Finite Group Databasebeta

← #37560006

Subgroups — #37560006

Order 23872.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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#26045786C211yesmaximal, characteristic, complemented
2#26045785C211yesmaximal, characteristic, complemented
3#26045781C211yesmaximal, characteristic, complemented
4#26045776C211yesmaximal, characteristic, complemented
5#26045774C211yesmaximal, characteristic, complemented
6#26045768C211yesmaximal, characteristic, complemented
7#26045765C211yesmaximal, characteristic, complemented
8#26045765C211yesmaximal, characteristic, complemented
9#26045764C211yesmaximal, characteristic, complemented
10#26045762C211yesmaximal, characteristic, complemented
11#26045760C211yesmaximal, characteristic, complemented
12C4×D1492C211yesmaximal, characteristic, complemented
13#26045752C211yesmaximal, characteristic, complemented
14C22⋊C4×D373C211yesmaximal, characteristic, complemented
15#124×C373C211yesmaximal, characteristic, complemented
16C23×D373C2211yescharacteristic
17C2×D1492C2211yescharacteristic
18C2×D1492C2211yescharacteristic
19C2×C22⋊D373C2211yescharacteristic
20C2×C22⋊D373C2211yescharacteristic
21C2×C22⋊D373C2211yescharacteristic
22C2×C22⋊D373C2211yescharacteristic, complemented
23C22×Dic373C2211yescharacteristic, complemented
24C2×C4×D373C2211yescharacteristic
25C2×C4×D373C2211yescharacteristic
26C2×C4×D373C2211yescharacteristic, complemented
27C4⋊Dic373C2211yescharacteristic, complemented
28C4⋊Dic373C2211yescharacteristic, complemented
29#20238589C2211yescharacteristic, complemented
30#20238589C2211yescharacteristic, complemented
31#20238589C2211yescharacteristic, complemented
32#20238589C2211yescharacteristic, complemented
33C22⋊Dic373C2211yescharacteristic
34C22⋊Dic373C2211yescharacteristic
35#20238587C2211yescharacteristic
36#20238587C2211yescharacteristic
37#20238587C2211yescharacteristic
38#20238587C2211yescharacteristic
39#20238587C2211yescharacteristic
40#20238587C2211yescharacteristic
41#20238587C2211yescharacteristic, complemented
42#20238587C2211yescharacteristic, complemented
43C4×Dic373C2211yescharacteristic, complemented
44C2×Q8×C373C2211yescharacteristic, complemented
45C2×D4×C373C2211yescharacteristic
46C22⋊C4×C373C2211yescharacteristic
47C22⋊C4×C373C2211yescharacteristic
48C22⋊C4×C373C2211yescharacteristic, complemented
49C22⋊C4×C373C2211yescharacteristic, complemented
50C42×C373C2211yescharacteristic, complemented
51C22×D373—22no—
52C22×D373—41no—
53C22×D373—41no—
54C22×D373C2311yescharacteristic
55C22×D373C2311yescharacteristic
56C22×D373C2311yescharacteristic
57D1492—22no—
58D1492—41no—
59C22⋊D373—22no—
60C22⋊D373—41no—
61C22⋊D373—41no—
62C22⋊D373—41no—
63C2×Dic373—41no—
64C2×Dic373C2311yescharacteristic
65C2×Dic373C2311yescharacteristic
66C2×Dic373C2311yescharacteristic
67C2×Dic373C2311yescharacteristic
68C2×Dic373C2311yescharacteristic
69C4×D373—41no—
70C4×D373—41no—
71C4×D373—41no—
72C23×C373C2311yescharacteristic
73C23×C373C2311yescharacteristic
74Q8×C373—41no—
75D4×C373—41no—
76C2×C4×C373C2311yescharacteristic
77C2×C4×C373C2311yescharacteristic
78C2×C4×C373C2311yescharacteristic
79C2×C4×C373C2311yescharacteristic
80C2×C4×C373C2311yescharacteristic
81C22×C373—21no—
82C22×C373—21no—
83C22×C373—21no—
84C22×C373—21no—
85C22×C373—21no—
86C22×C373—21no—
87C22×C373C2411yescharacteristic
88C2×D373—21no—
89C2×D373—21no—
90C2×D373—21no—
91C2×D373—21no—
92C2×D373—21no—
93C2×D373—21no—
94C2×D373—21no—
95C2×D373—21no—
96C2×D373—44no—
97C2×D373#4112yes—
98C4×C373—21no—
99C4×C373—21no—
100C4×C373—21no—
101C4×C373—21no—
102C4×C373—21no—
103Dic373—21no—
104Dic373—21no—
105Dic373—21no—
106Dic373—21no—
107Dic373—21no—
108C2×C373—41no—
109C2×C373—41no—
110C2×C373C2×#4111yescharacteristic
