Finite Group Databasebeta

← #17997749

Subgroups — #17997749

Order 4736.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924509C211yesmaximal, characteristic
2#14557435C211yesmaximal, characteristic, complemented
3#14557433C211yesmaximal, characteristic, complemented
4#14557425C211yesmaximal, characteristic, complemented
5#14557420C211yesmaximal, characteristic, complemented
6#14557417C211yesmaximal, characteristic, complemented
7#14557411C211yesmaximal, characteristic, complemented
8#14557410C211yesmaximal, characteristic, complemented
9#14557410C211yesmaximal, characteristic, complemented
10#14557409C211yesmaximal, characteristic, complemented
11#14557408C211yesmaximal, characteristic, complemented
12#14557404C211yesmaximal, characteristic, complemented
13#14557395C211yesmaximal, characteristic, complemented
14#14557390C211yesmaximal, characteristic, complemented
15C37×#388C211yesmaximal, characteristic, complemented
16C22×D148C2211yescharacteristic
17C22×C22⋊D37C2211yescharacteristic
18C22×C22⋊D37C2211yescharacteristic
19C23×Dic37C2211yescharacteristic
20C22×C4×D37C2211yescharacteristic
21C22×C4×D37C2211yescharacteristic
22#11924509C418yescomplemented
23C2×C4⋊Dic37C2211yescharacteristic
24C2×C4⋊Dic37C2211yescharacteristic
25C2×#10596936C2211yescharacteristic
26C2×#10596936C2211yescharacteristic
27C2×C22⋊Dic37C2211yescharacteristic
28C2×#10596938C2211yescharacteristic
29C2×#10596938C2211yescharacteristic
30C2×#10596938C2211yescharacteristic
31C2×#10596938C2211yescharacteristic
32C2×#10596938C2211yescharacteristic
33C2×#10596938C2211yescharacteristic
34C2×#10596938C2211yescharacteristic
35C2×#10596938C2211yescharacteristic
36C2×C4×Dic37C2211yescharacteristic
37#11924443C2211yescharacteristic
38#11924443C2211yescharacteristic
39#11924443C2211yescharacteristic
40#11924443C2211yescharacteristic
41#11924443C2211yescharacteristic
42#11924443C2211yescharacteristic
43#11924443C2211yescharacteristic
44#11924443C2211yescharacteristic
45C2×C4⋊C4×C37C2211yescharacteristic
46C2×C22⋊C4×C37C2211yescharacteristic
47C2×C22⋊C4×C37C2211yescharacteristic
48C2×C22⋊C4×C37C2211yescharacteristic
49C2×C42×C37C2211yescharacteristic
50#95×C37C2211yescharacteristic
51#95×C37C2211yescharacteristic
52C24×C37C2311yescharacteristic
53C23×D37C2311yescharacteristic
54C23×D37C2311yescharacteristic
55C22×C4×C37C2311yescharacteristic
56C22×C4×C37C2311yescharacteristic
57C22×C4×C37C2311yescharacteristic
58C22×C4×C37C2311yescharacteristic
59C22×C4×C37C2311yescharacteristic
60C22×C4×C37C2311yescharacteristic
61C2×C22⋊D37—41no—
62C2×C22⋊D37—41no—
63C2×C22⋊D37—41no—
64C2×C22⋊D37—41no—
65C2×C22⋊D37C2×C414yes—
66C2×C22⋊D37C2×C414yescomplemented
67C22×Dic37—41no—
68C22×Dic37—41no—
69C22×Dic37—41no—
70C22×Dic37C2311yescharacteristic
71C22×Dic37C2311yescharacteristic
72C22×Dic37C2311yescharacteristic
73C22×Dic37C2311yescharacteristic
74C22×Dic37C2311yescharacteristic
75C22×Dic37C2311yescharacteristic
76C2×D148—41no—
77C2×D148—41no—
78C2×D148C2×C414yescomplemented
79C2×C4×D37—41no—
80C2×C4×D37—41no—
81C2×C4×D37—41no—
82C2×C4×D37—41no—
83C2×C4×D37—41no—
84C2×C4×D37C2×C414yescomplemented
85C4⋊C4×C37—41no—
86C22⋊C4×C37—41no—
87C22⋊C4×C37—41no—
88C22⋊C4×C37C2×C414yescomplemented
89C42×C37—41no—
90C22⋊Dic37—41no—
91#10596938—41no—
92#10596938—41no—
93#10596938—41no—
94#10596938—41no—
95#10596938—41no—
96#10596938—41no—
97#10596938—41no—
98#10596938C2×C414yescomplemented
99C4⋊Dic37—41no—
100C4⋊Dic37—41no—
101#10596936—41no—
