Finite Group Databasebeta

← #28348066

Subgroups — #28348066

Order 13696.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic, complemented
2#20293271C211yesmaximal, characteristic, complemented
3#20293265C211yesmaximal, characteristic, complemented
4#20293008C211yesmaximal, characteristic, complemented
5#20292873C211yesmaximal, characteristic, complemented
6#20292870C211yesmaximal, characteristic, complemented
7#20292844C211yesmaximal, characteristic, complemented
8#20292840C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292786C211yesmaximal, characteristic, complemented
11#20292761C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13D8×Dic107C211yesmaximal, characteristic, complemented
14#20292703C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic, complemented
18#16812133—24no—
19C2×#13133808C2211yescharacteristic, complemented
20C2×#13133807C2211yescharacteristic, complemented
21C2×#13133806C2211yescharacteristic, complemented
22C2×#13133803C2211yescharacteristic
23C2×#13133803C2211yescharacteristic
24C2×#13133801C2211yescharacteristic, complemented
25#16812098C2211yescharacteristic, complemented
26#16812096C2211yescharacteristic, complemented
27#16812092—21no—
28#16812090—21no—
29#16812088C2211yescharacteristic, complemented
30#16812085C2211yescharacteristic, complemented
31#16812083C2211yescharacteristic
32#16812081—21no—
33#16812081—21no—
34#16812080C2211yescharacteristic
35#16812078—21no—
36#16812077—21no—
37D4×Dic107C2211yescharacteristic, complemented
38D4×Dic107C2211yescharacteristic, complemented
39C4×C22⋊D107C2211yescharacteristic
40C2×C22⋊Dic107—21no—
41C2×(C2×D107)⋊C4—21no—
42#16812043C2211yescharacteristic, complemented
43Dic107⋊C8C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic, complemented
48#16812030C2211yescharacteristic, complemented
49#16812030C2211yescharacteristic, complemented
50C22⋊(C107⋊C8)C2211yescharacteristic, complemented
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52C8×Dic107C2211yescharacteristic, complemented
53C2×#41×C107C2211yescharacteristic
54C22×D4×C107C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C107—24no—
56C2×SD16×C107C2211yescharacteristic, complemented
57C2×SD16×C107C2211yescharacteristic, complemented
58C2×D8×C107C2211yescharacteristic
59C2×D8×C107C2211yescharacteristic
60C2×M16×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133817—22no—
63#13133817—41no—
64C2×Dic214C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107D411yescharacteristic
71D4×D107—22no—
72D4×D107—41no—
73C22×Dic107—21no—
74C22×Dic107—21no—
75C22×Dic107—21no—
76C2×C4×D107C2311yescharacteristic
77#13133808—41no—
78#13133807—22no—
79#13133806—22no—
80#13133803—22no—
81#13133803—41no—
82#13133801—22no—
83C2×C107⋊C8C2311yescharacteristic
84C2×C107⋊C8C2311yescharacteristic
85C4⋊Dic107C2311yescharacteristic
86C4⋊Dic107C2311yescharacteristic
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4D411yescharacteristic
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92C22⋊Dic107—21no—
93C22⋊Dic107—21no—
94C22⋊Dic107—21no—
95C22⋊Dic107—21no—
96C22⋊Dic107D411yescharacteristic
97(C2×D107)⋊C4—21no—
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4D411yescharacteristic
102C4×Dic107C2311yescharacteristic
103C24×C107—21no—
104#41×C107—22no—
105#41×C107—41no—
106C2×Q8×C107C2311yescharacteristic
107C2×D4×C107—22no—
108C2×D4×C107—21no—
109C2×D4×C107—21no—
110C2×D4×C107—41no—
111C2×D4×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116SD16×C107—22no—
117SD16×C107—22no—
118D8×C107—22no—
