Finite Group Databasebeta

← #37545612

Subgroups — #37545612

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#26041971C211yesmaximal, characteristic, complemented
2#26041940C211yesmaximal, characteristic
3#26041635C211yesmaximal, characteristic, complemented
4#26041619C211yesmaximal, characteristic, complemented
5#26041593C211yesmaximal, characteristic, complemented
6#26041588C211yesmaximal, characteristic, complemented
7#26041517C211yesmaximal, characteristic, complemented
8#26041510C211yesmaximal, characteristic, complemented
9#26041503C211yesmaximal, characteristic, complemented
10#26041440C211yesmaximal, characteristic, complemented
11Q16×Dic181C211yesmaximal, characteristic, complemented
12#26041342C211yesmaximal, characteristic, complemented
13#26041274C211yesmaximal, characteristic, complemented
14C2×#19153740C211yesmaximal, characteristic, complemented
15#451×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668436C2211yescharacteristic
18C22×#13089678C2211yescharacteristic
19C2×C8×D181C2211yescharacteristic
20#19153811—21no—
21#19153807—21no—
22#19153807C2211yescharacteristic
23#19153805—21no—
24#19153805C2211yescharacteristic
25#19153800—21no—
26#19153792—21no—
27#19153792C2211yescharacteristic
28#19153792C2211yescharacteristic
29#19153790—21no—
30#19153790C2211yescharacteristic
31Q8×Dic181C2211yescharacteristic
32Q8×Dic181C2211yescharacteristic
33C4×C22⋊D181—21no—
34C4×C22⋊D181C2211yescharacteristic
35#19153774C2211yescharacteristic
36#19153772—21no—
37C2×C4⋊Dic181C2211yescharacteristic
38#19153750C2211yescharacteristic
39#19153750C2211yescharacteristic
40#19153747C2211yescharacteristic
41#19153745C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153738C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C22×Q8×C181C2211yescharacteristic
53C2×Q16×C181C2211yescharacteristic
54C2×Q16×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59#15668450—22no—
60#15668450—41no—
61C2×D724—21no—
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181D411yescharacteristic
65C22×Dic181—21no—
66C2×C4×D181—21no—
67C2×C4×D181C2311yescharacteristic
68#15668436—41no—
69C2×#13089678—41no—
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C8×D181—41no—
73C4⋊Dic181—21no—
74C4⋊Dic181—21no—
75C4⋊Dic181—21no—
76C4⋊Dic181—21no—
77C4⋊Dic181C2311yescharacteristic
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84C22⋊Dic181D411yescharacteristic
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426D411yescharacteristic
92C4×Dic181—21no—
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96C2×Q8×C181—21no—
97C2×Q8×C181—22no—
98C2×Q8×C181—41no—
99C2×Q8×C181C2311yescharacteristic
100C2×Q8×C181C2311yescharacteristic
101C2×Q8×C181C2311yescharacteristic
102C22×C4×C181—21no—
103C22×C4×C181C2311yescharacteristic
104Q16×C181—22no—
105Q16×C181—41no—
106C2×C8×C181C2311yescharacteristic
107C2×C8×C181C2311yescharacteristic
108C4⋊C4×C181—21no—
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181—21no—
112C22×D181C2×D411yescharacteristic
113D724—21no—
114D724—21no—
115D724—42no—
116C22⋊D181—22no—
117C22⋊D181—42no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—42no—
122C2×Dic181C2×D411yescharacteristic
123C2×Dic181C2×D411yescharacteristic
124C2×Dic181C2×D411yescharacteristic
125C4×D181—21no—
126C4×D181—21no—
127C4×D181—42no—
128#13089678—21no—
129#13089678—21no—
130C23×C181C2×D411yescharacteristic
131Q8×C181—21no—
132Q8×C181—21no—
133Q8×C181—21no—
134Q8×C181—21no—
135Q8×C181—21no—
136Q8×C181—42no—
137Q8×C181#4112yes—
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—42no—
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2411yescharacteristic
147C8×C181—21no—
148C8×C181—21no—
149C22×C181—21no—
150C22×C181—21no—
151C22×C181#13512yes—
152C22×C181C22×D411yescharacteristic
153C2×D181—21no—
154C2×D181—21no—
155C2×D181—21no—
156C2×D181—41no—
157C2×D181—41no—
158C2×D181—41no—
159C4×C181—21no—
160C4×C181—22no—
161C4×C181—41no—
162C4×C181—41no—
163C4×C181—41no—
164C4×C181C2×#4111yescharacteristic
