Finite Group Databasebeta

← #49170618

Subgroups — #49170618

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#34036814C211yesmaximal, characteristic, complemented
2#34036770C211yesmaximal, characteristic
3#34036485C211yesmaximal, characteristic, complemented
4#34036455C211yesmaximal, characteristic, complemented
5#34036428C211yesmaximal, characteristic, complemented
6#34036368C211yesmaximal, characteristic, complemented
7#34036359C211yesmaximal, characteristic, complemented
8#34036354C211yesmaximal, characteristic, complemented
9#34036294C211yesmaximal, characteristic, complemented
10#34036282C211yesmaximal, characteristic, complemented
11#34036216C211yesmaximal, characteristic, complemented
12#34036192C211yesmaximal, characteristic, complemented
13#34036124C211yesmaximal, characteristic, complemented
14C2×#22627544C211yesmaximal, characteristic, complemented
15#483×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18C2×#17989792C2211yescharacteristic, complemented
19C2×#17989791C2211yescharacteristic
20C2×#17989791C2211yescharacteristic
21C22×#14556873C2211yescharacteristic
22C2×C8×D293C2211yescharacteristic, complemented
23#22627619—21no—
24#22627615C2211yescharacteristic, complemented
25#22627614—21no—
26#22627613C2211yescharacteristic
27#22627609—21no—
28#22627607—21no—
29#22627604C2211yescharacteristic, complemented
30#22627603C2211yescharacteristic
31#22627599—21no—
32#22627599C2211yescharacteristic
33#22627594—21no—
34#22627589C2211yescharacteristic, complemented
35#22627586C2211yescharacteristic
36C4×C22⋊D293C2211yescharacteristic
37#22627583—21no—
38#22627582C2211yescharacteristic, complemented
39C22⋊C4×D293—21no—
40#22627557C2211yescharacteristic, complemented
41#22627553C2211yescharacteristic, complemented
42#22627548C2211yescharacteristic, complemented
43#22627547C2211yescharacteristic, complemented
44#22627546C2211yescharacteristic
45#22627546C2211yescharacteristic
46#22627544C2211yescharacteristic
47#22627544C2211yescharacteristic
48#22627544C2211yescharacteristic
49#22627544C2211yescharacteristic
50#22627536C2211yescharacteristic, complemented
51C8×Dic293C2211yescharacteristic, complemented
52C22⋊Q8×C293C2211yescharacteristic
53C2×C4⋊C4×C293C2211yescharacteristic
54#106×C293C2211yescharacteristic, complemented
55#106×C293C2211yescharacteristic, complemented
56#103×C293C2211yescharacteristic, complemented
57#103×C293C2211yescharacteristic, complemented
58C22⋊C8×C293C2211yescharacteristic
59C23×D293—21no—
60#17989803—22no—
61#17989803—41no—
62C2×Dic586C2311yescharacteristic
63C2×D1172—22no—
64C2×D1172—21no—
65C2×D1172—21no—
66C2×D1172—21no—
67C2×D1172—41no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×C22⋊D293D411yescharacteristic
71C2×C4×D293—21no—
72C2×C4×D293—21no—
73C2×C4×D293C2311yescharacteristic
74#17989792—41no—
75#17989791—22no—
76#17989791—41no—
77C2×#14556873—41no—
78C2×#14556873C2311yescharacteristic
79C2×#14556873C2311yescharacteristic
80C8×D293—41no—
81C4⋊Dic293—21no—
82C4⋊Dic293—21no—
83C4⋊Dic293C2311yescharacteristic
84#17989781—21no—
85#17989781—21no—
86#17989781—21no—
87#17989781D411yescharacteristic
88C22⋊Dic293—21no—
89C22⋊Dic293D411yescharacteristic
90#17989779—21no—
91#17989779—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95#17989779—21no—
