Finite Group Databasebeta

← #17997583

Subgroups — #17997583

Order 4736.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924521C211yesmaximal, characteristic
2C2×#11924498C211yesmaximal, characteristic
3C2×#11924494C211yesmaximal, characteristic, complemented
4#14557425C211yesmaximal, characteristic, complemented
5#14557423C211yesmaximal, characteristic, complemented
6#14557419C211yesmaximal, characteristic, complemented
7#14557414C211yesmaximal, characteristic, complemented
8#14557411C211yesmaximal, characteristic
9#14557404C211yesmaximal, characteristic, complemented
10#14557404C211yesmaximal, characteristic, complemented
11#14557384C211yesmaximal, characteristic, complemented
12C4×#10596938C211yesmaximal, characteristic, complemented
13C4×#10596938C211yesmaximal, characteristic, complemented
14#95×D37C211yesmaximal, characteristic, complemented
15C37×#389C211yesmaximal, characteristic, complemented
16C22×D148C2211yescharacteristic
17C22×C4×D37C2211yescharacteristic
18C22×C4×D37C2211yescharacteristic
19C22×C4×D37C2211yescharacteristic
20C22×C4×D37C2211yescharacteristic
21#11924521C418yescomplemented
22#11924498—24no—
23#11924494—24no—
24C2×C4⋊Dic37C2211yescharacteristic
25C2×#10596936C2211yescharacteristic
26C2×#10596936C2211yescharacteristic
27C2×#10596938C2211yescharacteristic
28C2×#10596938C2211yescharacteristic
29C2×#10596938C2211yescharacteristic
30C2×#10596938C2211yescharacteristic
31C2×#10596938C2211yescharacteristic
32C2×#10596938C2211yescharacteristic
33C2×#10596938C2211yescharacteristic
34C2×#10596938C2211yescharacteristic
35C2×C4×Dic37C2211yescharacteristic
36C2×C4×Dic37C2211yescharacteristic
37C2×C4×Dic37C2211yescharacteristic
38C2×C4×Dic37C2211yescharacteristic
39#11924443C2211yescharacteristic
40#11924443C2211yescharacteristic
41#11924443C2211yescharacteristic
42#11924443C2211yescharacteristic
43#11924443C2211yescharacteristic
44#11924443C2211yescharacteristic
45#11924443C2211yescharacteristic
46#11924443C2211yescharacteristic
47C2×C4⋊C4×C37C2211yescharacteristic
48C2×C4⋊C4×C37C2211yescharacteristic
49C2×C4⋊C4×C37C2211yescharacteristic
50C2×C4⋊C4×C37C2211yescharacteristic
51C2×C42×C37C2211yescharacteristic
52#95×C37C2211yescharacteristic
53#95×C37C2211yescharacteristic
54C23×D37C2311yescharacteristic
55C23×D37C2311yescharacteristic
56C22×C4×C37C2311yescharacteristic
57C22×C4×C37C2311yescharacteristic
58C22×C4×C37C2311yescharacteristic
59C22×C4×C37C2311yescharacteristic
60C22×C4×C37C2311yescharacteristic
61C22×C4×C37C2311yescharacteristic
62C22×C4×C37C2311yescharacteristic
63C22×Dic37C2311yescharacteristic
64C22×Dic37C2311yescharacteristic
65C22×Dic37C2311yescharacteristic
66C22×Dic37C2311yescharacteristic
67C22×Dic37C2311yescharacteristic
68C22×Dic37C2311yescharacteristic
69C2×D148—22no—
70C2×D148—22no—
71C2×D148C2×C414yes—
72C2×C4×D37—22no—
73C2×C4×D37—22no—
74C2×C4×D37—22no—
75C2×C4×D37—22no—
76C2×C4×D37—22no—
77C2×C4×D37—22no—
78C2×C4×D37—22no—
79C2×C4×D37—22no—
80C2×C4×D37—41no—
81C2×C4×D37—41no—
82C2×C4×D37—41no—
83C2×C4×D37C2×C414yescomplemented
84C4⋊C4×C37—22no—
85C4⋊C4×C37—22no—
86C4⋊C4×C37—41no—
87C4⋊C4×C37C2×C414yescomplemented
88C42×C37—41no—
89#10596938—22no—
90#10596938—22no—
91#10596938—22no—
92#10596938—41no—
93#10596938C2×C414yescomplemented
94#10596938C2×C414yescomplemented
95#10596938C2×C414yescomplemented
96#10596938D414yes—
97C4⋊Dic37—22no—
98#10596936—22no—
99#10596936C2×C414yescomplemented
