Finite Group Databasebeta

← #18006174

Subgroups — #18006174

Order 4736.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924565C211yesmaximal, characteristic, complemented
2C2×#11924556C211yesmaximal, characteristic, complemented
3#14558201C211yesmaximal, characteristic, complemented
4#14558190C211yesmaximal, characteristic, complemented
5#14557921C211yesmaximal, characteristic, complemented
6#14557785C211yesmaximal, characteristic, complemented
7#14557760C211yesmaximal, characteristic, complemented
8#14557754C211yesmaximal, characteristic, complemented
9#14557731C211yesmaximal, characteristic, complemented
10#14557697C211yesmaximal, characteristic, complemented
11#14557675C211yesmaximal, characteristic, complemented
12#14557673C211yesmaximal, characteristic, complemented
13#14557637C211yesmaximal, characteristic, complemented
14#14557601C211yesmaximal, characteristic, complemented
15C37×#449C211yesmaximal, characteristic, complemented
16C2×#10596962C2211yescharacteristic
17C2×#10596964C2211yescharacteristic
18C2×Q8×D37C2211yescharacteristic
19#11924565—24no—
20#11924556—24no—
21C2×Dic148C2211yescharacteristic
22C2×#10596942C2211yescharacteristic
23C2×#10596930C2211yescharacteristic
24C2×#10596940C2211yescharacteristic
25C2×#10596940C2211yescharacteristic
26C2×#10596941C2211yescharacteristic
27C2×#10596939C2211yescharacteristic
28C2×#10596929C2211yescharacteristic
29C22×#64436C2211yescharacteristic
30#11924532—21no—
31#11924532C2211yescharacteristic
32#11924523—21no—
33#11924523C2211yescharacteristic
34#11924515—21no—
35#11924515C2211yescharacteristic
36#11924511C2211yescharacteristic
37#11924509—21no—
38#11924508—21no—
39#11924507—21no—
40#11924507C2211yescharacteristic
41Q8×Dic37C2211yescharacteristic
42#11924501C2211yescharacteristic
43C4×C22⋊D37—21no—
44C4×C22⋊D37C2211yescharacteristic
45#11924496—21no—
46#11924467C2211yescharacteristic
47#11924465C2211yescharacteristic
48#11924461C2211yescharacteristic
49#11924459C2211yescharacteristic
50#11924458C2211yescharacteristic
51#11924457C2211yescharacteristic
52#11924448C2211yescharacteristic
53#11924446C2211yescharacteristic
54C2×#41×C37C2211yescharacteristic
55C2×SD16×C37C2211yescharacteristic
56C2×D8×C37C2211yescharacteristic
57#121×C37C2211yescharacteristic
58#103×C37C2211yescharacteristic
59#102×C37C2211yescharacteristic
60C22⋊C8×C37C2211yescharacteristic
61#41×C37—22no—
62#41×C37—41no—
63C2×Q8×C37C2311yescharacteristic
64C2×D4×C37—21no—
65C2×D4×C37—21no—
66C2×D4×C37C2311yescharacteristic
67C2×D4×C37C2311yescharacteristic
68C22×C4×C37—21no—
69C22×C4×C37C2311yescharacteristic
70C2×C22⋊D37—21no—
71C2×C22⋊D37—21no—
72C2×C22⋊D37—21no—
73C2×C22⋊D37—21no—
74C2×C22⋊D37—21no—
75C2×C22⋊D37D411yescharacteristic
76C22×Dic37—21no—
77Q8×D37—22no—
78Q8×D37—41no—
79#10596964—22no—
80#10596964—41no—
81#10596962—22no—
82#10596962—41no—
83C2×D148—21no—
84C2×D148C2311yescharacteristic
85C2×C4×D37—21no—
86C2×C4×D37—21no—
87C2×C4×D37C2311yescharacteristic
88C2×Dic74—21no—
89C2×Dic74C2311yescharacteristic
90C2×Dic74C2311yescharacteristic
91SD16×C37—22no—
92D8×C37—41no—
93C2×C8×C37C2311yescharacteristic
94C2×C8×C37C2311yescharacteristic
95C4⋊C4×C37C2311yescharacteristic
96C22⋊C4×C37—21no—
97C22⋊Dic37—21no—
98C22⋊Dic37—21no—
99C22⋊Dic37—21no—
100C22⋊Dic37—21no—
