Finite Group Databasebeta

← #37526952

Subgroups — #37526952

Order 22976.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#19153341C211yesmaximal, characteristic, complemented
2C22×C4×D359C211yesmaximal, characteristic, complemented
3#26040000C211yesmaximal, characteristic, complemented
4#26039997C211yesmaximal, characteristic, complemented
5#26039993C211yesmaximal, characteristic, complemented
6#26039992C211yesmaximal, characteristic, complemented
7#26039988C211yesmaximal, characteristic, complemented
8#26039987C211yesmaximal, characteristic, complemented
9#26039985C211yesmaximal, characteristic, complemented
10#26039983C211yesmaximal, characteristic, complemented
11C4×D1436C211yesmaximal, characteristic, complemented
12C4×C22⋊D359C211yesmaximal, characteristic, complemented
13C4×C22⋊D359C211yesmaximal, characteristic, complemented
14#26039972C211yesmaximal, characteristic, complemented
15C4×D4×C359C211yesmaximal, characteristic, complemented
16C23×D359C2211yescharacteristic
17#19153341—24no—
18C2×Dic718C2211yescharacteristic, complemented
19C2×D1436C2211yescharacteristic
20C2×D1436C2211yescharacteristic
21C2×C22⋊D359C2211yescharacteristic
22C2×C22⋊D359C2211yescharacteristic
23C2×C22⋊D359C2211yescharacteristic
24C2×C22⋊D359C2211yescharacteristic, complemented
25C22×Dic359C2211yescharacteristic
26C2×C4×D359—24no—
27C2×C4×D359C2211yescharacteristic
28C2×C4×D359C2211yescharacteristic
29C2×C4×D359C2211yescharacteristic
30C2×C4×D359C2211yescharacteristic
31C2×C4×D359C2211yescharacteristic, complemented
32C4⋊Dic359C2211yescharacteristic, complemented
33Dic359⋊C4C2211yescharacteristic, complemented
34Dic359⋊C4C2211yescharacteristic, complemented
35Dic359⋊C4C2211yescharacteristic, complemented
36Dic359⋊C4C2211yescharacteristic, complemented
37C22⋊Dic359C2211yescharacteristic
38C22⋊Dic359C2211yescharacteristic, complemented
39(C2×D359)⋊C4C2211yescharacteristic
40(C2×D359)⋊C4C2211yescharacteristic
41(C2×D359)⋊C4C2211yescharacteristic
42(C2×D359)⋊C4C2211yescharacteristic
43(C2×D359)⋊C4C2211yescharacteristic, complemented
44(C2×D359)⋊C4C2211yescharacteristic, complemented
45C4×Dic359C2211yescharacteristic, complemented
46C2×D4×C359C2211yescharacteristic
47C22×C4×C359C2211yescharacteristic
48C22×C4×C359C2211yescharacteristic, complemented
49C4⋊C4×C359C2211yescharacteristic, complemented
50C22⋊C4×C359C2211yescharacteristic
51C22⋊C4×C359C2211yescharacteristic, complemented
52C42×C359C2211yescharacteristic, complemented
53C22×D359—22no—
54C22×D359—22no—
55C22×D359—22no—
56C22×D359C2311yescharacteristic
57C22×D359C2311yescharacteristic
58C22×D359C2311yescharacteristic
59Dic718—22no—
60D1436—22no—
61D1436—22no—
62C22⋊D359—22no—
63C22⋊D359—22no—
64C22⋊D359—22no—
65C22⋊D359—22no—
66C2×Dic359—22no—
67C2×Dic359C2311yescharacteristic
68C2×Dic359C2311yescharacteristic
69C2×Dic359C2311yescharacteristic
70C2×Dic359C2311yescharacteristic
71C2×Dic359C2311yescharacteristic
72C4×D359—22no—
73C4×D359—22no—
74C4×D359—22no—
75C4×D359—22no—
76C4×D359D414yescomplemented
77C23×C359C2311yescharacteristic
78C23×C359C2311yescharacteristic
79D4×C359—22no—
80C2×C4×C359—22no—
81C2×C4×C359—22no—
82C2×C4×C359C2311yescharacteristic
83C2×C4×C359C2311yescharacteristic
84C2×C4×C359C2311yescharacteristic
85C2×C4×C359C2311yescharacteristic
86C2×C4×C359C2311yescharacteristic
87C22×C359—21no—
88C22×C359—21no—
89C22×C359—21no—
90C22×C359—21no—
91C22×C359—21no—
92C22×C359#4112yescomplemented
93C22×C359C2411yescharacteristic
94C2×D359—28no—
95C2×D359—21no—
96C2×D359—21no—
97C2×D359—21no—
98C2×D359—21no—
99C2×D359—21no—
100C2×D359—21no—
101C2×D359—21no—
102C2×D359C2×D412yes—
103C2×D359#4112yescomplemented
104C4×C359—21no—
105C4×C359—21no—
106C4×C359—21no—
107C4×C359—21no—
108C4×C359C2×D412yes—
109Dic359—21no—
110Dic359—21no—
111Dic359—21no—
112Dic359—21no—
113Dic359C2×D412yes—
114C2×C359—22no—
115C2×C359—41no—
116C2×C359C22×D411yescharacteristic
117C2×C359C2×#4111yescharacteristic
118C2×C359C2×#4111yescharacteristic
119D359—22no—
120D359—22no—
121D359—41no—
