Finite Group Databasebeta

← #17997733

Subgroups — #17997733

Order 4736.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#11924504C211yesmaximal, characteristic
2#14557433C212yesmaximal, complemented
3#14557424C212yesmaximal, complemented
4#14557423C212yesmaximal, complemented
5#14557421C212yesmaximal, complemented
6#14557413C211yesmaximal, characteristic, complemented
7#14557410C212yesmaximal, complemented
8#14557388C212yesmaximal, complemented
9C37×#390C211yesmaximal, characteristic, complemented
10C24×D37C2211yescharacteristic
11C22×C22⋊D37C2212yes—
12C23×Dic37C2211yescharacteristic, complemented
13#11924504C418yescomplemented
14C2×#10596936C2212yescomplemented
15C2×C22⋊Dic37C2212yes—
16C2×C22⋊Dic37C2211yescharacteristic
17C2×C22⋊Dic37C2211yescharacteristic, complemented
18C2×#10596938C2212yes—
19C2×#10596938C2212yes—
20C2×#10596938C2212yes—
21C2×#10596938C2212yes—
22C2×#10596938C2212yes—
23C2×C4×Dic37C2212yescomplemented
24#11924443C2212yescomplemented
25#11924443C2212yescomplemented
26#11924443C2212yescomplemented
27C22×D4×C37C2211yescharacteristic
28C2×C4⋊C4×C37C2211yescharacteristic, complemented
29C2×C22⋊C4×C37C2212yes—
30C2×C22⋊C4×C37C2212yescomplemented
31C2×C42×C37C2211yescharacteristic, complemented
32C24×C37C2311yescharacteristic
33C24×C37C2311yescharacteristic
34C23×D37—41no—
35C23×D37—41no—
36C23×D37—42no—
37C23×D37—42no—
38C23×D37C2×C414yescomplemented
39C23×D37C2312yes—
40C2×D4×C37—41no—
41C2×D4×C37—41no—
42C2×D4×C37C2×C414yes—
43C22×C4×C37C2312yes—
44C22×C4×C37C2312yes—
45C22×C4×C37C2311yescharacteristic
46C2×C22⋊D37—42no—
47C2×C22⋊D37—42no—
48C2×C22⋊D37C2×C418yes—
49C22×Dic37—41no—
50C22×Dic37—42no—
51C22×Dic37C2312yes—
52C22×Dic37C2312yes—
53C22×Dic37C2312yes—
54C4⋊C4×C37—41no—
55C22⋊C4×C37—42no—
56C22⋊C4×C37—42no—
57C42×C37—41no—
58C22⋊Dic37—42no—
59C22⋊Dic37—41no—
60C22⋊Dic37C2×C414yescomplemented
61#10596938—42no—
62#10596938—42no—
63#10596938—42no—
64#10596938—42no—
65#10596938C2×C418yescomplemented
66#10596936—42no—
67C4×Dic37—42no—
68C23×C37—21no—
69C23×C37—22no—
70C23×C37—22no—
71C23×C37—22no—
72C23×C37—22no—
73C23×C37—21no—
74C23×C37—21no—
75C23×C37—21no—
76C23×C37C22×C412yes—
77C23×C37C22×C412yes—
78C23×C37C2411yescharacteristic
79C22×D37—22no—
80C22×D37—22no—
81C22×D37—22no—
82C22×D37—22no—
83C22×D37—22no—
84C22×D37—22no—
85C22×D37—48no—
86C22×D37—44no—
87C22×D37—44no—
88C22×D37—48no—
89C22×D37—44no—
90C22×D37C22×C414yes—
91D4×C37—44no—
92C2×C4×C37—21no—
93C2×C4×C37—21no—
94C2×C4×C37—22no—
95C2×C4×C37—22no—
96C2×C4×C37—22no—
97C2×C4×C37—22no—
98C2×C4×C37—22no—
99C2×C4×C37—22no—
100C2×C4×C37C22×C412yes—
101C22⋊D37—48no—
102C2×Dic37—22no—
103C2×Dic37—22no—
104C2×Dic37—22no—
105C2×Dic37—22no—
106C2×Dic37—22no—
107C2×Dic37—22no—
108C2×Dic37—22no—
109C2×Dic37—22no—
110C2×Dic37—44no—
111C2×Dic37C22×C414yes—
112C22×C37—24no—
113C22×C37—22no—
114C22×C37—22no—
115C22×C37—24no—
116C22×C37—41no—
117C22×C37—42no—
118C22×C37—41no—
119C22×C37—42no—
120C22×C37—41no—
121C22×C37—41no—
122C22×C37C23×C411yescharacteristic
123C22×C37H(5,𝔽2)12yes—
124C22×C37H(5,𝔽2)11yescharacteristic
125C22×C37H(5,𝔽2)11yescharacteristic
126C22×C37H(5,𝔽2)12yes—
127C2×D37—24no—
128C2×D37—24no—
129C2×D37—24no—
130C2×D37—416no—
131C2×D37—42no—
132C2×D37—42no—
133C2×D37—42no—
134C2×D37—42no—
135C4×C37—22no—
136C4×C37—42no—
137C4×C37—42no—
138Dic37—24no—
139Dic37—42no—
140Dic37—42no—
141#2375—371nomaximal, Sylow
142C2×C37—42no—
143C2×C37—42no—
144C2×C37#51812yes—
145C2×C37#51811yescharacteristic
146C2×C37#56011yescharacteristic
147C2×C37#56012yes—
148C2×C37H(3,𝔽4)11yescharacteristic
149D37—44no—
150C2×#120—371no—
