Finite Group Databasebeta

← #28399933

Subgroups — #28399933

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#20299398C211yesmaximal, characteristic, complemented
3#20299396C211yesmaximal, characteristic, complemented
4#20299167C211yesmaximal, characteristic, complemented
5#20298777C214yesmaximal, complemented
6#20298767C214yesmaximal, complemented
7#20298616C211yesmaximal, characteristic, complemented
8C2×#16813538C211yesmaximal, characteristic, complemented
9#421×C113C211yesmaximal, characteristic, complemented
10#16813668—22no—
11#16813668—22no—
12#16813668—22no—
13#16813668C2214yes—
14C2×#14268648—21no—
15C2×#14268647—21no—
16C2×#14268647C2211yescharacteristic
17C2×#14268647C2211yescharacteristic, complemented
18C22×D452—21no—
19C22×D452C2211yescharacteristic
20C2×D4×D113—22no—
21C2×D4×D113—21no—
22#16813655—21no—
23#16813655C2212yescomplemented
24#16813653—21no—
25#16813652C2212yescomplemented
26#16813650—21no—
27#16813650C2212yescomplemented
28#16813648—21no—
29#16813647C2212yescomplemented
30#16813644—22no—
31#16813642—22no—
32C2×#14268631C2211yescharacteristic, complemented
33#16813619C2212yescomplemented
34#16813610C2212yescomplemented
35#16813597—22no—
36#16813592—22no—
37#16813550C2214yescomplemented
38#16813538C2214yescomplemented
39C2×#41×C113C2211yescharacteristic
40C2×M16×C113C2211yescharacteristic, complemented
41#117×C113C2211yescharacteristic, complemented
42#104×C113C2214yescomplemented
43C23×D113—22no—
44C23×D113D412yes—
45#14268648—24no—
46#14268648—24no—
47#14268647—24no—
48#14268647—21no—
49#14268647—21no—
50#14268647—21no—
51#14268647—21no—
52#14268647—21no—
53#14268647—21no—
54#14268647—24no—
55#14268647D412yescomplemented
56#14268647C2312yes—
57C2×Dic226D411yescharacteristic, complemented
58C2×Dic226C2311yescharacteristic, complemented
59C2×D452—22no—
60C2×D452—24no—
61C2×D452—21no—
62C2×D452—21no—
63C2×D452—21no—
64C2×D452—21no—
65C2×D452—24no—
66C2×D452—24no—
67C2×D452D412yes—
68C2×D452D411yescharacteristic, complemented
69C2×D452C2312yes—
70C2×D452C2311yescharacteristic, complemented
71C2×C22⋊D113—22no—
72C2×C22⋊D113—21no—
73C2×C22⋊D113—21no—
74D4×D113—28no—
75D4×D113—24no—
76D4×D113—28no—
77D4×D113—24no—
78C2×C4×D113—22no—
79C2×C4×D113—21no—
80C2×C4×D113—21no—
81Dic452—22no—
82#14268637—22no—
83#14268636—22no—
84#14268636—22no—
85#14268635—22no—
86#14268634—22no—
87D904—22no—
88#14268632—22no—
89#14268631—21no—
90#14268631C2312yescomplemented
91C2×#11744751—21no—
92#14268624—24no—
93#41×C113—21no—
94#41×C113—21no—
95#41×C113D412yescomplemented
96#41×C113C2312yes—
97C2×Q8×C113D411yescharacteristic, complemented
98C2×D4×C113—21no—
99C2×D4×C113D411yescharacteristic, complemented
100C22×C4×C113—21no—
101C22×C4×C113C2311yescharacteristic
102M16×C113—21no—
103M16×C113C2312yescomplemented
104C2×C8×C113—21no—
105C4⋊C4×C113—21no—
106C22⋊C4×C113—21no—
107C42×C113C2312yescomplemented
108C23×C113—21no—
109C23×C113C2×D411yescharacteristic
110C22×D113—22no—
111C22×D113—21no—
112C22×D113—28no—
113C22×D113—22no—
114C22×D113—21no—
115C22×D113—82no—
116C22×D113—84no—
117C22×D113C2×D414yes—
118Q8×C113—21no—
119Q8×C113C2×D412yes—
120D4×C113—22no—
121D4×C113—22no—
122D4×C113—21no—
123D4×C113C2×D412yes—
124C2×C4×C113—21no—
125C2×C4×C113—22no—
126C2×C4×C113—22no—
