Finite Group Databasebeta

← #28347990

Subgroups — #28347990

Order 13696.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812136C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293279C211yesmaximal, characteristic, complemented
4#20293265C211yesmaximal, characteristic, complemented
5#20293020C211yesmaximal, characteristic, complemented
6#20292884C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292845C211yesmaximal, characteristic, complemented
9#20292830C211yesmaximal, characteristic, complemented
10#20292788C211yesmaximal, characteristic, complemented
11#20292779C211yesmaximal, characteristic, complemented
12#20292774C211yesmaximal, characteristic, complemented
13#20292736C211yesmaximal, characteristic, complemented
14#20292695C211yesmaximal, characteristic, complemented
15#452×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818C2211yescharacteristic, complemented
18C2×D4×D107C2211yescharacteristic
19#16812136—24no—
20#16812129—24no—
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133807C2211yescharacteristic
23C2×#13133806C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25C2×#13133805C2211yescharacteristic, complemented
26C2×D856C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133801C2211yescharacteristic
29C22×C107⋊C8C2211yescharacteristic, complemented
30#16812107C2211yescharacteristic, complemented
31#16812098C2211yescharacteristic
32#16812094—21no—
33#16812092—21no—
34#16812091—21no—
35#16812089C2211yescharacteristic
36#16812085C2211yescharacteristic
37#16812083C2211yescharacteristic
38#16812082—21no—
39#16812081—21no—
40#16812077—21no—
41D4×Dic107C2211yescharacteristic
42#16812071C2211yescharacteristic, complemented
43C4×C22⋊D107C2211yescharacteristic
44C4⋊C4×D107—21no—
45C2×(C2×D107)⋊C4—21no—
46#16812040C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812033C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic
53#16812019C2211yescharacteristic, complemented
54C2×#41×C107C2211yescharacteristic
55C2×Q16×C107C2211yescharacteristic, complemented
56C2×SD16×C107C2211yescharacteristic
57C22⋊Q8×C107C2211yescharacteristic, complemented
58#103×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic
60C22⋊C8×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133819—22no—
63#13133819—41no—
64#13133818—22no—
65#13133818—41no—
66C2×Dic214C2311yescharacteristic
67C2×D428—21no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107D411yescharacteristic
74D4×D107—22no—
75D4×D107—41no—
76C22×Dic107—21no—
77C2×C4×D107—21no—
78C2×C4×D107—21no—
79C2×C4×D107—21no—
80C2×C4×D107C2311yescharacteristic
81#13133808—22no—
82#13133807—41no—
83#13133806—22no—
84#13133805—22no—
85#13133805—22no—
86D856—22no—
87#13133803D414yescomplemented
88#13133801—22no—
89C2×C107⋊C8—22no—
90C2×C107⋊C8C2311yescharacteristic
91C2×C107⋊C8C2311yescharacteristic
92C4⋊Dic107—21no—
93C4⋊Dic107C2311yescharacteristic
94Dic107⋊C4—21no—
95Dic107⋊C4—21no—
96Dic107⋊C4—21no—
97Dic107⋊C4D411yescharacteristic
98C22⋊Dic107—21no—
99C22⋊Dic107D411yescharacteristic
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4D411yescharacteristic
107C4×Dic107C2311yescharacteristic
108#41×C107—22no—
109#41×C107—41no—
110C2×Q8×C107C2311yescharacteristic
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107—21no—
