Finite Group Databasebeta

← #18004112

Subgroups — #18004112

Order 4736.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924565C211yesmaximal, characteristic, complemented
2C2×SD16×D37C211yesmaximal, characteristic, complemented
3#14558096C211yesmaximal, characteristic
4#14558009C211yesmaximal, characteristic, complemented
5#14557756C211yesmaximal, characteristic, complemented
6#14557755C211yesmaximal, characteristic, complemented
7#14557754C211yesmaximal, characteristic, complemented
8#14557753C211yesmaximal, characteristic, complemented
9#14557732C211yesmaximal, characteristic, complemented
10#14557731C211yesmaximal, characteristic, complemented
11#14557730C211yesmaximal, characteristic, complemented
12#14557558C211yesmaximal, characteristic, complemented
13#14557536C211yesmaximal, characteristic, complemented
14#14557526C211yesmaximal, characteristic, complemented
15C37×#460C211yesmaximal, characteristic, complemented
16C2×#10596964C2211yescharacteristic
17C2×Q8×D37C2211yescharacteristic
18C2×D4×D37C2211yescharacteristic
19#11924565—24no—
20C2×Dic148C2211yescharacteristic, complemented
21C2×#10596942C2211yescharacteristic, complemented
22C2×#10596930C2211yescharacteristic
23C2×#10596940C2211yescharacteristic
24C2×#10596940C2211yescharacteristic, complemented
25C2×#10596941C2211yescharacteristic
26SD16×D37—24no—
27C2×#10596929C2211yescharacteristic
28C2×C8×D37C2211yescharacteristic
29#11924532C2211yescharacteristic
30#11924524C2211yescharacteristic
31#11924523C2211yescharacteristic
32#11924508—21no—
33#11924508—21no—
34#11924504—21no—
35#11924504—21no—
36Q8×Dic37C2211yescharacteristic, complemented
37C4×Dic74—21no—
38C4×Dic74C2211yescharacteristic, complemented
39#11924501—21no—
40#11924501C2211yescharacteristic, complemented
41#11924494—21no—
42#11924494C2211yescharacteristic
43#11924493—21no—
44#11924493C2211yescharacteristic
45C4⋊#64436C2211yescharacteristic, complemented
46#11924465C2211yescharacteristic, complemented
47#11924465C2211yescharacteristic, complemented
48#11924461C2211yescharacteristic, complemented
49#11924459C2211yescharacteristic, complemented
50#11924458C2211yescharacteristic
51#11924457C2211yescharacteristic
52#11924457C2211yescharacteristic, complemented
53#11924456C2211yescharacteristic
54C2×SD16×C37C2211yescharacteristic
55C2×SD16×C37C2211yescharacteristic, complemented
56#127×C37C2211yescharacteristic
57C4×Q8×C37C2211yescharacteristic, complemented
58C4⋊C8×C37C2211yescharacteristic, complemented
59#102×C37C2211yescharacteristic
60#102×C37C2211yescharacteristic, complemented
61C23×D37—21no—
62C2×Q8×C37C2311yescharacteristic
63C2×D4×C37—21no—
64C2×D4×C37—21no—
65C2×D4×C37C2311yescharacteristic
66C2×D4×C37C2311yescharacteristic
67C2×C22⋊D37—21no—
68C2×C22⋊D37—21no—
69C2×C22⋊D37—21no—
70C2×C22⋊D37—21no—
71C22×Dic37—21no—
72Q8×D37—22no—
73Q8×D37D414yescomplemented
74#10596964—22no—
75#10596964—22no—
76D4×D37—22no—
77D4×D37—22no—
78C2×D148C2311yescharacteristic
79C2×C4×D37—21no—
80C2×C4×D37C2311yescharacteristic
81C2×Dic74—21no—
82C2×Dic74C2311yescharacteristic
83C2×Dic74C2311yescharacteristic
84SD16×C37—22no—
85SD16×C37—22no—
86C2×C8×C37C2311yescharacteristic
87C2×C8×C37C2311yescharacteristic
88C4⋊C4×C37—21no—
89C4⋊C4×C37C2311yescharacteristic
90C42×C37—21no—
91C42×C37C2311yescharacteristic
92C22⋊Dic37—21no—
93C22⋊Dic37—21no—
94#10596942—22no—
95#10596941—22no—
96#10596940—22no—
97#10596940—22no—
98#10596938—21no—
99#10596938—21no—
100#10596938—21no—
