Finite Group Databasebeta

← #28399929

Subgroups — #28399929

Order 14464.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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, complemented
2#20299396C211yesmaximal, characteristic, complemented
3#20299162C211yesmaximal, characteristic, complemented
4#20298774C214yesmaximal, complemented
5#20298764C214yesmaximal, complemented
6#20298615C211yesmaximal, characteristic, complemented
7#20298614C211yesmaximal, characteristic, complemented
8C2×#16813538C211yesmaximal, characteristic, complemented
9#575×C113C211yesmaximal, characteristic, complemented
10#16813668—22no—
11#16813668—22no—
12#16813668—22no—
13#16813668C2214yes—
14C2×#14268648—21no—
15C2×#14268647—21no—
16C2×#14268647C2211yescharacteristic
17C22×D452—21no—
18C22×D452C2211yescharacteristic
19C2×D4×D113—22no—
20C2×D4×D113—21no—
21#16813653C2212yescomplemented
22#16813652—21no—
23#16813648C2212yescomplemented
24#16813647—21no—
25#16813642—22no—
26C2×#14268631C2211yescharacteristic, complemented
27#16813618C2212yescomplemented
28#16813611C2212yescomplemented
29#16813596—22no—
30#16813590—22no—
31#16813581C2211yescharacteristic, complemented
32#16813549C2214yescomplemented
33#16813548C2214yescomplemented
34#16813538C2214yescomplemented
35C2×#41×C113C2211yescharacteristic
36C2×#41×C113C2211yescharacteristic, complemented
37#137×C113—21no—
38#137×C113C2212yescomplemented
39AGL(1,ℤ/8)×C113—21no—
40AGL(1,ℤ/8)×C113C2212yescomplemented
41#135×C113—22no—
42C2×M16×C113C2211yescharacteristic, complemented
43C23×D113—22no—
44C23×D113D412yes—
45#14268648—24no—
46#14268648—24no—
47#14268647—24no—
48#14268647—21no—
49#14268647—21no—
50#14268647—24no—
51#14268647D412yescomplemented
52#14268647C2312yes—
53C2×Dic226D411yescharacteristic, complemented
54C2×D452—22no—
55C2×D452—24no—
56C2×D452—21no—
57C2×D452—21no—
58C2×D452—21no—
59C2×D452—21no—
60C2×D452—24no—
61C2×D452D412yes—
62C2×D452D411yescharacteristic, complemented
63C2×D452C2312yes—
64C2×C22⋊D113—22no—
65C2×C22⋊D113—21no—
66C2×C22⋊D113—24no—
67D4×D113—28no—
68D4×D113—24no—
69D4×D113—28no—
70D4×D113—24no—
71C2×C4×D113—22no—
72C2×C4×D113—21no—
73#14268637—22no—
74#14268635—22no—
75#14268634—22no—
76#14268632—22no—
77#14268631—21no—
78#14268631C2312yescomplemented
79C2×#11744751—21no—
80C4⋊Dic113—21no—
81C22⋊Dic113—21no—
82#14268624—24no—
83C4×Dic113C2312yescomplemented
84#41×C113—21no—
85#41×C113—21no—
86#41×C113—21no—
87#41×C113—21no—
88#41×C113—21no—
89#41×C113—21no—
90#41×C113D412yescomplemented
91#41×C113C2312yes—
92C2×Q8×C113D411yescharacteristic, complemented
93C2×Q8×C113C2311yescharacteristic, complemented
94C2×D4×C113—21no—
95C2×D4×C113—21no—
96C2×D4×C113D411yescharacteristic, complemented
97C2×D4×C113C2311yescharacteristic, complemented
98C22×C4×C113—21no—
99C22×C4×C113—21no—
100C22×C4×C113C2311yescharacteristic
101Q16×C113—22no—
102SD16×C113—22no—
103SD16×C113—22no—
104D8×C113—22no—
105M16×C113—21no—
106M16×C113C2312yescomplemented
107C2×C8×C113—21no—
108C23×C113—21no—
109C23×C113—21no—
110C23×C113C2×D411yescharacteristic
111C22×D113—22no—
112C22×D113—21no—
113C22×D113—28no—
114C22×D113—22no—
115C22×D113—82no—
116C22×D113—84no—
117C22×D113C2×D414yes—
118Q8×C113—21no—
119Q8×C113—21no—
120Q8×C113—21no—
121Q8×C113C2×D412yes—
122D4×C113—22no—
123D4×C113—22no—