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116Q16×C107—41no—
117SD16×C107—22no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107—21no—
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—81no—
130C22×D107C2×D411yescharacteristic
131Q8×C107—21no—
132Q8×C107—21no—
133Q8×C107—21no—
134Q8×C107—21no—
135D4×C107—21no—
136D4×C107—42no—
137D4×C107C2×D412yes—
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—41no—
147C22⋊D107—42no—
148C22⋊D107—42no—
149C22⋊D107—81no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107—81no—
153C2×Dic107C2×D411yescharacteristic
154C2×Dic107C2×D411yescharacteristic
155C2×Dic107C2×D411yescharacteristic
156D428—21no—
157D428—21no—
158D428—21no—
159D428—42no—
160D428C2×D412yes—
161C4×D107—21no—
162C4×D107—21no—
163C4×D107—42no—
164C4×D107—42no—
165C4×D107—42no—
166Dic214—21no—
167Dic214—21no—
168C8×C107—21no—
169C8×C107—21no—
170C107⋊C8—21no—
171C107⋊C8C2×D412yes—
172C22×C107—21no—
173C22×C107—22no—
174C22×C107—21no—
175C22×C107—21no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107C22×D411yescharacteristic
179C2×D107—21no—
180C2×D107—22no—
181C2×D107—21no—
182C2×D107—21no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—84no—
189C4×C107—21no—
190C4×C107—41no—
191C4×C107—41no—
192C4×C107—41no—
193C4×C107C22×D411yescharacteristic
194C4×C107H(5,𝔽2)11yescharacteristic
195Dic107—21no—
196Dic107—21no—
197Dic107—21no—
198Dic107—41no—
199Dic107—41no—
200C2×C107—42no—
201C2×C107—41no—
202C2×C107D4211yescharacteristic
203C2×C107#57511yescharacteristic
204C2×C107#57711yescharacteristic
205D107—41no—
206D107—42no—
207D107—81no—
208#3267—1071nomaximal, Sylow
209C107#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1071no—
211C2×#135—1071no—
212#547—1071no—
213#546—1071no—
214#497—1071no—
215#470—1071no—
216#465—1071no—
217#463—1071no—
218#461—1071no—
219#452—1071no—
220#452D10711yescharacteristic, complemented
221#450—1071no—
222#449—1071no—
223#442—1071no—
224#435—1071no—
225C2×#41—1071no—
226C2×#41—1071no—
227C2×#41C2×D10711yescharacteristic
228C22×D4—1071no—
229AGL(1,ℤ/8)—2144no—
230#135—2144no—
231C2×Q16—1071no—
232C2×Q16C2×D10711yescharacteristic, complemented
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—1071no—
236C2×SD16—1071no—
237C2×SD16C2×D10711yescharacteristic
238C2×D8—1071no—
239C2×D8—1071no—
240C2×M16—1071no—
241C22×C8—1071no—
242C4⋊Q8—1071no—
243#124—1071no—
244#123—2141no—
245#123—2141no—
246#123—2141no—
247C22⋊Q8—1071no—
248C22⋊Q8—1071no—
249C22⋊Q8C2×D10711yescharacteristic, complemented
250#121—1071no—
251#121—2141no—
252#121—2141no—
253#120—2141no—
254C4×D4—1071no—
255C4×D4—1071no—
256C4×D4—1071no—
257C2×C4⋊C4—2141no—
258C2×C22⋊C4—2141no—
259#106—1071no—
260C4⋊C8—1071no—
261#103—1071no—
262#103—1071no—
263#103C2×D10711yescharacteristic, complemented
264#102—1071no—
265#102—1071no—
266#102C2×D10711yescharacteristic
267C22⋊C8—1071no—
268C22⋊C8C2×D10711yescharacteristic, complemented
269#97—1071no—
270C24—2141no—
271#41—22no—
272#41—41no—
273#41—4281no—
274#41—4281no—
275#41—2142no—
276#41—2142no—
277C2×Q8—1071no—
278C2×Q8C22×D10711yescharacteristic
279C2×Q8C22×D10711yescharacteristic
280C2×D4—21no—
281C2×D4—1071no—
282C2×D4—1071no—
283C2×D4—1071no—
284C2×D4—4281no—
285C2×D4—2141no—
286C2×D4—2141no—
287C2×D4—2141no—
288C2×D4—2141no—
289C2×D4—2142no—
290C2×D4C22×D10711yescharacteristic
