Finite Group Databasebeta

← #28347974

Subgroups — #28347974

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×#16812140C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3Q8×C22⋊D107C211yesmaximal, characteristic, complemented
4#20293274C211yesmaximal, characteristic
5#20293018C211yesmaximal, characteristic, complemented
6#20292871C211yesmaximal, characteristic, complemented
7#20292867C211yesmaximal, characteristic, complemented
8#20292847C211yesmaximal, characteristic, complemented
9#20292831C211yesmaximal, characteristic, complemented
10#20292791C211yesmaximal, characteristic, complemented
11#20292784C211yesmaximal, characteristic, complemented
12#20292776C211yesmaximal, characteristic, complemented
13#20292730C211yesmaximal, characteristic, complemented
14#20292701C211yesmaximal, characteristic, complemented
15#448×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812140—24no—
20C2×Dic428C2211yescharacteristic, complemented
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic
24C2×#13133806C2211yescharacteristic
25C2×#13133805C2211yescharacteristic
26C2×#13133805C2211yescharacteristic
27SD16×D107—24no—
28C2×#13133802C2211yescharacteristic, complemented
29C2×C8×D107C2211yescharacteristic
30#16812107—21no—
31#16812107C2211yescharacteristic, complemented
32#16812096C2211yescharacteristic
33#16812091—21no—
34#16812089C2211yescharacteristic
35#16812088—21no—
36#16812088C2211yescharacteristic, complemented
37#16812085—21no—
38#16812085C2211yescharacteristic
39#16812080—21no—
40#16812077—21no—
41Q8×Dic107C2211yescharacteristic, complemented
42#16812071C2211yescharacteristic
43C4×C22⋊D107—21no—
44C4×C22⋊D107C2211yescharacteristic
45C22⋊C4×D107—21no—
46#16812042C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic
51#16812030C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic
53#16812019C2211yescharacteristic, complemented
54C22×Q8×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic
56C2×SD16×C107C2211yescharacteristic
57#121×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60C22⋊C8×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133818—22no—
63#13133818—41no—
64C2×Dic214—21no—
65C2×Dic214C2311yescharacteristic
66C2×Dic214C2311yescharacteristic
67Q8×D107—22no—
68Q8×D107—41no—
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107D411yescharacteristic
73D4×D107—22no—
74D4×D107—41no—
75C2×C4×D107—21no—
76C2×C4×D107—21no—
77C2×C4×D107—21no—
78C2×C4×D107C2311yescharacteristic
79Dic428—41no—
80#13133808—22no—
81#13133808—22no—
82#13133807—22no—
83#13133806—22no—
84#13133805—22no—
85#13133805D414yescomplemented
86#13133802—22no—
87C2×C107⋊C8C2311yescharacteristic
88C2×C107⋊C8C2311yescharacteristic
89C8×D107—22no—
90C4⋊Dic107—21no—
91C4⋊Dic107—21no—
92C4⋊Dic107C2311yescharacteristic
93Dic107⋊C4—21no—
94Dic107⋊C4—21no—
95Dic107⋊C4—21no—
96Dic107⋊C4—21no—
97Dic107⋊C4D411yescharacteristic
98C22⋊Dic107—21no—
99C22⋊Dic107—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107D411yescharacteristic
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4D411yescharacteristic
106C4×Dic107—21no—
107C4×Dic107C2311yescharacteristic
108C2×Q8×C107—21no—
109C2×Q8×C107—22no—
110C2×Q8×C107—41no—
111C2×Q8×C107C2311yescharacteristic
112C2×Q8×C107C2311yescharacteristic
113C2×D4×C107—21no—
