Finite Group Databasebeta

← #28348008

Subgroups — #28348008

Order 13696.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812130C211yesmaximal, characteristic, complemented
3#20293295C211yesmaximal, characteristic, complemented
4#20293268C211yesmaximal, characteristic, complemented
5#20293013C211yesmaximal, characteristic, complemented
6#20292864C211yesmaximal, characteristic, complemented
7#20292852C211yesmaximal, characteristic, complemented
8#20292821C211yesmaximal, characteristic, complemented
9#20292787C211yesmaximal, characteristic, complemented
10#20292775C211yesmaximal, characteristic, complemented
11#20292773C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13#20292726C211yesmaximal, characteristic, complemented
14#20292696C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818C2211yescharacteristic
18C2×#13133817C2211yescharacteristic, complemented
19C22×D428C2211yescharacteristic
20#16812137—24no—
21#16812130—24no—
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic
24C2×#13133807C2211yescharacteristic
25C2×#13133805C2211yescharacteristic, complemented
26C2×D856C2211yescharacteristic
27C2×D856C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133803C2211yescharacteristic
30C2×C8×D107C2211yescharacteristic, complemented
31#16812105—21no—
32#16812105C2211yescharacteristic, complemented
33#16812098C2211yescharacteristic
34#16812094—21no—
35#16812084—21no—
36#16812084C2211yescharacteristic
37#16812084C2211yescharacteristic, complemented
38#16812082—21no—
39#16812081—21no—
40#16812079—21no—
41#16812074C2211yescharacteristic, complemented
42#16812071C2211yescharacteristic
43C4×C22⋊D107C2211yescharacteristic
44(C22⋊D107)⋊C4—21no—
45C22⋊C4×D107—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic
50#16812029C2211yescharacteristic, complemented
51C22⋊(C107⋊C8)C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812019C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×M16×C107C2211yescharacteristic
56C22⋊Q8×C107C2211yescharacteristic
57#121×C107C2211yescharacteristic
58#107×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133819—22no—
63#13133819—41no—
64#13133818—22no—
65#13133818—41no—
66#13133817—22no—
67#13133817—41no—
68C2×Dic214C2311yescharacteristic
69C2×D428—22no—
70C2×D428—21no—
71C2×D428—21no—
72C2×D428—21no—
73C2×D428—21no—
74C2×D428—41no—
75C2×D428C2311yescharacteristic
76C2×D428C2311yescharacteristic
77C2×D428C2311yescharacteristic
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80C2×C22⋊D107—21no—
81C2×C22⋊D107—21no—
82C2×C22⋊D107D411yescharacteristic
83C22×Dic107—21no—
84C2×C4×D107—21no—
85C2×C4×D107—21no—
86C2×C4×D107—21no—
87C2×C4×D107C2311yescharacteristic
88#13133808—22no—
89#13133807—22no—
90#13133807D414yescomplemented
91#13133805—41no—
92D856—22no—
93D856—22no—
94#13133803—22no—
95#13133803—41no—
96C2×C107⋊C8C2311yescharacteristic
97C2×C107⋊C8C2311yescharacteristic
98C8×D107—22no—
99Dic107⋊C4—21no—
100Dic107⋊C4D411yescharacteristic
101C22⋊Dic107D411yescharacteristic
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4D411yescharacteristic
108C4×Dic107—21no—
109C4×Dic107C2311yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107C2311yescharacteristic
114SD16×C107—22no—
115M16×C107—22no—
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107—21no—
119C4⋊C4×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—81no—
131C22×D107C2×D411yescharacteristic
132Q8×C107—21no—
133Q8×C107—21no—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—81no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107C2×D411yescharacteristic
143C2×C4×C107C2411yescharacteristic
144C22⋊D107—41no—
145C22⋊D107—42no—
146C22⋊D107—42no—
147C22⋊D107—42no—
148C22⋊D107—81no—
149C2×Dic107—21no—
150C2×Dic107—42no—
151C2×Dic107C2×D411yescharacteristic
152C2×Dic107C2×D411yescharacteristic
153C2×Dic107C2×D411yescharacteristic
154D428—21no—
155D428—21no—
156D428—21no—
157D428—21no—
158D428—21no—
159D428—42no—
160D428—42no—
161D428—81no—
162D428—81no—
163D428C2×D412yes—
164C4×D107—21no—
165C4×D107—21no—
166C4×D107—42no—
167C4×D107—42no—
168C4×D107—81no—
169Dic214—21no—
170Dic214C2×D412yes—
171C8×C107—21no—
172C8×C107C2×D412yes—
173C107⋊C8—21no—
174C107⋊C8—21no—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—21no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C22×D411yescharacteristic
182C2×D107—21no—
183C2×D107—22no—
184C2×D107—21no—
185C2×D107—21no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—84no—
195C4×C107—21no—
196C4×C107—41no—
197C4×C107—41no—
198C4×C107—41no—
199C4×C107C22×D411yescharacteristic
200C4×C107H(5,𝔽2)11yescharacteristic
201Dic107—22no—
202Dic107—21no—
203Dic107—21no—
204Dic107—21no—
205C2×C107—41no—
206C2×C107—81no—
207C2×C107D4211yescharacteristic
208C2×C107#57511yescharacteristic
209C2×C107#57611yescharacteristic
210D107—41no—
211D107—42no—
212D107—81no—
213D107—81no—
214#3268—1071nomaximal, Sylow
215C107#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1071no—
217C2×#135—1071no—
218#550—1071no—
