Finite Group Databasebeta

← #28347967

Subgroups — #28347967

Order 13696.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812139C211yesmaximal, characteristic, complemented
2C22×#13133806C211yesmaximal, characteristic, complemented
3D4×Dic214C211yesmaximal, characteristic, complemented
4#20293022C211yesmaximal, characteristic, complemented
5#20292871C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292846C211yesmaximal, characteristic, complemented
8#20292832C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292782C211yesmaximal, characteristic, complemented
11#20292778C211yesmaximal, characteristic, complemented
12#20292765C211yesmaximal, characteristic, complemented
13#20292727C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15#546×C107C211yesmaximal, characteristic
16C22×Dic214C2211yescharacteristic
17C22×Dic214C2211yescharacteristic, complemented
18#16812139—24no—
19C2×#13133808C2211yescharacteristic, complemented
20C2×#13133808C2211yescharacteristic, complemented
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133806—24no—
23C2×#13133806C2211yescharacteristic
24C2×#13133806C2211yescharacteristic
25C2×#13133806C2211yescharacteristic
26C2×#13133806C2211yescharacteristic
27C2×#13133806C2211yescharacteristic
28C2×#13133806C2211yescharacteristic
29C2×#13133802C2211yescharacteristic, complemented
30C22×C107⋊C8C2211yescharacteristic
31#16812109C2211yescharacteristic, complemented
32#16812108—21no—
33#16812088C2211yescharacteristic
34#16812088C2211yescharacteristic, complemented
35#16812086—21no—
36#16812086—21no—
37C4×Dic214C2211yescharacteristic, complemented
38D4×Dic107—21no—
39#16812042C2211yescharacteristic, complemented
40C4⋊(C107⋊C8)C2211yescharacteristic, complemented
41#16812033C2211yescharacteristic, complemented
42#16812032C2211yescharacteristic, complemented
43#16812032C2211yescharacteristic, complemented
44#16812032C2211yescharacteristic, complemented
45#16812030C2211yescharacteristic
46#16812030C2211yescharacteristic
47C22⋊(C107⋊C8)C2211yescharacteristic
48C22⋊(C107⋊C8)C2211yescharacteristic, complemented
49#16812020C2211yescharacteristic, complemented
50C2×#41×C107C2211yescharacteristic
51C22×D4×C107C2211yescharacteristic
52#124×C107C2211yescharacteristic
53#123×C107—21no—
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107—21no—
56#121×C107C2211yescharacteristic
57#120×C107—21no—
58C4×D4×C107C2211yescharacteristic
59C4×D4×C107C2211yescharacteristic
60C2×C22⋊C4×C107—21no—
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—41no—
75#13133806—22no—
76#13133806—22no—
77#13133806—22no—
78#13133806—22no—
79#13133806—41no—
80#13133806D414yescomplemented
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—
96C24×C107—21no—
97#41×C107—22no—
98#41×C107—41no—
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—22no—
101C2×D4×C107—21no—
102C2×D4×C107—21no—
103C2×D4×C107—21no—
104C2×D4×C107—41no—
105C2×D4×C107D411yescharacteristic
106C2×D4×C107C2311yescharacteristic
107C2×D4×C107C2311yescharacteristic
108C22×C4×C107—21no—
109C22×C4×C107—21no—
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
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×C107—21no—
120C22⋊C4×C107D411yescharacteristic
121C22⋊C4×C107D411yescharacteristic
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—42no—
126C23×C107—42no—
127C23×C107—81no—
128C23×C107C2×D411yescharacteristic
129C23×C107C2×D411yescharacteristic
130Q8×C107—21no—
131Q8×C107—21no—
132D4×C107—22no—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—42no—
137D4×C107—42no—
138D4×C107—81no—
139D4×C107C2×D412yes—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—42no—
147C2×C4×C107—81no—
148C2×C4×C107C2×D411yescharacteristic
149C2×C4×C107C2×D411yescharacteristic
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2×D411yescharacteristic
152C2×C4×C107C2411yescharacteristic
153C2×Dic107—21no—
154C2×Dic107—21no—
155C2×Dic107—21no—
156C2×Dic107—21no—
157C2×Dic107—42no—
158C2×Dic107—42no—
159Dic214—21no—
160Dic214—21no—
161Dic214—21no—
162Dic214—21no—
163Dic214—21no—
164Dic214—42no—
165Dic214—42no—
166Dic214C2×D412yes—
167C107⋊C8—21no—
168C107⋊C8—21no—
169C107⋊C8—21no—
170C107⋊C8C2×D412yes—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—22no—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107—84no—
183C22×C107#13712yescomplemented
184C22×C107C22×D411yescharacteristic
185C4×C107—21no—
186C4×C107—21no—
187C4×C107—21no—
188C4×C107—21no—
189C4×C107—41no—
190C4×C107—41no—
191C4×C107C22×D411yescharacteristic
192C4×C107H(5,𝔽2)11yescharacteristic
193Dic107—22no—
194Dic107—41no—
195Dic107—41no—
196Dic107—41no—
197C2×C107—22no—
198C2×C107—41no—
199C2×C107—42no—
200C2×C107—81no—
201C2×C107D4211yescharacteristic
