Finite Group Databasebeta

← #18006154

Subgroups — #18006154

Order 4736.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924567C211yesmaximal, characteristic, complemented
2C2×#11924559C211yesmaximal, characteristic, complemented
3#14558194C211yesmaximal, characteristic, complemented
4#14558185C211yesmaximal, characteristic, complemented
5#14557926C211yesmaximal, characteristic, complemented
6#14557781C211yesmaximal, characteristic, complemented
7#14557773C211yesmaximal, characteristic, complemented
8#14557751C211yesmaximal, characteristic, complemented
9#14557741C211yesmaximal, characteristic, complemented
10#14557698C211yesmaximal, characteristic, complemented
11#14557694C211yesmaximal, characteristic, complemented
12#14557680C211yesmaximal, characteristic, complemented
13#14557636C211yesmaximal, characteristic, complemented
14#14557607C211yesmaximal, characteristic, complemented
15C37×#451C211yesmaximal, characteristic, complemented
16C2×#10596966C2211yescharacteristic
17C2×#10596966C2211yescharacteristic
18C2×#10596962C2211yescharacteristic
19#11924567—24no—
20#11924559—24no—
21C2×#10596942C2211yescharacteristic
22C2×#10596942C2211yescharacteristic
23C2×#10596930C2211yescharacteristic
24C2×#10596941C2211yescharacteristic
25C2×#10596941C2211yescharacteristic
26C2×#10596941C2211yescharacteristic
27C2×D296C2211yescharacteristic
28C2×#10596934C2211yescharacteristic
29C2×C8×D37C2211yescharacteristic
30#11924534C2211yescharacteristic
31#11924525—21no—
32#11924525C2211yescharacteristic
33#11924521—21no—
34#11924518—21no—
35#11924516C2211yescharacteristic
36#11924512—21no—
37#11924512C2211yescharacteristic
38#11924510—21no—
39#11924510C2211yescharacteristic
40#11924509—21no—
41Q8×Dic37C2211yescharacteristic
42#11924498C2211yescharacteristic
43C4×C22⋊D37—21no—
44C4×C22⋊D37C2211yescharacteristic
45C4⋊C4×D37—21no—
46#11924469C2211yescharacteristic
47#11924465C2211yescharacteristic
48#11924461C2211yescharacteristic
49#11924460C2211yescharacteristic
50#11924458C2211yescharacteristic
51#11924456C2211yescharacteristic
52#11924448C2211yescharacteristic
53#11924446C2211yescharacteristic
54C22×Q8×C37C2211yescharacteristic
55C2×Q16×C37C2211yescharacteristic
56C2×Q16×C37C2211yescharacteristic
57C22⋊Q8×C37C2211yescharacteristic
58#103×C37C2211yescharacteristic
59#103×C37C2211yescharacteristic
60C22⋊C8×C37C2211yescharacteristic
61C2×Q8×C37—21no—
62C2×Q8×C37—22no—
63C2×Q8×C37—41no—
64C2×Q8×C37C2311yescharacteristic
65C2×Q8×C37C2311yescharacteristic
66C2×Q8×C37C2311yescharacteristic
67C22×C4×C37—21no—
68C22×C4×C37C2311yescharacteristic
69C2×C22⋊D37—21no—
70C2×C22⋊D37—21no—
71C2×C22⋊D37D411yescharacteristic
72#10596966—22no—
73#10596966—22no—
74#10596966—41no—
75#10596966—41no—
76#10596962—22no—
77#10596962—41no—
78C2×D148—21no—
79C2×D148—21no—
80C2×D148C2311yescharacteristic
81C2×D148C2311yescharacteristic
82C2×C4×D37—21no—
83C2×C4×D37—21no—
84C2×C4×D37—21no—
85C2×C4×D37—21no—
86C2×C4×D37C2311yescharacteristic
87C2×Dic74C2311yescharacteristic
88Q16×C37—22no—
89Q16×C37—41no—
90C2×C8×C37C2311yescharacteristic
91C2×C8×C37C2311yescharacteristic
92C4⋊C4×C37—21no—
93C4⋊C4×C37C2311yescharacteristic
94C22⋊C4×C37—21no—
95C22⋊Dic37D411yescharacteristic
