Finite Group Databasebeta

← #49170615

Subgroups — #49170615

Order 37504.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036813C211yesmaximal, characteristic, complemented
2#34036776C211yesmaximal, characteristic
3#34036482C211yesmaximal, characteristic, complemented
4#34036461C211yesmaximal, characteristic, complemented
5#34036450C211yesmaximal, characteristic, complemented
6#34036428C211yesmaximal, characteristic, complemented
7#34036369C211yesmaximal, characteristic, complemented
8#34036349C211yesmaximal, characteristic, complemented
9#34036289C211yesmaximal, characteristic, complemented
10#34036278C211yesmaximal, characteristic, complemented
11#34036214C211yesmaximal, characteristic, complemented
12C8×C22⋊D293C211yesmaximal, characteristic, complemented
13#34036126C211yesmaximal, characteristic, complemented
14#102×D293C211yesmaximal, characteristic, complemented
15#465×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C2×D4×D293C2211yescharacteristic
18C2×#17989792C2211yescharacteristic, complemented
19C2×#17989789C2211yescharacteristic
20C2×C8×D293C2211yescharacteristic
21#22627617—21no—
22#22627615C2211yescharacteristic, complemented
23#22627614—21no—
24#22627611C2211yescharacteristic
25#22627606—21no—
26#22627606—21no—
27#22627604C2211yescharacteristic
28#22627600C2211yescharacteristic
29#22627595—21no—
30#22627592—21no—
31Q8×Dic293C2211yescharacteristic, complemented
32#22627586C2211yescharacteristic
33C4×C22⋊D293C2211yescharacteristic
34#22627583C2211yescharacteristic, complemented
35#22627582—21no—
36C4⋊C4×D293C2211yescharacteristic
37C22⋊C4×D293—21no—
38#22627557C2211yescharacteristic, complemented
39#22627555C2211yescharacteristic, complemented
40#22627553C2211yescharacteristic, complemented
41#22627549C2211yescharacteristic, complemented
42#22627549C2211yescharacteristic, complemented
43#22627547C2211yescharacteristic, complemented
44#22627546C2211yescharacteristic
45#22627546C2211yescharacteristic
46#22627545C2211yescharacteristic
47#22627545C2211yescharacteristic
48#22627544C2211yescharacteristic
49C22⋊#14556873C2211yescharacteristic, complemented
50#22627536C2211yescharacteristic
51C8×Dic293C2211yescharacteristic, complemented
52C2×SD16×C293C2211yescharacteristic
53C22×C8×C293C2211yescharacteristic, complemented
54C22⋊Q8×C293C2211yescharacteristic, complemented
55#121×C293C2211yescharacteristic
56#106×C293C2211yescharacteristic, complemented
57#103×C293C2211yescharacteristic, complemented
58#102×C293C2211yescharacteristic
59C23×D293—21no—
60#17989802—22no—
61#17989802—41no—
62C2×Dic586C2311yescharacteristic
63C2×D1172C2311yescharacteristic
64C2×C22⋊D293—21no—
65C2×C22⋊D293—21no—
66C2×C22⋊D293D411yescharacteristic
67D4×D293—22no—
68D4×D293—41no—
69C2×C4×D293—21no—
70C2×C4×D293—21no—
71C2×C4×D293—21no—
72C2×C4×D293C2311yescharacteristic
73#17989792—41no—
74#17989789—22no—
75C2×#14556873C2311yescharacteristic
76C2×#14556873C2311yescharacteristic
77C8×D293—41no—
78C4⋊Dic293—21no—
79C4⋊Dic293—21no—
80C4⋊Dic293—21no—
81C4⋊Dic293C2311yescharacteristic
82C4⋊Dic293C2311yescharacteristic
83#17989781—21no—
84#17989781—21no—
85#17989781—21no—
86#17989781D411yescharacteristic
87C22⋊Dic293—21no—
88C22⋊Dic293—21no—
89C22⋊Dic293—21no—
90C22⋊Dic293D411yescharacteristic
91#17989779—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779D411yescharacteristic
95C4×Dic293—21no—
96C4×Dic293—21no—
