Finite Group Databasebeta

← #28348027

Subgroups — #28348027

Order 13696.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812139C211yesmaximal, characteristic, complemented
2C22×#13133808C211yesmaximal, characteristic, complemented
3D4×Dic214C211yesmaximal, characteristic, complemented
4#20293022C211yesmaximal, characteristic, complemented
5#20292883C211yesmaximal, characteristic, complemented
6#20292854C211yesmaximal, characteristic, complemented
7#20292846C211yesmaximal, characteristic, complemented
8#20292837C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292783C211yesmaximal, characteristic, complemented
11#20292782C211yesmaximal, characteristic, complemented
12#20292765C211yesmaximal, characteristic, complemented
13#20292732C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15#548×C107C211yesmaximal, characteristic
16C22×Dic214C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18#16812139—24no—
19C2×#13133808—24no—
20C2×#13133808C2211yescharacteristic
21C2×#13133808C2211yescharacteristic
22C2×#13133808C2211yescharacteristic
23C2×#13133808C2211yescharacteristic
24C2×#13133808C2211yescharacteristic
25C2×#13133808C2211yescharacteristic
26C2×#13133806C2211yescharacteristic
27C2×#13133806C2211yescharacteristic
28C2×#13133806C2211yescharacteristic
29C2×#13133802C2211yescharacteristic
30C22×C107⋊C8C2211yescharacteristic
31#16812109C2211yescharacteristic
32#16812108—21no—
33#16812088C2211yescharacteristic
34#16812088C2211yescharacteristic
35#16812086—21no—
36#16812086—21no—
37C4×Dic214C2211yescharacteristic
38D4×Dic107—21no—
39#16812040C2211yescharacteristic
40C4⋊(C107⋊C8)C2211yescharacteristic
41#16812033C2211yescharacteristic
42#16812033C2211yescharacteristic
43#16812032C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812030C2211yescharacteristic
47C22⋊(C107⋊C8)C2211yescharacteristic
48C22⋊(C107⋊C8)C2211yescharacteristic
49#16812020C2211yescharacteristic
50C2×#41×C107C2211yescharacteristic
51C22×Q8×C107C2211yescharacteristic
52#124×C107—21no—
53#124×C107C2211yescharacteristic
54C22⋊Q8×C107—21no—
55C22⋊Q8×C107C2211yescharacteristic
56#121×C107—21no—
57#121×C107C2211yescharacteristic
58C4×Q8×C107C2211yescharacteristic
59C4×D4×C107—21no—
60C4×D4×C107C2211yescharacteristic
61C2×Dic214—22no—
62C2×Dic214—22no—
63C2×Dic214—21no—
64C2×Dic214—21no—
65C2×Dic214—41no—
66C2×Dic214—41no—
67C2×Dic214C2311yescharacteristic
68C2×Dic214C2311yescharacteristic
69C2×Dic214C2311yescharacteristic
70C22×Dic107—21no—
71C22×Dic107—21no—
72#13133808—22no—
73#13133808—22no—
74#13133808—22no—
75#13133808—22no—
76#13133808—41no—
77#13133808D414yescomplemented
78#13133806—22no—
79#13133806—22no—
80#13133806—41no—
81#13133802—22no—
82C2×C107⋊C8—22no—
83C2×C107⋊C8C2311yescharacteristic
84C2×C107⋊C8C2311yescharacteristic
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C4⋊Dic107—21no—
88C4⋊Dic107—21no—
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91Dic107⋊C4—21no—
92Dic107⋊C4—21no—
93C22⋊Dic107—21no—
94C22⋊Dic107—21no—
95C4×Dic107—21no—
96#41×C107—22no—
97#41×C107—41no—
98C2×Q8×C107—21no—
99C2×Q8×C107—22no—
100C2×Q8×C107—41no—
101C2×Q8×C107C2311yescharacteristic
