Finite Group Databasebeta

← #49170352

Subgroups — #49170352

Order 37504.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627653C211yesmaximal, characteristic, complemented
2C22×Dic1172C211yesmaximal, characteristic, complemented
3#34036792C211yesmaximal, characteristic, complemented
4#34036785C211yesmaximal, characteristic
5#34036520C211yesmaximal, characteristic, complemented
6#34036388C211yesmaximal, characteristic, complemented
7#34036350C211yesmaximal, characteristic, complemented
8#34036338C211yesmaximal, characteristic, complemented
9#34036292C211yesmaximal, characteristic, complemented
10#34036286C211yesmaximal, characteristic, complemented
11#34036284C211yesmaximal, characteristic, complemented
12#34036267C211yesmaximal, characteristic, complemented
13#34036235C211yesmaximal, characteristic, complemented
14#34036200C211yesmaximal, characteristic, complemented
15#465×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C22×Dic586C2211yescharacteristic
18C22×Dic586C2211yescharacteristic
19C2×Q8×D293C2211yescharacteristic
20#22627653—24no—
21C2×Dic1172—24no—
22C2×Dic1172C2211yescharacteristic
23C2×Dic1172C2211yescharacteristic
24C2×Dic1172C2211yescharacteristic
25C2×Dic1172C2211yescharacteristic
26C2×#17989792C2211yescharacteristic
27C2×#17989792C2211yescharacteristic
28C2×#17989790C2211yescharacteristic
29C2×#17989790C2211yescharacteristic
30C2×#17989785C2211yescharacteristic
31#22627623—21no—
32#22627622C2211yescharacteristic
33#22627611C2211yescharacteristic
34#22627610—21no—
35#22627604C2211yescharacteristic
36#22627604C2211yescharacteristic
37#22627602—21no—
38#22627601—21no—
39#22627601—21no—
40#22627596—21no—
41#22627589C2211yescharacteristic
42#22627589C2211yescharacteristic
43D4×Dic293—21no—
44C4×C22⋊D293C2211yescharacteristic
45#22627584—21no—
46#22627553C2211yescharacteristic
47#22627547C2211yescharacteristic
48#22627547C2211yescharacteristic
49#22627547C2211yescharacteristic
50#22627547C2211yescharacteristic
51C22⋊#14556873C2211yescharacteristic
52#22627536C2211yescharacteristic
53#22627534C2211yescharacteristic
54C2×SD16×C293C2211yescharacteristic
55C22×C8×C293C2211yescharacteristic
56C22⋊Q8×C293C2211yescharacteristic
57#121×C293C2211yescharacteristic
58#106×C293C2211yescharacteristic
59#103×C293C2211yescharacteristic
60#102×C293C2211yescharacteristic
61#17989801—22no—
62#17989801—41no—
63C2×Dic586—21no—
64C2×Dic586—22no—
65C2×Dic586—22no—
66C2×Dic586—21no—
67C2×Dic586—21no—
68C2×Dic586—41no—
69C2×Dic586—41no—
70C2×Dic586C2311yescharacteristic
71C2×Dic586C2311yescharacteristic
72C2×Dic586C2311yescharacteristic
73C2×Dic586C2311yescharacteristic
74Q8×D293—22no—
75Q8×D293—41no—
76C2×C22⋊D293—21no—
77C2×C22⋊D293D411yescharacteristic
78C22×Dic293—21no—
79C22×Dic293—21no—
80C22×Dic293—21no—
81C2×C4×D293—21no—
82C2×C4×D293C2311yescharacteristic
83Dic1172—22no—
84Dic1172—22no—
85Dic1172—22no—
86Dic1172D414yescomplemented
87#17989792—22no—
88#17989792—41no—
89#17989790—22no—
90#17989790—41no—
91#17989785—22no—
92C2×#14556873C2311yescharacteristic
93C2×#14556873C2311yescharacteristic
94C4⋊Dic293—21no—
95C4⋊Dic293—21no—
96#17989781—21no—
97#17989781—21no—
98#17989781—21no—
99#17989781—21no—
100#17989781D411yescharacteristic
101C22⋊Dic293—21no—
102C22⋊Dic293—21no—
103C22⋊Dic293D411yescharacteristic
104#17989779—21no—
105#17989779—21no—
