Finite Group Databasebeta

← #18005357

Subgroups — #18005357

Order 4736.

472 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924510C211yesmaximal, characteristic, complemented
2#14557866C211yesmaximal, characteristic, complemented
3#14557857C212yesmaximal, complemented
4#14557849C211yesmaximal, characteristic, complemented
5#14557767C212yesmaximal, complemented
6#14557764C212yesmaximal, complemented
7#14557683C211yesmaximal, characteristic, complemented
8#14557678C212yesmaximal, complemented
9#14557666C211yesmaximal, characteristic, complemented
10C2×C22⋊#64436C211yesmaximal, characteristic, complemented
11C37×#525C211yesmaximal, characteristic, complemented
12C22×D148C2211yescharacteristic
13C22×C22⋊D37—21no—
14C2×#10596941C2212yescomplemented
15C2×#10596939C2212yescomplemented
16C22×#64436C2212yescomplemented
17#11924510—21no—
18#11924510—21no—
19#11924510C2212yes—
20#11924510C2212yes—
21C2×C4⋊Dic37C2211yescharacteristic
22C2×#10596938—21no—
23#11924469C2212yescomplemented
24#11924467C2212yescomplemented
25#11924460C2212yescomplemented
26#11924457C2212yescomplemented
27#11924456C2212yescomplemented
28#11924456C2212yescomplemented
29C22⋊#64436C2211yescharacteristic
30C22⋊#64436C2211yescharacteristic
31C22⋊#64436C2212yescomplemented
32C2×#41×C37C2211yescharacteristic, complemented
33C23×C4×C37C2211yescharacteristic
34#123×C37—21no—
35C22⋊Q8×C37C2211yescharacteristic
36C22⋊Q8×C37C2211yescharacteristic, complemented
37#121×C37C2211yescharacteristic
38#121×C37C2211yescharacteristic, complemented
39#120×C37—21no—
40C4×D4×C37—21no—
41C4×D4×C37—21no—
42#117×C37C2211yescharacteristic, complemented
43C24×C37D411yescharacteristic
44C23×D37—21no—
45#41×C37—42no—
46C2×Q8×C37C2311yescharacteristic
47C2×D4×C37—21no—
48C2×D4×C37—21no—
49C2×D4×C37—21no—
50C2×D4×C37C2311yescharacteristic
51C22×C4×C37—21no—
52C22×C4×C37—42no—
53C22×C4×C37D411yescharacteristic
54C22×C4×C37D411yescharacteristic
55C22×C4×C37D411yescharacteristic
56C22×C4×C37C2311yescharacteristic
57C22×C4×C37C2311yescharacteristic
58C22×C4×C37C2311yescharacteristic
59C2×C22⋊D37—22no—
60C2×C22⋊D37—22no—
61C2×C22⋊D37—44no—
62C22×Dic37—21no—
63C2×D148—21no—
64C2×D148—42no—
65C2×D148C2312yes—
66C4⋊C4×C37—21no—
67C4⋊C4×C37—21no—
68C4⋊C4×C37C2311yescharacteristic
69C4⋊C4×C37C2311yescharacteristic
70C22⋊C4×C37—21no—
71C22⋊C4×C37—21no—
72C22⋊C4×C37—21no—
73C22⋊C4×C37—21no—
74C22⋊C4×C37—21no—
75C42×C37—21no—
76#10596941—42no—
77#10596939—42no—
78#10596938—22no—
79#10596938—22no—
80C4⋊Dic37—21no—
81C4⋊Dic37C2312yes—
82C2×#64436—42no—
83C2×#64436C2312yes—
84C2×#64436C2312yes—
85C23×C37—22no—
86C23×C37—22no—
87C23×C37—21no—
88C23×C37—41no—
89C23×C37C2×D411yescharacteristic
90C23×C37C2×D411yescharacteristic
91C23×C37C2×D411yescharacteristic
92C22×D37—22no—
93C22×D37—42no—
94C22×D37—42no—
95C22×D37—42no—
96Q8×C37—22no—
97D4×C37—22no—
98D4×C37—42no—
99D4×C37—42no—
100D4×C37—42no—
101C2×C4×C37—22no—
102C2×C4×C37—22no—
103C2×C4×C37—22no—
104C2×C4×C37—21no—
105C2×C4×C37—21no—
106C2×C4×C37—21no—
107C2×C4×C37—41no—
108C2×C4×C37—41no—
109C2×C4×C37—41no—
110C2×C4×C37—42no—
111C2×C4×C37C2×D411yescharacteristic
112C2×C4×C37C2×D411yescharacteristic
113C2×C4×C37C2×D411yescharacteristic
114C2×C4×C37C2411yescharacteristic
115C22⋊D37—44no—
116C22⋊D37—44no—
117C2×Dic37—22no—
118C2×Dic37—42no—
119D148—22no—
120D148—22no—
121D148—42no—
122#64436—22no—
123#64436—22no—
124C22×C37—21no—
125C22×C37—21no—
126C22×C37—21no—
127C22×C37—21no—
128C22×C37—21no—
129C22×C37—21no—
130C22×C37—21no—
131C22×C37—44no—
132C22×C37—41no—
133C22×C37—41no—
134C22×C37—41no—
135C22×C37#13512yes—
136C22×C37AGL(1,ℤ/8)12yes—
137C22×C37C22×D411yescharacteristic
138C2×D37—42no—
139C2×D37—42no—
140C2×D37—42no—
141C2×D37—84no—
142C4×C37—21no—
143C4×C37—21no—
144C4×C37—21no—
145C4×C37—41no—
146C4×C37—41no—
147C4×C37—41no—
148C4×C37H(5,𝔽2)12yes—
149Dic37—42no—
150#3188—371nomaximal, Sylow
151C2×C37—22no—
152C2×C37—22no—
153C2×C37—41no—
154C2×C37—81no—
155C2×C37#53411yescharacteristic
156C2×C37C2×#13511yescharacteristic
157C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
158D37—82no—