151#390—372no—
152#390D3711yescharacteristic, complemented
153#388—372no—
154#388—372no—
155#388—372no—
156#386—371no—
157#386—372no—
158#379—372no—
159C37#237511yescharacteristic, complemented, Sylow
160C25—371no—
161C22×D4—372no—
162C22×D4C2×D3711yescharacteristic
163C23×C4—371no—
164#120—378no—
165C2×C4⋊C4—372no—
166C2×C4⋊C4C2×D3711yescharacteristic, complemented
167C2×C22⋊C4—371no—
168C2×C22⋊C4—372no—
169C2×C22⋊C4—372no—
170C2×C22⋊C4—371no—
171C2×C22⋊C4—372no—
172C2×C22⋊C4—372no—
173C2×C22⋊C4—372no—
174C2×C22⋊C4—372no—
175C2×C22⋊C4C2×D3712yes—
176C2×C22⋊C4C2×D3712yescomplemented
177C2×C42—372no—
178C2×C42C2×D3711yescharacteristic, complemented
179#95—372no—
180#95—372no—
181#95—372no—
182C24—372no—
183C24—374no—
184C24—1482no—
185C24—1481no—
186C24—1481no—
187C24—1482no—
188C24C22×D3711yescharacteristic
189C24C22×D3711yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—378no—
193C2×D4—1482no—
194C2×D4—1482no—
195C2×D4C4×D3714yes—
196C22×C4—372no—
197C22×C4—372no—
198C22×C4—372no—
199C22×C4—1481no—
200C22×C4—1482no—
201C22×C4C22×D3712yes—
202C22×C4C22×D3712yes—
203C22×C4C22×D3711yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—1482no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—374no—
209C22⋊C4—378no—
210C22⋊C4—1482no—
211C22⋊C4—1482no—
212C22⋊C4—1482no—
213C22⋊C4—1482no—
214C22⋊C4—1481no—
215C22⋊C4—1482no—
216C42—41no—
217C42—1482no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—374no—
227C23—742no—
228C23—742no—
229C23—742no—
230C23—742no—
231C23—742no—
232C23—742no—
233C23—1488no—
234C23—1484no—
235C23—1484no—
236C23—1488no—
237C23—1484no—
238C23C2×C4×D3712yes—
239C23C2×C4×D3712yes—
240C23C23×D3711yescharacteristic
241D4—44no—
242D4—1488no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—374no—
252C2×C4—742no—
253C2×C4—742no—
254C2×C4—742no—
255C2×C4—742no—
256C2×C4—742no—
257C2×C4—742no—
258C2×C4—742no—
259C2×C4—742no—
260C2×C4—1484no—
261C2×C4C2×C4×D3712yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—744no—
273C22—744no—
274C22—744no—
275C22—14816no—
276C22—1482no—
277C22—1482no—
278C22—1482no—
279C22—1482no—
280C22C22×C4×D3711yescharacteristic
281C22#1192458112yes—
282C22#1192458111yescharacteristic
283C22#1192458111yescharacteristic
284C22#1192458212yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—744no—
289C4—1482no—
290C4—1482no—
291C2—42no—
292C2—42no—
293C2—1484no—
294C2#1455800512yes—
295C2#1455800611yescharacteristic
296C2#1455827011yescharacteristic
297C2#1455827112yes—
298C2#1455827511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#390C2
socle, cosocleC23×C37C24
prime radicalC2×#11924504C2
hypercenter#390D37
hypocenterC37#2375
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #14557433C2semidirect
3. #14557424C2semidirect
4. #14557423C2semidirect
5. #14557421C2semidirect
6. #14557413C2semidirect
7. #14557410C2semidirect
8. #14557388C2semidirect
9. C37×#390C2semidirect
12. C23×Dic37C22semidirect
13. #11924504C4semidirect
14. C2×#10596936C22semidirect
17. C2×C22⋊Dic37C22semidirect
23. C2×C4×Dic37C22semidirect
24. #11924443C22semidirect
25. #11924443C22semidirect
26. #11924443C22semidirect
28. C2×C4⋊C4×C37C22semidirect
30. C2×C22⋊C4×C37C22semidirect
31. C2×C42×C37C22semidirect
38. C23×D37C2×C4semidirect
60. C22⋊Dic37C2×C4semidirect
65. #10596938C2×C4semidirect
152. #390D37semidirect
159. C37#2375semidirect
166. C2×C4⋊C4C2×D37semidirect
176. C2×C22⋊C4C2×D37semidirect
178. C2×C42C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245042
#145574332
#145574242
#145574232
#145574212
#145574132
#145574102
#145573882
C37×#3902
#237537