100C4×Dic37—22no—
101C4×Dic37—22no—
102C4×Dic37—22no—
103C4×Dic37—22no—
104C23×C37C2411yescharacteristic
105C22×D37—21no—
106C22×D37—21no—
107C22×D37—21no—
108C22×D37—21no—
109C22×D37—21no—
110C22×D37—21no—
111C22×D37—21no—
112C22×D37—21no—
113C22×D37—21no—
114C22×D37—21no—
115C22×D37C22×C412yes—
116C22×D37C22×C412yes—
117C22×D37C2×D412yes—
118C22×D37#4112yes—
119C2×C4×C37—21no—
120C2×C4×C37—21no—
121C2×C4×C37—21no—
122C2×C4×C37—21no—
123C2×C4×C37—21no—
124C2×C4×C37—21no—
125C2×C4×C37—21no—
126C2×C4×C37—21no—
127C2×C4×C37—21no—
128C2×C4×C37—21no—
129C2×C4×C37—21no—
130C2×C4×C37—21no—
131C2×C4×C37—21no—
132C2×C4×C37—21no—
133C2×C4×C37—21no—
134C2×C4×C37—21no—
135C2×C4×C37C22×C412yes—
136C2×C4×C37C22×C412yes—
137C2×C4×C37C22×C412yes—
138C2×C4×C37C2×D412yes—
139C2×C4×C37#4112yes—
140C2×Dic37—21no—
141C2×Dic37—21no—
142C2×Dic37—21no—
143C2×Dic37—21no—
144C2×Dic37—21no—
145C2×Dic37—21no—
146C2×Dic37—21no—
147C2×Dic37—21no—
148C2×Dic37—21no—
149C2×Dic37—21no—
150C2×Dic37—21no—
151C2×Dic37—21no—
152C2×Dic37—21no—
153C2×Dic37—21no—
154C2×Dic37C22×C412yes—
155C2×Dic37C22×C412yes—
156C2×Dic37C2×D412yes—
157C2×Dic37#4112yes—
158D148—28no—
159C4×D37—28no—
160C4×D37—44no—
161C4×D37—44no—
162C4×D37—44no—
163C22×C37C23×C411yescharacteristic
164C22×C37C22×D411yescharacteristic
165C22×C37C2×#4111yescharacteristic
166C22×C37C2×#4111yescharacteristic
167C22×C37C2×#4111yescharacteristic
168C22×C37H(5,𝔽2)11yescharacteristic
169C22×C372−1+411yescharacteristic
170C2×D37—22no—
171C2×D37—22no—
172C2×D37—22no—
173C2×D37—22no—
174C2×D37—22no—
175C2×D37—22no—
176C2×D37—22no—
177C2×D37—22no—
178C2×D37—22no—
179C2×D37—41no—
180C2×D37—41no—
181C2×D37—41no—
182C2×D37—41no—
183C2×D37C4×D414yes—
184C4×C37—22no—
185C4×C37—22no—
186C4×C37—22no—
187C4×C37—22no—
188C4×C37—22no—
189C4×C37—41no—
190C4×C37—41no—
191Dic37—22no—
192Dic37—22no—
193Dic37—22no—
194Dic37—22no—
195Dic37—41no—
196Dic37—41no—
197#2359—371nomaximal, Sylow
198C2×C37C2×C4×D411yescharacteristic
199C2×C37C4×#4111yescharacteristic
200C2×C37#52011yescharacteristic
201C2×C37#54611yescharacteristic
202C2×C37#54711yescharacteristic
203C2×C37#55111yescharacteristic
204C2×C37#55211yescharacteristic
205D37—24no—
206D37—42no—
207C2×#123—371no—
208C2×C4×D4—371no—
209C2×#117—371no—
210#389D3711yescharacteristic, complemented
211#388—371no—
212#388—371no—
213#387—371no—
214#387—371no—
215#386—371no—
216#385—371no—
217#385—371no—
218C4×C4⋊C4—371no—
219C4×C22⋊C4—371no—
220C4×C22⋊C4—371no—
221C2×#95—371no—
222C37#235911yescharacteristic, complemented, Sylow
223C22×D4—371no—
224C23×C4—371no—
225C23×C4—371no—
226C23×C4—371no—
227C23×C4—371no—
228#123—378no—
229C4×D4—744no—
230#117—744no—
231C2×C4⋊C4—371no—
232C2×C4⋊C4—371no—
233C2×C4⋊C4—371no—
234C2×C4⋊C4C2×D3711yescharacteristic
235C2×C4⋊C4C2×D3711yescharacteristic
236C2×C4⋊C4C2×D3711yescharacteristic
237C2×C4⋊C4C2×D3711yescharacteristic
238C2×C22⋊C4—371no—
239C2×C22⋊C4—371no—
240C2×C22⋊C4—371no—
241C2×C22⋊C4—371no—
242C2×C22⋊C4—371no—
243C2×C22⋊C4—371no—
244C2×C22⋊C4—371no—
245C2×C22⋊C4—371no—
246C2×C42—371no—
247C2×C42—371no—
248C2×C42—371no—
249C2×C42—371no—
250C2×C42C2×D3711yescharacteristic
251#95—371no—
252#95—371no—
253#95—371no—
254#95—371no—
255#95—371no—
256#95—371no—
257#95—371no—
258#95—371no—
259#95C2×D3711yescharacteristic
260#95C2×D3711yescharacteristic
261C24—371no—
262C24—371no—
263C2×D4—374no—
264C2×D4—742no—
265C2×D4—742no—
266C22×C4—371no—
267C22×C4—371no—