101#10596938D411yescharacteristic
102#10596938D411yescharacteristic
103C4⋊Dic37—21no—
104C4⋊Dic37C2311yescharacteristic
105#10596936—21no—
106#10596936—21no—
107#10596936D411yescharacteristic
108#10596936D411yescharacteristic
109C4×Dic37—21no—
110C4×Dic37—21no—
111C4×Dic37—21no—
112C4×Dic37—21no—
113C4×Dic37C2311yescharacteristic
114C2×#64436C2311yescharacteristic
115C2×#64436C2311yescharacteristic
116Dic148—22no—
117#10596930—22no—
118#10596929—22no—
119C8×D37—22no—
120C23×C37—21no—
121C23×C37—21no—
122C22×D37—21no—
123C22×D37—42no—
124C22×D37—81no—
125C22×D37—81no—
126C22×D37C2×D411yescharacteristic
127Q8×C37—21no—
128Q8×C37C2×D412yes—
129D4×C37—21no—
130D4×C37—21no—
131D4×C37—21no—
132D4×C37—21no—
133D4×C37—81no—
134D4×C37—81no—
135C2×C4×C37—21no—
136C2×C4×C37—21no—
137C2×C4×C37C2×D411yescharacteristic
138C2×C4×C37C2×D411yescharacteristic
139C2×C4×C37C2411yescharacteristic
140C22⋊D37—42no—
141C22⋊D37—42no—
142C22⋊D37—81no—
143C22⋊D37—81no—
144C2×Dic37—21no—
145C2×Dic37—21no—
146C2×Dic37—21no—
147C2×Dic37—42no—
148C2×Dic37C2×D411yescharacteristic
149C2×Dic37C2×D411yescharacteristic
150C2×Dic37C2×D411yescharacteristic
151D148—21no—
152D148—21no—
153C4×D37—24no—
154C4×D37C2×D412yes—
155C4×D37#4112yes—
156Dic74—21no—
157Dic74—24no—
158Dic74—21no—
159Dic74—21no—
160Dic74C2×D412yes—
161C8×C37—21no—
162C8×C37—21no—
163#64436—21no—
164#64436—21no—
165C22×C37—41no—
166C22×C37—41no—
167C22×C37—41no—
168C22×C37—41no—
169C22×C37—41no—
170C22×C37—41no—
171C22×C37C22×D411yescharacteristic
172C2×D37—21no—
173C2×D37—21no—
174C2×D37—41no—
175C2×D37—41no—
176C2×D37—41no—
177C2×D37—84no—
178C2×D37#12112yes—
179C4×C37—22no—
180C4×C37—21no—
181C4×C37—21no—
182C4×C37—41no—
183C4×C37C22×D411yescharacteristic
184C4×C37C2×#4111yescharacteristic
185Dic37—22no—
186Dic37—21no—
187Dic37—21no—
188Dic37—41no—
189Dic37—41no—
190Dic37#12112yes—
191#3043—371nomaximal, Sylow
192C2×C37—81no—
193C2×C37—81no—
194C2×C37C2×#12111yescharacteristic
195C2×C37#57711yescharacteristic
196C2×C37#57711yescharacteristic
197D37—22no—
198D37—81no—
199C2×#137—371no—
200C22×SD16—371no—
201#535—371no—
202#519—371no—
203#464—371no—
204#464—371no—
205#464—371no—
206#464—371no—
207#460—371no—
208#460—371no—
209#460—371no—
210#460D3711yescharacteristic, complemented
211#424—371no—
212#418—371no—
213#416—371no—
214C37#304311yescharacteristic, complemented, Sylow
215C2×#41—371no—
216C22×Q8—371no—
217C22×D4—371no—
218#137—744no—
219C2×Q16—371no—
220C2×Q16—371no—
221C2×SD16—371no—
222C2×SD16—371no—
223C2×SD16—371no—
224C2×SD16—371no—
225C2×SD16—744no—
226C2×SD16C2×D3711yescharacteristic
227C2×SD16C2×D3711yescharacteristic, complemented
228C2×M16—371no—
229C22×C8—371no—
230#127—371no—
231#127C2×D3711yescharacteristic
232#124—371no—
233#124—371no—
234#121—741no—
235#121—741no—
236#120—741no—
237#120—741no—
238C4×Q8—371no—
239C4×Q8—371no—
240C4×Q8—371no—
241C4×Q8—741no—
242C4×Q8—741no—
243C4×Q8C2×D3711yescharacteristic, complemented
244#117—371no—
245#117—371no—
246#117—741no—
247#117—741no—
248C4⋊C8—371no—
249C4⋊C8—371no—
250C4⋊C8—371no—
251C4⋊C8C2×D3711yescharacteristic, complemented
252#103—371no—
253#103—371no—
254#102—371no—
255#102—371no—
256#102—371no—
257#102—371no—
258#102C2×D3711yescharacteristic
259#102C2×D3711yescharacteristic, complemented
260C24—741no—
261#41—742no—
262#41—742no—
263C2×Q8—374no—
264C2×Q8—371no—
265C2×Q8—371no—
266C2×Q8—741no—
267C2×Q8—742no—
268C2×Q8C22×D3711yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—371no—
