Finite Group Databasebeta

← #37543545

Subgroups — #37543545

Order 23168.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041906C211yesmaximal, characteristic, complemented
2#26041751C211yesmaximal, characteristic
3#26041582C211yesmaximal, characteristic, complemented
4#26041580C211yesmaximal, characteristic, complemented
5#26041579C211yesmaximal, characteristic, complemented
6#26041577C211yesmaximal, characteristic, complemented
7#26041567C211yesmaximal, characteristic, complemented
8#26041566C211yesmaximal, characteristic, complemented
9#26041558C211yesmaximal, characteristic, complemented
10#26041320C211yesmaximal, characteristic, complemented
11#26041316C211yesmaximal, characteristic, complemented
12#26041304C211yesmaximal, characteristic, complemented
13#26041282C211yesmaximal, characteristic
14#102×D181C211yesmaximal, characteristic
15#474×C181C211yesmaximal, characteristic, complemented
16C2×D4×D181C2211yescharacteristic
17C2×#15668432C2211yescharacteristic
18C2×C8×D181C2211yescharacteristic
19#19153815C2211yescharacteristic, complemented
20#19153814—21no—
21#19153808C2211yescharacteristic, complemented
22#19153807C2211yescharacteristic, complemented
23#19153806—21no—
24#19153794—21no—
25#19153792—21no—
26C4×D724C2211yescharacteristic
27D4×Dic181C2211yescharacteristic
28#19153778C2211yescharacteristic
29C4×C22⋊D181—21no—
30#19153776—21no—
31#19153774C2211yescharacteristic
32#19153774C2211yescharacteristic
33#19153773C2211yescharacteristic
34C4⋊C4×D181C2211yescharacteristic
35C2×#15668426—21no—
36C22⋊C4×D181—21no—
37#19153750C2211yescharacteristic, complemented
38#19153749C2211yescharacteristic, complemented
39#19153748C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153747C2211yescharacteristic, complemented
42#19153747C2211yescharacteristic, complemented
43C4⋊#13089678C2211yescharacteristic, complemented
44#19153745C2211yescharacteristic, complemented
45#19153745C2211yescharacteristic, complemented
46#19153738C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153736C2211yescharacteristic
51#19153736C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic, complemented
53C4×D4×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic, complemented
55#107×C181C2211yescharacteristic, complemented
56C4⋊C8×C181C2211yescharacteristic, complemented
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C23×D181—21no—
60C2×Dic362—21no—
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63D4×D181—22no—
64D4×D181Q814yescomplemented
65C22×Dic181—21no—
66C2×C4×D181—21no—
67C2×C4×D181—21no—
68C2×C4×D181—21no—
69C2×C4×D181C2311yescharacteristic
70#15668432—22no—
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C8×D181—22no—
74C4⋊Dic181—21no—
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84#15668428D411yescharacteristic
85C22⋊Dic181—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426D411yescharacteristic
91#15668426D411yescharacteristic
92C4×Dic181—21no—
93C4×Dic181—21no—
94C4×Dic181C2311yescharacteristic
95C2×Q8×C181—21no—
96C2×D4×C181C2311yescharacteristic
97C22×C4×C181—21no—
98C2×C8×C181C2311yescharacteristic
99C2×C8×C181C2311yescharacteristic
100C4⋊C4×C181—21no—
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C22⋊C4×C181—21no—
105C42×C181C2311yescharacteristic
106C22×D181—21no—
107C22×D181—24no—
108C22×D181—81no—
109C22×D181—81no—
110C22×D181C2×D411yescharacteristic
111Dic362—81no—
112D724—21no—
113D724C2×Q812yes—
114C22⋊D181—24no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C2×Dic181—81no—
119C2×Dic181C2×D411yescharacteristic
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122C4×D181—81no—
123C4×D181—81no—
124C4×D181—81no—
125C4×D181C2×Q812yes—
126C4×D181#4112yes—
127#13089678—21no—
128#13089678—21no—
129C23×C181—21no—
130Q8×C181—81no—
131D4×C181—21no—
132D4×C181C2×Q812yes—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—81no—
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2411yescharacteristic
140C8×C181—21no—
141C8×C181—21no—
142C22×C181—22no—
143C22×C181—41no—
144C22×C181—41no—
145C22×C181C22×D411yescharacteristic
146C2×D181—21no—
147C2×D181—21no—
148C2×D181—22no—
149C2×D181—41no—
150C2×D181—41no—
151C2×D181—84no—
152C2×D181C22⋊Q812yes—
153C4×C181—21no—
154C4×C181—21no—
155C4×C181—41no—
156C4×C181—41no—
157C4×C181—41no—
158C4×C181C22×Q811yescharacteristic
159C4×C181C2×#4111yescharacteristic
160Dic181—21no—
161Dic181—21no—
162Dic181—41no—
163Dic181—41no—
164Dic181—41no—
165Dic181C22⋊Q812yes—
166C2×C181—42no—
167C2×C181C2×C22⋊Q811yescharacteristic
168C2×C181#57611yescharacteristic
169C2×C181#57711yescharacteristic
170D181—22no—
171D181—42no—
