Finite Group Databasebeta

← #49170349

Subgroups — #49170349

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×Q16×D293C211yesmaximal, characteristic, complemented
2Q8×C22⋊D293C211yesmaximal, characteristic, complemented
3#34036786C211yesmaximal, characteristic
4#34036520C211yesmaximal, characteristic, complemented
5#34036387C211yesmaximal, characteristic, complemented
6#34036355C211yesmaximal, characteristic, complemented
7#34036349C211yesmaximal, characteristic, complemented
8#34036337C211yesmaximal, characteristic, complemented
9#34036291C211yesmaximal, characteristic, complemented
10#34036285C211yesmaximal, characteristic, complemented
11#34036283C211yesmaximal, characteristic, complemented
12#34036266C211yesmaximal, characteristic, complemented
13#34036236C211yesmaximal, characteristic, complemented
14#34036198C211yesmaximal, characteristic, complemented
15C2×#137×C293C211yesmaximal, characteristic, complemented
16C2×Q8×D293C2211yescharacteristic
17C2×Q8×D293C2211yescharacteristic
18C2×Dic1172C2211yescharacteristic
19C2×#17989792C2211yescharacteristic
20C2×#17989792C2211yescharacteristic
21C2×#17989792C2211yescharacteristic
22Q16×D293—24no—
23C2×#17989790C2211yescharacteristic
24C2×C8×D293C2211yescharacteristic
25#22627622—21no—
26#22627622C2211yescharacteristic
27#22627611—21no—
28#22627611C2211yescharacteristic
29#22627603—21no—
30#22627603—21no—
31#22627603C2211yescharacteristic
32#22627603C2211yescharacteristic
33#22627600—21no—
34#22627600C2211yescharacteristic
35#22627595—21no—
36#22627595C2211yescharacteristic
37Q8×Dic293C2211yescharacteristic
38Q8×Dic293C2211yescharacteristic
39C4×C22⋊D293—21no—
40C4×C22⋊D293—21no—
41C4×C22⋊D293C2211yescharacteristic
42#22627556C2211yescharacteristic
43#22627553C2211yescharacteristic
44#22627549C2211yescharacteristic
45#22627549C2211yescharacteristic
46#22627548C2211yescharacteristic
47#22627548C2211yescharacteristic
48#22627548C2211yescharacteristic
49#22627545C2211yescharacteristic
50C22⋊#14556873C2211yescharacteristic
51#22627536C2211yescharacteristic
52#22627534C2211yescharacteristic
53C2×#41×C293C2211yescharacteristic
54C22×Q8×C293C2211yescharacteristic
55#137×C293—24no—
56C2×Q16×C293C2211yescharacteristic
57C2×Q16×C293C2211yescharacteristic
58C2×SD16×C293C2211yescharacteristic
59C2×SD16×C293C2211yescharacteristic
60C2×M16×C293C2211yescharacteristic
61C2×Dic586—21no—
62C2×Dic586—21no—
63C2×Dic586C2311yescharacteristic
64C2×Dic586C2311yescharacteristic
65Q8×D293—22no—
66Q8×D293—22no—
67Q8×D293—41no—
68Q8×D293—41no—
69C2×C22⋊D293—21no—
70C2×C22⋊D293D411yescharacteristic
71C2×C4×D293—21no—
72C2×C4×D293—21no—
73C2×C4×D293C2311yescharacteristic
74Dic1172—22no—
75#17989792—22no—
76#17989792—22no—
77#17989792—41no—
78#17989790—41no—
79C2×#14556873C2311yescharacteristic
80C2×#14556873C2311yescharacteristic
81C8×D293—22no—
82C4⋊Dic293—21no—
83C4⋊Dic293—21no—
84C4⋊Dic293C2311yescharacteristic
85C4⋊Dic293C2311yescharacteristic
86#17989781—21no—
87#17989781—21no—
88#17989781—21no—
89#17989781—21no—
90#17989781—21no—
91#17989781D411yescharacteristic
92C22⋊Dic293—21no—
93C22⋊Dic293—21no—
94C22⋊Dic293—21no—
95C22⋊Dic293—21no—
96C22⋊Dic293—21no—
97C22⋊Dic293D411yescharacteristic