96#17989779D411yescharacteristic
97C4×Dic293—21no—
98C4×Dic293—21no—
99C4×Dic293C2311yescharacteristic
100C2×Q8×C293C2311yescharacteristic
101C22×C4×C293—21no—
102C22×C4×C293C2311yescharacteristic
103C2×C8×C293C2311yescharacteristic
104C2×C8×C293C2311yescharacteristic
105C4⋊C4×C293—21no—
106C4⋊C4×C293—21no—
107C4⋊C4×C293C2311yescharacteristic
108C4⋊C4×C293C2311yescharacteristic
109C4⋊C4×C293C2311yescharacteristic
110C22⋊C4×C293—21no—
111C22×D293—21no—
112C22×D293—21no—
113C22×D293—42no—
114C22×D293—42no—
115C22×D293—81no—
116C22×D293C2×D411yescharacteristic
117Dic586—21no—
118Dic586—21no—
119D1172—21no—
120D1172—21no—
121D1172—21no—
122D1172—42no—
123D1172—42no—
124D1172—81no—
125D1172#4112yes—
126C22⋊D293—22no—
127C2×Dic293—21no—
128C2×Dic293—21no—
129C2×Dic293C2×D411yescharacteristic
130C2×Dic293C2×D411yescharacteristic
131C2×Dic293C2×D411yescharacteristic
132C4×D293—21no—
133C4×D293—21no—
134C4×D293—42no—
135C4×D293—81no—
136#14556873—21no—
137#14556873—21no—
138C23×C293C2×D411yescharacteristic
139Q8×C293—21no—
140Q8×C293—21no—
141C2×C4×C293—21no—
142C2×C4×C293—21no—
143C2×C4×C293—21no—
144C2×C4×C293—21no—
145C2×C4×C293—21no—
146C2×C4×C293—21no—
147C2×C4×C293—42no—
148C2×C4×C293C2×D411yescharacteristic
149C2×C4×C293C2411yescharacteristic
150C8×C293—21no—
151C8×C293—21no—
152C22×C293—21no—
153C22×C293—21no—
154C22×C293#13512yes—
155C22×C293C22×D411yescharacteristic
156C2×D293—21no—
157C2×D293—22no—
158C2×D293—21no—
159C2×D293—21no—
160C2×D293—41no—
161C2×D293—41no—
162C2×D293—41no—
163C2×D293—41no—
164C2×D293—41no—
165C2×D293—84no—
166C4×C293—21no—
167C4×C293—41no—
168C4×C293—41no—
169C4×C293—41no—
170C4×C293—41no—
171C4×C293C2×#4111yescharacteristic
172C4×C293H(5,𝔽2)11yescharacteristic
173Dic293—21no—
174Dic293—21no—
175Dic293—21no—
176Dic293—41no—
177Dic293—41no—
178C2×C293—22no—
179C2×C293#54611yescharacteristic
180C2×C293C2×#13511yescharacteristic
181C2×C293#57711yescharacteristic
182D293—41no—
183D293—42no—
184D293—81no—
185C293#329611yescharacteristic, complemented, Sylow
186#3296—2931nomaximal, Sylow
187#552—2931no—
188#546—2931no—
189#488—2931no—
190#483D29311yescharacteristic, complemented
191#482—2931no—
192#478—2931no—
193#467—2931no—
194#465—2931no—
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×SD16—2931no—
207C22×C8—2931no—
208C22×C8—2931no—
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
218#121—2931no—
219#121—5861no—
220#120—5861no—
221C4×Q8—2931no—
222C4×D4—2931no—
223C4×D4—2931no—
224#117—2931no—
225#117—5861no—
226C2×C4⋊C4C2×D29311yescharacteristic
227C2×C22⋊C4—5861no—
228#107—2931no—
229#106C2×D29311yescharacteristic, complemented
230#106C2×D29311yescharacteristic, complemented
231C4⋊C8—2931no—
232#103—2931no—
233#103—2931no—
234#103C2×D29311yescharacteristic, complemented
235#103C2×D29311yescharacteristic, complemented
236#102—2931no—
237#102—2931no—
238#102—2931no—
239#102—2931no—
240#102—2931no—
241#102—2931no—
242C22⋊C8—2931no—
243C22⋊C8C2×D29311yescharacteristic
244C4×C8—2931no—
245C24—5861no—
246#41—5862no—
247#41—11721no—
248C2×Q8—2931no—
249C2×Q8C22×D29311yescharacteristic
250C2×D4—2931no—
251C2×D4—2931no—
252C2×D4—2931no—
253C2×D4—5862no—
254C2×D4—5861no—
255C2×D4—5861no—
256C2×D4—5861no—
