Finite Group Databasebeta

← #49170299

Subgroups — #49170299

Order 37504.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627653C211yesmaximal, characteristic, complemented
2C22×#17989790C211yesmaximal, characteristic, complemented
3#34036792C211yesmaximal, characteristic, complemented
4#34036773C211yesmaximal, characteristic
5#34036520C211yesmaximal, characteristic, complemented
6#34036374C211yesmaximal, characteristic, complemented
7#34036368C211yesmaximal, characteristic, complemented
8#34036350C211yesmaximal, characteristic, complemented
9#34036332C211yesmaximal, characteristic, complemented
10#34036292C211yesmaximal, characteristic, complemented
11#34036287C211yesmaximal, characteristic, complemented
12#34036266C211yesmaximal, characteristic, complemented
13#34036226C211yesmaximal, characteristic, complemented
14#34036197C211yesmaximal, characteristic, complemented
15#450×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C22×Dic586C2211yescharacteristic
18C2×Q8×D293C2211yescharacteristic, complemented
19#22627653—24no—
20C2×Dic1172C2211yescharacteristic, complemented
21C2×Dic1172C2211yescharacteristic, complemented
22C2×#17989792C2211yescharacteristic, complemented
23C2×#17989790—24no—
24C2×#17989790C2211yescharacteristic
25C2×#17989790C2211yescharacteristic
26C2×#17989790C2211yescharacteristic
27C2×#17989790C2211yescharacteristic
28C2×#17989785C2211yescharacteristic, complemented
29C22×#14556873C2211yescharacteristic
30#22627623—21no—
31#22627622C2211yescharacteristic, complemented
32#22627611C2211yescharacteristic
33#22627605—21no—
34#22627604C2211yescharacteristic, complemented
35#22627603C2211yescharacteristic
36#22627602—21no—
37#22627601—21no—
38#22627596—21no—
39#22627595C2211yescharacteristic
40#22627593—21no—
41#22627589C2211yescharacteristic, complemented
42D4×Dic293C2211yescharacteristic
43C4×C22⋊D293C2211yescharacteristic
44#22627584—21no—
45C2×C22⋊Dic293—21no—
46#22627557C2211yescharacteristic, complemented
47#22627553C2211yescharacteristic, complemented
48#22627549C2211yescharacteristic, complemented
49#22627549C2211yescharacteristic, complemented
50#22627548C2211yescharacteristic, complemented
51#22627547C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53#22627534C2211yescharacteristic, complemented
54C22×D4×C293C2211yescharacteristic
55C2×SD16×C293C2211yescharacteristic
56C2×SD16×C293C2211yescharacteristic
57C22⋊Q8×C293C2211yescharacteristic
58#102×C293C2211yescharacteristic
59#102×C293C2211yescharacteristic
60C22⋊C8×C293C2211yescharacteristic
61#17989801—22no—
62#17989801—41no—
63C2×Dic586—21no—
64C2×Dic586—22no—
65C2×Dic586—21no—
66C2×Dic586—41no—
67C2×Dic586C2311yescharacteristic
68C2×Dic586C2311yescharacteristic
69C2×Dic586C2311yescharacteristic
70Q8×D293—22no—
71Q8×D293—41no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293—21no—
74C2×C22⋊D293D411yescharacteristic
75C22×Dic293—21no—
76C22×Dic293—21no—
77C22×Dic293—21no—
78C2×C4×D293—21no—
79C2×C4×D293C2311yescharacteristic
80Dic1172—22no—
81Dic1172—41no—
82#17989792—22no—
83#17989790—22no—
84#17989790—22no—
85#17989790—22no—
86#17989790D414yescomplemented
87#17989785—22no—
88C2×#14556873—22no—
89C2×#14556873C2311yescharacteristic
90C2×#14556873C2311yescharacteristic
91C4⋊Dic293—21no—
92C4⋊Dic293C2311yescharacteristic
93#17989781—21no—
94#17989781—21no—
95#17989781—21no—
96#17989781—21no—
97#17989781D411yescharacteristic
98C22⋊Dic293—21no—
99C22⋊Dic293—21no—
100C22⋊Dic293—21no—
101C22⋊Dic293—21no—
102C22⋊Dic293—21no—
103C22⋊Dic293D411yescharacteristic
104#17989779—21no—
105#17989779D411yescharacteristic