172C181#308111yescharacteristic, complemented, Sylow
173#3081—1811nomaximal, Sylow
174#544—1811no—
175#518—1811no—
176#478—1811no—
177#478—1811no—
178#478—1811no—
179#478—1811no—
180#476—1811no—
181#476—1811no—
182#474—1811no—
183#474D18111yescharacteristic, complemented
184#428—1811no—
185#427—1811no—
186#424—1811no—
187#418—1811no—
188C2×#102—1811no—
189C22×D4—1811no—
190C2×M16—1811no—
191C22×C8—1811no—
192C4⋊Q8—3621no—
193C4⋊Q8—1811no—
194C4⋊Q8C2×D18111yescharacteristic, complemented
195#125—3621no—
196#125—1811no—
197#125—1811no—
198C22⋊Q8—3621no—
199C22⋊Q8—3621no—
200C4×D4—3621no—
201C4×D4—3621no—
202C4×D4—1811no—
203C4×D4—1811no—
204C4×D4—1811no—
205C4×D4C2×D18111yescharacteristic
206#117—1811no—
207#117—1811no—
208#117—1811no—
209C2×C4⋊C4—1811no—
210C2×C22⋊C4—3621no—
211C2×C22⋊C4—3621no—
212#107—1811no—
213#107—1811no—
214#107C2×D18111yescharacteristic, complemented
215#107C2×D18111yescharacteristic, complemented
216#106—1811no—
217#106—1811no—
218#106—1811no—
219#106—1811no—
220C4⋊C8—1811no—
221C4⋊C8—1811no—
222C4⋊C8—1811no—
223C4⋊C8C2×D18111yescharacteristic, complemented
224#102—1811no—
225#102—1811no—
226#102—1811no—
227#102—1811no—
228#102—1811no—
229#102—1811no—
230#102C2×D18111yescharacteristic
231#102C2×D18111yescharacteristic
232C24—3621no—
233C2×Q8—21no—
234C2×Q8—3621no—
235C2×D4—3622no—
236C2×D4—3621no—
237C2×D4—1814no—
238C2×D4—1811no—
239C2×D4C22×D18111yescharacteristic
240C22×C4—21no—
241C22×C4—3621no—
242C22×C4—3621no—
243C22×C4—3621no—
244C22×C4—3621no—
245C22×C4—1811no—
246M16—3622no—
247C2×C8—3622no—
248C2×C8—1811no—
249C2×C8—1811no—
250C2×C8C22×D18111yescharacteristic
251C2×C8C22×D18111yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—3621no—
256C4⋊C4—3621no—
257C4⋊C4—3621no—
258C4⋊C4—3621no—
259C4⋊C4—1811no—
260C4⋊C4—1811no—
261C4⋊C4—1811no—
262C4⋊C4—1811no—
263C4⋊C4—1811no—
264C4⋊C4C22×D18111yescharacteristic
265C4⋊C4C22×D18111yescharacteristic
266C4⋊C4C22×D18111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3621no—
269C22⋊C4—3621no—
270C22⋊C4—3621no—
271C22⋊C4—3621no—
272C22⋊C4—3621no—
273C22⋊C4—1811no—
274C22⋊C4—1811no—
275C42—3621no—
276C42—3621no—
277C42—1811no—
278C42C22×D18111yescharacteristic
279C23—21no—
280C23—3624no—
281C23—3621no—
282C23—14481no—
283C23—14481no—
284C23—1811no—
285Q8—81no—
286Q8—14481no—
287D4—21no—
288D4—3624no—
289D4—3621no—
290D4—1812no—
291D4Q8×D18112yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—3621no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—14481no—
300C2×C4—14481no—
301C2×C4—14481no—
302C2×C4—14481no—
303C2×C4—1811no—
304C2×C4—1811no—
305C2×C4—1811no—
306C2×C4—1812no—
307C2×C4—1812no—
308C2×C4D4×D18111yescharacteristic
309C2×C4D4×D18111yescharacteristic
310C2×C4C23×D18111yescharacteristic
311C8—21no—
312C8—21no—
313C8—3621no—
314C8—3621no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—3621no—
319C22—3621no—
320C22—3622no—
321C22—14484no—
322C22—1812no—
323C22—7241no—
324C22—7241no—
325C22C2×D4×D18111yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—3621no—
332C4—3621no—
333C4—1812no—
334C4—7241no—
335C4—7241no—
336C4—7241no—
337C4C2×Q8×D18111yescharacteristic
338C4#41×D18111yescharacteristic
339C2—42no—
340C2—3622no—
341C2—7242no—
342C2C22⋊Q8×D18111yescharacteristic
343C2#2604208011yescharacteristic
344C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#474×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×D181C22
hypercenter#474D181
hypocenterC181#3081
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #26041906C2semidirect
3. #26041582C2semidirect
4. #26041580C2semidirect
5. #26041579C2semidirect
6. #26041577C2semidirect
7. #26041567C2semidirect
8. #26041566C2semidirect
9. #26041558C2semidirect
10. #26041320C2semidirect
11. #26041316C2semidirect
12. #26041304C2semidirect
15. #474×C181C2semidirect
19. #19153815C22semidirect
21. #19153808C22semidirect
22. #19153807C22semidirect
37. #19153750C22semidirect
38. #19153749C22semidirect
39. #19153748C22semidirect
40. #19153747C22semidirect
41. #19153747C22semidirect
42. #19153747C22semidirect
43. C4⋊#13089678C22semidirect
44. #19153745C22semidirect
45. #19153745C22semidirect
52. C4⋊Q8×C181C22semidirect
54. #107×C181C22semidirect
55. #107×C181C22semidirect
56. C4⋊C8×C181C22semidirect
64. D4×D181Q8semidirect
172. C181#3081semidirect
183. #474D181semidirect
194. C4⋊Q8C2×D181semidirect
214. #107C2×D181semidirect
215. #107C2×D181semidirect
223. C4⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419062
#260417512
#260415822
#260415802
#260415792
#260415772
#260415672
#260415662
#260415582
#260413202
#260413162
#260413042
#260412822
#102×D1812
#474×C1812
#3081181