127C2×C4×C113—22no—
128C2×C4×C113C2×D411yescharacteristic
129C2×C4×C113C2×D411yescharacteristic
130C2×C4×C113C2×D411yescharacteristic
131C2×C4×C113C2×D411yescharacteristic
132C2×C4×C113C2×D411yescharacteristic
133C2×C4×C113C2411yescharacteristic
134C22⋊D113—28no—
135C22⋊D113—22no—
136C22⋊D113—22no—
137C22⋊D113—41no—
138C22⋊D113—41no—
139C2×Dic113—21no—
140C2×Dic113—21no—
141D452—28no—
142D452—22no—
143D452—22no—
144D452—21no—
145D452—22no—
146D452—22no—
147D452—21no—
148D452—21no—
149D452—84no—
150D452C2×D412yes—
151C4×D113—28no—
152C4×D113—22no—
153C4×D113—22no—
154C4×D113—41no—
155C4×D113—41no—
156Dic226—21no—
157Dic226—21no—
158Dic226—21no—
159Dic226C2×D412yes—
160C8×C113—22no—
161#11744751—22no—
162C22×C113—22no—
163C22×C113—41no—
164C22×C113—42no—
165C22×C113#12011yescharacteristic
166C22×C113#12011yescharacteristic
167C22×C113C22×D411yescharacteristic
168C2×D113—24no—
169C2×D113—22no—
170C2×D113—44no—
171C2×D113—42no—
172C2×D113—44no—
173C2×D113—42no—
174C2×D113—41no—
175C2×D113—82no—
176C2×D113—88no—
177C4×C113—22no—
178C4×C113—42no—
179C4×C113#12011yescharacteristic
180C4×C113#12011yescharacteristic
181C4×C113C22×D411yescharacteristic
182C4×C113C22×D411yescharacteristic
183Dic113—22no—
184Dic113—41no—
185C2×C113—21no—
186C2×C113—21no—
187C2×C113—21no—
188C2×C113—42no—
189C2×C113C2×#12011yescharacteristic
190D113—44no—
191D113—42no—
192D113—81no—
193#3019—1131nomaximal, Sylow
194C113#301911yescharacteristic, complemented, Sylow
195C2×H(5,𝔽2)—1131no—
196#575—1131no—
197#575—1131no—
198#535—1131no—
199#455—1134no—
200#453—1134no—
201#421—1131no—
202#421D11311yescharacteristic, complemented
203C2×#100—1131no—
204H(5,𝔽2)—1134no—
205H(5,𝔽2)—2262no—
206H(5,𝔽2)—2262no—
207H(5,𝔽2)—2262no—
208C2×#41—1131no—
209C2×#41—1131no—
210C2×#41—2261no—
211C2×#41—2261no—
212C2×#41C2×D11311yescharacteristic
213C22×D4—1131no—
214C22×D4—2261no—
215C22×D4—2262no—
216C22×D4—2261no—
217#137—1132no—
218#137—1132no—
219#137—2261no—
220#137—2261no—
221AGL(1,ℤ/8)—1132no—
222AGL(1,ℤ/8)—1132no—
223AGL(1,ℤ/8)—2261no—
224AGL(1,ℤ/8)—2261no—
225#135—2262no—
226#135—2262no—
227C2×M16—1131no—
228C2×M16C2×D11311yescharacteristic, complemented
229#127—1132no—
230#124—1132no—
231#121—2262no—
232#120—2262no—
233#117C2×D11311yescharacteristic, complemented
234#104—1134no—
235#104C2×D11314yescomplemented
236#100—1134no—
237C24—1132no—
238C24—2262no—
239#41—21no—
240#41—21no—
241#41—1132no—
242#41—1132no—
243#41—2264no—
244#41—2264no—
245#41—2261no—
246#41—2261no—
247#41—2261no—
248#41—2261no—
249#41—2264no—
250#41—2264no—
251#41—2261no—
252#41—2261no—
253#41D45212yescomplemented
254#41C22×D11312yes—
255C2×Q8—1131no—
256C2×Q8—1131no—
257C2×Q8D45211yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—1131no—
260C2×D4—1132no—
261C2×D4—1132no—
262C2×D4—1131no—
263C2×D4—2264no—
264C2×D4—2264no—
265C2×D4—2262no—
266C2×D4—2264no—
267C2×D4—2268no—
268C2×D4—2268no—
269C2×D4—2264no—
270C2×D4—2261no—
271C2×D4—2262no—
272C2×D4—2261no—
273C2×D4—2264no—
274C2×D4—2261no—
275C2×D4—2261no—
276C2×D4—2261no—
277C2×D4—2261no—
278C2×D4D45211yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—2262no—
281C22×C4—2261no—
282C22×C4—2261no—
283C22×C4C22×D11311yescharacteristic
284Q16—2262no—
285Q16—2262no—
286SD16—2262no—
287SD16—2262no—
288SD16—2262no—
