Finite Group Databasebeta

← #37554924

Subgroups — #37554924

Order 23488.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#26044257C211yesmaximal, characteristic, complemented
2#26044256C211yesmaximal, characteristic, complemented
3#26044254C211yesmaximal, characteristic, complemented
4#26044246C211yesmaximal, characteristic, complemented
5#26044245C211yesmaximal, characteristic, complemented
6#26044244C211yesmaximal, characteristic, complemented
7#26044235C211yesmaximal, characteristic, complemented
8#26044234C211yesmaximal, characteristic, complemented
9#26044233C211yesmaximal, characteristic, complemented
10#26044233C211yesmaximal, characteristic, complemented
11#26044232C211yesmaximal, characteristic, complemented
12#26044231C211yesmaximal, characteristic, complemented
13#26044229C211yesmaximal, characteristic, complemented
14#26044229C211yesmaximal, characteristic, complemented
15#121×C367C211yesmaximal, characteristic, complemented
16C23×D367C2211yescharacteristic
17C2×D1468C2211yescharacteristic
18C2×D1468C2211yescharacteristic
19C2×C22⋊D367C2211yescharacteristic
20C2×C22⋊D367C2211yescharacteristic
21C2×C22⋊D367C2211yescharacteristic
22C2×C22⋊D367C2211yescharacteristic
23C2×C22⋊D367C2211yescharacteristic
24C2×C22⋊D367C2211yescharacteristic, complemented
25C2×C22⋊D367C2211yescharacteristic, complemented
26C22×Dic367C2211yescharacteristic, complemented
27C2×C4×D367C2211yescharacteristic, complemented
28C4⋊Dic367C2211yescharacteristic, complemented
29C4⋊Dic367C2211yescharacteristic, complemented
30Dic367⋊C4C2211yescharacteristic, complemented
31Dic367⋊C4C2211yescharacteristic, complemented
32Dic367⋊C4C2211yescharacteristic, complemented
33C22⋊Dic367C2211yescharacteristic
34C22⋊Dic367C2211yescharacteristic
35C22⋊Dic367C2211yescharacteristic, complemented
36(C2×D367)⋊C4C2211yescharacteristic
37(C2×D367)⋊C4C2211yescharacteristic
38(C2×D367)⋊C4C2211yescharacteristic
39(C2×D367)⋊C4C2211yescharacteristic
40(C2×D367)⋊C4C2211yescharacteristic
41(C2×D367)⋊C4C2211yescharacteristic
42(C2×D367)⋊C4C2211yescharacteristic, complemented
43C4×Dic367C2211yescharacteristic, complemented
44C2×D4×C367C2211yescharacteristic
45C2×D4×C367C2211yescharacteristic
46C2×D4×C367C2211yescharacteristic, complemented
47C22×C4×C367C2211yescharacteristic, complemented
48C4⋊C4×C367C2211yescharacteristic, complemented
49C22⋊C4×C367C2211yescharacteristic
50C22⋊C4×C367C2211yescharacteristic, complemented
51C22×D367—41no—
52C22×D367—41no—
53C22×D367—41no—
54C22×D367C2311yescharacteristic
55C22×D367C2311yescharacteristic
56C22×D367C2311yescharacteristic
57D1468—41no—
58D1468—41no—
59C22⋊D367—41no—
60C22⋊D367—41no—
61C22⋊D367—41no—
62C22⋊D367—41no—
63C22⋊D367—41no—
64C22⋊D367—41no—
65C22⋊D367—41no—
66C2×Dic367—41no—
67C2×Dic367C2311yescharacteristic
68C2×Dic367C2311yescharacteristic
69C2×Dic367C2311yescharacteristic
70C2×Dic367C2311yescharacteristic
71C2×Dic367C2311yescharacteristic
72C4×D367—41no—
73C23×C367C2311yescharacteristic
74C23×C367C2311yescharacteristic
75C23×C367C2311yescharacteristic
76D4×C367—41no—
77D4×C367—41no—
78D4×C367—41no—
79C2×C4×C367—41no—
80C2×C4×C367C2311yescharacteristic
81C2×C4×C367C2311yescharacteristic
82C2×C4×C367C2311yescharacteristic
83C2×C4×C367C2311yescharacteristic
84C22×C367—21no—
85C22×C367—21no—
86C22×C367—21no—
87C22×C367—21no—
88C22×C367—21no—
89C22×C367—21no—
90C22×C367—21no—
91C22×C367—21no—
92C22×C367—21no—
93C22×C367C2411yescharacteristic
94C2×D367—21no—
95C2×D367—21no—
96C2×D367—21no—
97C2×D367—21no—
98C2×D367—21no—
99C2×D367—21no—
100C2×D367—21no—
101C2×D367—21no—
102C2×D367—21no—
103C2×D367—44no—
104C4×C367—21no—
105C4×C367—21no—
106C4×C367—21no—
107C4×C367—21no—
108Dic367—21no—
109Dic367—21no—
110Dic367—21no—
111Dic367—21no—
112Dic367—21no—
113C2×C367—41no—
114C2×C367—41no—
115C2×C367—41no—
116C2×C367H(5,𝔽2)11yescharacteristic
