Finite Group Databasebeta

← #38763107

Subgroups — #38763107

Order 25408.

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#27163089C211yesmaximal, characteristic, complemented
2#27163089C211yesmaximal, characteristic, complemented
3#27163083C211yesmaximal, characteristic, complemented
4#27163082C211yesmaximal, characteristic, complemented
5#27163078C211yesmaximal, characteristic, complemented
6#27163077C211yesmaximal, characteristic, complemented
7#27163073C211yesmaximal, characteristic, complemented
8#27163068C211yesmaximal, characteristic, complemented
9#27163068C211yesmaximal, characteristic, complemented
10#27163067C211yesmaximal, characteristic, complemented
11#27163065C211yesmaximal, characteristic, complemented
12D4×Dic397C211yesmaximal, characteristic, complemented
13#27163056C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic397C211yesmaximal, characteristic, complemented
15#120×C397C211yesmaximal, characteristic, complemented
16C2×Dic794C2211yescharacteristic, complemented
17C2×C22⋊D397C2211yescharacteristic
18C2×C22⋊D397C2211yescharacteristic
19C2×C22⋊D397C2211yescharacteristic
20C2×C22⋊D397C2211yescharacteristic, complemented
21C22×Dic397C2211yescharacteristic
22C22×Dic397C2211yescharacteristic
23C22×Dic397C2211yescharacteristic, complemented
24C2×C4×D397C2211yescharacteristic, complemented
25C4⋊Dic397C2211yescharacteristic, complemented
26C4⋊Dic397C2211yescharacteristic, complemented
27#20263710C2211yescharacteristic, complemented
28#20263710C2211yescharacteristic, complemented
29#20263710C2211yescharacteristic, complemented
30#20263710C2211yescharacteristic, complemented
31C22⋊Dic397C2211yescharacteristic
32C22⋊Dic397C2211yescharacteristic
33C22⋊Dic397C2211yescharacteristic
34C22⋊Dic397C2211yescharacteristic
35C22⋊Dic397C2211yescharacteristic
36C22⋊Dic397C2211yescharacteristic
37C22⋊Dic397C2211yescharacteristic
38C22⋊Dic397C2211yescharacteristic, complemented
39C22⋊Dic397C2211yescharacteristic, complemented
40#20263708C2211yescharacteristic, complemented
41#20263708C2211yescharacteristic, complemented
42C4×Dic397C2211yescharacteristic, complemented
43C4×Dic397C2211yescharacteristic, complemented
44C24×C397C2211yescharacteristic
45C2×D4×C397C2211yescharacteristic
46C2×D4×C397C2211yescharacteristic
47C2×D4×C397C2211yescharacteristic
48C22⋊C4×C397C2211yescharacteristic
49C22⋊C4×C397C2211yescharacteristic
50C22⋊C4×C397C2211yescharacteristic
51C22×D397C2311yescharacteristic
52Dic794—41no—
53C22⋊D397—22no—
54C22⋊D397—41no—
55C22⋊D397—41no—
56C22⋊D397—41no—
57C2×Dic397—41no—
58C2×Dic397—41no—
59C2×Dic397—41no—
60C2×Dic397C2311yescharacteristic
61C2×Dic397C2311yescharacteristic
62C2×Dic397C2311yescharacteristic
63C2×Dic397C2311yescharacteristic
64C2×Dic397C2311yescharacteristic
65C2×Dic397C2311yescharacteristic
66C2×Dic397C2311yescharacteristic
67C4×D397—41no—
68C23×C397—22no—
69C23×C397—41no—
70C23×C397—41no—
71C23×C397C2311yescharacteristic
72C23×C397C2311yescharacteristic
73C23×C397C2311yescharacteristic
74C23×C397C2311yescharacteristic
75D4×C397—22no—
76D4×C397—41no—
77D4×C397—41no—
78C2×C4×C397C2311yescharacteristic
79C2×C4×C397C2311yescharacteristic
80C2×C4×C397C2311yescharacteristic
81C22×C397—21no—
82C22×C397—21no—
83C22×C397—21no—
84C22×C397—21no—
85C22×C397—21no—
86C22×C397—21no—
87C22×C397—21no—
88C22×C397—21no—
89C22×C397—21no—
90C22×C397—21no—
91C22×C397—21no—
92C22×C397—44no—
93C22×C397#4112yes—
94C22×C397C2411yescharacteristic
95C2×D397—21no—
96C2×D397—21no—
97C2×D397—21no—
98C4×C397—21no—
99C4×C397—21no—
100C4×C397—21no—
101Dic397—21no—
102Dic397—21no—
103Dic397—21no—
104Dic397—21no—
105Dic397—21no—
106Dic397—21no—
