Finite Group Databasebeta

← #17997175

Subgroups — #17997175

Order 4736.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×D4×D37C211yesmaximal, characteristic
2C2×#11924494C211yesmaximal, characteristic, complemented
3C2×#11924493C211yesmaximal, characteristic, complemented
4#14557436C211yesmaximal, characteristic, complemented
5#14557435C211yesmaximal, characteristic, complemented
6#14557434C211yesmaximal, characteristic, complemented
7#14557433C214yesmaximal, complemented
8#14557388C214yesmaximal, complemented
9C37×#390C211yesmaximal, characteristic, complemented
10C24×D37C2212yes—
11C22×D148C2211yescharacteristic
12C22×C22⋊D37C2212yes—
13C2×D4×D37—24no—
14C2×D4×D37—24no—
15C2×D4×D37C418yescomplemented
16C22×C4×D37C2211yescharacteristic
17#11924494—24no—
18#11924493—24no—
19C2×C4⋊Dic37C2211yescharacteristic, complemented
20C2×#10596936C2212yescomplemented
21C2×C22⋊Dic37C2214yescomplemented
22C2×#10596938C2212yes—
23C2×#10596938C2212yes—
24C2×#10596938C2214yescomplemented
25C2×#10596938C2214yescomplemented
26C2×C4×Dic37C2211yescharacteristic, complemented
27C2×C4×Dic37C2212yescomplemented
28C22×D4×C37C2211yescharacteristic
29C2×C4⋊C4×C37C2211yescharacteristic, complemented
30C2×C22⋊C4×C37C2214yescomplemented
31C2×C42×C37C2211yescharacteristic, complemented
32C24×C37C2312yes—
33C23×D37—24no—
34C23×D37—24no—
35C23×D37—44no—
36C23×D37—44no—
37C23×D37C2×C418yes—
38C23×D37C2312yes—
39C23×D37C2311yescharacteristic
40C2×D4×C37—22no—
41C2×D4×C37—22no—
42C2×D4×C37C2×C414yes—
43C22×C4×C37C2312yes—
44C22×C4×C37C2312yes—
45C22×C4×C37C2311yescharacteristic
46C2×C22⋊D37—24no—
47C2×C22⋊D37—24no—
48C2×C22⋊D37C2×C418yes—
49C22×Dic37C2312yes—
50C22×Dic37C2312yes—
51C22×Dic37C2311yescharacteristic
52D4×D37—232no—
53C2×D148—22no—
54C2×D148—22no—
55C2×D148C2×C414yes—
56C2×C4×D37C2×C414yes—
57C2×C4×D37D414yes—
58C2×C4×D37D414yes—
59C4⋊C4×C37—22no—
60C22⋊C4×C37—44no—
61C42×C37—22no—
62C22⋊Dic37—44no—
63#10596938—24no—
64#10596938—24no—
65#10596938—44no—
66#10596938—44no—
67C4⋊Dic37—22no—
68#10596936—24no—
69C4×Dic37—22no—
70C4×Dic37—24no—
71C23×C37—22no—
72C23×C37—22no—
73C23×C37—24no—
74C23×C37—24no—
75C23×C37C22×C414yes—
76C23×C37C2411yescharacteristic
77C22×D37—22no—
78C22×D37—22no—
79C22×D37—22no—
80C22×D37—22no—
81C22×D37—24no—
82C22×D37—24no—
83C22×D37—216no—
84C22×D37—416no—
85C22×D37—48no—
86C22×D37—416no—
87C22×D37—48no—
88C22×D37C22×C414yes—
89C22×D37C22×C412yes—
90C22×D37C2×D412yes—
91C22×D37C2×D412yes—
92D4×C37—28no—
93C2×C4×C37—22no—
94C2×C4×C37—22no—
95C2×C4×C37—22no—
96C2×C4×C37—24no—
97C2×C4×C37—22no—
98C2×C4×C37C22×C412yes—
99C2×C4×C37C2×D412yes—
100C2×C4×C37C2×D412yes—
101C22⋊D37—216no—
102C2×Dic37—22no—
103C2×Dic37—22no—
104C2×Dic37—22no—
105C2×Dic37—24no—
106C2×Dic37—22no—
107C2×Dic37C22×C412yes—
108C2×Dic37C2×D412yes—
109C2×Dic37C2×D412yes—
110D148—28no—
111C4×D37C22⋊C4116yescomplemented
112C22×C37—24no—
113C22×C37—28no—
114C22×C37—42no—
115C22×C37—44no—
116C22×C37—42no—
117C22×C37C23×C411yescharacteristic
118C22×C37C22×D411yescharacteristic
119C22×C37C22×D411yescharacteristic
120C22×C37H(5,𝔽2)12yes—
121C22×C37H(5,𝔽2)12yes—
122C2×D37—24no—
123C2×D37—22no—
124C2×D37—24no—
125C2×D37—22no—
126C2×D37—24no—
127C2×D37—28no—
128C2×D37—432no—
129C2×D37—42no—
130C2×D37—44no—
131C2×D37—42no—
132C2×D37C2×C22⋊C414yes—
133C4×C37—42no—
134C4×C37—42no—
135C4×C37C2×C22⋊C414yes—
136Dic37—42no—
137Dic37—42no—
138Dic37C2×C22⋊C414yes—
139#2309—371nomaximal, Sylow
140C2×C37—44no—
141C2×C37C22×C22⋊C411yescharacteristic
142C2×C37#51812yes—
143C2×C37#53511yescharacteristic
144C2×C37#53512yes—
145C2×C37#53511yescharacteristic
146D37—24no—
147D37—44no—
148C23×D4—371no—
149C2×#117—371no—
