirect
5. #26041576C2semidirect
6. #26041536C2semidirect
7. #26041519C2semidirect
8. #26041486C2semidirect
9. #26041444C2semidirect
10. #26041444C2semidirect
11. #26041381C2semidirect
12. #26041352C2semidirect
13. #26041279C2semidirect
14. #26041274C2semidirect
15. #491×C181C2semidirect
178. C181#3315semidirect
182. #491D181semidirect

Maximal subgroups

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

subgroupindex
#260419672
#260419322
#260416192
#260416132
#260415762
#260415362
#260415192
#260414862
#260414442
#260413812
#260413522
#260412792
#260412742
#491×C1812
#3315181