124D4×C113—21no—
125D4×C113—21no—
126D4×C113—21no—
127D4×C113—41no—
128D4×C113—41no—
129D4×C113C2×D412yes—
130C2×C4×C113—21no—
131C2×C4×C113—22no—
132C2×C4×C113—22no—
133C2×C4×C113—21no—
134C2×C4×C113—41no—
135C2×C4×C113—41no—
136C2×C4×C113C2×D411yescharacteristic
137C2×C4×C113C2×D411yescharacteristic
138C2×C4×C113C2×D411yescharacteristic
139C2×C4×C113C2×D411yescharacteristic
140C2×C4×C113C2×D411yescharacteristic
141C2×C4×C113C2411yescharacteristic
142C22⋊D113—28no—
143C22⋊D113—22no—
144C22⋊D113—22no—
145C22⋊D113—84no—
146C2×Dic113—21no—
147C2×Dic113—22no—
148D452—28no—
149D452—22no—
150D452—22no—
151D452—21no—
152D452—22no—
153D452—22no—
154D452C2×D412yes—
155C4×D113—28no—
156C4×D113—22no—
157C4×D113—22no—
158Dic226—21no—
159Dic226C2×D412yes—
160C8×C113—22no—
161#11744751—22no—
162C22×C113—22no—
163C22×C113—41no—
164C22×C113—42no—
165C22×C113—42no—
166C22×C113—41no—
167C22×C113#12011yescharacteristic
168C22×C113#12011yescharacteristic
169C22×C113C22×D411yescharacteristic
170C2×D113—24no—
171C2×D113—22no—
172C2×D113—44no—
173C2×D113—42no—
174C2×D113—44no—
175C2×D113—82no—
176C2×D113—88no—
177C4×C113—22no—
178C4×C113—41no—
179C4×C113#12011yescharacteristic
180C4×C113#12011yescharacteristic
181C4×C113C22×D411yescharacteristic
182C4×C113C22×D411yescharacteristic
183Dic113—22no—
184Dic113—42no—
185C2×C113—21no—
186C2×C113—21no—
187C2×C113—21no—
188C2×C113—42no—
189C2×C113—81no—
190C2×C113C2×#12011yescharacteristic
191D113—44no—
192D113—42no—
193#3019—1131nomaximal, Sylow
194C113#301911yescharacteristic, complemented, Sylow
195C2×H(5,𝔽2)—1131no—
196#575—1131no—
197#575D11311yescharacteristic, complemented
198#535—1131no—
199#455—1134no—
200#453—1134no—
201#421—1131no—
202#421—1131no—
203C2×#100—1131no—
204H(5,𝔽2)—1134no—
205H(5,𝔽2)—2262no—
206H(5,𝔽2)—2262no—
207H(5,𝔽2)—2262no—
208C2×#41—1131no—
209C2×#41—2261no—
210C2×#41—2261no—
211C2×#41C2×D11311yescharacteristic
212C2×#41C2×D11311yescharacteristic, complemented
213C22×D4—1131no—
214C22×D4—2261no—
215C22×D4—2262no—
216C22×D4—2261no—
217#137—21no—
218#137—1132no—
219#137—2261no—
220#137C2×D11312yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—1132no—
223AGL(1,ℤ/8)—2261no—
224AGL(1,ℤ/8)C2×D11312yescomplemented
225#135—22no—
226#135—2262no—
227C2×M16—1131no—
228C2×M16C2×D11311yescharacteristic, complemented
229#127—1132no—
230#124—1132no—
231#121—2262no—
232#120—2262no—
233#117—1131no—
234#104—1134no—
235#104—1134no—
236#100—1134no—
237C24—1132no—
238C24—2262no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—1132no—
246#41—1132no—
247#41—2264no—
248#41—2264no—
249#41—2264no—
250#41—2264no—
251#41—2261no—
252#41—2261no—
253#41C22⋊D11312yescomplemented
254#41C22×D11312yes—
255C2×Q8—1131no—
256C2×Q8C22⋊D11311yescharacteristic, complemented
257C2×Q8C22×D11311yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—1131no—
261C2×D4—1132no—
262C2×D4—1132no—
263C2×D4—2264no—
264C2×D4—2262no—
265C2×D4—2264no—
266C2×D4—2268no—
267C2×D4—2268no—
268C2×D4—2264no—
269C2×D4—2264no—
270C2×D4—2262no—
271C2×D4—2261no—
272C2×D4—2264no—
273C2×D4—2261no—
274C2×D4—2261no—
275C2×D4—2261no—
276C2×D4—2261no—
277C2×D4C22⋊D11311yescharacteristic, complemented
278C2×D4C22×D11311yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2262no—
282C22×C4—2261no—