159#525D3711yescharacteristic, complemented
160C2×#121—371no—
161#484—371no—
162#482—372no—
163#481—371no—
164#466—372no—
165#465—372no—
166#450—371no—
167#449—372no—
168#447—371no—
169C2×C22⋊C8—371no—
170C37#318811yescharacteristic, complemented, Sylow
171C2×#41C2×D3711yescharacteristic, complemented
172C22×D4—371no—
173C22×D4—741no—
174C23×C4C2×D3711yescharacteristic
175C2×SD16—372no—
176C2×D8—372no—
177C22×C8—372no—
178#123—21no—
179C22⋊Q8C2×D3711yescharacteristic
180C22⋊Q8C2×D3711yescharacteristic, complemented
181#121—372no—
182#121—372no—
183#121—741no—
184#121—741no—
185#121C2×D3711yescharacteristic
186#121C2×D3711yescharacteristic, complemented
187#120—21no—
188C4×D4—21no—
189C4×D4—21no—
190#117C2×D3711yescharacteristic, complemented
191C2×C4⋊C4—371no—
192C2×C22⋊C4—741no—
193#107—372no—
194#106—372no—
195#103—372no—
196#102—372no—
197#102—372no—
198#102—372no—
199C22⋊C8—371no—
200C22⋊C8—372no—
201C22⋊C8—371no—
202C24—741no—
203C24C22⋊D3711yescharacteristic
204#41—42no—
205C2×Q8C22×D3711yescharacteristic
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—372no—
210C2×D4—742no—
211C2×D4—742no—
212C2×D4—741no—
213C2×D4—1482no—
214C2×D4—1484no—
215C2×D4C22×D3711yescharacteristic
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—741no—
219C22×C4C22⋊D3711yescharacteristic
220C22×C4C22⋊D3711yescharacteristic
221C22×C4C22⋊D3711yescharacteristic
222C22×C4C22×D3711yescharacteristic
223C22×C4C22×D3711yescharacteristic
224C22×C4C22×D3711yescharacteristic
225SD16—1482no—
226D8—1482no—
227C2×C8—372no—
228C2×C8—372no—
229C2×C8—1482no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—372no—
233C4⋊C4—741no—
234C4⋊C4C22×D3711yescharacteristic
235C4⋊C4C22×D3711yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—21no—
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—742no—
242C22⋊C4—742no—
243C42—21no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—41no—
248C23—742no—
249C23—1482no—
250C23—1482no—
251C23—1482no—
252C23C2×C22⋊D3711yescharacteristic
253C23C2×C22⋊D3711yescharacteristic
254C23C2×C22⋊D3711yescharacteristic
255Q8—22no—
256D4—22no—
257D4—42no—
258D4—42no—
259D4—42no—
260D4—742no—
261D4—742no—
262D4—1482no—
263D4—1484no—
264D4—1484no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—742no—
276C2×C4—1482no—
277C2×C4C2×C22⋊D3711yescharacteristic
278C2×C4C2×C22⋊D3711yescharacteristic
279C2×C4C2×C22⋊D3711yescharacteristic
280C2×C4C23×D3711yescharacteristic
281C8—742no—
282C8—742no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—44no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—2964no—
295C22—1482no—
296C22—1482no—
297C22—1482no—
298C22#1192455612yes—
299C22#1192456112yes—
300C22C22×C22⋊D3711yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—1482no—
308C4#1192458112yes—
309C2—22no—
310C2—22no—
311C2—41no—
312C2—81no—
313C2—2962no—
314C2#1455809211yescharacteristic
315C2C2×#1192455611yescharacteristic
316C2C2×#1192456111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#525C2
socleC22×C37C22×D4
hypercenter#525D37
hypocenterC37#3188
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#11924510C2semidirect
2. #14557866C2semidirect
3. #14557857C2semidirect
4. #14557849C2semidirect
5. #14557767C2semidirect
6. #14557764C2semidirect
7. #14557683C2semidirect
8. #14557678C2semidirect
9. #14557666C2semidirect
10. C2×C22⋊#64436C2semidirect
11. C37×#525C2semidirect
14. C2×#10596941C22semidirect
15. C2×#10596939C22semidirect
16. C22×#64436C22semidirect
23. #11924469C22semidirect
24. #11924467C22semidirect
25. #11924460C22semidirect
26. #11924457C22semidirect
27. #11924456C22semidirect
28. #11924456C22semidirect
31. C22⋊#64436C22semidirect
32. C2×#41×C37C22semidirect
36. C22⋊Q8×C37C22semidirect
38. #121×C37C22semidirect
42. #117×C37C22semidirect
159. #525D37semidirect
170. C37#3188semidirect
171. C2×#41C2×D37semidirect
180. C22⋊Q8C2×D37semidirect
186. #121C2×D37semidirect
190. #117C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245102
#145578662
#145578572
#145578492
#145577672
#145577642
#145576832
#145576782
#145576662
C2×C22⋊#644362
C37×#5252
#318837