165C4×C181H(5,𝔽2)11yescharacteristic
166Dic181—21no—
167Dic181—21no—
168Dic181—21no—
169Dic181—41no—
170Dic181—41no—
171Dic181—41no—
172C2×C181—22no—
173C2×C181#54611yescharacteristic
174C2×C181C2×#13511yescharacteristic
175C2×C181#57811yescharacteristic
176D181—41no—
177D181—81no—
178C181#330011yescharacteristic, complemented, Sylow
179#3300—1811nomaximal, Sylow
180#552—1811no—
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#451D18111yescharacteristic, complemented
191C4×Q16—1811no—
192C8×D4—1811no—
193#416—1811no—
194C2×#103—1811no—
195C2×#41—1811no—
196C22×Q8C2×D18111yescharacteristic
197C2×Q16C2×D18111yescharacteristic
198C2×Q16C2×D18111yescharacteristic
199C2×SD16—1811no—
200C22×C8—1811no—
201C22×C8—1811no—
202#126—3621no—
203#125—3621no—
204#125—1811no—
205#124—3621no—
206#124—1811no—
207#123—3621no—
208C22⋊Q8—3621no—
209C22⋊Q8—1811no—
210C22⋊Q8—1811no—
211C22⋊Q8C2×D18111yescharacteristic
212#121—3621no—
213#121—1811no—
214C4×Q8—1811no—
215C4×Q8—1811no—
216C4×D4—3621no—
217C4×D4—1811no—
218#117—3621no—
219#117—1811no—
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#103—1811no—
229#103—1811no—
230#103C2×D18111yescharacteristic
231#103C2×D18111yescharacteristic
232#102—1811no—
233#102—1811no—
234#102—1811no—
235C22⋊C8—1811no—
236C22⋊C8C2×D18111yescharacteristic
237C4×C8—1811no—
238#41—3622no—
239#41—7241no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8C22×D18111yescharacteristic
244C2×Q8C22×D18111yescharacteristic
245C2×Q8C22×D18111yescharacteristic
246C2×D4—3621no—
247C2×D4—3621no—
248C2×D4—1811no—
249C2×D4—1811no—
250C22×C4—21no—
251C22×C4—3621no—
252C22×C4—3621no—
253C22×C4—1811no—
254C22×C4C22×D18111yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—7241no—
258C2×C8—1811no—
259C2×C8—1811no—
260C2×C8—7241no—
261C2×C8—7241no—
262C2×C8C22×D18111yescharacteristic
263C2×C8C22×D18111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
267C4⋊C4—3621no—
268C4⋊C4—3621no—
269C4⋊C4—3621no—
270C4⋊C4—3621no—
271C4⋊C4—3621no—
272C4⋊C4—1811no—
273C4⋊C4—1811no—
274C4⋊C4—1811no—
275C4⋊C4—1811no—
276C4⋊C4C22×D18111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C22⋊C4—1811no—
285C22⋊C4—1811no—
286C42—3621no—
287C42—3621no—
288C42—3621no—
289C42—1811no—
290C23—3621no—
291C23—1811no—
292C23D4×D18111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8#1566844812yes—
300D4—3622no—
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—42no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—1811no—
318C2×C4—1811no—
319C2×C4—1811no—
320C2×C4—7242no—
321C2×C4—7242no—
322C2×C4D4×D18111yescharacteristic
323C2×C4C23×D18111yescharacteristic
324C8—21no—
325C8—21no—
326C8—3621no—
327C8—3621no—
328C22—21no—
329C22—21no—
330C22—3621no—
331C22—3621no—
332C22—3621no—
333C22—7241no—
334C22—7241no—
335C22—7241no—
336C22#1915383912yes—
337C22C2×D4×D18111yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3621no—
344C4—3621no—
345C4—3621no—
346C4—7241no—
347C4—7241no—
348C4—7241no—
349C4C2×#1566844811yescharacteristic
350C4#1915386111yescharacteristic
351C2—22no—
352C2—14481no—
353C2—7241no—
354C2#2604191611yescharacteristic
355C2C2×#1915383911yescharacteristic
356C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#451×C181C2
socleC22×C181C22×D4
prime radical#26041940C2
hypercenter#451D181
hypocenterC181#3300
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #26041971C2semidirect
3. #26041635C2semidirect
4. #26041619C2semidirect
5. #26041593C2semidirect
6. #26041588C2semidirect
7. #26041517C2semidirect
8. #26041510C2semidirect
9. #26041503C2semidirect
10. #26041440C2semidirect
11. Q16×Dic181C2semidirect
12. #26041342C2semidirect
13. #26041274C2semidirect
14. C2×#19153740C2semidirect
15. #451×C181C2semidirect
178. C181#3300semidirect
190. #451D181semidirect

Maximal subgroups

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

subgroupindex
#260419712
#260419402
#260416352
#260416192
#260415932
#260415882
#260415172
#260415102
#260415032
#260414402
Q16×Dic1812
#260413422
#260412742
C2×#191537402
#451×C1812
#3300181