98#17989779—21no—
99#17989779D411yescharacteristic
100C4×Dic293—21no—
101C4×Dic293—21no—
102C4×Dic293C2311yescharacteristic
103#41×C293—22no—
104#41×C293—41no—
105C2×Q8×C293—21no—
106C2×Q8×C293—22no—
107C2×Q8×C293—41no—
108C2×Q8×C293C2311yescharacteristic
109C2×Q8×C293C2311yescharacteristic
110C2×Q8×C293C2311yescharacteristic
111C2×D4×C293—21no—
112C2×D4×C293C2311yescharacteristic
113C22×C4×C293—21no—
114C22×C4×C293—21no—
115C22×C4×C293C2311yescharacteristic
116Q16×C293—22no—
117Q16×C293D414yescomplemented
118SD16×C293—22no—
119SD16×C293—22no—
120M16×C293—22no—
121C2×C8×C293C2311yescharacteristic
122C2×C8×C293C2311yescharacteristic
123C22×D293C2×D411yescharacteristic
124Dic586—21no—
125Dic586—21no—
126Dic586—21no—
127Dic586—21no—
128Dic586—42no—
129Dic586—42no—
130C22⋊D293—22no—
131C22⋊D293—42no—
132C2×Dic293—21no—
133C2×Dic293—21no—
134C2×Dic293—21no—
135C2×Dic293—21no—
136C2×Dic293C2×D411yescharacteristic
137C2×Dic293C2×D411yescharacteristic
138C2×Dic293C2×D411yescharacteristic
139C4×D293—21no—
140C4×D293—21no—
141C4×D293—42no—
142C4×D293—42no—
143#14556873—21no—
144#14556873—21no—
145C23×C293—21no—
146C23×C293C2×D411yescharacteristic
147Q8×C293—21no—
148Q8×C293—21no—
149Q8×C293—21no—
150Q8×C293—21no—
151Q8×C293—42no—
152Q8×C293C2×D412yes—
153Q8×C293C2×D412yes—
154D4×C293—21no—
155D4×C293—21no—
156D4×C293—42no—
157C2×C4×C293—21no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—21no—
162C2×C4×C293—42no—
163C2×C4×C293—42no—
164C2×C4×C293C2×D411yescharacteristic
165C2×C4×C293C2411yescharacteristic
166C8×C293—21no—
167C8×C293C2×D412yes—
168C22×C293—21no—
169C22×C293—21no—
170C22×C293—21no—
171C22×C293—41no—
172C22×C293—41no—
173C22×C293—41no—
174C22×C293C22×D411yescharacteristic
175C2×D293—21no—
176C2×D293—21no—
177C2×D293#13712yescomplemented
178C4×C293—21no—
179C4×C293—22no—
180C4×C293—22no—
181C4×C293—41no—
182C4×C293C22×D411yescharacteristic
183C4×C293H(5,𝔽2)11yescharacteristic
184Dic293—21no—
185Dic293—21no—
186Dic293—21no—
187Dic293—41no—
188Dic293—41no—
189Dic293—41no—
190Dic293—41no—
191C2×C293—41no—
192C2×C293—81no—
193C2×C293D4211yescharacteristic
194C2×C293C2×#13711yescharacteristic
195C2×C293#57811yescharacteristic
196D293—22no—
197C293#327011yescharacteristic, complemented, Sylow
198#3270—2931nomaximal, Sylow
199C2×#137D29311yescharacteristic, complemented
200C22×Q16—2931no—
201D4×Q8—2931no—
202#548—2931no—
203#497—2931no—
204#470—2931no—
205#465—2931no—
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×#41C2×D29311yescharacteristic
215C22×Q8—2931no—
216C22×Q8—2931no—
217C22×Q8C2×D29311yescharacteristic
218#137—24no—
219C2×Q16—2931no—
220C2×Q16—2931no—
221C2×Q16—2931no—
222C2×Q16—2931no—
223C2×Q16—5864no—
224C2×Q16C2×D29311yescharacteristic
225C2×Q16C2×D29311yescharacteristic
226C2×SD16—2931no—
227C2×SD16C2×D29311yescharacteristic
228C2×SD16C2×D29311yescharacteristic
229C2×M16C2×D29311yescharacteristic
230C22×C8—2931no—
231C4⋊Q8—2931no—
232C4⋊Q8—5861no—
233#124—2931no—
234#124—5861no—
235C22⋊Q8—2931no—
236C22⋊Q8—2931no—
237C22⋊Q8—2931no—
238C22⋊Q8—5861no—
239C22⋊Q8—5861no—
240C22⋊Q8—5861no—
241#121—2931no—
242#121—5861no—
243C4×Q8—2931no—
244C4×Q8—2931no—
245C4×D4—2931no—
246C4×D4—5861no—