102C2×Q8×C107C2311yescharacteristic
103C2×D4×C107—21no—
104C2×D4×C107—21no—
105C2×D4×C107D411yescharacteristic
106C2×D4×C107C2311yescharacteristic
107C22×C4×C107—21no—
108C22×C4×C107—21no—
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C4⋊C4×C107—21no—
112C4⋊C4×C107—21no—
113C4⋊C4×C107D411yescharacteristic
114C4⋊C4×C107C2311yescharacteristic
115C22⋊C4×C107—21no—
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107D411yescharacteristic
120C22⋊C4×C107D411yescharacteristic
121C42×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C23×C107C2×D411yescharacteristic
126Q8×C107—21no—
127Q8×C107—21no—
128Q8×C107—21no—
129Q8×C107—42no—
130Q8×C107C2×D412yes—
131D4×C107—22no—
132D4×C107—21no—
133D4×C107—21no—
134D4×C107—42no—
135D4×C107—42no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—42no—
144C2×C4×C107—42no—
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2×D411yescharacteristic
147C2×C4×C107C2×D411yescharacteristic
148C2×C4×C107C2×D411yescharacteristic
149C2×C4×C107C2411yescharacteristic
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107—21no—
153C2×Dic107—21no—
154C2×Dic107—42no—
155C2×Dic107—42no—
156Dic214—21no—
157Dic214—21no—
158Dic214—21no—
159Dic214—21no—
160Dic214—21no—
161Dic214—42no—
162Dic214—42no—
163Dic214C2×D412yes—
164C107⋊C8—21no—
165C107⋊C8—21no—
166C107⋊C8—21no—
167C107⋊C8C2×D412yes—
168C22×C107—21no—
169C22×C107—21no—
170C22×C107—21no—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107#13712yescomplemented
177C22×C107C22×D411yescharacteristic
178C4×C107—21no—
179C4×C107—21no—
180C4×C107—22no—
181C4×C107—21no—
182C4×C107—21no—
183C4×C107—41no—
184C4×C107—41no—
185C4×C107C22×D411yescharacteristic
186C4×C107H(5,𝔽2)11yescharacteristic
187Dic107—22no—
188Dic107—41no—
189Dic107—41no—
190Dic107—41no—
191C2×C107—22no—
192C2×C107—41no—
193C2×C107—81no—
194C2×C107D4211yescharacteristic
195C2×C107C2×#13711yescharacteristic
196C2×C107#57811yescharacteristic
197#3270—1071nomaximal, Sylow
198C107#327011yescharacteristic, complemented, Sylow
199C2×#137—1071no—
200C22×Q16—1071no—
201D4×Q8—1071no—
202#548D10711yescharacteristic
203#497—1071no—
204#470—1071no—
205#465—1071no—
206#464—1071no—
207#462—1071no—
208#452—1071no—
209#451—1071no—
210#451—1071no—
211#448—1071no—
212#441—1071no—
213#435—1071no—
214C2×#41C2×D10711yescharacteristic
215C22×Q8—1071no—
216C22×Q8—1071no—
217C22×Q8C2×D10711yescharacteristic
218#137—2144no—
219C2×Q16—1071no—
220C2×Q16—1071no—
221C2×Q16—1071no—
222C2×Q16—1071no—
223C2×Q16—1071no—
224C2×Q16—1071no—
225C2×Q16—2144no—
226C2×SD16—1071no—
227C2×SD16—1071no—
228C2×SD16—1071no—
229C2×M16—1071no—
230C22×C8—1071no—
231C4⋊Q8—1071no—
232C4⋊Q8—2141no—
233#124—21no—
234#124C2×D10711yescharacteristic
235C22⋊Q8—21no—
236C22⋊Q8—1071no—
237C22⋊Q8—1071no—
238C22⋊Q8—2141no—
239C22⋊Q8—2141no—
240C22⋊Q8C2×D10711yescharacteristic
241#121—21no—
242#121C2×D10711yescharacteristic
243C4×Q8—1071no—
244C4×Q8C2×D10711yescharacteristic
245C4×D4—21no—
246C4×D4—2141no—
247C4×D4C2×D10711yescharacteristic
248#106—1071no—
249C4⋊C8—1071no—
250#103—1071no—
251#103—1071no—
252#103—1071no—
253#103—1071no—
254#103—1071no—
255#102—1071no—
256C22⋊C8—1071no—
