Finite Group Databasebeta

← #37546451

Subgroups — #37546451

Order 23168.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042005C211yesmaximal, characteristic, complemented
2#26041950C211yesmaximal, characteristic
3#26041644C211yesmaximal, characteristic, complemented
4#26041642C211yesmaximal, characteristic, complemented
5#26041624C211yesmaximal, characteristic, complemented
6#26041582C211yesmaximal, characteristic, complemented
7#26041579C211yesmaximal, characteristic, complemented
8#26041558C211yesmaximal, characteristic, complemented
9#26041479C211yesmaximal, characteristic, complemented
10#26041477C211yesmaximal, characteristic, complemented
11#26041477C211yesmaximal, characteristic, complemented
12#26041407C211yesmaximal, characteristic, complemented
13#26041382C211yesmaximal, characteristic, complemented
14#26041364C211yesmaximal, characteristic, complemented
15#440×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17C2×#15668436C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19#19153815—21no—
20#19153815C2211yescharacteristic
21#19153813C2211yescharacteristic
22#19153812C2211yescharacteristic
23#19153812C2211yescharacteristic
24#19153808—21no—
25#19153808C2211yescharacteristic
26#19153807—21no—
27#19153807C2211yescharacteristic
28#19153790—21no—
29#19153789—21no—
30#19153789—21no—
31Q8×Dic181C2211yescharacteristic
32Q8×Dic181C2211yescharacteristic
33C4×Dic362—21no—
34C4×Dic362C2211yescharacteristic
35#19153778C2211yescharacteristic
36#19153778C2211yescharacteristic
37#19153749C2211yescharacteristic
38#19153747C2211yescharacteristic
39C4⋊#13089678C2211yescharacteristic
40#19153745C2211yescharacteristic
41#19153745C2211yescharacteristic
42#19153738C2211yescharacteristic
43#19153738C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54C4×D4×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59#15668449—44no—
60C2×Dic362—21no—
61C2×Dic362D411yescharacteristic
62C2×D724—21no—
63C2×D724—21no—
64C2×D724C2311yescharacteristic
65C2×D724C2311yescharacteristic
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72#15668436—22no—
73#15668436—22no—
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181—21no—
79C4⋊Dic181D411yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428D411yescharacteristic
87#15668428D411yescharacteristic
88#15668426—21no—
89#15668426—21no—
90C4×Dic181—21no—
91C4×Dic181—21no—
92C4×Dic181C2311yescharacteristic
93C4×Dic181C2311yescharacteristic
94C2×Q8×C181C2311yescharacteristic
95C2×D4×C181C2311yescharacteristic
96C22×C4×C181—21no—
97SD16×C181—22no—
98C2×C8×C181C2311yescharacteristic
99C2×C8×C181C2311yescharacteristic
100C4⋊C4×C181—21no—
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C22⋊C4×C181—21no—
104C42×C181—21no—
105C42×C181C2311yescharacteristic
106C22×D181—21no—
107C22×D181—21no—
108Dic362—22no—
109Dic362—42no—
110D724—21no—
111D724—21no—
112D724—21no—
113D724—21no—
114D724—42no—
115D724—81no—
116C22⋊D181—42no—
117C22⋊D181—42no—
118C22⋊D181—81no—
119C22⋊D181—81no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181C2×D411yescharacteristic
123C2×Dic181C2×D411yescharacteristic
124C2×Dic181C2×D411yescharacteristic
125C2×Dic181C2×D411yescharacteristic
126C4×D181—42no—
127C4×D181—42no—
128#13089678—21no—
129#13089678—21no—
130C23×C181—21no—
131Q8×C181—21no—
132Q8×C181#4112yescomplemented
133D4×C181—21no—
134D4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—42no—
139C2×C4×C181C2×D411yescharacteristic
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2411yescharacteristic
142C8×C181—21no—
143C8×C181—21no—
144C22×C181—41no—
145C22×C181—41no—
146C22×C181—41no—
147C22×C181C22×D411yescharacteristic
148C2×D181—41no—
149C2×D181—41no—
150C2×D181—41no—
151C2×D181—41no—
152C2×D181—41no—
153C2×D181—41no—
154C4×C181—22no—
155C4×C181—21no—
156C4×C181—21no—
157C4×C181—41no—
158C4×C181—41no—
159C4×C181C2×#4111yescharacteristic
160C4×C1812−1+411yescharacteristic
161Dic181—21no—
162Dic181—21no—
163Dic181—21no—
164Dic181—41no—
165Dic181—41no—
166Dic181AGL(1,ℤ/8)12yes—
167C2×C181—81no—
168C2×C181#54811yescharacteristic
169C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C181#57711yescharacteristic