97C4×Dic293C2311yescharacteristic
98C2×Q8×C293C2311yescharacteristic
99C2×D4×C293—21no—
100C2×D4×C293C2311yescharacteristic
101C22×C4×C293C2311yescharacteristic
102SD16×C293—41no—
103C2×C8×C293—41no—
104C2×C8×C293C2311yescharacteristic
105C2×C8×C293C2311yescharacteristic
106C4⋊C4×C293—21no—
107C4⋊C4×C293C2311yescharacteristic
108C4⋊C4×C293C2311yescharacteristic
109C22⋊C4×C293—21no—
110C22⋊C4×C293—21no—
111C22×D293—21no—
112C22×D293—42no—
113C22×D293—42no—
114C22×D293—81no—
115C22×D293C2×D411yescharacteristic
116Dic586—21no—
117Dic586—21no—
118D1172—21no—
119D1172#4112yes—
120C22⋊D293—22no—
121C22⋊D293—42no—
122C22⋊D293—42no—
123C2×Dic293—21no—
124C2×Dic293—21no—
125C2×Dic293—21no—
126C2×Dic293C2×D411yescharacteristic
127C2×Dic293C2×D411yescharacteristic
128C2×Dic293C2×D411yescharacteristic
129C4×D293—21no—
130C4×D293—21no—
131C4×D293—42no—
132C4×D293—42no—
133C4×D293—81no—
134#14556873—21no—
135#14556873—21no—
136C23×C293—21no—
137C23×C293C2×D411yescharacteristic
138Q8×C293—21no—
139Q8×C293—21no—
140D4×C293—21no—
141D4×C293—21no—
142D4×C293—81no—
143C2×C4×C293—21no—
144C2×C4×C293—21no—
145C2×C4×C293—21no—
146C2×C4×C293—21no—
147C2×C4×C293—21no—
148C2×C4×C293C2×D411yescharacteristic
149C2×C4×C293C2411yescharacteristic
150C8×C293—21no—
151C8×C293—21no—
152C22×C293—21no—
153C22×C293—21no—
154C22×C293—21no—
155C22×C293—41no—
156C22×C293—41no—
157C22×C293—41no—
158C22×C293C22×D411yescharacteristic
159C2×D293—21no—
160C2×D293—21no—
161C2×D293—22no—
162C2×D293—41no—
163C2×D293—41no—
164C2×D293—84no—
165C2×D293#13512yes—
166C4×C293—21no—
167C4×C293—41no—
168C4×C293—41no—
169C4×C293—41no—
170C4×C293C2×#4111yescharacteristic
171C4×C293H(5,𝔽2)11yescharacteristic
172Dic293—21no—
173Dic293—21no—
174Dic293—21no—
175Dic293—41no—
176Dic293—41no—
177Dic293—41no—
178C2×C293—41no—
179C2×C293—81no—
180C2×C293#54611yescharacteristic
181C2×C293C2×#13511yescharacteristic
182C2×C293#57711yescharacteristic
183D293—22no—
184D293—42no—
185C293#329611yescharacteristic, complemented, Sylow
186#3296—2931nomaximal, Sylow
187#552—2931no—
188#546—2931no—
189#488—2931no—
190#483—2931no—
191#482—2931no—
192#478—2931no—
193#467—2931no—
194#465D29311yescharacteristic, complemented
195#464—2931no—
196#452—2931no—
197#450—2931no—
198C4×SD16—2931no—
199C8×D4—2931no—
200#416—2931no—
201C2×#102—2931no—
202C2×#41—2931no—
203C22×D4—2931no—
204C2×Q16—2931no—
205C2×SD16—2931no—
206C2×SD16C2×D29311yescharacteristic
207C22×C8—2931no—
208C22×C8C2×D29311yescharacteristic, complemented
209#126—5861no—
210#125—2931no—
211#125—5861no—
212#124—2931no—
213#123—5861no—
214#123—5861no—
215C22⋊Q8—2931no—
216C22⋊Q8—2931no—
217C22⋊Q8C2×D29311yescharacteristic, complemented
218#121—5861no—
219#121C2×D29311yescharacteristic
220#120—5861no—
221C4×Q8—2931no—
222C4×D4—2931no—
223C4×D4—2931no—
224#117—2931no—
225#117—5861no—
226C2×C4⋊C4—2931no—
227C2×C22⋊C4—5861no—
228#107—2931no—
229#106—2931no—
230#106C2×D29311yescharacteristic, complemented
231C4⋊C8—2931no—
232#103—2931no—
233#103—2931no—
234#103—2931no—
235#103C2×D29311yescharacteristic, complemented
236#102—2931no—
237#102—2931no—
238#102—2931no—
239#102—2931no—
240#102—2931no—
241#102C2×D29311yescharacteristic
242C22⋊C8—2931no—
243C22⋊C8—2931no—
244C4×C8—2931no—
245C24—5861no—
246#41—5862no—
247#41—11721no—
248C2×Q8—2931no—
249C2×Q8C22×D29311yescharacteristic
250C2×D4—21no—
251C2×D4—2931no—
252C2×D4—2931no—
253C2×D4—5861no—
254C2×D4—5862no—