257C2×D4—11721no—
258C22×C4—21no—
259C22×C4—2931no—
260C22×C4—5861no—
261C22×C4—5861no—
262C22×C4C22×D29311yescharacteristic
263Q16—11721no—
264SD16—5862no—
265SD16—11721no—
266C2×C8—2931no—
267C2×C8—2931no—
268C2×C8—11721no—
269C2×C8—11721no—
270C2×C8C22×D29311yescharacteristic
271C2×C8C22×D29311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—2931no—
275C4⋊C4—2931no—
276C4⋊C4—5861no—
277C4⋊C4—5861no—
278C4⋊C4—5861no—
279C4⋊C4—5861no—
280C4⋊C4—5861no—
281C4⋊C4C22×D29311yescharacteristic
282C4⋊C4C22×D29311yescharacteristic
283C4⋊C4C22×D29311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2931no—
286C22⋊C4—2931no—
287C22⋊C4—5861no—
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—2931no—
298C23—23441no—
299C23—5861no—
300C23—5861no—
301C23—11722no—
302C23—11722no—
303C23D4×D29311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5861no—
307Q8—5861no—
308D4—2932no—
309D4—23441no—
310D4—5862no—
311D4—5861no—
312D4—5861no—
313D4—5861no—
314D4—11722no—
315D4—11722no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—2931no—
324C2×C4—2931no—
325C2×C4—2931no—
326C2×C4—23441no—
327C2×C4—5861no—
328C2×C4—5861no—
329C2×C4—5861no—
330C2×C4—5861no—
331C2×C4—11722no—
332C2×C4D4×D29311yescharacteristic
333C2×C4C23×D29311yescharacteristic
334C8—21no—
335C8—21no—
336C8—5861no—
337C8—5861no—
338C22—21no—
339C22—21no—
340C22—23444no—
341C22—5861no—
342C22—5862no—
343C22—5861no—
344C22—5861no—
345C22—11721no—
346C22—11721no—
347C22—11721no—
348C22—11721no—
349C22—11721no—
350C22#2262764512yes—
351C22C2×D4×D29311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5861no—
358C4—5861no—
359C4—5861no—
360C4—11721no—
361C4—11721no—
362C4#41×D29311yescharacteristic
363C4#2262767011yescharacteristic
364C2—22no—
365C2—23441no—
366C2—11722no—
367C2—11721no—
368C2#3403677111yescharacteristic
369C2C2×#2262764511yescharacteristic
370C2#3403693811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#483×C293C2
socleC22×C293C22×D4
prime radical#34036770C2
hypercenter#483D293
hypocenterC293#3296
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #34036814C2semidirect
3. #34036485C2semidirect
4. #34036455C2semidirect
5. #34036428C2semidirect
6. #34036368C2semidirect
7. #34036359C2semidirect
8. #34036354C2semidirect
9. #34036294C2semidirect
10. #34036282C2semidirect
11. #34036216C2semidirect
12. #34036192C2semidirect
13. #34036124C2semidirect
14. C2×#22627544C2semidirect
15. #483×C293C2semidirect
18. C2×#17989792C22semidirect
22. C2×C8×D293C22semidirect
24. #22627615C22semidirect
29. #22627604C22semidirect
34. #22627589C22semidirect
38. #22627582C22semidirect
40. #22627557C22semidirect
41. #22627553C22semidirect
42. #22627548C22semidirect
43. #22627547C22semidirect
50. #22627536C22semidirect
51. C8×Dic293C22semidirect
54. #106×C293C22semidirect
55. #106×C293C22semidirect
56. #103×C293C22semidirect
57. #103×C293C22semidirect
185. C293#3296semidirect
190. #483D293semidirect
229. #106C2×D293semidirect
230. #106C2×D293semidirect
234. #103C2×D293semidirect
235. #103C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340368142
#340367702
#340364852
#340364552
#340364282
#340363682
#340363592
#340363542
#340362942
#340362822
#340362162
#340361922
#340361242
C2×#226275442
#483×C2932
#3296293