102#10596936C2×C414yescomplemented
103C4×Dic37—41no—
104C23×C37—21no—
105C23×C37—21no—
106C23×C37—21no—
107C23×C37—21no—
108C23×C37—21no—
109C23×C37—21no—
110C23×C37C22×C412yes—
111C23×C37C2411yescharacteristic
112C22×D37—21no—
113C22×D37—21no—
114C22×D37—21no—
115C22×D37—21no—
116C22×D37—21no—
117C22×D37—21no—
118C22×D37—21no—
119C22×D37—21no—
120C22×D37—21no—
121C22×D37—21no—
122C22×D37—21no—
123C22×D37—21no—
124C22×D37C22×C412yes—
125C22×D37C22×C412yes—
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×C37—21no—
136C2×C4×C37—21no—
137C2×C4×C37—21no—
138C2×C4×C37—21no—
139C2×C4×C37—21no—
140C2×C4×C37—21no—
141C2×C4×C37—21no—
142C2×C4×C37C22×C412yes—
143C2×C4×C37C22×C412yes—
144C22⋊D37—44no—
145C22⋊D37—44no—
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×Dic37—21no—
155C2×Dic37—21no—
156C2×Dic37—21no—
157C2×Dic37—21no—
158C2×Dic37—21no—
159C2×Dic37—21no—
160C2×Dic37—21no—
161C2×Dic37—21no—
162C2×Dic37—44no—
163C2×Dic37C22×C412yes—
164C2×Dic37C22×C412yes—
165D148—44no—
166C4×D37—44no—
167C4×D37—44no—
168C22×C37—22no—
169C22×C37—22no—
170C22×C37—22no—
171C22×C37—41no—
172C22×C37—41no—
173C22×C37—41no—
174C22×C37—41no—
175C22×C37C23×C411yescharacteristic
176C22×C37H(5,𝔽2)11yescharacteristic
177C22×C37H(5,𝔽2)11yescharacteristic
178C22×C37H(5,𝔽2)11yescharacteristic
179C22×C37H(5,𝔽2)11yescharacteristic
180C22×C37H(5,𝔽2)11yescharacteristic
181C22×C372−1+411yescharacteristic
182C2×D37—22no—
183C2×D37—22no—
184C2×D37—22no—
185C2×D37—22no—
186C2×D37—22no—
187C2×D37—22no—
188C2×D37—41no—
189C2×D37—41no—
190C2×D37—41no—
191C2×D37—41no—
192C2×D37—41no—
193C2×D37—41no—
194C2×D37—41no—
195C2×D37—41no—
196C4×C37—22no—
197C4×C37—22no—
198C4×C37—41no—
199C4×C37—41no—
200C4×C37—41no—
201C4×C37—41no—
202Dic37—22no—
203Dic37—22no—
204Dic37—41no—
205Dic37—41no—
206Dic37—41no—
207Dic37—41no—
208#2377—371nomaximal, Sylow
209C2×C37—42no—
210C2×C37#51811yescharacteristic
211C2×C37#51811yescharacteristic
212C2×C37#52011yescharacteristic
213C2×C37#56011yescharacteristic
214C2×C37#56011yescharacteristic
215C2×C37#56211yescharacteristic
216C2×C37#56211yescharacteristic
217D37—42no—
218D37—42no—
219C2×#121—371no—
220#390—371no—
221#390—371no—
222#388—371no—
223#388D3711yescharacteristic, complemented
224#387—371no—
225#387—371no—
226#386—371no—
227#386—371no—
228#386—371no—
229#386—371no—
230#385—371no—
231#385—371no—
232#381—371no—
233#380—371no—
234C37#237711yescharacteristic, complemented, Sylow
235C22×D4—371no—
236C22×D4—371no—
237C22×D4—371no—
238C23×C4—371no—
239C23×C4—371no—
240C23×C4—371no—
241#121—378no—
242C2×C4⋊C4—371no—
243C2×C4⋊C4—371no—
244C2×C4⋊C4—371no—
245C2×C4⋊C4—371no—
246C2×C4⋊C4C2×D3711yescharacteristic
247C2×C22⋊C4—371no—
248C2×C22⋊C4—371no—
249C2×C22⋊C4—371no—
250C2×C22⋊C4—371no—
251C2×C22⋊C4—371no—
252C2×C22⋊C4—371no—
253C2×C22⋊C4—371no—
254C2×C22⋊C4—371no—
255C2×C22⋊C4—371no—
256C2×C22⋊C4C2×D3711yescharacteristic
257C2×C22⋊C4C2×D3711yescharacteristic
258C2×C22⋊C4C2×D3711yescharacteristic
259C2×C42—371no—
260C2×C42C2×D3711yescharacteristic
261#95—371no—
262#95—371no—
263#95—371no—
264#95—371no—
265#95—371no—
266#95—371no—
267#95—371no—
268#95—371no—
269#95C2×D3711yescharacteristic
270#95C2×D3711yescharacteristic
271C24—371no—
272C24—371no—
273C24C22×D3711yescharacteristic
274C2×D4—374no—
275C2×D4—374no—
276C2×D4—374no—
277C2×D4—1481no—
278C2×D4—1481no—
279C2×D4—1481no—
280C2×D4—1481no—
281C2×D4—1481no—
282C2×D4—1481no—
283C22×C4—371no—
284C22×C4—371no—
285C22×C4—374no—