268C22×C4—371no—
269C22×C4—371no—
270C22×C4—374no—
271C22×C4—371no—
272C22×C4—371no—
273C22×C4—742no—
274C22×C4—742no—
275C22×C4—742no—
276C22×C4—742no—
277C22×C4—742no—
278C22×C4—742no—
279C22×C4—742no—
280C22×C4—742no—
281C22×C4—1481no—
282C22×C4—1481no—
283C22×C4—1481no—
284C22×C4C22×D3711yescharacteristic
285C22×C4C22×D3711yescharacteristic
286C22×C4C22×D3711yescharacteristic
287C22×C4C22×D3711yescharacteristic
288C22×C4C22×D3711yescharacteristic
289C22×C4C22×D3711yescharacteristic
290C22×C4C22×D3711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—374no—
295C4⋊C4—742no—
296C4⋊C4—742no—
297C4⋊C4C4×D3714yescomplemented
298C22⋊C4—374no—
299C22⋊C4—374no—
300C22⋊C4—374no—
301C22⋊C4—374no—
302C22⋊C4—742no—
303C22⋊C4—742no—
304C22⋊C4—742no—
305C22⋊C4—1481no—
306C42—41no—
307C42—742no—
308C42—742no—
309C42—742no—
310C42—742no—
311C23—372no—
312C23—372no—
313C23—372no—
314C23—372no—
315C23—741no—
316C23—741no—
317C23—741no—
318C23—741no—
319C23—741no—
320C23—741no—
321C23—741no—
322C23—741no—
323C23—741no—
324C23—741no—
325C23C23×D3711yescharacteristic
326D4—748no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—372no—
344C2×C4—372no—
345C2×C4—372no—
346C2×C4—372no—
347C2×C4—748no—
348C2×C4—741no—
349C2×C4—741no—
350C2×C4—741no—
351C2×C4—741no—
352C2×C4—741no—
353C2×C4—741no—
354C2×C4—741no—
355C2×C4—741no—
356C2×C4—741no—
357C2×C4—741no—
358C2×C4—741no—
359C2×C4—741no—
360C2×C4—741no—
361C2×C4—741no—
362C2×C4—1484no—
363C2×C4—1484no—
364C2×C4—1484no—
365C2×C4C2×C4×D3712yes—
366C2×C4C2×C4×D3712yes—
367C2×C4C2×C4×D3712yes—
368C2×C4D4×D3712yes—
369C2×C4#1059696612yes—
370C22—374no—
371C22—742no—
372C22—742no—
373C22—742no—
374C22—742no—
375C22—742no—
376C22—742no—
377C22—742no—
378C22—742no—
379C22—742no—
380C22—1481no—
381C22—1481no—
382C22—1481no—
383C22—1481no—
384C22C22×C4×D3711yescharacteristic
385C22C2×D4×D3711yescharacteristic
386C22#41×D3711yescharacteristic
387C22#41×D3711yescharacteristic
388C22C2×#1059696611yescharacteristic
389C22#1192458111yescharacteristic
390C22#1192458411yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—41no—
397C4—41no—
398C4—742no—
399C4—742no—
400C4—742no—
401C4—742no—
402C4—1481no—
403C4—1481no—
404C2—744no—
405C2—1482no—
406C2C4×D4×D3711yescharacteristic
407C2C4×#1059696611yescharacteristic
408C2#1455801411yescharacteristic
409C2#1455817211yescharacteristic
410C2#1455818611yescharacteristic
411C2#1455821011yescharacteristic
412C2#1455821511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#389C2
socle, cosocleC23×C37C24
prime radicalC22×D148C22
hypercenter#389D37
hypocenterC37#2359
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
3. C2×#11924494C2semidirect
4. #14557425C2semidirect
5. #14557423C2semidirect
6. #14557419C2semidirect
7. #14557414C2semidirect
9. #14557404C2semidirect
10. #14557404C2semidirect
11. #14557384C2semidirect
12. C4×#10596938C2semidirect
13. C4×#10596938C2semidirect
14. #95×D37C2semidirect
15. C37×#389C2semidirect
21. #11924521C4semidirect
83. C2×C4×D37C2×C4semidirect
87. C4⋊C4×C37C2×C4semidirect
93. #10596938C2×C4semidirect
94. #10596938C2×C4semidirect
95. #10596938C2×C4semidirect
99. #10596936C2×C4semidirect
210. #389D37semidirect
222. C37#2359semidirect
297. C4⋊C4C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245212
C2×#119244982
C2×#119244942
#145574252
#145574232
#145574192
#145574142
#145574112
#145574042
#145573842
C4×#105969382
#95×D372
C37×#3892
#235937