202C2×C107C2×#13711yescharacteristic
203C2×C107#57711yescharacteristic
204#3266—1071nomaximal, Sylow
205C107#326611yescharacteristic, complemented, Sylow
206C2×#137—1071no—
207C22×SD16—1071no—
208D4×Q8—1071no—
209#546D10711yescharacteristic
210#497—1071no—
211#468—1071no—
212#467—1071no—
213#464—1071no—
214#461—1071no—
215#452—1071no—
216#451—1071no—
217#450—1071no—
218#448—1071no—
219#440—1071no—
220#435—1071no—
221C2×#41C2×D10711yescharacteristic
222C22×Q8—1071no—
223C22×Q8—1071no—
224C22×D4C2×D10711yescharacteristic
225#137—2144no—
226C2×Q16—1071no—
227C2×Q16—1071no—
228C2×Q16—1071no—
229C2×SD16—1071no—
230C2×SD16—1071no—
231C2×SD16—1071no—
232C2×SD16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—2144no—
236C2×M16—1071no—
237C22×C8—1071no—
238C4⋊Q8—1071no—
239C4⋊Q8—2141no—
240#124C2×D10711yescharacteristic
241#123—21no—
242C22⋊Q8—1071no—
243C22⋊Q8—1071no—
244C22⋊Q8—2141no—
245C22⋊Q8—2141no—
246C22⋊Q8C2×D10711yescharacteristic
247#121—21no—
248#121C2×D10711yescharacteristic
249#120—21no—
250C4×Q8—1071no—
251C4×D4—2141no—
252C4×D4C2×D10711yescharacteristic
253C4×D4C2×D10711yescharacteristic
254C2×C22⋊C4—21no—
255#107—1071no—
256C4⋊C8—1071no—
257#103—1071no—
258#103—1071no—
259#103—1071no—
260#103—1071no—
261#102—1071no—
262#102—1071no—
263C22⋊C8—1071no—
264C22⋊C8—1071no—
265#97—1071no—
266C24—21no—
267#41—22no—
268#41—41no—
269C2×Q8—1071no—
270C2×Q8—1071no—
271C2×Q8—1071no—
272C2×Q8—4281no—
273C2×Q8—4281no—
274C2×Q8—2142no—
275C2×Q8—2142no—
276C2×Q8—2141no—
277C2×Q8—2141no—
278C2×Q8C22×D10711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D10711yescharacteristic
285C2×D4C22×D10711yescharacteristic
286C2×D4C22×D10711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2141no—
290C22×C4—2141no—
291C22×C4C22×D10711yescharacteristic
292C22×C4C22×D10711yescharacteristic
293Q16—4281no—
294Q16—2142no—
295Q16—2142no—
296SD16—1074no—
297SD16—4281no—
298SD16—2142no—
299SD16—2142no—
300SD16—2142no—
301SD16—2142no—
302M16—2142no—
303C2×C8—1071no—
304C2×C8—1071no—
305C2×C8—1071no—
306C2×C8—1071no—
307C2×C8—2142no—
308C4⋊C4—21no—
309C4⋊C4—1071no—
310C4⋊C4—2141no—
311C4⋊C4—2141no—
312C4⋊C4—2141no—
313C4⋊C4—2141no—
314C4⋊C4—2141no—
315C4⋊C4C22⋊D10711yescharacteristic
316C4⋊C4C22×D10711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4C22⋊D10711yescharacteristic
325C22⋊C4C22⋊D10711yescharacteristic
326C42—2141no—
327C42C22×D10711yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D10711yescharacteristic
334C23C2×C22⋊D10711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—1072no—
338Q8—4282no—
339Q8—4282no—
340Q8—2141no—
341Q8—2141no—
342Q8—2141no—
343Q8—2141no—
344Q8—2141no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D10712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—4282no—
362C2×C4—4282no—
363C2×C4—2141no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4C2×C22⋊D10711yescharacteristic
368C2×C4C2×C22⋊D10711yescharacteristic
369C2×C4C2×C22⋊D10711yescharacteristic
370C2×C4C2×C22⋊D10711yescharacteristic
371C2×C4C23×D10711yescharacteristic
372C8—1072no—
373C8—2141no—
374C8—2141no—
375C8—2141no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#1681213912yescomplemented
389C22C22×C22⋊D10711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—4281no—
397C4—4281no—
398C4—4281no—
399C4—2142no—
400C4C2×D4×D10711yescharacteristic
401C4#1681215411yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D10711yescharacteristic
407C2C2×#1681213911yescharacteristic
408C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#546×C107C2
socleC22×C107C22×D4
hypercenter#546D107
hypocenterC107#3266
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. C22×#13133806C2semidirect
3. D4×Dic214C2semidirect
4. #20293022C2semidirect
5. #20292871C2semidirect
6. #20292866C2semidirect
7. #20292846C2semidirect
8. #20292832C2semidirect
9. #20292789C2semidirect
10. #20292782C2semidirect
11. #20292778C2semidirect
12. #20292765C2semidirect
13. #20292727C2semidirect
14. #20292699C2semidirect
17. C22×Dic214C22semidirect
19. C2×#13133808C22semidirect
20. C2×#13133808C22semidirect
21. C2×#13133808C22semidirect
29. C2×#13133802C22semidirect
31. #16812109C22semidirect
34. #16812088C22semidirect
37. C4×Dic214C22semidirect
39. #16812042C22semidirect
40. C4⋊(C107⋊C8)C22semidirect
41. #16812033C22semidirect
42. #16812032C22semidirect
43. #16812032C22semidirect
44. #16812032C22semidirect
48. C22⋊(C107⋊C8)C22semidirect
49. #16812020C22semidirect
80. #13133806D4semidirect
183. C22×C107#137semidirect
205. C107#3266semidirect
388. 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×#131338062
D4×Dic2142
#202930222
#202928712
#202928662
#202928462
#202928322
#202927892
#202927822
#202927782
#202927652
#202927272
#202926992
#546×C1072
#3266107