171D181—81no—
172D181—81no—
173C181#336811yescharacteristic, complemented, Sylow
174#3368—1811nomaximal, Sylow
175#557—1811no—
176#550—1811no—
177#490—1811no—
178#490—1811no—
179#486—1811no—
180#478—1811no—
181#478—1811no—
182#474—1811no—
183#460—1811no—
184#460—1811no—
185#460—1811no—
186#446—1811no—
187#440—1811no—
188#440D18111yescharacteristic, complemented
189C4×SD16—1811no—
190C2×#41—3621no—
191C2×SD16—1811no—
192C2×SD16—1811no—
193C2×SD16C2×D18111yescharacteristic
194C4⋊Q8—3621no—
195C4⋊Q8—1811no—
196#127—1811no—
197#127—1811no—
198#127—1811no—
199#125—3621no—
200#125—3621no—
201#125—1811no—
202#125—1811no—
203#121—3621no—
204#121—3621no—
205#121—3621no—
206C4×Q8—3621no—
207C4×Q8—1811no—
208C4×Q8—1811no—
209C4×Q8—1811no—
210C4×Q8C2×D18111yescharacteristic
211C4×D4—1811no—
212C4×D4—1811no—
213C4×D4C2×D18111yescharacteristic
214#107—1811no—
215#107C2×D18111yescharacteristic
216#106—1811no—
217C4⋊C8—1811no—
218C4⋊C8—1811no—
219C4⋊C8—1811no—
220#103C2×D18111yescharacteristic
221#102—1811no—
222#102—1811no—
223#102—1811no—
224#102—1811no—
225#102—1811no—
226#102—1811no—
227#102—1811no—
228#102—1811no—
229#102C2×D18111yescharacteristic
230#97—1811no—
231#97C2×D18111yescharacteristic
232C4×C8—1811no—
233#41—7244no—
234C2×Q8—3621no—
235C2×Q8—1811no—
236C2×Q8C22×D18111yescharacteristic
237C2×D4—3621no—
238C2×D4—3621no—
239C2×D4—3621no—
240C2×D4—3621no—
241C2×D4—3621no—
242C2×D4—3621no—
243C2×D4—1811no—
244C2×D4—1811no—
245C2×D4C22×D18111yescharacteristic
246C22×C4—21no—
247C22×C4—3621no—
248C22×C4—3621no—
249SD16—22no—
250SD16—3622no—
251SD16—3622no—
252C2×C8—1811no—
253C2×C8—1811no—
254C2×C8C22×D18111yescharacteristic
255C2×C8C22×D18111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3621no—
258C4⋊C4—3621no—
259C4⋊C4—3621no—
260C4⋊C4—3621no—
261C4⋊C4—3621no—
262C4⋊C4—3621no—
263C4⋊C4—3621no—
264C4⋊C4—1811no—
265C4⋊C4—1811no—
266C4⋊C4—1811no—
267C4⋊C4—1811no—
268C4⋊C4—1811no—
269C4⋊C4C22×D18111yescharacteristic
270C4⋊C4C22×D18111yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—3621no—
273C22⋊C4—3621no—
274C42—21no—
275C42—3621no—
276C42—3621no—
277C42—1811no—
278C42—1811no—
279C42C22×D18111yescharacteristic
280C23—21no—
281C23—3621no—
282C23—3621no—
283Q8—21no—
284Q8—3622no—
285Q8—7242no—
286Q8#1566844912yescomplemented
287D4—21no—
288D4—21no—
289D4—3621no—
290D4—3621no—
291D4—3621no—
292D4—3621no—
293D4—14481no—
294D4—14481no—
295D4—14481no—
296D4—7242no—
297D4—7242no—
298D4—7242no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—3621no—
304C2×C4—3621no—
305C2×C4—1811no—
306C2×C4—1811no—
307C2×C4—1811no—
308C2×C4—1811no—
309C2×C4—7242no—
310C2×C4—7242no—
311C2×C4D4×D18111yescharacteristic
312C2×C4D4×D18111yescharacteristic
313C2×C4C23×D18111yescharacteristic
314C8—21no—
315C8—21no—
316C8—3621no—
317C8—3621no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—7241no—
322C22—7241no—
323C22—7241no—
324C22—7241no—
325C22—7241no—
326C22—7241no—
327C22C2×D4×D18111yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—3621no—
334C4—3621no—
335C4—3621no—
336C4—1812no—
337C4—7241no—
338C4—7241no—
339C4C2×#1566844911yescharacteristic
340C4#1915386311yescharacteristic
341C2—81no—
342C2—14481no—
343C2—14481no—
344C2#2604193811yescharacteristic
345C2AGL(1,ℤ/8)×D18111yescharacteristic
346C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#440×C181C2
socleC22×C181C22×D4
prime radical#26041950C2
hypercenter#440D181
hypocenterC181#3368
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #26042005C2semidirect
3. #26041644C2semidirect
4. #26041642C2semidirect
5. #26041624C2semidirect
6. #26041582C2semidirect
7. #26041579C2semidirect
8. #26041558C2semidirect
9. #26041479C2semidirect
10. #26041477C2semidirect
11. #26041477C2semidirect
12. #26041407C2semidirect
13. #26041382C2semidirect
14. #26041364C2semidirect
15. #440×C181C2semidirect
132. Q8×C181#41semidirect
173. C181#3368semidirect
188. #440D181semidirect
286. Q8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260420052
#260419502
#260416442
#260416422
#260416242
#260415822
#260415792
#260415582
#260414792
#260414772
#260414072
#260413822
#260413642
#440×C1812
#3368181