Finite Group Databasebeta

← #72259384

Subgroups — #72259384

Order 8112.

312 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 55 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
1D13×C22⋊D13C311yesmaximal, characteristic, complemented, Hall
2#72229261C211yesmaximal, characteristic, complemented
3#72229237C211yesmaximal, characteristic, complemented
4C22×#11759817C211yesmaximal, characteristic, complemented
5#72229013C211yesmaximal, characteristic, complemented
6#72214094C211yesmaximal, characteristic, complemented
7#72214085C211yesmaximal, characteristic, complemented
8C2×#11759815⋊C2C211yesmaximal, characteristic, complemented
9C22×#92788C2211yescharacteristic, complemented
10C2×#11759817—22no—
11C2×#11759817C2211yescharacteristic
12C2×#11759817C2211yescharacteristic, complemented
13C2×#11759817C2211yescharacteristic, complemented
14C2×#11759815C2211yescharacteristic, complemented
15#11759815⋊C2—22no—
16#14523130C2211yescharacteristic, complemented
17#14523128C2211yescharacteristic, complemented
18C22×C13×D13C2×C311yescharacteristic, complemented
19C2×D132C2×C311yescharacteristic, complemented
20C22⋊(C13⋊D13)C2×C311yescharacteristic, complemented
21C13×C22⋊D13C2×C311yescharacteristic, complemented
22#13062579C2×C311yescharacteristic, complemented
23#13062578C2×C311yescharacteristic, complemented
24D13×Dic13C2×C311yescharacteristic, complemented
25C2×#92788—21no—
26C2×#92788C2311yescharacteristic
27#11759817—21no—
28#11759817—21no—
29#11759817D412yescomplemented
30#11759815—21no—
31C22×C132C22×C311yescharacteristic, complemented
32C2×C13⋊D13C22×C311yescharacteristic, complemented
33C2×C13×D13—22no—
34C2×C13×D13C22×C311yescharacteristic
35C2×C13×D13C22×C311yescharacteristic, complemented
36C2×C13×D13C22×C311yescharacteristic, complemented
37D132—22no—
38#10621725C22×C311yescharacteristic, complemented
39C13×Dic13C22×C311yescharacteristic, complemented
40C2×(C22⋊D13)⋊C3—131nomaximal
41D4×D13⋊C3—131nomaximal
42#92788C2×D411yescharacteristic, complemented, Hall
43C2×C132—21no—
44C2×C132C23×C311yescharacteristic
45C13⋊D13—21no—
46C13×D13—21no—
47C13×D13—21no—
48C13×D13C3×D412yescomplemented
49C23×C13⋊C3—131no—
50C22×D13⋊C3—131no—
51C22×D13⋊C3—131no—
52C22×D13⋊C3—261no—
53D4×C13⋊C3—131no—
54(C22⋊D13)⋊C3—131no—
55(C22⋊D13)⋊C3—131no—
56(C22⋊D13)⋊C3—131no—
57(C22⋊D13)⋊C3—131no—
58(C22⋊D13)⋊C3—131no—
59(C22⋊D13)⋊C3—262no—
60(C22⋊D13)⋊C3—264no—
61(C22⋊D13)⋊C3—261no—
62C2×Dic13⋊C3—131no—
63C4⋊(D13⋊C3)—131no—
64C4×D13⋊C3—131no—
65C2×C22⋊D13—131no—
66D4×D13—131no—
67C132C2×C3×D411yescharacteristic, complemented, Sylow
68C22×C13⋊C3—131no—
69C22×C13⋊C3—131no—
70C22×C13⋊C3—131no—
71C22×C13⋊C3—262no—
72C22×C13⋊C3—262no—
73C22×C13⋊C3—261no—
74C22×C13⋊C3—261no—
75C22×C13⋊C3—524no—
76C2×D13⋊C3—131no—
77C2×D13⋊C3—131no—
78C2×D13⋊C3—262no—
79C2×D13⋊C3—262no—
80C2×D13⋊C3—262no—
81C2×D13⋊C3—262no—
82C2×D13⋊C3—264no—
83C2×D13⋊C3—261no—
84C2×D13⋊C3—261no—
85C2×D13⋊C3—261no—
86C4×C13⋊C3—131no—
87Dic13⋊C3—131no—
88Dic13⋊C3—131no—
89Dic13⋊C3—262no—
90Dic13⋊C3—264no—
91Dic13⋊C3—261no—
92C23×C13—131no—
93C22×D13—131no—
94C22×D13—131no—
95C22×D13D13⋊C311yescharacteristic, complemented
96D4×C13—131no—
97C22⋊D13—131no—
98C22⋊D13—131no—
99C22⋊D13—131no—
100C22⋊D13—131no—
101C22⋊D13—131no—
102C22⋊D13—262no—
103C22⋊D13—264no—
104C22⋊D13D13⋊C311yescharacteristic, complemented
105C2×Dic13—131no—
106D52—131no—
107C4×D13—131no—
108C2×C13⋊C3—132no—
109C2×C13⋊C3—261no—
110C2×C13⋊C3—262no—
111C2×C13⋊C3—261no—
112C2×C13⋊C3—261no—
113C2×C13⋊C3—261no—
114C2×C13⋊C3—261no—
115C2×C13⋊C3—521no—
116C2×C13⋊C3—524no—
117C2×C13⋊C3—522no—
118C2×C13⋊C3—524no—
119D13⋊C3—261no—
120D13⋊C3—261no—
121D13⋊C3—261no—
122D13⋊C3—262no—
123D13⋊C3—522no—
