Finite Group Databasebeta

← #28399850

Subgroups — #28399850

Order 14464.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16813668C211yesmaximal, characteristic
2C22×#14268634C211yesmaximal, characteristic, complemented
3#20299172C211yesmaximal, characteristic, complemented
4#20298759C214yesmaximal, complemented
5#20298747C214yesmaximal, complemented
6#20298604C211yesmaximal, characteristic, complemented
7#20298594C211yesmaximal, characteristic, complemented
8#20298566C211yesmaximal, characteristic, complemented
9C2×#137×C113C211yesmaximal, characteristic, complemented
10#16813668—24no—
11#16813668—24no—
12C2×#14268648C2212yes—
13C2×#14268647—21no—
14C2×#14268647C2211yescharacteristic
15C22×D452—21no—
16C22×D452C2211yescharacteristic
17C2×D4×D113—22no—
18C2×D4×D113C2212yes—
19C2×#14268634—24no—
20C2×#14268634C2214yescomplemented
21C22×#11744751C2211yescharacteristic, complemented
22#16813620—21no—
23#16813611—21no—
24#16813605—22no—
25#16813598C2214yescomplemented
26#16813581C2211yescharacteristic, complemented
27#16813546C2212yescomplemented
28#16813544C2214yescomplemented
29#16813543C2212yescomplemented
30#16813534C2214yescomplemented
31C2×#41×C113C2211yescharacteristic
32C22×Q8×C113C2211yescharacteristic, complemented
33#137×C113—24no—
34C2×Q16×C113C2212yescomplemented
35C2×SD16×C113C2212yescomplemented
36C2×M16×C113C2211yescharacteristic, complemented
37C23×D113—21no—
38C23×D113—22no—
39#14268648—24no—
40#14268648—24no—
41#14268647—28no—
42#14268647—22no—
43#14268647—22no—
44C2×Dic226D411yescharacteristic
45C2×D452—28no—
46C2×D452—22no—
47C2×D452—21no—
48C2×D452—22no—
49C2×D452D411yescharacteristic
50C2×D452D414yes—
51C2×D452C2312yes—
52C2×C22⋊D113—22no—
53C2×C22⋊D113D412yes—
54D4×D113—216no—
55D4×D113—24no—
56D4×D113—24no—
57C2×C4×D113—22no—
58C2×C4×D113C2312yes—
59#14268634—28no—
60C2×#11744751—22no—
61C2×#11744751C2312yes—
62C4⋊Dic113C2312yes—
63#14268626—24no—
64C22⋊Dic113—21no—
65#14268624—24no—
66C4×Dic113—21no—
67#41×C113—22no—
68#41×C113D414yes—
69C2×Q8×C113—21no—
70C2×Q8×C113—22no—
71C2×Q8×C113—22no—
72C2×Q8×C113C2312yes—
73C2×Q8×C113C2311yescharacteristic
74C2×D4×C113—21no—
75C2×D4×C113C2311yescharacteristic
76C22×C4×C113—21no—
77C22×C4×C113—21no—
78C22×C4×C113C2311yescharacteristic
79Q16×C113—24no—
80SD16×C113—24no—
81M16×C113—22no—
82C2×C8×C113C2312yes—
83C23×C113—21no—
84C23×C113C2×D411yescharacteristic
85C22×D113—22no—
86C22×D113—28no—
87C22×D113—24no—
88C22×D113—84no—
89C22×D113—82no—
90C22×D113C2×D412yes—
91Q8×C113—21no—
92Q8×C113—22no—
93Q8×C113—22no—
94Q8×C113—42no—
95Q8×C113C2×D412yes—
96D4×C113—24no—
97D4×C113—21no—
98D4×C113C2×D412yes—
99C2×C4×C113—21no—
100C2×C4×C113—24no—
101C2×C4×C113—22no—
102C2×C4×C113—42no—
103C2×C4×C113C2×D411yescharacteristic
104C2×C4×C113C2×D412yes—
105C2×C4×C113C2×D412yes—
106C2×C4×C113C2411yescharacteristic
107C22⋊D113—28no—
108C22⋊D113—24no—
109C2×Dic113—22no—
110C2×Dic113C2×D412yes—
111D452—28no—
112D452—22no—
113D452—24no—
114D452—22no—
115D452C2×D414yes—
116C4×D113—28no—
117C4×D113—22no—
118C4×D113—22no—
119Dic226—22no—
120C8×C113—22no—
121#11744751—22no—
122C22×C113—22no—
123C22×C113—22no—
124C22×C113—42no—
125C22×C113#12012yes—
126C22×C113C22×D411yescharacteristic
127C2×D113—22no—
128C2×D113—24no—
129C2×D113—24no—
130C2×D113—44no—
131C2×D113—84no—
132C2×D113—88no—
133C4×C113—22no—
134C4×C113—42no—
135C4×C113#12012yes—
136C4×C113C22×D411yescharacteristic
137C4×C113C22×D411yescharacteristic
138Dic113—22no—
139Dic113—42no—
140C2×C113—22no—
141C2×C113—42no—
142C2×C113C2×#12011yescharacteristic
143C2×C113#57712yes—
144D113—42no—
145D113—44no—
146#3009—1131nomaximal, Sylow
147C113#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—1131no—