101C22⋊Dic37D411yescharacteristic
102#10596942D414yescomplemented
103#10596941—22no—
104#10596940—22no—
105#10596940—22no—
106#10596939—22no—
107#10596938—21no—
108#10596938D411yescharacteristic
109C4⋊Dic37—21no—
110C4⋊Dic37C2311yescharacteristic
111#10596936—21no—
112#10596936—21no—
113#10596936D411yescharacteristic
114C4×Dic37—21no—
115C4×Dic37—21no—
116C4×Dic37C2311yescharacteristic
117C2×#64436—22no—
118C2×#64436C2311yescharacteristic
119C2×#64436C2311yescharacteristic
120Dic148—22no—
121#10596930—41no—
122#10596929—22no—
123C23×C37—21no—
124C23×C37—21no—
125C23×C37C2×D411yescharacteristic
126C22×D37—21no—
127C22×D37C2×D411yescharacteristic
128Q8×C37—21no—
129Q8×C37C2×D412yes—
130D4×C37—21no—
131D4×C37—21no—
132D4×C37—21no—
133D4×C37—21no—
134D4×C37—42no—
135D4×C37—81no—
136C2×C4×C37—21no—
137C2×C4×C37—21no—
138C2×C4×C37—21no—
139C2×C4×C37—21no—
140C2×C4×C37—42no—
141C2×C4×C37C2×D411yescharacteristic
142C2×C4×C37C2411yescharacteristic
143C22⋊D37—41no—
144C22⋊D37—42no—
145C22⋊D37—42no—
146C22⋊D37—42no—
147C22⋊D37—42no—
148C22⋊D37—81no—
149C2×Dic37—21no—
150C2×Dic37—21no—
151C2×Dic37—21no—
152C2×Dic37—42no—
153C2×Dic37C2×D411yescharacteristic
154C2×Dic37C2×D411yescharacteristic
155C2×Dic37C2×D411yescharacteristic
156D148—21no—
157D148—21no—
158D148—81no—
159C4×D37—21no—
160C4×D37—21no—
161C4×D37—42no—
162C4×D37—42no—
163Dic74—21no—
164Dic74—21no—
165Dic74—21no—
166Dic74—42no—
167Dic74C2×D412yes—
168C8×C37—21no—
169C8×C37—21no—
170#64436—21no—
171#64436C2×D412yes—
172C22×C37—21no—
173C22×C37—21no—
174C22×C37—21no—
175C22×C37—41no—
176C22×C37—41no—
177C22×C37—41no—
178C22×C37—41no—
179C22×C37—41no—
180C22×C37—41no—
181C22×C37C22×D411yescharacteristic
182C2×D37—21no—
183C2×D37—21no—
184C2×D37—21no—
185C2×D37—41no—
186C2×D37—41no—
187C2×D37—41no—
188C4×C37—22no—
189C4×C37—21no—
190C4×C37—41no—
191C4×C37C22×D411yescharacteristic
192C4×C37H(5,𝔽2)11yescharacteristic
193Dic37—22no—
194Dic37—21no—
195Dic37—21no—
196Dic37—21no—
197Dic37—41no—
198Dic37—41no—
199#3271—371nomaximal, Sylow
200C2×C37—41no—
201C2×C37—81no—
202C2×C37—81no—
203C2×C37D4211yescharacteristic
204C2×C37#57511yescharacteristic
205C2×C37#57711yescharacteristic
206D37—41no—
207D37—81no—
208C2×#137—371no—
209C2×#135—371no—
210#550—371no—
211#548—371no—
212#496—371no—
213#469—371no—
214#465—371no—
215#464—371no—
216#460—371no—
217#452—371no—
218#449—371no—
219#449D3711yescharacteristic, complemented
220#448—371no—
221#441—371no—
222#435—371no—
223C37#327111yescharacteristic, complemented, Sylow
224C2×#41—371no—
225C2×#41—371no—
226C2×#41C2×D3711yescharacteristic
227C22×Q8—371no—
228#137—744no—
229#135—744no—
230C2×Q16—371no—
231C2×Q16—371no—
232C2×SD16—371no—
233C2×SD16—371no—
234C2×SD16—371no—
235C2×SD16—371no—
236C2×SD16C2×D3711yescharacteristic
237C2×D8—371no—
238C2×D8C2×D3711yescharacteristic
239C2×M16—371no—
240C22×C8—371no—
241#127—371no—
242#127—741no—
243#124—371no—
244#124—741no—
245C22⋊Q8—371no—
246C22⋊Q8—741no—
247#121—371no—
248#121—371no—
249#121—741no—
250#121—741no—
251#121—741no—
252#121C2×D3711yescharacteristic
253C4×Q8—371no—
254C4×Q8—371no—
255C4×D4—371no—
256C4×D4—741no—
257C4×D4—741no—
258#106—371no—
259C4⋊C8—371no—
260#103—371no—
261#103—371no—
262#103C2×D3711yescharacteristic
263#102—371no—
264#102—371no—
265#102C2×D3711yescharacteristic