107Dic397—21no—
108C2×C397—22no—
109C2×C397—41no—
110C2×C397—41no—
111C2×C397—41no—
112C2×C397C2×#4111yescharacteristic
113C2×C397H(5,𝔽2)11yescharacteristic
114C2×C397H(5,𝔽2)11yescharacteristic
115D397—41no—
116C397#53811yescharacteristic, complemented, Sylow
117#538—3971nomaximal, Sylow
118#126—3971no—
119#126—3971no—
120#124—3971no—
121#124—3971no—
122#123—3971no—
123#123—3971no—
124C22⋊Q8—3971no—
125#121—3971no—
126#121—3971no—
127#121—3971no—
128#120—3971no—
129#120D39711yescharacteristic, complemented
130C4×D4—3971no—
131C4×D4—3971no—
132C2×C22⋊C4—3971no—
133C24C2×D39711yescharacteristic
134C2×Q8—3971no—
135C2×D4—3971no—
136C2×D4—3971no—
137C2×D4—3971no—
138C2×D4—3971no—
139C2×D4C2×D39711yescharacteristic
140C2×D4C2×D39711yescharacteristic
141C2×D4C2×D39711yescharacteristic
142C22×C4—3971no—
143C22×C4—3971no—
144C22×C4—3971no—
145C22×C4—3971no—
146C4⋊C4—3971no—
147C4⋊C4—3971no—
148C4⋊C4—3971no—
149C4⋊C4—3971no—
150C4⋊C4—3971no—
151C4⋊C4—3971no—
152C22⋊C4—3971no—
153C22⋊C4—3971no—
154C22⋊C4—3971no—
155C22⋊C4—3971no—
156C22⋊C4—3971no—
157C22⋊C4—3971no—
158C22⋊C4—3971no—
159C22⋊C4—3971no—
160C22⋊C4—3971no—
161C22⋊C4—3971no—
162C22⋊C4—3971no—
163C22⋊C4C2×D39711yescharacteristic
164C22⋊C4C2×D39711yescharacteristic
165C22⋊C4C2×D39711yescharacteristic
166C42—3971no—
167C42—3971no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—3971no—
172C23C22×D39711yescharacteristic
173C23C22×D39711yescharacteristic
174C23C22×D39711yescharacteristic
175C23C22×D39711yescharacteristic
176Q8—15881no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—7942no—
181D4—15881no—
182D4—15881no—
183D4—15881no—
184C2×C4—15881no—
185C2×C4—15881no—
186C2×C4—15881no—
187C2×C4—15881no—
188C2×C4—3971no—
189C2×C4—3971no—
190C2×C4—3971no—
191C2×C4—3971no—
192C2×C4—3971no—
193C2×C4—3971no—
194C2×C4—3971no—
195C2×C4C22×D39711yescharacteristic
196C2×C4C22×D39711yescharacteristic
197C2×C4C22×D39711yescharacteristic
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—7941no—
211C22—7941no—
212C22—7941no—
213C22#2026373012yes—
214C22C23×D39711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—7941no—
219C4—7941no—
220C4—7941no—
221C4—7941no—
222C4—7941no—
223C4—7941no—
224C4—7941no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—15881no—
230C2C2×#2026373011yescharacteristic
231C2#2716314111yescharacteristic
232C2#2716314111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
Fitting#120×C397C2
hypercenter#120D397
hypocenterC397#538
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #27163089C2semidirect
2. #27163089C2semidirect
3. #27163083C2semidirect
4. #27163082C2semidirect
5. #27163078C2semidirect
6. #27163077C2semidirect
7. #27163073C2semidirect
8. #27163068C2semidirect
9. #27163068C2semidirect
10. #27163067C2semidirect
11. #27163065C2semidirect
12. D4×Dic397C2semidirect
13. #27163056C2semidirect
14. C2×C22⋊Dic397C2semidirect
15. #120×C397C2semidirect
16. C2×Dic794C22semidirect
20. C2×C22⋊D397C22semidirect
23. C22×Dic397C22semidirect
24. C2×C4×D397C22semidirect
25. C4⋊Dic397C22semidirect
26. C4⋊Dic397C22semidirect
27. #20263710C22semidirect
28. #20263710C22semidirect
29. #20263710C22semidirect
30. #20263710C22semidirect
38. C22⋊Dic397C22semidirect
39. C22⋊Dic397C22semidirect
40. #20263708C22semidirect
41. #20263708C22semidirect
42. C4×Dic397C22semidirect
43. C4×Dic397C22semidirect
116. C397#538semidirect
129. #120D397semidirect

Maximal subgroups

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

subgroupindex
#271630892
#271630832
#271630822
#271630782
#271630772
#271630732
#271630682
#271630672
#271630652
D4×Dic3972
#271630562
C2×C22⋊Dic3972
#120×C3972
#538397