149C2×#137D11311yescharacteristic, complemented
150C22×SD16—1131no—
151#536—1131no—
152#452—1134no—
153#450—1134no—
154#418—1131no—
155#416—1131no—
156#408—1131no—
157H(5,𝔽2)—2264no—
158H(5,𝔽2)—2264no—
159C2×#41—1131no—
160C2×#41—1132no—
161C2×#41—2261no—
162C2×#41C2×D11311yescharacteristic
163C22×Q8C2×D11311yescharacteristic, complemented
164C22×D4—1132no—
165C22×D4—1131no—
166C22×D4—2261no—
167C22×D4—2262no—
168#137—24no—
169C2×Q16C2×D11312yescomplemented
170C2×SD16—1134no—
171C2×SD16—2264no—
172C2×SD16C2×D11312yescomplemented
173C2×M16C2×D11311yescharacteristic, complemented
174C22×C8—1131no—
175C4⋊Q8—2261no—
176#124—2261no—
177#123—2262no—
178C22⋊Q8—1134no—
179#117—1131no—
180#103—1132no—
181#102—1132no—
182#102—1134no—
183C22⋊C8—1134no—
184C24—2262no—
185C24—2261no—
186#41—22no—
187#41—2268no—
188#41—2264no—
189#41—2264no—
190#41—2262no—
191#41—2262no—
192#41C22⋊D11314yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1131no—
197C2×Q8C22×D11312yes—
198C2×Q8C22×D11311yescharacteristic
199C2×D4—21no—
200C2×D4—1132no—
201C2×D4—1134no—
202C2×D4—1131no—
203C2×D4—1132no—
204C2×D4—2262no—
205C2×D4—2268no—
206C2×D4—2262no—
207C2×D4—2264no—
208C2×D4—22616no—
209C2×D4—2264no—
210C2×D4—2261no—
211C2×D4—2262no—
212C2×D4C22×D11311yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—1132no—
216C22×C4—2262no—
217C22×C4C22×D11311yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—2268no—
221M16—22no—
222C2×C8—1132no—
223C2×C8—2262no—
224C2×C8C22×D11312yes—
225C4⋊C4—1132no—
226C4⋊C4—2264no—
227C22⋊C4—2264no—
228C22⋊C4—2261no—
229C42—2261no—
230C23—21no—
231C23—1132no—
232C23—9042no—
233C23—9044no—
234C23—2268no—
235C23—2264no—
236C23—2262no—
237C23D4×D11311yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2262no—
243Q8C2×C22⋊D11312yes—
244D4—24no—
245D4—21no—
246D4—1134no—
247D4—2268no—
248D4—2268no—
249D4—2264no—
250D4—2264no—
251D4—2262no—
252D4—2262no—
253D4C2×C22⋊D11312yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—1132no—
259C2×C4—2268no—
260C2×C4—2262no—
261C2×C4—2262no—
262C2×C4—2262no—
263C2×C4D4×D11311yescharacteristic
264C2×C4D4×D11312yes—
265C2×C4C2×C22⋊D11312yes—
266C2×C4C23×D11311yescharacteristic
267C8—22no—
268C8—2262no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—9044no—
273C22—9048no—
274C22—4524no—
275C22—2264no—
276C22—2262no—
277C22—2264no—
278C22#1681359012yes—
279C22C2×D4×D11311yescharacteristic
280C4—22no—
281C4—42no—
282C4—4522no—
283C4—2262no—
284C4#1681359012yes—
285C4C2×D4×D11311yescharacteristic
286C4C22×C22⋊D11311yescharacteristic
287C2—22no—
288C2—42no—
289C2—4524no—
290C2—4522no—
291C2C2×#1681359011yescharacteristic
292C2#2029940712yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×#137×C113C2
socleC22×C113C22×D4
prime radicalC2×#16813668C2
hypercenterC2×#137D113
hypocenterC113#3009
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14268634C2semidirect
3. #20299172C2semidirect
4. #20298759C2semidirect
5. #20298747C2semidirect
6. #20298604C2semidirect
7. #20298594C2semidirect
8. #20298566C2semidirect
9. C2×#137×C113C2semidirect
20. C2×#14268634C22semidirect
21. C22×#11744751C22semidirect
25. #16813598C22semidirect
26. #16813581C22semidirect
27. #16813546C22semidirect
28. #16813544C22semidirect
29. #16813543C22semidirect
30. #16813534C22semidirect
32. C22×Q8×C113C22semidirect
34. C2×Q16×C113C22semidirect
35. C2×SD16×C113C22semidirect
36. C2×M16×C113C22semidirect
147. C113#3009semidirect
149. C2×#137D113semidirect
163. C22×Q8C2×D113semidirect
169. C2×Q16C2×D113semidirect
172. C2×SD16C2×D113semidirect
173. C2×M16C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136682
C22×#142686342
#202991722
#202987592
#202987472
#202986042
#202985942
#202985662
C2×#137×C1132
#3009113