117C2×C367H(5,𝔽2)11yescharacteristic
118C2×C367H(5,𝔽2)11yescharacteristic
119D367—41no—
120D367—41no—
121D367—41no—
122C367#56011yescharacteristic, complemented, Sylow
123#560—3671nomaximal, Sylow
124#127—3671no—
125#126—3671no—
126#126—3671no—
127#123—3671no—
128#123—3671no—
129#123—3671no—
130#121—3671no—
131#121—3671no—
132#121—3671no—
133#121—3671no—
134#121—3671no—
135#121D36711yescharacteristic, complemented
136#120—3671no—
137#120—3671no—
138#120—3671no—
139C24—3671no—
140C2×D4—3671no—
141C2×D4—3671no—
142C2×D4—3671no—
143C2×D4—3671no—
144C2×D4—3671no—
145C2×D4—3671no—
146C2×D4—3671no—
147C2×D4—3671no—
148C2×D4—3671no—
149C2×D4C2×D36711yescharacteristic
150C2×D4C2×D36711yescharacteristic
151C2×D4C2×D36711yescharacteristic, complemented
152C22×C4—3671no—
153C22×C4—3671no—
154C22×C4C2×D36711yescharacteristic, complemented
155C4⋊C4—3671no—
156C4⋊C4—3671no—
157C4⋊C4—3671no—
158C4⋊C4—3671no—
159C4⋊C4—3671no—
160C4⋊C4C2×D36711yescharacteristic, complemented
161C22⋊C4—3671no—
162C22⋊C4—3671no—
163C22⋊C4—3671no—
164C22⋊C4—3671no—
165C22⋊C4—3671no—
166C22⋊C4—3671no—
167C22⋊C4—3671no—
168C22⋊C4—3671no—
169C22⋊C4—3671no—
170C22⋊C4—3671no—
171C22⋊C4C2×D36711yescharacteristic
172C22⋊C4C2×D36711yescharacteristic, complemented
173C42—3671no—
174C23—3671no—
175C23—3671no—
176C23—3671no—
177C23—14681no—
178C23—14681no—
179C23—14681no—
180C23C22×D36711yescharacteristic
181C23C22×D36711yescharacteristic
182C23C22×D36711yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—14681no—
187D4—14681no—
188D4—14681no—
189D4—14681no—
190D4—14681no—
191D4—14681no—
192D4—14681no—
193D4—14681no—
194D4—14681no—
195C2×C4—41no—
196C2×C4—3671no—
197C2×C4—3671no—
198C2×C4—3671no—
199C2×C4—3671no—
200C2×C4—3671no—
201C2×C4—14681no—
202C2×C4—14681no—
203C2×C4C22×D36711yescharacteristic
204C2×C4C22×D36711yescharacteristic
205C2×C4C22×D36711yescharacteristic
206C2×C4C22×D36711yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—14684no—
217C22—7341no—
218C22—7341no—
219C22—7341no—
220C22—7341no—
221C22—7341no—
222C22—7341no—
223C22—7341no—
224C22—7341no—
225C22—7341no—
226C22C23×D36711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—7341no—
232C4—7341no—
233C4—7341no—
234C4—7341no—
235C4—7341no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—14681no—
240C2—14681no—
241C2—14681no—
242C2#2604430611yescharacteristic
243C2#2604430611yescharacteristic
244C2#2604430711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D367
commutator, socle, cosocleC22×C367C24
Fitting#121×C367C2
hypercenter#121D367
hypocenterC367#560
LCSG ▷ C22×C367 ▷ C367
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #26044257C2semidirect
2. #26044256C2semidirect
3. #26044254C2semidirect
4. #26044246C2semidirect
5. #26044245C2semidirect
6. #26044244C2semidirect
7. #26044235C2semidirect
8. #26044234C2semidirect
9. #26044233C2semidirect
10. #26044233C2semidirect
11. #26044232C2semidirect
12. #26044231C2semidirect
13. #26044229C2semidirect
14. #26044229C2semidirect
15. #121×C367C2semidirect
24. C2×C22⋊D367C22semidirect
25. C2×C22⋊D367C22semidirect
26. C22×Dic367C22semidirect
27. C2×C4×D367C22semidirect
28. C4⋊Dic367C22semidirect
29. C4⋊Dic367C22semidirect
30. Dic367⋊C4C22semidirect
31. Dic367⋊C4C22semidirect
32. Dic367⋊C4C22semidirect
35. C22⋊Dic367C22semidirect
42. (C2×D367)⋊C4C22semidirect
43. C4×Dic367C22semidirect
46. C2×D4×C367C22semidirect
47. C22×C4×C367C22semidirect
48. C4⋊C4×C367C22semidirect
50. C22⋊C4×C367C22semidirect
122. C367#560semidirect
135. #121D367semidirect
151. C2×D4C2×D367semidirect
154. C22×C4C2×D367semidirect
160. C4⋊C4C2×D367semidirect
172. C22⋊C4C2×D367semidirect

Maximal subgroups

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

subgroupindex
#260442572
#260442562
#260442542
#260442462
#260442452
#260442442
#260442352
#260442342
#260442332
#260442322
#260442312
#260442292
#121×C3672
#560367