106C4×Dic293—21no—
107C4×Dic293C2311yescharacteristic
108C24×C293—21no—
109C2×Q8×C293C2311yescharacteristic
110C2×D4×C293—22no—
111C2×D4×C293—21no—
112C2×D4×C293—41no—
113C2×D4×C293C2311yescharacteristic
114C2×D4×C293C2311yescharacteristic
115C22×C4×C293C2311yescharacteristic
116SD16×C293—22no—
117SD16×C293—41no—
118C2×C8×C293C2311yescharacteristic
119C2×C8×C293C2311yescharacteristic
120C4⋊C4×C293—21no—
121C4⋊C4×C293C2311yescharacteristic
122C22⋊C4×C293—21no—
123C22×D293C2×D411yescharacteristic
124Dic586—21no—
125Dic586—21no—
126Dic586—21no—
127Dic586—21no—
128Dic586—21no—
129Dic586—42no—
130Dic586—42no—
131Dic586C2×D412yes—
132C22⋊D293—22no—
133C22⋊D293—42no—
134C22⋊D293—81no—
135C2×Dic293—21no—
136C2×Dic293—21no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C2×Dic293—42no—
140C2×Dic293—42no—
141C2×Dic293—81no—
142C2×Dic293C2×D411yescharacteristic
143C2×Dic293C2×D411yescharacteristic
144C2×Dic293C2×D411yescharacteristic
145C4×D293—21no—
146C4×D293—21no—
147C4×D293—42no—
148#14556873—21no—
149#14556873C2×D412yes—
150C23×C293—21no—
151C23×C293—21no—
152C23×C293—42no—
153C23×C293—42no—
154C23×C293—81no—
155C23×C293C2×D411yescharacteristic
156Q8×C293—21no—
157Q8×C293—21no—
158D4×C293—21no—
159D4×C293—21no—
160D4×C293—21no—
161D4×C293—42no—
162D4×C293C2×D412yes—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293C2×D411yescharacteristic
168C2×C4×C293C2411yescharacteristic
169C8×C293—21no—
170C8×C293—21no—
171C22×C293—21no—
172C22×C293—21no—
173C22×C293—22no—
174C22×C293—41no—
175C22×C293—41no—
176C22×C293—41no—
177C22×C293—41no—
178C22×C293—41no—
179C22×C293—84no—
180C22×C293#13712yescomplemented
181C22×C293C22×D411yescharacteristic
182C2×D293—21no—
183C2×D293—21no—
184C2×D293—21no—
185C4×C293—21no—
186C4×C293—41no—
187C4×C293—41no—
188C4×C293C22×D411yescharacteristic
189C4×C293H(5,𝔽2)11yescharacteristic
190Dic293—21no—
191Dic293—22no—
192Dic293—21no—
193Dic293—21no—
194Dic293—41no—
195Dic293—41no—
196Dic293—41no—
197C2×C293—22no—
198C2×C293—42no—
199C2×C293—81no—
200C2×C293D4211yescharacteristic
201C2×C293C2×#13711yescharacteristic
202C2×C293#57711yescharacteristic
203D293—41no—
204C293#326611yescharacteristic, complemented, Sylow
205#3266—2931nomaximal, Sylow
206C2×#137—2931no—
207C22×SD16—2931no—
208D4×Q8—2931no—
209#546—2931no—
210#497—2931no—
211#468—2931no—
212#467—2931no—
213#464—2931no—
214#461—2931no—
215#452—2931no—
216#451—2931no—
217#450D29311yescharacteristic, complemented
218#448—2931no—
219#440—2931no—
220#435—2931no—
221C2×#41—2931no—
222C22×Q8—2931no—
223C22×Q8—2931no—
224C22×D4C2×D29311yescharacteristic
225#137—5864no—
226C2×Q16—2931no—
227C2×Q16—2931no—
228C2×Q16—2931no—
229C2×SD16—2931no—
230C2×SD16—2931no—
231C2×SD16—2931no—
232C2×SD16—2931no—
233C2×SD16—5864no—
234C2×SD16C2×D29311yescharacteristic
235C2×SD16C2×D29311yescharacteristic
236C2×M16—2931no—
237C22×C8—2931no—
238C4⋊Q8—2931no—
239C4⋊Q8—5861no—
240#124—2931no—
241#123—5861no—
242C22⋊Q8—2931no—
243C22⋊Q8—2931no—
244C22⋊Q8—5861no—
245C22⋊Q8—5861no—
246C22⋊Q8C2×D29311yescharacteristic
247#121—2931no—
248#121—5861no—
249#120—5861no—
250C4×Q8—2931no—
251C4×D4—2931no—
252C4×D4—2931no—
253C4×D4—5861no—
254C2×C22⋊C4—5861no—
255#107—2931no—
256C4⋊C8—2931no—
257#103—2931no—
258#103—2931no—
259#103—2931no—
260#103—2931no—
261#102C2×D29311yescharacteristic
262#102C2×D29311yescharacteristic
263C22⋊C8—2931no—
264C22⋊C8C2×D29311yescharacteristic
265#97—2931no—
266C24—21no—
267#41—5862no—
268#41—11721no—
269C2×Q8—2931no—
270C2×Q8—2931no—
271C2×Q8—2931no—
272C2×Q8—5861no—
273C2×Q8—5862no—
274C2×Q8—5861no—
275C2×Q8—5862no—