124D13⋊C3—524no—
125D13⋊C3—521no—
126C22×C13—22no—
127C22×C13—24no—
128C22×C13—131no—
129C22×C13—131no—
130C22×C13—131no—
131C22×C13—262no—
132C22×C13C2×D13⋊C311yescharacteristic, complemented
133C22×C13C2×D13⋊C311yescharacteristic, complemented
134C2×D13—22no—
135C2×D13—131no—
136C2×D13—131no—
137C2×D13—262no—
138C2×D13—262no—
139C2×D13—262no—
140C2×D13—264no—
141C2×D13C2×D13⋊C311yescharacteristic
142C2×D13C2×D13⋊C311yescharacteristic, complemented
143C2×D13C2×D13⋊C311yescharacteristic, complemented
144C4×C13—131no—
145Dic13—131no—
146Dic13—131no—
147Dic13—262no—
148Dic13—264no—
149Dic13C2×D13⋊C311yescharacteristic, complemented
150C2×C3×D4—1691noHall
151C13⋊C3—262no—
152C13⋊C3—261no—
153C13⋊C3—261no—
154C13⋊C3—524no—
155C2×C13—21no—
156C2×C13—21no—
157C2×C13—22no—
158C2×C13—24no—
159C2×C13—42no—
160C2×C13—44no—
161C2×C13—132no—
162C2×C13—261no—
163C2×C13—261no—
164C2×C13C22×D13⋊C311yescharacteristic
165C2×C13C22×D13⋊C311yescharacteristic
166D13—21no—
167D13—21no—
168D13—261no—
169D13—261no—
170D13—522no—
171D13—524no—
172D13(C22⋊D13)⋊C312yescomplemented
173C23×C3—3381no—
174C23×C3—1691no—
175C3×D4—3381no—
176C3×D4—1691no—
177C3×D4—1691no—
178C3×D4—1691no—
179C2×C3×C4—1691no—
180C2×D4—1691noSylow
181C13—22no—
182C13—24no—
183C13D4×D13⋊C311yescharacteristic, complemented
184C13C2×(C22⋊D13)⋊C311yescharacteristic, complemented
185C22×C3—3381no—
186C22×C3—3381no—
187C22×C3—3381no—
188C22×C3—3382no—
189C22×C3—3382no—
190C22×C3—6761no—
191C22×C3—1691no—
192C3×C4—3381no—
193C3×C4—1691no—
194C23—131no—
195C23—1691no—
196D4—131no—
197D4—1691no—
198D4—1691no—
199D4—1691no—
200C2×C4—1691no—
201C2×C3—3382no—
202C2×C3—3381no—
203C2×C3—3381no—
204C2×C3—6761no—
205C2×C3—6761no—
206C2×C3—6761no—
207C22—131no—
208C22—131no—
209C22—131no—
210C22—262no—
211C22—3382no—
212C22—1691no—
213C22#11759815⋊C211yescharacteristic, complemented
214C4—131no—
215C4—1691no—
216C3—6761noSylow
217C2—21no—
218C2—132no—
219C2—261no—
220C2—261no—
221C2—3381no—
222C2C2×#11759815⋊C211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#11759815⋊C2
commutator, socle, cosocleC2×C132C23×C3
FittingC22×C132C22×C3
hypercenterC22#11759815⋊C2
hypocenterC132C2×C3×D4
LCSG ▷ C2×C132 ▷ C132
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. D13×C22⋊D13C3semidirect
2. #72229261C2semidirect
3. #72229237C2semidirect
4. C22×#11759817C2semidirect
5. #72229013C2semidirect
6. #72214094C2semidirect
7. #72214085C2semidirect
8. C2×#11759815⋊C2C2semidirect
9. C22×#92788C22semidirect
12. C2×#11759817C22semidirect
13. C2×#11759817C22semidirect
14. C2×#11759815C22semidirect
16. #14523130C22semidirect
17. #14523128C22semidirect
18. C22×C13×D13C2×C3semidirect
19. C2×D132C2×C3semidirect
20. C22⋊(C13⋊D13)C2×C3semidirect
21. C13×C22⋊D13C2×C3semidirect
22. #13062579C2×C3semidirect
23. #13062578C2×C3semidirect
24. D13×Dic13C2×C3semidirect
29. #11759817D4semidirect
31. C22×C132C22×C3semidirect
32. C2×C13⋊D13C22×C3semidirect
35. C2×C13×D13C22×C3semidirect
36. C2×C13×D13C22×C3semidirect
38. #10621725C22×C3semidirect
39. C13×Dic13C22×C3semidirect
42. #92788C2×D4semidirect
48. C13×D13C3×D4semidirect
67. C132C2×C3×D4semidirect
95. C22×D13D13⋊C3semidirect
104. C22⋊D13D13⋊C3semidirect
132. C22×C13C2×D13⋊C3semidirect
133. C22×C13C2×D13⋊C3semidirect
142. C2×D13C2×D13⋊C3semidirect
143. C2×D13C2×D13⋊C3semidirect
149. Dic13C2×D13⋊C3semidirect
172. D13(C22⋊D13)⋊C3semidirect
183. C13D4×D13⋊C3semidirect
184. C13C2×(C22⋊D13)⋊C3semidirect
213. C22#11759815⋊C2semidirect

Maximal subgroups

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

subgroupindex
D13×C22⋊D133
#722292612
#722292372
C22×#117598172
#722290132
#722140942
#722140852
C2×#11759815⋊C22
C2×(C22⋊D13)⋊C313
D4×D13⋊C313