96#10596942—22no—
97#10596942—22no—
98#10596941—22no—
99#10596941—22no—
100#10596941D414yescomplemented
101#10596938—21no—
102#10596938—21no—
103#10596938—21no—
104#10596938—21no—
105#10596938—21no—
106#10596938—21no—
107#10596938D411yescharacteristic
108C4⋊Dic37—21no—
109C4⋊Dic37—21no—
110C4⋊Dic37C2311yescharacteristic
111#10596936—21no—
112#10596936—21no—
113#10596936—21no—
114#10596936D411yescharacteristic
115C4×Dic37—21no—
116C4×Dic37C2311yescharacteristic
117#10596934—22no—
118C2×#64436C2311yescharacteristic
119C2×#64436C2311yescharacteristic
120D296—22no—
121#10596930—41no—
122C8×D37—22no—
123C23×C37C2×D411yescharacteristic
124C22×D37—21no—
125C22×D37—21no—
126C22×D37C2×D411yescharacteristic
127Q8×C37—21no—
128Q8×C37—21no—
129Q8×C37—21no—
130Q8×C37—21no—
131Q8×C37—21no—
132Q8×C37—42no—
133Q8×C37C2×D412yes—
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—42no—
141C2×C4×C37C2×D411yescharacteristic
142C2×C4×C37C2411yescharacteristic
143C22⋊D37—41no—
144C22⋊D37—42no—
145C22⋊D37—42no—
146C2×Dic37—21no—
147C2×Dic37—21no—
148C2×Dic37C2×D411yescharacteristic
149C2×Dic37C2×D411yescharacteristic
150C2×Dic37C2×D411yescharacteristic
151D148—21no—
152D148—21no—
153D148—21no—
154D148—42no—
155D148—42no—
156D148C2×D412yes—
157C4×D37—21no—
158C4×D37—21no—
159C4×D37—42no—
160C4×D37—42no—
161C4×D37—42no—
162C4×D37—42no—
163Dic74—21no—
164Dic74—21no—
165C8×C37—21no—
166C8×C37—21no—
167#64436—21no—
168#64436C2×D412yes—
169C22×C37—21no—
170C22×C37—21no—
171C22×C37—21no—
172C22×C37C22×D411yescharacteristic
173C2×D37—21no—
174C2×D37—22no—
175C2×D37—21no—
176C2×D37—21no—
177C2×D37—41no—
178C2×D37—41no—
179C2×D37—41no—
180C2×D37—41no—
181C2×D37—41no—
182C4×C37—21no—
183C4×C37—22no—
184C4×C37—41no—
185C4×C37—41no—
186C4×C37—41no—
187C4×C37C22×D411yescharacteristic
188C4×C37H(5,𝔽2)11yescharacteristic
189Dic37—21no—
190Dic37—21no—
191Dic37—21no—
192Dic37—41no—
193Dic37—41no—
194#3269—371nomaximal, Sylow
195C2×C37—41no—
196C2×C37D4211yescharacteristic
197C2×C37#57511yescharacteristic
198C2×C37#57811yescharacteristic
199D37—41no—
200D37—42no—
201D37—81no—
202C2×#137—371no—
203C2×#135—371no—
204#548—371no—
205#547—371no—
206#497—371no—
207#468—371no—
208#467—371no—
209#463—371no—
210#462—371no—
211#452—371no—
212#452—371no—
213#451D3711yescharacteristic, complemented
214#449—371no—
215#440—371no—
216#435—371no—
217C37#326911yescharacteristic, complemented, Sylow
218C2×#41—371no—
219C2×#41—371no—
220C2×#41—371no—
221C22×Q8C2×D3711yescharacteristic
222#137—744no—
223#135—744no—
224C2×Q16—371no—
225C2×Q16—371no—
226C2×Q16C2×D3711yescharacteristic
227C2×Q16C2×D3711yescharacteristic
228C2×SD16—371no—
229C2×SD16—371no—
230C2×SD16—371no—
231C2×SD16—371no—
232C2×D8—371no—
233C2×M16—371no—
234C22×C8—371no—
235C4⋊Q8—371no—
236#124—371no—
237#124—741no—
238#123—741no—
239#123—741no—
240C22⋊Q8—371no—
241C22⋊Q8—371no—
242C22⋊Q8—741no—
243C22⋊Q8C2×D3711yescharacteristic
244#121—371no—
245#121—741no—
246#121—741no—
247C4×Q8—371no—
248C4×D4—371no—
249C4×D4—371no—
250C4×D4—741no—
251C2×C4⋊C4—741no—
252#107—371no—
253C4⋊C8—371no—
254#103—371no—
255#103—371no—
256#103C2×D3711yescharacteristic
257#103C2×D3711yescharacteristic
258#102—371no—
259#102—371no—
260C22⋊C8—371no—
