Finite Group Databasebeta

← #37527083

Subgroups — #37527083

Order 22976.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×C22⋊D359C211yesmaximal, characteristic, complemented
3#26040001C211yesmaximal, characteristic, complemented
4#26039998C211yesmaximal, characteristic, complemented
5#26039997C211yesmaximal, characteristic, complemented
6#26039992C211yesmaximal, characteristic, complemented
7#26039989C211yesmaximal, characteristic, complemented
8#26039988C211yesmaximal, characteristic, complemented
9#26039988C211yesmaximal, characteristic, complemented
10#26039985C211yesmaximal, characteristic, complemented
11#26039984C211yesmaximal, characteristic, complemented
12C4×C22⋊D359C211yesmaximal, characteristic, complemented
13(C22⋊D359)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D359C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C359C211yesmaximal, characteristic, complemented
16C23×D359C2211yescharacteristic
17#19153341—24no—
18C2×Dic718C2211yescharacteristic, complemented
19C2×D1436C2211yescharacteristic, complemented
20C2×D1436C2211yescharacteristic, complemented
21C2×C22⋊D359—24no—
22C2×C22⋊D359C2211yescharacteristic
23C2×C22⋊D359C2211yescharacteristic
24C2×C22⋊D359C2211yescharacteristic
25C2×C22⋊D359C2211yescharacteristic
26C2×C22⋊D359C2211yescharacteristic
27C2×C22⋊D359C2211yescharacteristic, complemented
28C2×C22⋊D359C2211yescharacteristic, complemented
29C22×Dic359C2211yescharacteristic
30C2×C4×D359C2211yescharacteristic, complemented
31C2×C4×D359C2211yescharacteristic, complemented
32C2×C4×D359C2211yescharacteristic, complemented
33C4⋊Dic359C2211yescharacteristic, complemented
34Dic359⋊C4C2211yescharacteristic, complemented
35Dic359⋊C4C2211yescharacteristic, complemented
36Dic359⋊C4C2211yescharacteristic, complemented
37C22⋊Dic359C2211yescharacteristic
38C22⋊Dic359C2211yescharacteristic, complemented
39C22⋊Dic359C2211yescharacteristic, complemented
40(C2×D359)⋊C4C2211yescharacteristic, complemented
41(C2×D359)⋊C4C2211yescharacteristic, complemented
42(C2×D359)⋊C4C2211yescharacteristic, complemented
43(C2×D359)⋊C4C2211yescharacteristic, complemented
44(C2×D359)⋊C4C2211yescharacteristic, complemented
45C4×Dic359C2211yescharacteristic, complemented
46C24×C359C2211yescharacteristic
47C22×C4×C359C2211yescharacteristic, complemented
48C22×C4×C359C2211yescharacteristic, complemented
49C22⋊C4×C359C2211yescharacteristic
50C22⋊C4×C359C2211yescharacteristic
51C22⋊C4×C359C2211yescharacteristic, complemented
52C22⋊C4×C359C2211yescharacteristic, complemented
53C22×D359—22no—
54C22×D359—22no—
55C22×D359—41no—
56C22×D359C2311yescharacteristic
57C22×D359C2311yescharacteristic
58C22×D359C2311yescharacteristic
59Dic718—22no—
60D1436—22no—
61D1436—41no—
62C22⋊D359—22no—
63C22⋊D359—22no—
64C22⋊D359—22no—
65C22⋊D359—22no—
66C22⋊D359—22no—
67C22⋊D359—41no—
68C22⋊D359D414yescomplemented
69C2×Dic359—22no—
70C2×Dic359C2311yescharacteristic
71C2×Dic359C2311yescharacteristic
72C2×Dic359C2311yescharacteristic
73C2×Dic359C2311yescharacteristic
74C2×Dic359C2311yescharacteristic
75C4×D359—22no—
76C4×D359—22no—
77C4×D359—41no—
78C23×C359—22no—
79C23×C359—22no—
80C23×C359—41no—
81C23×C359C2311yescharacteristic
82C23×C359C2311yescharacteristic
83C23×C359C2311yescharacteristic
84C2×C4×C359—22no—
85C2×C4×C359—41no—
86C2×C4×C359C2311yescharacteristic
87C2×C4×C359C2311yescharacteristic
88C2×C4×C359C2311yescharacteristic
89C2×C4×C359C2311yescharacteristic
90C22×C359—21no—
91C22×C359—21no—
92C22×C359—21no—
93C22×C359—21no—
94C22×C359—21no—
95C22×C359—21no—
96C22×C359—21no—
97C22×C359—44no—
98C22×C359C2×D412yes—
99C22×C359#4112yescomplemented
100C22×C359C2411yescharacteristic
101C2×D359—21no—
102C2×D359—21no—
103C2×D359—21no—
104C2×D359—21no—
105C2×D359—21no—
106C2×D359—21no—
107C2×D359—21no—
108C2×D359—21no—
109C2×D359—44no—
110C2×D359C2×D412yes—
111C4×C359—21no—
112C4×C359—21no—
113C4×C359—21no—
114C4×C359—21no—
115Dic359—21no—
116Dic359—21no—
117Dic359—21no—
118Dic359—21no—
119Dic359C2×D412yes—
120C2×C359—22no—
121C2×C359—22no—
122C2×C359—41no—
123C2×C359C22×D411yescharacteristic
124C2×C359C2×#4111yescharacteristic
125C2×C359H(5,𝔽2)11yescharacteristic
126D359—22no—
127D359—41no—
128D359—41no—