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107—21no—
116C22×C4×C107C2311yescharacteristic
117SD16×C107—22no—
118SD16×C107—41no—
119C2×C8×C107C2311yescharacteristic
120C2×C8×C107C2311yescharacteristic
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—42no—
127C22×D107—42no—
128C22×D107—81no—
129C22×D107C2×D411yescharacteristic
130Q8×C107—21no—
131Q8×C107—21no—
132Q8×C107—21no—
133Q8×C107—42no—
134Q8×C107C2×D412yes—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—81no—
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×C107C2×D411yescharacteristic
145C2×C4×C107C2411yescharacteristic
146C22⋊D107—22no—
147C22⋊D107—42no—
148C22⋊D107—42no—
149C2×Dic107—21no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107C2×D411yescharacteristic
153C2×Dic107C2×D411yescharacteristic
154C2×Dic107C2×D411yescharacteristic
155D428—21no—
156D428C2×D412yes—
157C4×D107—21no—
158C4×D107—21no—
159C4×D107—42no—
160C4×D107—42no—
161C4×D107—81no—
162Dic214—21no—
163Dic214—21no—
164Dic214—21no—
165Dic214—21no—
166Dic214—42no—
167C8×C107—21no—
168C8×C107—21no—
169C107⋊C8—21no—
170C107⋊C8C2×D412yes—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107C22×D411yescharacteristic
178C2×D107—21no—
179C2×D107—21no—
180C2×D107—22no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107—84no—
184C2×D107#13712yescomplemented
185C4×C107—22no—
186C4×C107—21no—
187C4×C107—41no—
188C4×C107—41no—
189C4×C107C22×D411yescharacteristic
190C4×C107H(5,𝔽2)11yescharacteristic
191Dic107—21no—
192Dic107—21no—
193Dic107—21no—
194Dic107—41no—
195Dic107—41no—
196Dic107—41no—
197C2×C107—41no—
198C2×C107—81no—
199C2×C107D4211yescharacteristic
200C2×C107C2×#13711yescharacteristic
201C2×C107#57711yescharacteristic
202D107—22no—
203D107—42no—
204#3266—1071nomaximal, Sylow
205C107#326611yescharacteristic, complemented, Sylow
206C2×#137—1071no—
207C22×SD16—1071no—
208D4×Q8—1071no—
209#546—1071no—
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#448D10711yescharacteristic, complemented
219#440—1071no—
220#435—1071no—
221C2×#41—1071no—
222C22×Q8—1071no—
223C22×Q8C2×D10711yescharacteristic, complemented
224C22×D4—1071no—
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—2144no—
234C2×SD16C2×D10711yescharacteristic
235C2×SD16C2×D10711yescharacteristic
236C2×M16—1071no—
237C22×C8—1071no—
238C4⋊Q8—1071no—
239C4⋊Q8—2141no—
240#124—1071no—
241#123—2141no—
242C22⋊Q8—1071no—
243C22⋊Q8—1071no—
244C22⋊Q8—1071no—
245C22⋊Q8—2141no—
246C22⋊Q8—2141no—
247#121—2141no—
248#121C2×D10711yescharacteristic
249#120—2141no—
250C4×Q8—1071no—
251C4×D4—1071no—
252C4×D4—1071no—
253C4×D4—2141no—
254C2×C22⋊C4—2141no—
255#107—1071no—
256C4⋊C8—1071no—
257#103—1071no—
258#103—1071no—
259#103C2×D10711yescharacteristic, complemented
260#103C2×D10711yescharacteristic, complemented
261#102—1071no—
262#102—1071no—
263C22⋊C8—1071no—
264C22⋊C8C2×D10711yescharacteristic, complemented
265#97—1071no—
266C24—2141no—
267#41—4281no—
268#41—2142no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1071no—
273C2×Q8—1071no—
274C2×Q8—4281no—
275C2×Q8—2141no—
276C2×Q8—2142no—
277C2×Q8C22×D10711yescharacteristic
278C2×Q8C22×D10711yescharacteristic
279C2×D4—21no—
280C2×D4—1071no—
281C2×D4—1071no—
282C2×D4—4281no—
283C2×D4—2141no—
284C2×D4—2142no—
285C2×D4—2141no—
286C2×D4C22×D10711yescharacteristic
287C22×C4—21no—
288C22×C4—1071no—