150C2×#117—371no—
151#390—371no—
152#390—371no—
153#390—374no—
154#390—371no—
155#390D3711yescharacteristic, complemented
156#379—374no—
157C37#230911yescharacteristic, complemented, Sylow
158C25—372no—
159C22×D4—372no—
160C22×D4—378no—
161C22×D4—371no—
162C22×D4—744no—
163C22×D4—744no—
164C22×D4C2×D3711yescharacteristic
165C23×C4—371no—
166#117—744no—
167#117—744no—
168C2×C4⋊C4—372no—
169C2×C4⋊C4—371no—
170C2×C4⋊C4C2×D3711yescharacteristic, complemented
171C2×C22⋊C4—372no—
172C2×C22⋊C4—372no—
173C2×C22⋊C4—374no—
174C2×C22⋊C4—374no—
175C2×C22⋊C4—374no—
176C2×C22⋊C4C2×D3714yescomplemented
177C2×C42—372no—
178C2×C42—371no—
179C2×C42C2×D3711yescharacteristic, complemented
180C24—371no—
181C24—372no—
182C24—378no—
183C24—744no—
184C24—744no—
185C24—1484no—
186C24—1484no—
187C24C22×D3712yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—378no—
191C2×D4—374no—
192C2×D4—7432no—
193C2×D4—744no—
194C2×D4—744no—
195C2×D4—742no—
196C2×D4—742no—
197C2×D4C4×D3714yes—
198C22×C4—371no—
199C22×C4—374no—
200C22×C4—374no—
201C22×C4—374no—
202C22×C4—372no—
203C22×C4—372no—
204C22×C4C22×D3712yes—
205C22×C4C22×D3712yes—
206C22×C4C22×D3711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—744no—
209C4⋊C4—742no—
210C22⋊C4—44no—
211C22⋊C4—744no—
212C22⋊C4—744no—
213C22⋊C4—1484no—
214C22⋊C4—1484no—
215C22⋊C4—1484no—
216C42—22no—
217C42—744no—
218C42—742no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—24no—
223C23—372no—
224C23—372no—
225C23—372no—
226C23—374no—
227C23—7416no—
228C23—742no—
229C23—742no—
230C23—742no—
231C23—742no—
232C23—744no—
233C23—744no—
234C23—14816no—
235C23—1488no—
236C23—14816no—
237C23—1488no—
238C23C2×C4×D3714yes—
239C23C23×D3711yescharacteristic
240D4—28no—
241D4—7416no—
242D4—748no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—372no—
249C2×C4—372no—
250C2×C4—372no—
251C2×C4—3716no—
252C2×C4—742no—
253C2×C4—742no—
254C2×C4—742no—
255C2×C4—742no—
256C2×C4—744no—
257C2×C4C2×C4×D3712yes—
258C2×C4D4×D3712yes—
259C2×C4D4×D3712yes—
260C22—24no—
261C22—28no—
262C22—42no—
263C22—44no—
264C22—42no—
265C22—374no—
266C22—744no—
267C22—742no—
268C22—744no—
269C22—742no—
270C22—744no—
271C22—748no—
272C22—14832no—
273C22—1482no—
274C22—1484no—
275C22—1482no—
276C22C22×C4×D3711yescharacteristic
277C22C2×D4×D3711yescharacteristic
278C22C2×D4×D3711yescharacteristic
279C22#1192458112yes—
280C22#1192458212yes—
281C4—42no—
282C4—42no—
283C4—374no—
284C4—1482no—
285C4—1482no—
286C4C22⋊C4×D3714yes—
287C2—44no—
288C2—744no—
289C2—1484no—
290C2C2×C22⋊C4×D3711yescharacteristic
291C2#1455800512yes—
292C2#1455809611yescharacteristic
293C2#1455809912yes—
294C2#1455810111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#390C2
socle, cosocleC23×C37C24
prime radicalC22×D4×D37C2
hypercenter#390D37
hypocenterC37#2309
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#11924494C2semidirect
3. C2×#11924493C2semidirect
4. #14557436C2semidirect
5. #14557435C2semidirect
6. #14557434C2semidirect
7. #14557433C2semidirect
8. #14557388C2semidirect
9. C37×#390C2semidirect
15. C2×D4×D37C4semidirect
19. C2×C4⋊Dic37C22semidirect
20. C2×#10596936C22semidirect
21. C2×C22⋊Dic37C22semidirect
24. C2×#10596938C22semidirect
25. C2×#10596938C22semidirect
26. C2×C4×Dic37C22semidirect
27. C2×C4×Dic37C22semidirect
29. C2×C4⋊C4×C37C22semidirect
30. C2×C22⋊C4×C37C22semidirect
31. C2×C42×C37C22semidirect
111. C4×D37C22⋊C4semidirect
155. #390D37semidirect
157. C37#2309semidirect
170. C2×C4⋊C4C2×D37semidirect
176. C2×C22⋊C4C2×D37semidirect
179. C2×C42C2×D37semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D372
C2×#119244942
C2×#119244932
#145574362
#145574352
#145574342
#145574332
#145573882
C37×#3902
#230937