266C22⋊C8—371no—
267C22⋊C8C2×D3711yescharacteristic
268#97—371no—
269#41—22no—
270#41—41no—
271#41—742no—
272#41—742no—
273#41—1481no—
274#41—1481no—
275C2×Q8—371no—
276C2×Q8—371no—
277C2×Q8—741no—
278C2×Q8—742no—
279C2×Q8—1481no—
280C2×Q8C22×D3711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—371no—
284C2×D4—371no—
285C2×D4—741no—
286C2×D4—741no—
287C2×D4—741no—
288C2×D4—741no—
289C2×D4—741no—
290C2×D4—741no—
291C2×D4C22×D3711yescharacteristic
292C2×D4C22×D3711yescharacteristic
293C22×C4—21no—
294C22×C4—371no—
295C22×C4—741no—
296C22×C4—741no—
297C22×C4—741no—
298C22×C4C22×D3711yescharacteristic
299Q16—374no—
300Q16—742no—
301SD16—22no—
302SD16—742no—
303SD16—742no—
304SD16—742no—
305SD16—1481no—
306D8—41no—
307D8—742no—
308M16—742no—
309C2×C8—371no—
310C2×C8—371no—
311C2×C8—742no—
312C2×C8C22×D3711yescharacteristic
313C2×C8C22×D3711yescharacteristic
314C4⋊C4—371no—
315C4⋊C4—371no—
316C4⋊C4—741no—
317C4⋊C4—741no—
318C4⋊C4—741no—
319C4⋊C4C22×D3711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—371no—
322C22⋊C4—371no—
323C22⋊C4—741no—
324C22⋊C4—741no—
325C22⋊C4—741no—
326C22⋊C4—741no—
327C22⋊C4—741no—
328C42—371no—
329C42—741no—
330C42—741no—
331C23—21no—
332C23—21no—
333C23—371no—
334C23—741no—
335C23D4×D3711yescharacteristic
336Q8—21no—
337Q8—372no—
338Q8—741no—
339Q8—741no—
340Q8—741no—
341Q8—1482no—
342Q8D4×D3712yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—81no—
349D4—2961no—
350D4—2961no—
351D4—741no—
352D4—741no—
353D4—1481no—
354D4—1482no—
355D4—1482no—
356D4—1482no—
357D4—1482no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—371no—
364C2×C4—371no—
365C2×C4—371no—
366C2×C4—741no—
367C2×C4—741no—
368C2×C4—741no—
369C2×C4—741no—
370C2×C4—741no—
371C2×C4—1482no—
372C2×C4—1482no—
373C2×C4—1482no—
374C2×C4D4×D3711yescharacteristic
375C2×C4C23×D3711yescharacteristic
376C8—21no—
377C8—21no—
378C8—372no—
379C8—741no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—741no—
390C22—741no—
391C22—741no—
392C22—1481no—
393C22—1481no—
394C22—1481no—
395C22C2×D4×D3711yescharacteristic
396C4—21no—
397C4—22no—
398C4—41no—
399C4—741no—
400C4—741no—
401C4—741no—
402C4—742no—
403C4—1481no—
404C4—1481no—
405C4C2×D4×D3711yescharacteristic
406C4#1192458111yescharacteristic
407C2—41no—
408C2—81no—
409C2—81no—
410C2—2961no—
411C2—1481no—
412C2#1455816511yescharacteristic
413C2#1455833011yescharacteristic
414C2#1455834311yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#449C2
socleC22×C37C22×D4
hypercenter#449D37
hypocenterC37#3271
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#11924565C2semidirect
2. C2×#11924556C2semidirect
3. #14558201C2semidirect
4. #14558190C2semidirect
5. #14557921C2semidirect
6. #14557785C2semidirect
7. #14557760C2semidirect
8. #14557754C2semidirect
9. #14557731C2semidirect
10. #14557697C2semidirect
11. #14557675C2semidirect
12. #14557673C2semidirect
13. #14557637C2semidirect
14. #14557601C2semidirect
15. C37×#449C2semidirect
102. #10596942D4semidirect
219. #449D37semidirect
223. C37#3271semidirect

Maximal subgroups

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

subgroupindex
C2×#119245652
C2×#119245562
#145582012
#145581902
#145579212
#145577852
#145577602
#145577542
#145577312
#145576972
#145576752
#145576732
#145576372
#145576012
C37×#4492
#327137