257C22⋊C8—1071no—
258#97—1071no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—1071no—
265C2×Q8—1071no—
266C2×Q8—1071no—
267C2×Q8—4281no—
268C2×Q8—4281no—
269C2×Q8—2142no—
270C2×Q8—2142no—
271C2×Q8—2141no—
272C2×Q8—2141no—
273C2×Q8C22×D10711yescharacteristic
274C2×Q8C22×D10711yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C22⋊D10711yescharacteristic
278C2×D4C22×D10711yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2141no—
282C22×C4—2141no—
283C22×C4C22×D10711yescharacteristic
284C22×C4C22×D10711yescharacteristic
285Q16—1074no—
286Q16—4281no—
287Q16—2142no—
288Q16—2142no—
289Q16—2142no—
290Q16—2142no—
291SD16—4281no—
292SD16—2142no—
293SD16—2142no—
294M16—2142no—
295C2×C8—1071no—
296C2×C8—1071no—
297C2×C8—1071no—
298C2×C8—1071no—
299C2×C8—2142no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—1071no—
303C4⋊C4—2141no—
304C4⋊C4—2141no—
305C4⋊C4—2141no—
306C4⋊C4—2141no—
307C4⋊C4—2141no—
308C4⋊C4C22⋊D10711yescharacteristic
309C4⋊C4C22×D10711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—2141no—
315C22⋊C4—2141no—
316C22⋊C4C22⋊D10711yescharacteristic
317C22⋊C4C22⋊D10711yescharacteristic
318C42—21no—
319C42—2141no—
320C42C22×D10711yescharacteristic
321C23—21no—
322C23C2×C22⋊D10711yescharacteristic
323C23C2×C22⋊D10711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—1072no—
329Q8—4282no—
330Q8—4282no—
331Q8—2141no—
332Q8—2141no—
333Q8—2141no—
334Q8—2141no—
335Q8—2141no—
336Q8D4×D10712yes—
337D4—22no—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—42no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—4282no—
352C2×C4—4282no—
353C2×C4—2141no—
354C2×C4—2141no—
355C2×C4—2141no—
356C2×C4—2141no—
357C2×C4C2×C22⋊D10711yescharacteristic
358C2×C4C2×C22⋊D10711yescharacteristic
359C2×C4C2×C22⋊D10711yescharacteristic
360C2×C4C2×C22⋊D10711yescharacteristic
361C2×C4C23×D10711yescharacteristic
362C8—1072no—
363C8—2141no—
364C8—2141no—
365C8—2141no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22#1681213912yescomplemented
375C22C22×C22⋊D10711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—4281no—
384C4—4281no—
385C4—4281no—
386C4—2142no—
387C4C2×D4×D10711yescharacteristic
388C4#1681215411yescharacteristic
389C2—22no—
390C2—41no—
391C2—81no—
392C2D4×C22⋊D10711yescharacteristic
393C2C2×#1681213911yescharacteristic
394C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#548×C107C2
socleC22×C107C22×D4
hypercenter#548D107
hypocenterC107#3270
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. C22×#13133808C2semidirect
3. D4×Dic214C2semidirect
4. #20293022C2semidirect
5. #20292883C2semidirect
6. #20292854C2semidirect
7. #20292846C2semidirect
8. #20292837C2semidirect
9. #20292789C2semidirect
10. #20292783C2semidirect
11. #20292782C2semidirect
12. #20292765C2semidirect
13. #20292732C2semidirect
14. #20292699C2semidirect
77. #13133808D4semidirect
176. C22×C107#137semidirect
198. C107#3270semidirect
374. C22#16812139semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
C22×#131338082
D4×Dic2142
#202930222
#202928832
#202928542
#202928462
#202928372
#202927892
#202927832
#202927822
#202927652
#202927322
#202926992
#548×C1072
#3270107