119D8×C107D414yescomplemented
120M16×C107—22no—
121C2×C8×C107C2311yescharacteristic
122C2×C8×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—42no—
127C23×C107—42no—
128C23×C107—81no—
129C23×C107C2×D411yescharacteristic
130C22×D107—21no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—81no—
134C22×D107C2×D411yescharacteristic
135Q8×C107—21no—
136Q8×C107—21no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—21no—
141D4×C107—42no—
142D4×C107—42no—
143D4×C107C2×D412yes—
144D4×C107C2×D412yes—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—42no—
149C2×C4×C107C2×D411yescharacteristic
150C2×C4×C107C2411yescharacteristic
151C22⋊D107—41no—
152C22⋊D107—42no—
153C22⋊D107—42no—
154C22⋊D107—81no—
155C22⋊D107—81no—
156C2×Dic107—21no—
157C2×Dic107—21no—
158C2×Dic107—21no—
159C2×Dic107—42no—
160C2×Dic107—81no—
161C2×Dic107—81no—
162C2×Dic107C2×D411yescharacteristic
163C2×Dic107C2×D411yescharacteristic
164C2×Dic107C2×D411yescharacteristic
165D428—21no—
166D428—21no—
167C4×D107—21no—
168C4×D107—21no—
169Dic214—21no—
170Dic214—21no—
171C8×C107—21no—
172C8×C107C2×D412yes—
173C107⋊C8—21no—
174C107⋊C8—21no—
175C22×C107—22no—
176C22×C107—22no—
177C22×C107—21no—
178C22×C107—21no—
179C22×C107—21no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—84no—
188C22×C107C22×D411yescharacteristic
189C2×D107—21no—
190C2×D107—21no—
191C2×D107—21no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—84no—
196C4×C107—21no—
197C4×C107—41no—
198C4×C107C22×D411yescharacteristic
199C4×C107H(5,𝔽2)11yescharacteristic
200Dic107—21no—
201Dic107—21no—
202Dic107—21no—
203Dic107—41no—
204Dic107—41no—
205Dic107—41no—
206C2×C107—42no—
207C2×C107—41no—
208C2×C107—42no—
209C2×C107—81no—
210C2×C107D4211yescharacteristic
211C2×C107#57511yescharacteristic
212C2×C107#57611yescharacteristic
213D107—41no—
214D107—81no—
215#3273—1071nomaximal, Sylow
216C107#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1071no—
218C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
219#546—1071no—
220#546—1071no—
221#495—1071no—
222#468—1071no—
223#468—1071no—
224#463—1071no—
225#463—1071no—
226#452—1071no—
227#452—1071no—
228#447—1071no—
229#447—1071no—
230C4×D8—1071no—
231#436—1071no—
232C2×#41—1071no—
233C2×#41C2×D10711yescharacteristic
234C22×D4—1071no—
235C22×D4C2×D10711yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—2144no—
238C2×Q16—1071no—
239C2×SD16—1071no—
240C2×SD16—1071no—
241C2×SD16C2×D10711yescharacteristic, complemented
242C2×SD16C2×D10711yescharacteristic, complemented
243C2×D8—1071no—
244C2×D8—1071no—
245C2×D8C2×D10711yescharacteristic
246C2×D8C2×D10711yescharacteristic
247C2×M16—1071no—
248C2×M16C2×D10711yescharacteristic, complemented
249#124—1071no—
250#124—1071no—
251#123—2141no—
252#123—2141no—
253C22⋊Q8—1071no—
254C22⋊Q8—1071no—
255#121—1071no—
256#121—1071no—
257#121—2141no—
258#121—2141no—
259#120—2141no—
260#120—2141no—
261C4×D4—1071no—
262C4×D4—1071no—
263C4×D4—1071no—
264C2×C22⋊C4—2141no—
265C2×C22⋊C4—2141no—
266#107—1071no—
267C4⋊C8—1071no—
268#103—1071no—
269#103—1071no—
270#102—1071no—
271#102—1071no—
272#102—1071no—
273#102—1071no—
274C22⋊C8—1071no—
275C22⋊C8—1071no—
276C4×C8—1071no—
277C24—21no—
278C24—2141no—
279#41—22no—
280#41—41no—
281#41—4281no—
282#41—2142no—
283C2×Q8—1071no—
284C2×Q8C22×D10711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—1071no—
290C2×D4—1071no—
291C2×D4—4281no—
292C2×D4—2141no—
293C2×D4—2141no—
294C2×D4—2141no—
295C2×D4—2141no—
296C2×D4—2142no—
297C2×D4C22×D10711yescharacteristic
298C2×D4C22×D10711yescharacteristic
299C2×D4C22×D10711yescharacteristic