276C2×Q8—11721no—
277C2×Q8—11721no—
278C2×Q8C22×D29311yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—2931no—
283C2×D4—5861no—
284C2×D4—5861no—
285C2×D4C22×D29311yescharacteristic
286C2×D4C22×D29311yescharacteristic
287C22×C4—2931no—
288C22×C4—5861no—
289C22×C4—5861no—
290C22×C4—5861no—
291C22×C4—5861no—
292C22×C4C22×D29311yescharacteristic
293Q16—5862no—
294Q16—5862no—
295Q16—11721no—
296SD16—22no—
297SD16—41no—
298SD16—2934no—
299SD16—5862no—
300SD16—5862no—
301SD16—5862no—
302M16—5862no—
303C2×C8—2931no—
304C2×C8—2931no—
305C2×C8—5862no—
306C2×C8C22×D29311yescharacteristic
307C2×C8C22×D29311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2931no—
310C4⋊C4—2931no—
311C4⋊C4—5861no—
312C4⋊C4—5861no—
313C4⋊C4—5861no—
314C4⋊C4—5861no—
315C4⋊C4—5861no—
316C4⋊C4C22×D29311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2931no—
319C22⋊C4—2931no—
320C22⋊C4—5861no—
321C22⋊C4—5861no—
322C22⋊C4—5861no—
323C22⋊C4—5861no—
324C22⋊C4—5861no—
325C22⋊C4—5861no—
326C42—2931no—
327C42—5861no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—2931no—
334C23D4×D29311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—2932no—
338Q8—5861no—
339Q8—5861no—
340Q8—5861no—
341Q8—5861no—
342Q8—5861no—
343Q8—11722no—
344Q8—11722no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—23441no—
350D4—5862no—
351D4—11722no—
352D4D4×D29312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—2931no—
358C2×C4—2931no—
359C2×C4—2931no—
360C2×C4—23441no—
361C2×C4—5861no—
362C2×C4—5861no—
363C2×C4—5861no—
364C2×C4—5861no—
365C2×C4—5861no—
366C2×C4—5861no—
367C2×C4—11722no—
368C2×C4—11722no—
369C2×C4—11722no—
370C2×C4D4×D29311yescharacteristic
371C2×C4C23×D29311yescharacteristic
372C8—21no—
373C8—21no—
374C8—2932no—
375C8—5861no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—5861no—
386C22—5861no—
387C22—5861no—
388C22#2262765312yescomplemented
389C22C2×D4×D29311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—5861no—
394C4—5862no—
395C4—5861no—
396C4—5861no—
397C4—11721no—
398C4—11721no—
399C4—11721no—
400C4C2×D4×D29311yescharacteristic
401C4#2262766911yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—11721no—
406C2#3403676111yescharacteristic
407C2C2×#2262765311yescharacteristic
408C2#3403693911yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#450×C293C2
socleC22×C293C22×D4
prime radical#34036773C2
hypercenter#450D293
hypocenterC293#3266
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22627653C2semidirect
2. C22×#17989790C2semidirect
3. #34036792C2semidirect
5. #34036520C2semidirect
6. #34036374C2semidirect
7. #34036368C2semidirect
8. #34036350C2semidirect
9. #34036332C2semidirect
10. #34036292C2semidirect
11. #34036287C2semidirect
12. #34036266C2semidirect
13. #34036226C2semidirect
14. #34036197C2semidirect
15. #450×C293C2semidirect
18. C2×Q8×D293C22semidirect
20. C2×Dic1172C22semidirect
21. C2×Dic1172C22semidirect
22. C2×#17989792C22semidirect
28. C2×#17989785C22semidirect
31. #22627622C22semidirect
34. #22627604C22semidirect
41. #22627589C22semidirect
46. #22627557C22semidirect
47. #22627553C22semidirect
48. #22627549C22semidirect
49. #22627549C22semidirect
50. #22627548C22semidirect
51. #22627547C22semidirect
52. #22627536C22semidirect
53. #22627534C22semidirect
86. #17989790D4semidirect
180. C22×C293#137semidirect
204. C293#3266semidirect
217. #450D293semidirect
388. 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×#179897902
#340367922
#340367732
#340365202
#340363742
#340363682
#340363502
#340363322
#340362922
#340362872
#340362662
#340362262
#340361972
#450×C2932
#3266293