129C359#54611yescharacteristic, complemented, Sylow
130#546—3591nomaximal, Sylow
131C2×#41—3591no—
132C22×D4—3591no—
133#124—3591no—
134#123—3591no—
135#123—3591no—
136C22⋊Q8—3591no—
137#121—3591no—
138#121—3591no—
139#121—3591no—
140#120—3591no—
141#120—3591no—
142C4×D4—3591no—
143C4×D4—3591no—
144C2×C22⋊C4—3591no—
145C2×C22⋊C4D35911yescharacteristic, complemented
146C24—3591no—
147C24C2×D35911yescharacteristic
148#41—7184no—
149C2×Q8—3591no—
150C2×D4—3591no—
151C2×D4—3591no—
152C2×D4—3591no—
153C2×D4—3591no—
154C2×D4—3591no—
155C2×D4—3591no—
156C2×D4—3591no—
157C2×D4—3591no—
158C2×D4—3591no—
159C2×D4—7184no—
160C22×C4—3591no—
161C22×C4—3591no—
162C22×C4—3591no—
163C22×C4—3591no—
164C22×C4C2×D35911yescharacteristic, complemented
165C22×C4C2×D35911yescharacteristic, complemented
166C4⋊C4—3591no—
167C4⋊C4—3591no—
168C4⋊C4—3591no—
169C4⋊C4—3591no—
170C22⋊C4—3591no—
171C22⋊C4—3591no—
172C22⋊C4—3591no—
173C22⋊C4—3591no—
174C22⋊C4—3591no—
175C22⋊C4—3591no—
176C22⋊C4—3591no—
177C22⋊C4—3591no—
178C22⋊C4C2×D35911yescharacteristic
179C22⋊C4C2×D35911yescharacteristic
180C22⋊C4C2×D35911yescharacteristic, complemented
181C22⋊C4C2×D35911yescharacteristic, complemented
182C42—3591no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—3591no—
187C23—3591no—
188C23—3591no—
189C23—14361no—
190C23—7182no—
191C23—7182no—
192C23C22×D35911yescharacteristic
193C23C22×D35911yescharacteristic
194C23C22×D35911yescharacteristic
195Q8—7182no—
196D4—3594no—
197D4—14361no—
198D4—14361no—
199D4—7182no—
200D4—7182no—
201D4—7182no—
202D4—7182no—
203D4—7182no—
204D4—7182no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—3591no—
208C2×C4—3591no—
209C2×C4—3591no—
210C2×C4—3591no—
211C2×C4—3591no—
212C2×C4—14361no—
213C2×C4—7182no—
214C2×C4—7182no—
215C2×C4—7182no—
216C2×C4C22×D35911yescharacteristic
217C2×C4C22×D35911yescharacteristic
218C2×C4C22×D35911yescharacteristic
219C2×C4C22×D35911yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—44no—
228C22—3592no—
229C22—14364no—
230C22—7181no—
231C22—7181no—
232C22—7181no—
233C22—7181no—
234C22—7181no—
235C22—7181no—
236C22—7181no—
237C22—7181no—
238C22D4×D35912yes—
239C22#1915334112yescomplemented
240C22C23×D35911yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—3592no—
246C4—7181no—
247C4—7181no—
248C4—7181no—
249C4—7181no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—14361no—
254C2—14361no—
255C2—7182no—
256C2C2×D4×D35911yescharacteristic
257C2C2×#1915334111yescharacteristic
258C2#2604006011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
FittingC2×C22⋊C4×C359C2
hypercenterC2×C22⋊C4D359
hypocenterC359#546
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19153341C2semidirect
2. C22×C22⋊D359C2semidirect
3. #26040001C2semidirect
4. #26039998C2semidirect
5. #26039997C2semidirect
6. #26039992C2semidirect
7. #26039989C2semidirect
8. #26039988C2semidirect
9. #26039988C2semidirect
10. #26039985C2semidirect
11. #26039984C2semidirect
12. C4×C22⋊D359C2semidirect
13. (C22⋊D359)⋊C4C2semidirect
14. C22⋊C4×D359C2semidirect
15. C2×C22⋊C4×C359C2semidirect
18. C2×Dic718C22semidirect
19. C2×D1436C22semidirect
20. C2×D1436C22semidirect
27. C2×C22⋊D359C22semidirect
28. C2×C22⋊D359C22semidirect
30. C2×C4×D359C22semidirect
31. C2×C4×D359C22semidirect
32. C2×C4×D359C22semidirect
33. C4⋊Dic359C22semidirect
34. Dic359⋊C4C22semidirect
35. Dic359⋊C4C22semidirect
36. Dic359⋊C4C22semidirect
38. C22⋊Dic359C22semidirect
39. C22⋊Dic359C22semidirect
40. (C2×D359)⋊C4C22semidirect
41. (C2×D359)⋊C4C22semidirect
42. (C2×D359)⋊C4C22semidirect
43. (C2×D359)⋊C4C22semidirect
44. (C2×D359)⋊C4C22semidirect
45. C4×Dic359C22semidirect
47. C22×C4×C359C22semidirect
48. C22×C4×C359C22semidirect
51. C22⋊C4×C359C22semidirect
52. C22⋊C4×C359C22semidirect
68. C22⋊D359D4semidirect
99. C22×C359#41semidirect
129. C359#546semidirect
145. C2×C22⋊C4D359semidirect
164. C22×C4C2×D359semidirect
165. C22×C4C2×D359semidirect
180. C22⋊C4C2×D359semidirect
181. C22⋊C4C2×D359semidirect
239. 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×C22⋊D3592
#260400012
#260399982
#260399972
#260399922
#260399892
#260399882
#260399852
#260399842
C4×C22⋊D3592
(C22⋊D359)⋊C42
C22⋊C4×D3592
C2×C22⋊C4×C3592
#546359