300C22×C4—21no—
301C22×C4—1071no—
302C22×C4—2141no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4C22×D10711yescharacteristic
306Q16—4281no—
307SD16—22no—
308SD16—22no—
309SD16—2142no—
310SD16—2142no—
311D8—22no—
312D8—4281no—
313D8—2142no—
314D8C22⋊D10714yescomplemented
315M16—22no—
316M16—2142no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8C22×D10711yescharacteristic
320C2×C8C22×D10711yescharacteristic
321C4⋊C4—1071no—
322C4⋊C4—1071no—
323C4⋊C4—1071no—
324C4⋊C4—2141no—
325C4⋊C4—2141no—
326C22⋊C4—1071no—
327C22⋊C4—1071no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C42—1071no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—8561no—
346C23—1071no—
347C23—4282no—
348C23—4282no—
349C23—2141no—
350C23D4×D10711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2141no—
354Q8—2141no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—8561no—
362D4—8561no—
363D4—4282no—
364D4—4282no—
365D4—4281no—
366D4—2141no—
367D4—2141no—
368D4C2×C22⋊D10712yes—
369D4C2×C22⋊D10712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—8561no—
375C2×C4—8561no—
376C2×C4—1071no—
377C2×C4—1071no—
378C2×C4—1071no—
379C2×C4—4282no—
380C2×C4—2141no—
381C2×C4—2141no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4—2141no—
385C2×C4D4×D10711yescharacteristic
386C2×C4C23×D10711yescharacteristic
387C8—21no—
388C8—2141no—
389C8—2141no—
390C8C2×C22⋊D10712yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—8564no—
405C22—4281no—
406C22—4281no—
407C22—4281no—
408C22—2141no—
409C22—2141no—
410C22—2141no—
411C22C2×D4×D10711yescharacteristic
412C4—21no—
413C4—41no—
414C4—4281no—
415C4—4281no—
416C4—4281no—
417C4—2141no—
418C4—2141no—
419C4—2141no—
420C4C22×C22⋊D10711yescharacteristic
421C4#1681215411yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—8561no—
427C2—4281no—
428C2D4×C22⋊D10711yescharacteristic
429C2#2029342311yescharacteristic
430C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3273
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. #20293271C2semidirect
3. #20293265C2semidirect
4. #20293008C2semidirect
5. #20292873C2semidirect
6. #20292870C2semidirect
7. #20292844C2semidirect
8. #20292840C2semidirect
9. #20292789C2semidirect
10. #20292786C2semidirect
11. #20292761C2semidirect
12. #20292759C2semidirect
13. D8×Dic107C2semidirect
14. #20292703C2semidirect
15. C2×AGL(1,ℤ/8)×C107C2semidirect
17. C2×D4×D107C22semidirect
19. C2×#13133808C22semidirect
20. C2×#13133807C22semidirect
21. C2×#13133806C22semidirect
24. C2×#13133801C22semidirect
25. #16812098C22semidirect
26. #16812096C22semidirect
29. #16812088C22semidirect
30. #16812085C22semidirect
37. D4×Dic107C22semidirect
38. D4×Dic107C22semidirect
42. #16812043C22semidirect
43. Dic107⋊C8C22semidirect
44. #16812034C22semidirect
45. #16812033C22semidirect
46. #16812031C22semidirect
47. #16812030C22semidirect
48. #16812030C22semidirect
49. #16812030C22semidirect
50. C22⋊(C107⋊C8)C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. C8×Dic107C22semidirect
54. C22×D4×C107C22semidirect
56. C2×SD16×C107C22semidirect
57. C2×SD16×C107C22semidirect
60. C2×M16×C107C22semidirect
119. D8×C107D4semidirect
216. C107#3273semidirect
218. C2×AGL(1,ℤ/8)D107semidirect
235. C22×D4C2×D107semidirect
241. C2×SD16C2×D107semidirect
242. C2×SD16C2×D107semidirect
248. C2×M16C2×D107semidirect
314. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202932712
#202932652
#202930082
#202928732
#202928702
#202928442
#202928402
#202927892
#202927862
#202927612
#202927592
D8×Dic1072
#202927032
C2×AGL(1,ℤ/8)×C1072
#3273107