286C22×C4—371no—
287C22×C4—371no—
288C22×C4—371no—
289C22×C4—371no—
290C22×C4—1481no—
291C22×C4—1481no—
292C22×C4—1481no—
293C22×C4—1481no—
294C22×C4—1481no—
295C22×C4—1481no—
296C22×C4—1481no—
297C22×C4—1481no—
298C22×C4C22×D3711yescharacteristic
299C22×C4C22×D3711yescharacteristic
300C22×C4C22×D3711yescharacteristic
301C22×C4C22×D3711yescharacteristic
302C22×C4C22×D3711yescharacteristic
303C22×C4C22×D3711yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—374no—
306C4⋊C4—1481no—
307C4⋊C4—1481no—
308C4⋊C4—1481no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—374no—
312C22⋊C4—1481no—
313C22⋊C4—1481no—
314C22⋊C4—1481no—
315C22⋊C4—1481no—
316C22⋊C4—1481no—
317C22⋊C4—1481no—
318C22⋊C4—1481no—
319C22⋊C4—1481no—
320C22⋊C4C4×D3714yescomplemented
321C42—41no—
322C42—1481no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—372no—
330C23—372no—
331C23—741no—
332C23—741no—
333C23—741no—
334C23—741no—
335C23—741no—
336C23—741no—
337C23—741no—
338C23—741no—
339C23—741no—
340C23—741no—
341C23—741no—
342C23—741no—
343C23C2×C4×D3712yes—
344C23C23×D3711yescharacteristic
345D4—1484no—
346D4—1484no—
347D4—1484no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—372no—
365C2×C4—372no—
366C2×C4—741no—
367C2×C4—741no—
368C2×C4—741no—
369C2×C4—741no—
370C2×C4—741no—
371C2×C4—741no—
372C2×C4—741no—
373C2×C4—741no—
374C2×C4—741no—
375C2×C4—741no—
376C2×C4—741no—
377C2×C4—741no—
378C2×C4—741no—
379C2×C4—741no—
380C2×C4—741no—
381C2×C4—741no—
382C2×C4—1484no—
383C2×C4—1484no—
384C2×C4—1484no—
385C2×C4C2×C4×D3712yes—
386C2×C4C2×C4×D3712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—742no—
395C22—742no—
396C22—742no—
397C22—742no—
398C22—742no—
399C22—742no—
400C22—1481no—
401C22—1481no—
402C22—1481no—
403C22—1481no—
404C22—1481no—
405C22—1481no—
406C22—1481no—
407C22—1481no—
408C22C22×C4×D3711yescharacteristic
409C22#1192458111yescharacteristic
410C22#1192458111yescharacteristic
411C22#1192458211yescharacteristic
412C22#1192458211yescharacteristic
413C22#1192458211yescharacteristic
414C22#1192458311yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—742no—
422C4—742no—
423C4—1481no—
424C4—1481no—
425C4—1481no—
426C4—1481no—
427C2—42no—
428C2—1482no—
429C2—1482no—
430C2#1455800511yescharacteristic
431C2#1455800511yescharacteristic
432C2#1455801311yescharacteristic
433C2#1455827111yescharacteristic
434C2#1455827411yescharacteristic
435C2#1455827711yescharacteristic
436C2#1455828011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#388C2
socle, cosocleC23×C37C24
prime radicalC2×#11924509C2
hypercenter#388D37
hypocenterC37#2377
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #14557435C2semidirect
3. #14557433C2semidirect
4. #14557425C2semidirect
5. #14557420C2semidirect
6. #14557417C2semidirect
7. #14557411C2semidirect
8. #14557410C2semidirect
9. #14557410C2semidirect
10. #14557409C2semidirect
11. #14557408C2semidirect
12. #14557404C2semidirect
13. #14557395C2semidirect
14. #14557390C2semidirect
15. C37×#388C2semidirect
22. #11924509C4semidirect
66. C2×C22⋊D37C2×C4semidirect
78. C2×D148C2×C4semidirect
84. C2×C4×D37C2×C4semidirect
88. C22⋊C4×C37C2×C4semidirect
98. #10596938C2×C4semidirect
102. #10596936C2×C4semidirect
223. #388D37semidirect
234. C37#2377semidirect
320. C22⋊C4C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245092
#145574352
#145574332
#145574252
#145574202
#145574172
#145574112
#145574102
#145574092
#145574082
#145574042
#145573952
#145573902
C37×#3882
#237737