261C22⋊C8C2×D3711yescharacteristic
262#97—371no—
263#41—742no—
264#41—742no—
265#41—742no—
266#41—1481no—
267#41—1481no—
268#41—1481no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—371no—
273C2×Q8C22×D3711yescharacteristic
274C2×Q8C22×D3711yescharacteristic
275C2×Q8C22×D3711yescharacteristic
276C2×D4—371no—
277C2×D4—371no—
278C2×D4—371no—
279C2×D4—741no—
280C2×D4—741no—
281C2×D4—741no—
282C2×D4—741no—
283C22×C4—21no—
284C22×C4—371no—
285C22×C4—741no—
286C22×C4—741no—
287C22×C4—741no—
288C22×C4—741no—
289C22×C4C22×D3711yescharacteristic
290Q16—22no—
291Q16—41no—
292Q16—742no—
293Q16—742no—
294SD16—374no—
295SD16—742no—
296SD16—742no—
297SD16—1481no—
298D8—742no—
299M16—742no—
300C2×C8—371no—
301C2×C8—371no—
302C2×C8—742no—
303C2×C8C22×D3711yescharacteristic
304C2×C8C22×D3711yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—371no—
307C4⋊C4—371no—
308C4⋊C4—741no—
309C4⋊C4—741no—
310C4⋊C4—741no—
311C4⋊C4—741no—
312C4⋊C4—741no—
313C4⋊C4C22×D3711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—371no—
316C22⋊C4—371no—
317C22⋊C4—741no—
318C22⋊C4—741no—
319C22⋊C4—741no—
320C22⋊C4—741no—
321C22⋊C4—741no—
322C22⋊C4—741no—
323C42—371no—
324C42—741no—
325C23—371no—
326C23—741no—
327C23—741no—
328C23D4×D3711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—42no—
335Q8—741no—
336Q8—741no—
337Q8D4×D3712yes—
338D4—372no—
339D4—741no—
340D4—741no—
341D4—741no—
342D4—1482no—
343D4—1482no—
344D4—1481no—
345D4—1482no—
346D4—1482no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—371no—
355C2×C4—371no—
356C2×C4—371no—
357C2×C4—741no—
358C2×C4—741no—
359C2×C4—741no—
360C2×C4—741no—
361C2×C4—1482no—
362C2×C4—1482no—
363C2×C4—1482no—
364C2×C4—1482no—
365C2×C4D4×D3711yescharacteristic
366C2×C4C23×D3711yescharacteristic
367C8—21no—
368C8—21no—
369C8—372no—
370C8—741no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—741no—
375C22—741no—
376C22—741no—
377C22—742no—
378C22—1481no—
379C22—1481no—
380C22—1481no—
381C22—1481no—
382C22—1481no—
383C22C2×D4×D3711yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—741no—
390C4—741no—
391C4—741no—
392C4—1481no—
393C4—1481no—
394C4C2×D4×D3711yescharacteristic
395C4#1192458111yescharacteristic
396C2—41no—
397C2—2961no—
398C2—1481no—
399C2—1482no—
400C2#1455816511yescharacteristic
401C2#1455833211yescharacteristic
402C2#1455834611yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#451C2
socleC22×C37C22×D4
hypercenter#451D37
hypocenterC37#3269
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#11924567C2semidirect
2. C2×#11924559C2semidirect
3. #14558194C2semidirect
4. #14558185C2semidirect
5. #14557926C2semidirect
6. #14557781C2semidirect
7. #14557773C2semidirect
8. #14557751C2semidirect
9. #14557741C2semidirect
10. #14557698C2semidirect
11. #14557694C2semidirect
12. #14557680C2semidirect
13. #14557636C2semidirect
14. #14557607C2semidirect
15. C37×#451C2semidirect
100. #10596941D4semidirect
213. #451D37semidirect
217. C37#3269semidirect

Maximal subgroups

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

subgroupindex
C2×#119245672
C2×#119245592
#145581942
#145581852
#145579262
#145577812
#145577732
#145577512
#145577412
#145576982
#145576942
#145576802
#145576362
#145576072
C37×#4512
#326937