247C4×D4—5861no—
248#106—2931no—
249C4⋊C8—2931no—
250#103—2931no—
251#103—2931no—
252#103—2931no—
253#103—2931no—
254#103—2931no—
255#102—2931no—
256C22⋊C8—2931no—
257C22⋊C8—2931no—
258#97—2931no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—2931no—
265C2×Q8—2931no—
266C2×Q8—5861no—
267C2×Q8—5861no—
268C2×Q8—5862no—
269C2×Q8—5862no—
270C2×Q8—11721no—
271C2×Q8—11721no—
272C2×Q8C22×D29311yescharacteristic
273C2×Q8C22×D29311yescharacteristic
274C2×Q8C22×D29311yescharacteristic
275C2×D4—21no—
276C2×D4—2931no—
277C2×D4—5861no—
278C2×D4C22×D29311yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2931no—
282C22×C4—5861no—
283C22×C4—5861no—
284C22×C4C22×D29311yescharacteristic
285Q16—22no—
286Q16—5862no—
287Q16—5862no—
288Q16—5862no—
289Q16—11721no—
290Q16C22⋊D29314yescomplemented
291SD16—22no—
292SD16—22no—
293SD16—11721no—
294M16—22no—
295C2×C8—2931no—
296C2×C8—2931no—
297C2×C8—5862no—
298C2×C8C22×D29311yescharacteristic
299C2×C8C22×D29311yescharacteristic
300C4⋊C4—2931no—
301C4⋊C4—2931no—
302C4⋊C4—2931no—
303C4⋊C4—5861no—
304C4⋊C4—5861no—
305C4⋊C4—5861no—
306C4⋊C4—5861no—
307C4⋊C4—5861no—
308C4⋊C4—5861no—
309C4⋊C4—5861no—
310C22⋊C4—2931no—
311C22⋊C4—2931no—
312C22⋊C4—5861no—
313C22⋊C4—5861no—
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—21no—
327Q8—21no—
328Q8—42no—
329Q8—5861no—
330Q8—5861no—
331Q8—5861no—
332Q8—5861no—
333Q8—11722no—
334Q8—11722no—
335Q8C2×C22⋊D29312yes—
336Q8C2×C22⋊D29312yes—
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—42no—
348C2×C4—42no—
349C2×C4—2931no—
350C2×C4—2931no—
351C2×C4—2931no—
352C2×C4—5861no—
353C2×C4—5861no—
354C2×C4—5861no—
355C2×C4—5861no—
356C2×C4—5861no—
357C2×C4—5861no—
358C2×C4—11722no—
359C2×C4—11722no—
360C2×C4D4×D29311yescharacteristic
361C2×C4C23×D29311yescharacteristic
362C8—21no—
363C8—5861no—
364C8—5861no—
365C8C2×C22⋊D29312yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—2932no—
373C22—5861no—
374C22—5861no—
375C22C2×D4×D29311yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—5861no—
381C4—5861no—
382C4—5861no—
383C4—11721no—
384C4—11721no—
385C4—11721no—
386C4—11721no—
387C4C22×C22⋊D29311yescharacteristic
388C4#2262766911yescharacteristic
389C2—41no—
390C2—81no—
391C2—5862no—
392C2D4×C22⋊D29311yescharacteristic
393C2#137×D29311yescharacteristic
394C2#3403694511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×#137×C293C2
socleC22×C293C22×D4
prime radical#34036786C2
hypercenterC2×#137D293
hypocenterC293#3270
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D293C2semidirect
2. Q8×C22⋊D293C2semidirect
4. #34036520C2semidirect
5. #34036387C2semidirect
6. #34036355C2semidirect
7. #34036349C2semidirect
8. #34036337C2semidirect
9. #34036291C2semidirect
10. #34036285C2semidirect
11. #34036283C2semidirect
12. #34036266C2semidirect
13. #34036236C2semidirect
14. #34036198C2semidirect
15. C2×#137×C293C2semidirect
117. Q16×C293D4semidirect
177. C2×D293#137semidirect
197. C293#3270semidirect
199. C2×#137D293semidirect
290. Q16C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D2932
Q8×C22⋊D2932
#340367862
#340365202
#340363872
#340363552
#340363492
#340363372
#340362912
#340362852
#340362832
#340362662
#340362362
#340361982
C2×#137×C2932
#3270293