255C2×D4—5861no—
256C2×D4—11721no—
257C2×D4C22×D29311yescharacteristic
258C22×C4—2931no—
259C22×C4—5861no—
260C22×C4—5861no—
261C22×C4—5861no—
262C22×C4C22×D29311yescharacteristic
263Q16—11721no—
264SD16—41no—
265SD16—5862no—
266C2×C8—41no—
267C2×C8—2931no—
268C2×C8—2931no—
269C2×C8—11721no—
270C2×C8C22×D29311yescharacteristic
271C2×C8C22×D29311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2931no—
274C4⋊C4—2931no—
275C4⋊C4—2931no—
276C4⋊C4—5861no—
277C4⋊C4—5861no—
278C4⋊C4—5861no—
279C4⋊C4—5861no—
280C4⋊C4—5861no—
281C4⋊C4—5861no—
282C4⋊C4C22×D29311yescharacteristic
283C4⋊C4C22×D29311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—2931no—
287C22⋊C4—2931no—
288C22⋊C4—5861no—
289C22⋊C4—5861no—
290C22⋊C4—5861no—
291C22⋊C4—5861no—
292C22⋊C4—5861no—
293C22⋊C4—5861no—
294C42—2931no—
295C42—5861no—
296C42—5861no—
297C23—21no—
298C23—2931no—
299C23—23441no—
300C23—5861no—
301C23—11722no—
302C23—11722no—
303C23D4×D29311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5861no—
307Q8—5861no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—2932no—
312D4—5862no—
313D4—5861no—
314D4—11722no—
315D4—11722no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2931no—
322C2×C4—2931no—
323C2×C4—2931no—
324C2×C4—23441no—
325C2×C4—5861no—
326C2×C4—5861no—
327C2×C4—5861no—
328C2×C4—5861no—
329C2×C4—5861no—
330C2×C4—11722no—
331C2×C4—11722no—
332C2×C4D4×D29311yescharacteristic
333C2×C4C23×D29311yescharacteristic
334C8—21no—
335C8—21no—
336C8—5861no—
337C8—5861no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—2932no—
345C22—23444no—
346C22—5861no—
347C22—5861no—
348C22—5862no—
349C22—11721no—
350C22—11721no—
351C22C2×D4×D29311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—5861no—
357C4—5861no—
358C4—5861no—
359C4—11721no—
360C4—11721no—
361C4—11721no—
362C4#41×D29311yescharacteristic
363C4#2262766911yescharacteristic
364C2—41no—
365C2—81no—
366C2—5862no—
367C2—11722no—
368C2#3403676811yescharacteristic
369C2#135×D29311yescharacteristic
370C2#3403693511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#465×C293C2
socleC22×C293C22×D4
prime radical#34036776C2
hypercenter#465D293
hypocenterC293#3296
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #34036813C2semidirect
3. #34036482C2semidirect
4. #34036461C2semidirect
5. #34036450C2semidirect
6. #34036428C2semidirect
7. #34036369C2semidirect
8. #34036349C2semidirect
9. #34036289C2semidirect
10. #34036278C2semidirect
11. #34036214C2semidirect
12. C8×C22⋊D293C2semidirect
13. #34036126C2semidirect
14. #102×D293C2semidirect
15. #465×C293C2semidirect
18. C2×#17989792C22semidirect
22. #22627615C22semidirect
31. Q8×Dic293C22semidirect
34. #22627583C22semidirect
38. #22627557C22semidirect
39. #22627555C22semidirect
40. #22627553C22semidirect
41. #22627549C22semidirect
42. #22627549C22semidirect
43. #22627547C22semidirect
49. C22⋊#14556873C22semidirect
51. C8×Dic293C22semidirect
53. C22×C8×C293C22semidirect
54. C22⋊Q8×C293C22semidirect
56. #106×C293C22semidirect
57. #103×C293C22semidirect
185. C293#3296semidirect
194. #465D293semidirect
208. C22×C8C2×D293semidirect
217. C22⋊Q8C2×D293semidirect
230. #106C2×D293semidirect
235. #103C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340368132
#340367762
#340364822
#340364612
#340364502
#340364282
#340363692
#340363492
#340362892
#340362782
#340362142
C8×C22⋊D2932
#340361262
#102×D2932
#465×C2932
#3296293