111C2×C373H(5,𝔽2)11yescharacteristic
112C2×C373H(5,𝔽2)11yescharacteristic
113D373—22no—
114D373—41no—
115D373—41no—
116C373#53811yescharacteristic, complemented, Sylow
117#538—3731nomaximal, Sylow
118#126—3731no—
119#126—3731no—
120#124—3731no—
121#124D37311yescharacteristic, complemented
122#123—3731no—
123#123—3731no—
124C22⋊Q8—3731no—
125#121—3731no—
126#121—3731no—
127#121—3731no—
128#120—3731no—
129#120—3731no—
130C4×D4—3731no—
131C4×D4—3731no—
132C2×C22⋊C4—3731no—
133C24—3731no—
134C2×Q8C2×D37311yescharacteristic, complemented
135C2×D4—3731no—
136C2×D4—3731no—
137C2×D4—3731no—
138C2×D4—3731no—
139C2×D4—3731no—
140C2×D4—3731no—
141C2×D4C2×D37311yescharacteristic
142C22×C4—3731no—
143C22×C4—3731no—
144C22×C4—3731no—
145C22×C4—3731no—
146C4⋊C4—3731no—
147C4⋊C4—3731no—
148C4⋊C4—3731no—
149C4⋊C4—3731no—
150C4⋊C4—3731no—
151C4⋊C4—3731no—
152C22⋊C4—3731no—
153C22⋊C4—3731no—
154C22⋊C4—3731no—
155C22⋊C4—3731no—
156C22⋊C4—3731no—
157C22⋊C4—3731no—
158C22⋊C4—3731no—
159C22⋊C4—3731no—
160C22⋊C4—3731no—
161C22⋊C4—3731no—
162C22⋊C4C2×D37311yescharacteristic
163C22⋊C4C2×D37311yescharacteristic
164C22⋊C4C2×D37311yescharacteristic, complemented
165C22⋊C4C2×D37311yescharacteristic, complemented
166C42—3731no—
167C42C2×D37311yescharacteristic, complemented
168C23—3731no—
169C23—3731no—
170C23—3731no—
171C23—14921no—
172C23—14921no—
173C23—7462no—
174C23C22×D37311yescharacteristic
175C23C22×D37311yescharacteristic
176Q8—41no—
177D4—41no—
178D4—14921no—
179D4—14921no—
180D4—14921no—
181D4—14921no—
182D4—7462no—
183D4—7462no—
184C2×C4—3731no—
185C2×C4—3731no—
186C2×C4—3731no—
187C2×C4—3731no—
188C2×C4—3731no—
189C2×C4—14921no—
190C2×C4—14921no—
191C2×C4—14921no—
192C2×C4—14921no—
193C2×C4C22×D37311yescharacteristic
194C2×C4C22×D37311yescharacteristic
195C2×C4C22×D37311yescharacteristic
196C2×C4C22×D37311yescharacteristic
197C2×C4C22×D37311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—3732no—
205C22—14924no—
206C22—7461no—
207C22—7461no—
208C22—7461no—
209C22—7461no—
210C22—7461no—
211C22—7461no—
212C22—7461no—
213C22—7461no—
214C22C23×D37311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—7461no—
221C4—7461no—
222C4—7461no—
223C4—7461no—
224C4—7461no—
225C2—41no—
226C2—41no—
227C2—14921no—
228C2—14921no—
229C2—7462no—
230C2#41×D37311yescharacteristic
231C2#2604583711yescharacteristic
232C2#2604583811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#124×C373C2
hypercenter#124D373
hypocenterC373#538
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26045786C2semidirect
2. #26045785C2semidirect
3. #26045781C2semidirect
4. #26045776C2semidirect
5. #26045774C2semidirect
6. #26045768C2semidirect
7. #26045765C2semidirect
8. #26045765C2semidirect
9. #26045764C2semidirect
10. #26045762C2semidirect
11. #26045760C2semidirect
12. C4×D1492C2semidirect
13. #26045752C2semidirect
14. C22⋊C4×D373C2semidirect
15. #124×C373C2semidirect
22. C2×C22⋊D373C22semidirect
23. C22×Dic373C22semidirect
26. C2×C4×D373C22semidirect
27. C4⋊Dic373C22semidirect
28. C4⋊Dic373C22semidirect
29. #20238589C22semidirect
30. #20238589C22semidirect
31. #20238589C22semidirect
32. #20238589C22semidirect
41. #20238587C22semidirect
42. #20238587C22semidirect
43. C4×Dic373C22semidirect
44. C2×Q8×C373C22semidirect
48. C22⋊C4×C373C22semidirect
49. C22⋊C4×C373C22semidirect
50. C42×C373C22semidirect
116. C373#538semidirect
121. #124D373semidirect
134. C2×Q8C2×D373semidirect
164. C22⋊C4C2×D373semidirect
165. C22⋊C4C2×D373semidirect
167. C42C2×D373semidirect

Maximal subgroups

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

subgroupindex
#260457862
#260457852
#260457812
#260457762
#260457742
#260457682
#260457652
#260457642
#260457622
#260457602
C4×D14922
#260457522
C22⋊C4×D3732
#124×C3732
#538373