106#17989779D411yescharacteristic
107C4×Dic293—21no—
108C4×Dic293—21no—
109C4×Dic293C2311yescharacteristic
110C2×Q8×C293C2311yescharacteristic
111C2×D4×C293—21no—
112C2×D4×C293C2311yescharacteristic
113C22×C4×C293C2311yescharacteristic
114SD16×C293—22no—
115C2×C8×C293—22no—
116C2×C8×C293C2311yescharacteristic
117C2×C8×C293C2311yescharacteristic
118C4⋊C4×C293—21no—
119C4⋊C4×C293C2311yescharacteristic
120C4⋊C4×C293C2311yescharacteristic
121C22⋊C4×C293—21no—
122C22⋊C4×C293—21no—
123C22×D293C2×D411yescharacteristic
124Dic586—21no—
125Dic586—21no—
126Dic586—21no—
127Dic586—21no—
128Dic586—21no—
129Dic586—21no—
130Dic586—42no—
131Dic586—42no—
132Dic586—42no—
133Dic586C2×D412yes—
134Dic586C2×D412yes—
135C22⋊D293—22no—
136C22⋊D293—42no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C2×Dic293—21no—
140C2×Dic293—21no—
141C2×Dic293—42no—
142C2×Dic293—42no—
143C2×Dic293—42no—
144C2×Dic293C2×D411yescharacteristic
145C2×Dic293C2×D411yescharacteristic
146C2×Dic293C2×D411yescharacteristic
147C4×D293—21no—
148C4×D293—21no—
149C4×D293—42no—
150#14556873—21no—
151#14556873—21no—
152C23×C293—21no—
153C23×C293C2×D411yescharacteristic
154Q8×C293—21no—
155Q8×C293—21no—
156D4×C293—21no—
157D4×C293—21no—
158D4×C293—42no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—21no—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293C2×D411yescharacteristic
165C2×C4×C293C2411yescharacteristic
166C8×C293—21no—
167C8×C293C2×D412yes—
168C22×C293—21no—
169C22×C293—21no—
170C22×C293—41no—
171C22×C293—41no—
172C22×C293—41no—
173C22×C293#13712yescomplemented
174C22×C293C22×D411yescharacteristic
175C2×D293—21no—
176C2×D293—21no—
177C2×D293—21no—
178C4×C293—21no—
179C4×C293—41no—
180C4×C293—41no—
181C4×C293—41no—
182C4×C293C22×D411yescharacteristic
183C4×C293H(5,𝔽2)11yescharacteristic
184Dic293—21no—
185Dic293—22no—
186Dic293—22no—
187Dic293—21no—
188Dic293—21no—
189Dic293—41no—
190Dic293—41no—
191C2×C293—22no—
192C2×C293—81no—
193C2×C293D4211yescharacteristic
194C2×C293C2×#13711yescharacteristic
195C2×C293#57811yescharacteristic
196D293—41no—
197C293#327011yescharacteristic, complemented, Sylow
198#3270—2931nomaximal, Sylow
199C2×#137—2931no—
200C22×Q16—2931no—
201D4×Q8—2931no—
202#548—2931no—
203#497—2931no—
204#470—2931no—
205#465D29311yescharacteristic, complemented
206#464—2931no—
207#462—2931no—
208#452—2931no—
209#451—2931no—
210#451—2931no—
211#448—2931no—
212#441—2931no—
213#435—2931no—
214C2×#41—2931no—
215C22×Q8—2931no—
216C22×Q8—2931no—
217C22×Q8—2931no—
218#137—5864no—
219C2×Q16—2931no—
220C2×Q16—2931no—
221C2×Q16—2931no—
222C2×Q16—2931no—
223C2×Q16—2931no—
224C2×Q16—2931no—
225C2×Q16—5864no—
226C2×SD16—2931no—
227C2×SD16—2931no—
228C2×SD16C2×D29311yescharacteristic
229C2×M16—2931no—
230C22×C8C2×D29311yescharacteristic
231C4⋊Q8—2931no—
232C4⋊Q8—5861no—
233#124—2931no—
234#124—5861no—
235C22⋊Q8—2931no—
236C22⋊Q8—2931no—
237C22⋊Q8—5861no—
238C22⋊Q8—5861no—
239C22⋊Q8—5861no—
240C22⋊Q8C2×D29311yescharacteristic
241#121—5861no—
242#121C2×D29311yescharacteristic
243C4×Q8—2931no—
244C4×Q8—2931no—
245C4×D4—2931no—
246C4×D4—5861no—
247C4×D4—5861no—
248#106C2×D29311yescharacteristic
249C4⋊C8—2931no—
250#103—2931no—
251#103—2931no—
252#103—2931no—
253#103—2931no—
254#103C2×D29311yescharacteristic
255#102C2×D29311yescharacteristic
256C22⋊C8—2931no—
257C22⋊C8—2931no—
258#97—2931no—
259#41—5862no—