291C22×C4—21no—
292C22×C4—1071no—
293C22×C4—2141no—
294C22×C4—2141no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4C22×D10711yescharacteristic
298Q16—41no—
299Q16—2142no—
300SD16—22no—
301SD16—4281no—
302SD16—2142no—
303SD16—2142no—
304SD16—2142no—
305D8—1074no—
306D8—2142no—
307M16—2142no—
308C2×C8—1071no—
309C2×C8—1071no—
310C2×C8—2142no—
311C2×C8C22×D10711yescharacteristic
312C2×C8C22×D10711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—1071no—
315C4⋊C4—1071no—
316C4⋊C4—2141no—
317C4⋊C4—2141no—
318C4⋊C4—2141no—
319C4⋊C4—2141no—
320C4⋊C4C22×D10711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—1071no—
323C22⋊C4—1071no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C42—1071no—
332C23—21no—
333C23—8561no—
334C23—1071no—
335C23—4282no—
336C23—4282no—
337C23—2141no—
338C23—2141no—
339C23D4×D10711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2141no—
345Q8—2141no—
346D4—21no—
347D4—42no—
348D4—8561no—
349D4—1072no—
350D4—4282no—
351D4—4282no—
352D4—4282no—
353D4—4281no—
354D4—2141no—
355D4—2141no—
356D4—2141no—
357D4D4×D10712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—8561no—
365C2×C4—1071no—
366C2×C4—1071no—
367C2×C4—1071no—
368C2×C4—4282no—
369C2×C4—4282no—
370C2×C4—4282no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4—2141no—
375C2×C4D4×D10711yescharacteristic
376C2×C4C23×D10711yescharacteristic
377C8—21no—
378C8—21no—
379C8—1072no—
380C8—2141no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—8564no—
388C22—4281no—
389C22—4281no—
390C22—4281no—
391C22—4281no—
392C22—4281no—
393C22—2141no—
394C22—2141no—
395C22—2142no—
396C22—2141no—
397C22C2×D4×D10711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—4281no—
403C4—4281no—
404C4—2141no—
405C4—2141no—
406C4—2141no—
407C4C2×D4×D10711yescharacteristic
408C4#1681215411yescharacteristic
409C2—41no—
410C2—42no—
411C2—8561no—
412C2—4281no—
413C2—4282no—
414C2#2029325911yescharacteristic
415C2#2029342411yescharacteristic
416C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#452×C107C2
socleC22×C107C22×D4
hypercenter#452D107
hypocenterC107#3267
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812129C2semidirect
3. #20293279C2semidirect
4. #20293265C2semidirect
5. #20293020C2semidirect
6. #20292884C2semidirect
7. #20292858C2semidirect
8. #20292845C2semidirect
9. #20292830C2semidirect
10. #20292788C2semidirect
11. #20292779C2semidirect
12. #20292774C2semidirect
13. #20292736C2semidirect
14. #20292695C2semidirect
15. #452×C107C2semidirect
17. C2×#13133818C22semidirect
21. C2×#13133808C22semidirect
25. C2×#13133805C22semidirect
29. C22×C107⋊C8C22semidirect
30. #16812107C22semidirect
42. #16812071C22semidirect
46. #16812040C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812033C22semidirect
50. #16812031C22semidirect
53. #16812019C22semidirect
55. C2×Q16×C107C22semidirect
57. C22⋊Q8×C107C22semidirect
58. #103×C107C22semidirect
60. C22⋊C8×C107C22semidirect
87. #13133803D4semidirect
209. C107#3267semidirect
220. #452D107semidirect
232. C2×Q16C2×D107semidirect
249. C22⋊Q8C2×D107semidirect
263. #103C2×D107semidirect
268. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121292
#202932792
#202932652
#202930202
#202928842
#202928582
#202928452
#202928302
#202927882
#202927792
#202927742
#202927362
#202926952
#452×C1072
#3267107