122C359#52511yescharacteristic, complemented, Sylow
123#525—3591nomaximal, Sylow
124C2×#41—3591no—
125C23×C4—3591no—
126#123—3591no—
127#123—3591no—
128C22⋊Q8—3591no—
129C22⋊Q8—3591no—
130#121—3591no—
131#121—3591no—
132#120—3591no—
133#120—3591no—
134C4×D4—3591no—
135C4×D4—3591no—
136C4×D4—3591no—
137C4×D4D35911yescharacteristic, complemented
138#117—3591no—
139C24—3591no—
140#41—7184no—
141C2×Q8—3591no—
142C2×D4—3591no—
143C2×D4—3591no—
144C2×D4—3591no—
145C2×D4—3591no—
146C2×D4—3591no—
147C2×D4—3591no—
148C2×D4C2×D35911yescharacteristic
149C22×C4—3591no—
150C22×C4—3591no—
151C22×C4—3591no—
152C22×C4—3591no—
153C22×C4—3591no—
154C22×C4—3591no—
155C22×C4—7184no—
156C22×C4C2×D35911yescharacteristic
157C22×C4C2×D35911yescharacteristic, complemented
158C4⋊C4—3591no—
159C4⋊C4—3591no—
160C4⋊C4—3591no—
161C4⋊C4—3591no—
162C4⋊C4—3591no—
163C4⋊C4C2×D35911yescharacteristic, complemented
164C22⋊C4—3591no—
165C22⋊C4—3591no—
166C22⋊C4—3591no—
167C22⋊C4—3591no—
168C22⋊C4—3591no—
169C22⋊C4—3591no—
170C22⋊C4—3591no—
171C22⋊C4—3591no—
172C22⋊C4C2×D35911yescharacteristic
173C22⋊C4C2×D35911yescharacteristic, complemented
174C42—3591no—
175C42C2×D35911yescharacteristic, complemented
176C23—3591no—
177C23—3591no—
178C23—3591no—
179C23—7182no—
180C23—7182no—
181C23—7182no—
182C23C22×D35911yescharacteristic
183C23C22×D35911yescharacteristic
184Q8—7182no—
185D4—22no—
186D4—7182no—
187D4—7182no—
188D4—7182no—
189D4—7182no—
190D4—7182no—
191D4—7182no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—3591no—
195C2×C4—3594no—
196C2×C4—3591no—
197C2×C4—3591no—
198C2×C4—3591no—
199C2×C4—3591no—
200C2×C4—7182no—
201C2×C4—7182no—
202C2×C4—7182no—
203C2×C4—7182no—
204C2×C4—7182no—
205C2×C4C22×D35911yescharacteristic
206C2×C4C22×D35911yescharacteristic
207C2×C4C22×D35911yescharacteristic
208C2×C4C22×D35911yescharacteristic
209C2×C4C22×D35911yescharacteristic
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—3592no—
216C22—3592no—
217C22—7188no—
218C22—7181no—
219C22—7181no—
220C22—7181no—
221C22—7181no—
222C22—7181no—
223C22—7181no—
224C22—7181no—
225C22#1915334112yescomplemented
226C22C23×D35911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—3592no—
232C4—7181no—
233C4—7181no—
234C4—7181no—
235C4—7181no—
236C4D4×D35912yes—
237C2—22no—
238C2—41no—
239C2—14361no—
240C2—7182no—
241C2—7182no—
242C2C2×D4×D35911yescharacteristic
243C2#41×D35911yescharacteristic
244C2C2×#1915334111yescharacteristic
subgroupG/H
centerC2×C4C22×D359
commutator, socle, cosocleC22×C359C24
FrattiniC22C23×D359
FittingC4×D4×C359C2
hypercenterC4×D4D359
hypocenterC359#525
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19153341C2semidirect
2. C22×C4×D359C2semidirect
3. #26040000C2semidirect
4. #26039997C2semidirect
5. #26039993C2semidirect
6. #26039992C2semidirect
7. #26039988C2semidirect
8. #26039987C2semidirect
9. #26039985C2semidirect
10. #26039983C2semidirect
11. C4×D1436C2semidirect
12. C4×C22⋊D359C2semidirect
13. C4×C22⋊D359C2semidirect
14. #26039972C2semidirect
15. C4×D4×C359C2semidirect
18. C2×Dic718C22semidirect
24. C2×C22⋊D359C22semidirect
31. C2×C4×D359C22semidirect
32. C4⋊Dic359C22semidirect
33. Dic359⋊C4C22semidirect
34. Dic359⋊C4C22semidirect
35. Dic359⋊C4C22semidirect
36. Dic359⋊C4C22semidirect
38. C22⋊Dic359C22semidirect
43. (C2×D359)⋊C4C22semidirect
44. (C2×D359)⋊C4C22semidirect
45. C4×Dic359C22semidirect
48. C22×C4×C359C22semidirect
49. C4⋊C4×C359C22semidirect
51. C22⋊C4×C359C22semidirect
52. C42×C359C22semidirect
76. C4×D359D4semidirect
92. C22×C359#41semidirect
103. C2×D359#41semidirect
122. C359#525semidirect
137. C4×D4D359semidirect
157. C22×C4C2×D359semidirect
163. C4⋊C4C2×D359semidirect
173. C22⋊C4C2×D359semidirect
175. C42C2×D359semidirect
225. C22#19153341semidirect

Maximal subgroups

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

subgroupindex
C2×#191533412
C22×C4×D3592
#260400002
#260399972
#260399932
#260399922
#260399882
#260399872
#260399852
#260399832
C4×D14362
C4×C22⋊D3592
#260399722
C4×D4×C3592
#525359