Finite Group Databasebeta

← #28347986

Subgroups — #28347986

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×#16812133C211yesmaximal, characteristic, complemented
2#20293276C211yesmaximal, characteristic, complemented
3#20293265C211yesmaximal, characteristic, complemented
4#20293020C211yesmaximal, characteristic, complemented
5#20292882C211yesmaximal, characteristic, complemented
6#20292856C211yesmaximal, characteristic, complemented
7#20292844C211yesmaximal, characteristic, complemented
8#20292830C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292786C211yesmaximal, characteristic, complemented
11#20292776C211yesmaximal, characteristic, complemented
12#20292772C211yesmaximal, characteristic, complemented
13#20292739C211yesmaximal, characteristic, complemented
14#20292698C211yesmaximal, characteristic, complemented
15C2×#135×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18#16812133—24no—
19C2×#13133808C2211yescharacteristic, complemented
20C2×#13133807C2211yescharacteristic
21C2×#13133806C2211yescharacteristic
22C2×#13133805C2211yescharacteristic
23C2×#13133803C2211yescharacteristic
24C2×#13133801C2211yescharacteristic
25#16812107C2211yescharacteristic, complemented
26#16812098C2211yescharacteristic
27#16812092—21no—
28#16812092—21no—
29#16812090—21no—
30#16812088C2211yescharacteristic, complemented
31#16812085C2211yescharacteristic
32#16812085C2211yescharacteristic
33#16812083C2211yescharacteristic
34#16812081—21no—
35#16812081—21no—
36#16812077—21no—
37D4×Dic107C2211yescharacteristic
38D4×Dic107C2211yescharacteristic, complemented
39C4×C22⋊D107C2211yescharacteristic
40C2×Dic107⋊C4—21no—
41C2×(C2×D107)⋊C4—21no—
42#16812041C2211yescharacteristic, complemented
43Dic107⋊C8C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812033C2211yescharacteristic, complemented
47#16812031C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812030C2211yescharacteristic, complemented
50C22⋊(C107⋊C8)C2211yescharacteristic, complemented
51(C2×D107)⋊C8C2211yescharacteristic
52#16812019C2211yescharacteristic, complemented
53C2×#41×C107C2211yescharacteristic
54C2×#41×C107C2211yescharacteristic, complemented
55#135×C107—24no—
56C2×Q16×C107C2211yescharacteristic, complemented
57C2×SD16×C107C2211yescharacteristic
58C2×SD16×C107C2211yescharacteristic, complemented
59C2×D8×C107C2211yescharacteristic
60C22×C8×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⋊D107D411yescharacteristic
70D4×D107—22no—
71D4×D107—41no—
72C22×Dic107—21no—
73C22×Dic107—21no—
74C22×Dic107—21no—
75C2×C4×D107C2311yescharacteristic
76#13133808—41no—
77#13133807—22no—
78#13133806—22no—
79#13133805—41no—
80#13133803—22no—
81#13133801—22no—
82C2×C107⋊C8C2311yescharacteristic
83C2×C107⋊C8C2311yescharacteristic
84C4⋊Dic107C2311yescharacteristic
85C4⋊Dic107C2311yescharacteristic
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4—21no—
90Dic107⋊C4—21no—
91Dic107⋊C4D411yescharacteristic
92C22⋊Dic107—21no—
93C22⋊Dic107—21no—
94C22⋊Dic107—21no—
95C22⋊Dic107D411yescharacteristic
96(C2×D107)⋊C4—21no—
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
103#41×C107—22no—
104#41×C107—22no—
105#41×C107—41no—
106#41×C107—41no—
107C2×Q8×C107C2311yescharacteristic
108C2×Q8×C107C2311yescharacteristic
109C2×D4×C107—21no—
110C2×D4×C107—21no—
111C2×D4×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116Q16×C107—22no—
117SD16×C107—22no—
118SD16×C107—22no—
119D8×C107D414yescomplemented
120C2×C8×C107—22no—
121C2×C8×C107C2311yescharacteristic
122C2×C8×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C23×C107C2×D411yescharacteristic
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—21no—
137D4×C107—42no—
138D4×C107—42no—
139D4×C107C2×D412yes—
140D4×C107C2×D412yes—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—42no—
146C2×C4×C107—42no—
147C2×C4×C107C2×D411yescharacteristic
148C2×C4×C107C2411yescharacteristic
149C22⋊D107—41no—
150C22⋊D107—42no—
151C22⋊D107—42no—
152C22⋊D107—81no—
153C2×Dic107—21no—
154C2×Dic107—21no—
155C2×Dic107—21no—
156C2×Dic107—42no—
157C2×Dic107—42no—
158C2×Dic107—81no—
159C2×Dic107C2×D411yescharacteristic
160C2×Dic107C2×D411yescharacteristic
161C2×Dic107C2×D411yescharacteristic
162D428—21no—
163D428—21no—
164C4×D107—21no—
165C4×D107—21no—
166Dic214—21no—
167Dic214—21no—
168C8×C107—21no—
169C8×C107C2×D412yes—
170C107⋊C8—21no—
171C107⋊C8—21no—
172C22×C107—22no—
173C22×C107—22no—
174C22×C107—21no—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C22×D411yescharacteristic
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—21no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—84no—
189C4×C107—21no—
190C4×C107—41no—