260#41—11721no—
261C2×Q8—2931no—
262C2×Q8—2931no—
263C2×Q8—2931no—
264C2×Q8—2931no—
265C2×Q8—5861no—
266C2×Q8—5862no—
267C2×Q8—5862no—
268C2×Q8—5861no—
269C2×Q8—5861no—
270C2×Q8—5862no—
271C2×Q8—11721no—
272C2×Q8—11721no—
273C2×Q8—11721no—
274C2×Q8C22×D29311yescharacteristic
275C2×D4—21no—
276C2×D4—2931no—
277C2×D4—5861no—
278C2×D4C22×D29311yescharacteristic
279C22×C4—2931no—
280C22×C4—5861no—
281C22×C4—5861no—
282C22×C4—5861no—
283C22×C4—5861no—
284C22×C4C22×D29311yescharacteristic
285Q16—2934no—
286Q16—5862no—
287Q16—5862no—
288Q16—5862no—
289Q16—5862no—
290Q16—11721no—
291SD16—22no—
292SD16—5862no—
293SD16—11721no—
294M16—5862no—
295C2×C8—22no—
296C2×C8—2931no—
297C2×C8—2931no—
298C2×C8C22×D29311yescharacteristic
299C2×C8C22×D29311yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—2931no—
302C4⋊C4—5861no—
303C4⋊C4—5861no—
304C4⋊C4—5861no—
305C4⋊C4—5861no—
306C4⋊C4—5861no—
307C4⋊C4—5861no—
308C4⋊C4C22×D29311yescharacteristic
309C4⋊C4C22×D29311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2931no—
313C22⋊C4—2931no—
314C22⋊C4—5861no—
315C22⋊C4—5861no—
316C22⋊C4—5861no—
317C22⋊C4—5861no—
318C42—2931no—
319C42—5861no—
320C42—5861no—
321C23—21no—
322C23—2931no—
323C23D4×D29311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—2932no—
327Q8—2932no—
328Q8—5861no—
329Q8—5861no—
330Q8—5861no—
331Q8—5861no—
332Q8—5861no—
333Q8—5861no—
334Q8—11722no—
335Q8—11722no—
336Q8—11722no—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—5862no—
341D4—11722no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—2931no—
348C2×C4—2931no—
349C2×C4—2931no—
350C2×C4—5861no—
351C2×C4—5861no—
352C2×C4—5861no—
353C2×C4—5861no—
354C2×C4—5861no—
355C2×C4—5861no—
356C2×C4—11722no—
357C2×C4—11722no—
358C2×C4—11722no—
359C2×C4—11722no—
360C2×C4D4×D29311yescharacteristic
361C2×C4C23×D29311yescharacteristic
362C8—21no—
363C8—5861no—
364C8—5861no—
365C8D4×D29312yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—5861no—
372C22—5861no—
373C22—5861no—
374C22#2262765312yescomplemented
375C22C2×D4×D29311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—5861no—
381C4—5862no—
382C4—5862no—
383C4—5861no—
384C4—5861no—
385C4—11721no—
386C4—11721no—
387C4C2×D4×D29311yescharacteristic
388C4#2262767011yescharacteristic
389C2—22no—
390C2—81no—
391C2—11721no—
392C2#3403676311yescharacteristic
393C2C2×#2262765311yescharacteristic
394C2#3403694311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#465×C293C2
socleC22×C293C22×D4
prime radical#34036785C2
hypercenter#465D293
hypocenterC293#3270
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. C2×#22627653C2semidirect
2. C22×Dic1172C2semidirect
3. #34036792C2semidirect
5. #34036520C2semidirect
6. #34036388C2semidirect
7. #34036350C2semidirect
8. #34036338C2semidirect
9. #34036292C2semidirect
10. #34036286C2semidirect
11. #34036284C2semidirect
12. #34036267C2semidirect
13. #34036235C2semidirect
14. #34036200C2semidirect
15. #465×C293C2semidirect
86. Dic1172D4semidirect
173. C22×C293#137semidirect
197. C293#3270semidirect
205. #465D293semidirect
374. C22#22627653semidirect

Maximal subgroups

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

subgroupindex
C2×#226276532
C22×Dic11722
#340367922
#340367852
#340365202
#340363882
#340363502
#340363382
#340362922
#340362862
#340362842
#340362672
#340362352
#340362002
#465×C2932
#3270293