289C22×C4—2141no—
290C22×C4—2141no—
291C22×C4—2141no—
292C22×C4C22×D10711yescharacteristic
293Q16—4281no—
294Q16—2142no—
295Q16—2142no—
296SD16—22no—
297SD16—41no—
298SD16—1074no—
299SD16—2142no—
300SD16—2142no—
301SD16—2142no—
302M16—2142no—
303C2×C8—1071no—
304C2×C8—1071no—
305C2×C8—2142no—
306C2×C8C22×D10711yescharacteristic
307C2×C8C22×D10711yescharacteristic
308C4⋊C4—1071no—
309C4⋊C4—1071no—
310C4⋊C4—2141no—
311C4⋊C4—2141no—
312C4⋊C4—2141no—
313C4⋊C4—2141no—
314C4⋊C4—2141no—
315C4⋊C4—2141no—
316C4⋊C4C22×D10711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—1071no—
319C22⋊C4—1071no—
320C22⋊C4—2141no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C42—1071no—
327C42—2141no—
328C23—21no—
329C23—8561no—
330C23—1071no—
331C23—4282no—
332C23—4282no—
333C23—2141no—
334C23D4×D10711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—4282no—
340Q8—2141no—
341Q8—2141no—
342Q8—2141no—
343Q8—2141no—
344Q8D4×D10712yes—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—1072no—
349D4—4282no—
350D4—4282no—
351D4—2142no—
352D4—2141no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—8561no—
360C2×C4—1071no—
361C2×C4—1071no—
362C2×C4—1071no—
363C2×C4—4282no—
364C2×C4—4282no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4—2141no—
368C2×C4—2141no—
369C2×C4—2141no—
370C2×C4D4×D10711yescharacteristic
371C2×C4C23×D10711yescharacteristic
372C8—21no—
373C8—21no—
374C8—1072no—
375C8—2141no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—8564no—
383C22—1072no—
384C22—4281no—
385C22—4281no—
386C22—2142no—
387C22—2141no—
388C22—2141no—
389C22C2×D4×D10711yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—4281no—
395C4—4281no—
396C4—4281no—
397C4—2141no—
398C4—2141no—
399C4—2141no—
400C4C2×D4×D10711yescharacteristic
401C4#1681215411yescharacteristic
402C2—41no—
403C2—81no—
404C2—4282no—
405C2—2142no—
406C2#2029325911yescharacteristic
407C2#137×D10711yescharacteristic
408C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#448×C107C2
socleC22×C107C22×D4
prime radical#20293274C2
hypercenter#448D107
hypocenterC107#3266
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C2×SD16×D107C2semidirect
3. Q8×C22⋊D107C2semidirect
5. #20293018C2semidirect
6. #20292871C2semidirect
7. #20292867C2semidirect
8. #20292847C2semidirect
9. #20292831C2semidirect
10. #20292791C2semidirect
11. #20292784C2semidirect
12. #20292776C2semidirect
13. #20292730C2semidirect
14. #20292701C2semidirect
15. #448×C107C2semidirect
20. C2×Dic428C22semidirect
21. C2×#13133808C22semidirect
22. C2×#13133808C22semidirect
28. C2×#13133802C22semidirect
31. #16812107C22semidirect
36. #16812088C22semidirect
41. Q8×Dic107C22semidirect
46. #16812042C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812032C22semidirect
53. #16812019C22semidirect
54. C22×Q8×C107C22semidirect
58. #103×C107C22semidirect
59. #103×C107C22semidirect
60. C22⋊C8×C107C22semidirect
85. #13133805D4semidirect
184. C2×D107#137semidirect
205. C107#3266semidirect
218. #448D107semidirect
223. C22×Q8C2×D107semidirect
259. #103C2×D107semidirect
260. #103C2×D107semidirect
264. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C2×SD16×D1072
Q8×C22⋊D1072
#202932742
#202930182
#202928712
#202928672
#202928472
#202928312
#202927912
#202927842
#202927762
#202927302
#202927012
#448×C1072
#3266107