Finite Group Databasebeta

← #37560007

Subgroups — #37560007

Order 23872.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#26045785C211yesmaximal, characteristic, complemented
2#26045785C211yesmaximal, characteristic, complemented
3#26045779C211yesmaximal, characteristic, complemented
4#26045778C211yesmaximal, characteristic, complemented
5#26045774C211yesmaximal, characteristic, complemented
6#26045773C211yesmaximal, characteristic, complemented
7#26045769C211yesmaximal, characteristic, complemented
8#26045764C211yesmaximal, characteristic, complemented
9#26045764C211yesmaximal, characteristic, complemented
10#26045763C211yesmaximal, characteristic, complemented
11#26045761C211yesmaximal, characteristic, complemented
12D4×Dic373C211yesmaximal, characteristic, complemented
13#26045752C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic373C211yesmaximal, characteristic, complemented
15#120×C373C211yesmaximal, characteristic, complemented
16C2×Dic746C2211yescharacteristic, complemented
17C2×C22⋊D373C2211yescharacteristic
18C2×C22⋊D373C2211yescharacteristic
19C2×C22⋊D373C2211yescharacteristic
20C2×C22⋊D373C2211yescharacteristic, complemented
21C22×Dic373C2211yescharacteristic
22C22×Dic373C2211yescharacteristic
23C22×Dic373C2211yescharacteristic, complemented
24C2×C4×D373C2211yescharacteristic, complemented
25C4⋊Dic373C2211yescharacteristic, complemented
26C4⋊Dic373C2211yescharacteristic, complemented
27#20238589C2211yescharacteristic, complemented
28#20238589C2211yescharacteristic, complemented
29#20238589C2211yescharacteristic, complemented
30#20238589C2211yescharacteristic, complemented
31C22⋊Dic373C2211yescharacteristic
32C22⋊Dic373C2211yescharacteristic
33C22⋊Dic373C2211yescharacteristic
34C22⋊Dic373C2211yescharacteristic
35C22⋊Dic373C2211yescharacteristic
36C22⋊Dic373C2211yescharacteristic
37C22⋊Dic373C2211yescharacteristic
38C22⋊Dic373C2211yescharacteristic, complemented
39C22⋊Dic373C2211yescharacteristic, complemented
40#20238587C2211yescharacteristic, complemented
41#20238587C2211yescharacteristic, complemented
42C4×Dic373C2211yescharacteristic, complemented
43C4×Dic373C2211yescharacteristic, complemented
44C24×C373C2211yescharacteristic
45C2×D4×C373C2211yescharacteristic
46C2×D4×C373C2211yescharacteristic
47C2×D4×C373C2211yescharacteristic
48C22⋊C4×C373C2211yescharacteristic
49C22⋊C4×C373C2211yescharacteristic
50C22⋊C4×C373C2211yescharacteristic
51C22×D373C2311yescharacteristic
52Dic746—41no—
53C22⋊D373—22no—
54C22⋊D373—41no—
55C22⋊D373—41no—
56C22⋊D373—41no—
57C2×Dic373—41no—
58C2×Dic373—41no—
59C2×Dic373—41no—
60C2×Dic373C2311yescharacteristic
61C2×Dic373C2311yescharacteristic
62C2×Dic373C2311yescharacteristic
63C2×Dic373C2311yescharacteristic
64C2×Dic373C2311yescharacteristic
65C2×Dic373C2311yescharacteristic
66C2×Dic373C2311yescharacteristic
67C4×D373—41no—
68C23×C373—22no—
69C23×C373—41no—
70C23×C373—41no—
71C23×C373C2311yescharacteristic
72C23×C373C2311yescharacteristic
73C23×C373C2311yescharacteristic
74C23×C373C2311yescharacteristic
75D4×C373—22no—
76D4×C373—41no—
77D4×C373—41no—
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×C373—21no—
88C22×C373—21no—
89C22×C373—21no—
90C22×C373—21no—
91C22×C373—21no—
92C22×C373—44no—
93C22×C373#4112yes—
94C22×C373C2411yescharacteristic
95C2×D373—21no—
96C2×D373—21no—
97C2×D373—21no—
98C4×C373—21no—
99C4×C373—21no—
100C4×C373—21no—