283C22×C4C22×D11311yescharacteristic
284Q16—22no—
285Q16—2262no—
286SD16—22no—
287SD16—22no—
288SD16—2262no—
289SD16—2262no—
290D8—22no—
291D8—2262no—
292M16—21no—
293M16—1132no—
294M16—2261no—
295M16C22×D11312yescomplemented
296C2×C8—21no—
297C2×C8—2261no—
298C4⋊C4—2261no—
299C22⋊C4—2264no—
300C22⋊C4—2261no—
301C42—1132no—
302C23—21no—
303C23—21no—
304C23—1134no—
305C23—9044no—
306C23—9042no—
307C23—2268no—
308C23—2262no—
309C23—2261no—
310C23—2262no—
311C23D4×D11311yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—1132no—
316Q8—2261no—
317Q8C2×C22⋊D11312yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—1132no—
326D4—9044no—
327D4—2268no—
328D4—2268no—
329D4—2262no—
330D4—2262no—
331D4—2261no—
332D4—2262no—
333D4—2262no—
334D4—2262no—
335D4—2262no—
336D4C2×C22⋊D11312yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—2268no—
344C2×C4—2262no—
345C2×C4—2261no—
346C2×C4—2262no—
347C2×C4—2262no—
348C2×C4D4×D11311yescharacteristic
349C2×C4D4×D11311yescharacteristic
350C2×C4D4×D11311yescharacteristic
351C2×C4C2×C22⋊D11311yescharacteristic
352C2×C4C2×C22⋊D11311yescharacteristic
353C2×C4C23×D11311yescharacteristic
354C8—22no—
355C8—2262no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—9042no—
362C22—9048no—
363C22—4524no—
364C22—4522no—
365C22—4524no—
366C22—2262no—
367C22—2264no—
368C22#1681359011yescharacteristic
369C22#1681359011yescharacteristic
370C22C2×D4×D11311yescharacteristic
371C4—22no—
372C4—41no—
373C4—4522no—
374C4—2262no—
375C4#1681359011yescharacteristic
376C4#1681359011yescharacteristic
377C4C2×D4×D11311yescharacteristic
378C4C22×C22⋊D11311yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—4522no—
385C2—4524no—
386C2C2×#1681359011yescharacteristic
subgroupG/H
centerC2C2×#16813590
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#575×C113C2
socleC2×C113C2×#120
hypercenter#575D113
hypocenterC113#3019
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#16813668C2semidirect
2. #20299396C2semidirect
3. #20299162C2semidirect
4. #20298774C2semidirect
5. #20298764C2semidirect
6. #20298615C2semidirect
7. #20298614C2semidirect
8. C2×#16813538C2semidirect
9. #575×C113C2semidirect
21. #16813653C22semidirect
23. #16813648C22semidirect
26. C2×#14268631C22semidirect
27. #16813618C22semidirect
28. #16813611C22semidirect
31. #16813581C22semidirect
32. #16813549C22semidirect
33. #16813548C22semidirect
34. #16813538C22semidirect
36. C2×#41×C113C22semidirect
38. #137×C113C22semidirect
40. AGL(1,ℤ/8)×C113C22semidirect
42. C2×M16×C113C22semidirect
51. #14268647D4semidirect
53. C2×Dic226D4semidirect
62. C2×D452D4semidirect
78. #14268631C23semidirect
83. C4×Dic113C23semidirect
90. #41×C113D4semidirect
92. C2×Q8×C113D4semidirect
93. C2×Q8×C113C23semidirect
96. C2×D4×C113D4semidirect
97. C2×D4×C113C23semidirect
106. M16×C113C23semidirect
194. C113#3019semidirect
197. #575D113semidirect
212. C2×#41C2×D113semidirect
220. #137C2×D113semidirect
224. AGL(1,ℤ/8)C2×D113semidirect
228. C2×M16C2×D113semidirect
253. #41C22⋊D113semidirect
256. C2×Q8C22⋊D113semidirect
257. C2×Q8C22×D113semidirect
277. C2×D4C22⋊D113semidirect
278. C2×D4C22×D113semidirect
295. M16C22×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136682
#202993962
#202991622
#202987742
#202987642
#202986152
#202986142
C2×#168135382
#575×C1132
#3019113