272C2×D4—742no—
273C2×D4—742no—
274C2×D4—741no—
275C2×D4—741no—
276C2×D4—741no—
277C2×D4—741no—
278C2×D4C22×D3711yescharacteristic
279C2×D4C22×D3711yescharacteristic
280C22×C4—371no—
281C22×C4—741no—
282C22×C4—741no—
283Q16—742no—
284Q16—742no—
285SD16—22no—
286SD16—22no—
287SD16—742no—
288SD16—742no—
289SD16—742no—
290SD16—742no—
291M16—742no—
292C2×C8—371no—
293C2×C8—371no—
294C2×C8—742no—
295C2×C8C22×D3711yescharacteristic
296C2×C8C22×D3711yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—371no—
299C4⋊C4—371no—
300C4⋊C4—371no—
301C4⋊C4—741no—
302C4⋊C4—741no—
303C4⋊C4—741no—
304C4⋊C4C22×D3711yescharacteristic
305C22⋊C4—371no—
306C22⋊C4—371no—
307C22⋊C4—741no—
308C22⋊C4—741no—
309C22⋊C4—741no—
310C22⋊C4—741no—
311C22⋊C4—741no—
312C42—21no—
313C42—371no—
314C42—741no—
315C42—741no—
316C42—741no—
317C42—741no—
318C42C22×D3711yescharacteristic
319C23—21no—
320C23—21no—
321C23—371no—
322C23—2961no—
323C23—2961no—
324C23—741no—
325C23—1482no—
326Q8—21no—
327Q8—372no—
328Q8—744no—
329Q8—741no—
330Q8—741no—
331Q8—741no—
332Q8D4×D3712yes—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—81no—
338D4—81no—
339D4—2961no—
340D4—2961no—
341D4—741no—
342D4—741no—
343D4—1482no—
344D4—1482no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—371no—
348C2×C4—372no—
349C2×C4—372no—
350C2×C4—371no—
351C2×C4—371no—
352C2×C4—744no—
353C2×C4—741no—
354C2×C4—741no—
355C2×C4—741no—
356C2×C4—1482no—
357C2×C4D4×D3711yescharacteristic
358C2×C4D4×D3711yescharacteristic
359C2×C4C23×D3711yescharacteristic
360C8—21no—
361C8—21no—
362C8—741no—
363C8—741no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—372no—
371C22—2964no—
372C22—741no—
373C22—741no—
374C22—1481no—
375C22—1481no—
376C22—1481no—
377C22C2×D4×D3711yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—372no—
383C4—742no—
384C4—741no—
385C4—741no—
386C4—1481no—
387C4—1481no—
388C4C2×D4×D3711yescharacteristic
389C4#41×D3711yescharacteristic
390C2—81no—
391C2—81no—
392C2—2961no—
393C2—742no—
394C2#121×D3711yescharacteristic
395C2#1455833911yescharacteristic
396C2#1455834311yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#460C2
socleC22×C37C22×D4
prime radical#14558096C2
hypercenter#460D37
hypocenterC37#3043
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#11924565C2semidirect
2. C2×SD16×D37C2semidirect
4. #14558009C2semidirect
5. #14557756C2semidirect
6. #14557755C2semidirect
7. #14557754C2semidirect
8. #14557753C2semidirect
9. #14557732C2semidirect
10. #14557731C2semidirect
11. #14557730C2semidirect
12. #14557558C2semidirect
13. #14557536C2semidirect
14. #14557526C2semidirect
15. C37×#460C2semidirect
20. C2×Dic148C22semidirect
21. C2×#10596942C22semidirect
24. C2×#10596940C22semidirect
36. Q8×Dic37C22semidirect
38. C4×Dic74C22semidirect
40. #11924501C22semidirect
45. C4⋊#64436C22semidirect
46. #11924465C22semidirect
47. #11924465C22semidirect
48. #11924461C22semidirect
49. #11924459C22semidirect
52. #11924457C22semidirect
55. C2×SD16×C37C22semidirect
57. C4×Q8×C37C22semidirect
58. C4⋊C8×C37C22semidirect
60. #102×C37C22semidirect
73. Q8×D37D4semidirect
210. #460D37semidirect
214. C37#3043semidirect
227. C2×SD16C2×D37semidirect
243. C4×Q8C2×D37semidirect
251. C4⋊C8C2×D37semidirect
259. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245652
C2×SD16×D372
#145580962
#145580092
#145577562
#145577552
#145577542
#145577532
#145577322
#145577312
#145577302
#145575582
#145575362
#145575262
C37×#4602
#304337