101Dic373—21no—
102Dic373—21no—
103Dic373—21no—
104Dic373—21no—
105Dic373—21no—
106Dic373—21no—
107Dic373—21no—
108C2×C373—22no—
109C2×C373—41no—
110C2×C373—41no—
111C2×C373—41no—
112C2×C373C2×#4111yescharacteristic
113C2×C373H(5,𝔽2)11yescharacteristic
114C2×C373H(5,𝔽2)11yescharacteristic
115D373—41no—
116C373#53811yescharacteristic, complemented, Sylow
117#538—3731nomaximal, Sylow
118#126—3731no—
119#126—3731no—
120#124—3731no—
121#124—3731no—
122#123—3731no—
123#123—3731no—
124C22⋊Q8—3731no—
125#121—3731no—
126#121—3731no—
127#121—3731no—
128#120—3731no—
129#120D37311yescharacteristic, complemented
130C4×D4—3731no—
131C4×D4—3731no—
132C2×C22⋊C4—3731no—
133C24C2×D37311yescharacteristic
134C2×Q8—3731no—
135C2×D4—3731no—
136C2×D4—3731no—
137C2×D4—3731no—
138C2×D4—3731no—
139C2×D4C2×D37311yescharacteristic
140C2×D4C2×D37311yescharacteristic
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⋊C4—3731no—
163C22⋊C4C2×D37311yescharacteristic
164C22⋊C4C2×D37311yescharacteristic
165C22⋊C4C2×D37311yescharacteristic
166C42—3731no—
167C42—3731no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—3731no—
172C23C22×D37311yescharacteristic
173C23C22×D37311yescharacteristic
174C23C22×D37311yescharacteristic
175C23C22×D37311yescharacteristic
176Q8—14921no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—14921no—
181D4—14921no—
182D4—14921no—
183D4—7462no—
184C2×C4—3731no—
185C2×C4—3731no—
186C2×C4—3731no—
187C2×C4—3731no—
188C2×C4—3731no—
189C2×C4—3731no—
190C2×C4—3731no—
191C2×C4—14921no—
192C2×C4—14921no—
193C2×C4—14921no—
194C2×C4—14921no—
195C2×C4C22×D37311yescharacteristic
196C2×C4C22×D37311yescharacteristic
197C2×C4C22×D37311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—7461no—
211C22—7461no—
212C22—7461no—
213C22#2023860912yes—
214C22C23×D37311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—7461no—
219C4—7461no—
220C4—7461no—
221C4—7461no—
222C4—7461no—
223C4—7461no—
224C4—7461no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—14921no—
230C2C2×#2023860911yescharacteristic
231C2#2604583711yescharacteristic
232C2#2604583711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#120×C373C2
hypercenter#120D373
hypocenterC373#538
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #26045785C2semidirect
2. #26045785C2semidirect
3. #26045779C2semidirect
4. #26045778C2semidirect
5. #26045774C2semidirect
6. #26045773C2semidirect
7. #26045769C2semidirect
8. #26045764C2semidirect
9. #26045764C2semidirect
10. #26045763C2semidirect
11. #26045761C2semidirect
12. D4×Dic373C2semidirect
13. #26045752C2semidirect
14. C2×C22⋊Dic373C2semidirect
15. #120×C373C2semidirect
16. C2×Dic746C22semidirect
20. C2×C22⋊D373C22semidirect
23. C22×Dic373C22semidirect
24. C2×C4×D373C22semidirect
25. C4⋊Dic373C22semidirect
26. C4⋊Dic373C22semidirect
27. #20238589C22semidirect
28. #20238589C22semidirect
29. #20238589C22semidirect
30. #20238589C22semidirect
38. C22⋊Dic373C22semidirect
39. C22⋊Dic373C22semidirect
40. #20238587C22semidirect
41. #20238587C22semidirect
42. C4×Dic373C22semidirect
43. C4×Dic373C22semidirect
116. C373#538semidirect
129. #120D373semidirect

Maximal subgroups

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

subgroupindex
#260457852
#260457792
#260457782
#260457742
#260457732
#260457692
#260457642
#260457632
#260457612
D4×Dic3732
#260457522
C2×C22⋊Dic3732
#120×C3732
#538373