289SD16—2262no—
290D8—2262no—
291D8—2262no—
292M16—21no—
293M16—1132no—
294M16—2261no—
295M16C22×D11312yescomplemented
296C2×C8—21no—
297C2×C8—2261no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—2264no—
301C42C22×D11312yescomplemented
302C23—21no—
303C23—1134no—
304C23—9044no—
305C23—9042no—
306C23—2268no—
307C23—2262no—
308C23—2261no—
309C23—2261no—
310C23—2262no—
311C23D4×D11311yescharacteristic
312Q8—21no—
313Q8—1132no—
314Q8—2261no—
315Q8—2261no—
316Q8—2261no—
317Q8C2×D45212yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—1132no—
322D4—9044no—
323D4—4521no—
324D4—4521no—
325D4—2268no—
326D4—2268no—
327D4—2262no—
328D4—2262no—
329D4—2261no—
330D4—2262no—
331D4—2262no—
332D4—2261no—
333D4—2262no—
334D4—2261no—
335D4—2262no—
336D4C2×D45212yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—4521no—
342C2×C4—4521no—
343C2×C4—2268no—
344C2×C4—2261no—
345C2×C4—2261no—
346C2×C4—2262no—
347C2×C4—2262no—
348C2×C4D4×D11311yescharacteristic
349C2×C4D4×D11311yescharacteristic
350C2×C4D4×D11311yescharacteristic
351C2×C4C2×D45211yescharacteristic
352C2×C4C2×D45211yescharacteristic
353C2×C4C23×D11311yescharacteristic
354C8—22no—
355C8—2262no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—9042no—
360C22—9048no—
361C22—4522no—
362C22—4524no—
363C22—4522no—
364C22—4524no—
365C22—4521no—
366C22—2262no—
367C22—2264no—
368C22#1681359211yescharacteristic
369C22#1681359211yescharacteristic
370C22C2×D4×D11311yescharacteristic
371C4—22no—
372C4—42no—
373C4—4521no—
374C4—2262no—
375C4#1681359211yescharacteristic
376C4#1681359211yescharacteristic
377C4C2×D4×D11311yescharacteristic
378C4C22×D45211yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—9041no—
384C2—4522no—
385C2—4524no—
386C2C2×#1681359211yescharacteristic
subgroupG/H
centerC2C2×#16813592
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#421×C113C2
socleC2×C113C2×#120
hypercenter#421D113
hypocenterC113#3019
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#16813668C2semidirect
2. #20299398C2semidirect
3. #20299396C2semidirect
4. #20299167C2semidirect
5. #20298777C2semidirect
6. #20298767C2semidirect
7. #20298616C2semidirect
8. C2×#16813538C2semidirect
9. #421×C113C2semidirect
17. C2×#14268647C22semidirect
23. #16813655C22semidirect
25. #16813652C22semidirect
27. #16813650C22semidirect
29. #16813647C22semidirect
32. C2×#14268631C22semidirect
33. #16813619C22semidirect
34. #16813610C22semidirect
37. #16813550C22semidirect
38. #16813538C22semidirect
40. C2×M16×C113C22semidirect
41. #117×C113C22semidirect
42. #104×C113C22semidirect
55. #14268647D4semidirect
57. C2×Dic226D4semidirect
58. C2×Dic226C23semidirect
68. C2×D452D4semidirect
70. C2×D452C23semidirect
90. #14268631C23semidirect
95. #41×C113D4semidirect
97. C2×Q8×C113D4semidirect
99. C2×D4×C113D4semidirect
103. M16×C113C23semidirect
107. C42×C113C23semidirect
194. C113#3019semidirect
202. #421D113semidirect
228. C2×M16C2×D113semidirect
233. #117C2×D113semidirect
235. #104C2×D113semidirect
253. #41D452semidirect
257. C2×Q8D452semidirect
278. C2×D4D452semidirect
295. M16C22×D113semidirect
301. C42C22×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136682
#202993982
#202993962
#202991672
#202987772
#202987672
#202986162
C2×#168135382
#421×C1132
#3019113