219#546—1071no—
220#496—1071no—
221#468D10711yescharacteristic, complemented
222#466—1071no—
223#464—1071no—
224#459—1071no—
225#452—1071no—
226#449—1071no—
227#449—1071no—
228#447—1071no—
229#440—1071no—
230#435—1071no—
231C2×#41—1071no—
232C2×#41—1071no—
233C2×#41—1071no—
234C22×D4—1071no—
235AGL(1,ℤ/8)—2144no—
236#135—2144no—
237C2×Q16—1071no—
238C2×SD16—1071no—
239C2×SD16—1071no—
240C2×SD16—1071no—
241C2×SD16C2×D10711yescharacteristic, complemented
242C2×D8—1071no—
243C2×D8—1071no—
244C2×D8—1071no—
245C2×D8—1071no—
246C2×M16C2×D10711yescharacteristic
247C22×C8—1071no—
248#127—1071no—
249#127—2141no—
250#124—1071no—
251#123—2141no—
252C22⋊Q8C2×D10711yescharacteristic
253#121—1071no—
254#121—1071no—
255#121—2141no—
256#121—2141no—
257#121—2141no—
258#121C2×D10711yescharacteristic
259#120—2141no—
260C4×Q8—1071no—
261C4×D4—1071no—
262C4×D4—1071no—
263C4×D4—2141no—
264C2×C22⋊C4—2141no—
265#107C2×D10711yescharacteristic, complemented
266C4⋊C8—1071no—
267#103—1071no—
268#103C2×D10711yescharacteristic, complemented
269#102—1071no—
270#102—1071no—
271#102—1071no—
272#102C2×D10711yescharacteristic, complemented
273C22⋊C8—1071no—
274C22⋊C8—1071no—
275#97—1071no—
276C24—2141no—
277#41—4281no—
278#41—4281no—
279#41—4281no—
280#41—2142no—
281#41—2142no—
282#41—2142no—
283C2×Q8—1071no—
284C2×Q8C22×D10711yescharacteristic
285C2×D4—21no—
286C2×D4—1071no—
287C2×D4—1071no—
288C2×D4—1071no—
289C2×D4—1071no—
290C2×D4—4281no—
291C2×D4—2141no—
292C2×D4—2141no—
293C2×D4—2142no—
294C2×D4—2141no—
295C2×D4—2141no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4C22×D10711yescharacteristic
301C22×C4—1071no—
302C22×C4—2141no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4—2141no—
306C22×C4C22×D10711yescharacteristic
307Q16—2142no—
308SD16—22no—
309SD16—1074no—
310SD16—4281no—
311SD16—2142no—
312D8—4281no—
313D8—2142no—
314D8—2142no—
315D8—2142no—
316M16—22no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8—2142no—
320C2×C8C22×D10711yescharacteristic
321C2×C8C22×D10711yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—1071no—
324C4⋊C4—2141no—
325C4⋊C4C22×D10711yescharacteristic
326C4⋊C4C22×D10711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—1071no—
330C22⋊C4—1071no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C42—1071no—
337C42—2141no—
338C23—21no—
339C23—8561no—
340C23—1071no—
341C23—4282no—
342C23—4282no—
343C23—2141no—
344C23—2141no—
345C23—2141no—
346C23D4×D10711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—1072no—
350Q8—2141no—
351D4—21no—
352D4—21no—
353D4—81no—
354D4—8561no—
355D4—8561no—
356D4—8561no—
357D4—1072no—
358D4—4282no—
359D4—4282no—
360D4—4282no—
361D4—4281no—
362D4—4282no—
363D4—4282no—
364D4—2141no—
365D4—2141no—
366D4—2141no—
367D4—2141no—
368D4—2141no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—8561no—
375C2×C4—1071no—
376C2×C4—1071no—
377C2×C4—1071no—
378C2×C4—4282no—
379C2×C4—4282no—
380C2×C4—4282no—
381C2×C4—2141no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4D4×D10711yescharacteristic
385C2×C4C23×D10711yescharacteristic
386C8—21no—
387C8—2141no—
388C8—2141no—
389C8D4×D10712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—8564no—
397C22—4281no—
398C22—4281no—
399C22—4281no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—4281no—
404C22—4281no—
405C22—2141no—
406C22—2141no—
407C22—2142no—
408C22—2141no—
409C22C2×D4×D10711yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—2141no—
415C4—2141no—
416C4—2141no—
417C4—2142no—
418C4C2×D4×D10711yescharacteristic
419C4#1681215511yescharacteristic
420C2—41no—
421C2—81no—
422C2—8561no—
423C2—8561no—
424C2—4282no—
425C2—4281no—
426C2#2029326111yescharacteristic
427C2#2029342611yescharacteristic
428C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
hypercenter#468D107
hypocenterC107#3268
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812130C2semidirect
3. #20293295C2semidirect
4. #20293268C2semidirect
5. #20293013C2semidirect
6. #20292864C2semidirect
7. #20292852C2semidirect
8. #20292821C2semidirect
9. #20292787C2semidirect
10. #20292775C2semidirect
11. #20292773C2semidirect
12. #20292760C2semidirect
13. #20292726C2semidirect
14. #20292696C2semidirect
15. #468×C107C2semidirect
18. C2×#13133817C22semidirect
22. C2×#13133808C22semidirect
25. C2×#13133805C22semidirect
30. C2×C8×D107C22semidirect
32. #16812105C22semidirect
37. #16812084C22semidirect
41. #16812074C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
50. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812019C22semidirect
54. C2×SD16×C107C22semidirect
58. #107×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
90. #13133807D4semidirect
215. C107#3268semidirect
221. #468D107semidirect
241. C2×SD16C2×D107semidirect
265. #107C2×D107semidirect
268. #103C2×D107semidirect
272. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121302
#202932952
#202932682
#202930132
#202928642
#202928522
#202928212
#202927872
#202927752
#202927732
#202927602
#202927262
#202926962
#468×C1072
#3268107