191C4×C107—41no—
192C4×C107C22×D411yescharacteristic
193C4×C107H(5,𝔽2)11yescharacteristic
194Dic107—21no—
195Dic107—21no—
196Dic107—21no—
197Dic107—41no—
198Dic107—41no—
199Dic107—41no—
200C2×C107—42no—
201C2×C107—41no—
202C2×C107—42no—
203C2×C107D4211yescharacteristic
204C2×C107#57511yescharacteristic
205C2×C107#57711yescharacteristic
206D107—41no—
207D107—81no—
208#3267—1071nomaximal, Sylow
209C107#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1071no—
211C2×#135D10711yescharacteristic, complemented
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#452—1071no—
221#450—1071no—
222#449—1071no—
223#442—1071no—
224#435—1071no—
225C2×#41—1071no—
226C2×#41C2×D10711yescharacteristic
227C2×#41C2×D10711yescharacteristic, complemented
228C22×D4—1071no—
229AGL(1,ℤ/8)—2144no—
230#135—24no—
231C2×Q16—1071no—
232C2×Q16C2×D10711yescharacteristic, complemented
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—1071no—
236C2×SD16C2×D10711yescharacteristic
237C2×SD16C2×D10711yescharacteristic, complemented
238C2×D8—1071no—
239C2×D8C2×D10711yescharacteristic
240C2×M16—1071no—
241C22×C8C2×D10711yescharacteristic, complemented
242C4⋊Q8—1071no—
243#124—1071no—
244#123—2141no—
245#123—2141no—
246#123—2141no—
247C22⋊Q8—1071no—
248C22⋊Q8—1071no—
249C22⋊Q8—1071no—
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#103—1071no—
264#102—1071no—
265#102—1071no—
266#102—1071no—
267C22⋊C8—1071no—
268C22⋊C8—1071no—
269#97—1071no—
270C24—2141no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—4281no—
276#41—2142no—
277C2×Q8—1071no—
278C2×Q8C22×D10711yescharacteristic
279C2×Q8C22×D10711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—1071no—
283C2×D4—1071no—
284C2×D4—4281no—
285C2×D4—2141no—
286C2×D4—2141no—
287C2×D4—2141no—
288C2×D4—2142no—
289C2×D4C22×D10711yescharacteristic
290C2×D4C22×D10711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—1071no—
294C22×C4—2141no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4C22×D10711yescharacteristic
298Q16—22no—
299Q16—4281no—
300SD16—22no—
301SD16—22no—
302SD16—4281no—
303SD16—2142no—
304SD16—2142no—
305D8—2142no—
306D8C22⋊D10714yescomplemented
307M16—2142no—
308C2×C8—22no—
309C2×C8—1071no—
310C2×C8—1071no—
311C2×C8C22×D10711yescharacteristic
312C2×C8C22×D10711yescharacteristic
313C4⋊C4—1071no—
314C4⋊C4—1071no—
315C4⋊C4—1071no—
316C4⋊C4—2141no—
317C4⋊C4—2141no—
318C4⋊C4—2141no—
319C4⋊C4—2141no—
320C4⋊C4—2141no—
321C22⋊C4—1071no—
322C22⋊C4—1071no—
323C22⋊C4—2141no—
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—21no—
334C23—8561no—
335C23—1071no—
336C23—4282no—
337C23—4282no—
338C23—2141no—
339C23D4×D10711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2141no—
345Q8—2141no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—8561no—
351D4—4282no—
352D4—4282no—
353D4—4281no—
354D4—2141no—
355D4—2141no—
356D4C2×C22⋊D10712yes—
357D4C2×C22⋊D10712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—8561no—
365C2×C4—1071no—
366C2×C4—1071no—
367C2×C4—1071no—
368C2×C4—4282no—
369C2×C4—4282no—
370C2×C4—2141no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4—2141no—
375C2×C4D4×D10711yescharacteristic
376C2×C4C23×D10711yescharacteristic
377C8—21no—
378C8—2141no—
379C8—2141no—
380C8C2×C22⋊D10712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—8564no—
391C22—4281no—
392C22—4281no—
393C22—4281no—
394C22—2141no—
395C22—2141no—
396C22—2141no—
397C22C2×D4×D10711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—4281no—
402C4—4281no—
403C4—4281no—
404C4—2141no—
405C4—2141no—
406C4—2141no—
407C4C22×C22⋊D10711yescharacteristic
408C4#1681215411yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—8561no—
413C2—4281no—
414C2D4×C22⋊D10711yescharacteristic
415C2#2029342411yescharacteristic
416C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#135×C107C2
socleC22×C107C22×D4
hypercenterC2×#135D107
hypocenterC107#3267
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. #20293276C2semidirect
3. #20293265C2semidirect
4. #20293020C2semidirect
5. #20292882C2semidirect
6. #20292856C2semidirect
7. #20292844C2semidirect
8. #20292830C2semidirect
9. #20292789C2semidirect
10. #20292786C2semidirect
11. #20292776C2semidirect
12. #20292772C2semidirect
13. #20292739C2semidirect
14. #20292698C2semidirect
15. C2×#135×C107C2semidirect
19. C2×#13133808C22semidirect
25. #16812107C22semidirect
30. #16812088C22semidirect
38. D4×Dic107C22semidirect
42. #16812041C22semidirect
43. Dic107⋊C8C22semidirect
44. #16812034C22semidirect
45. #16812033C22semidirect
46. #16812033C22semidirect
49. #16812030C22semidirect
50. C22⋊(C107⋊C8)C22semidirect
52. #16812019C22semidirect
54. C2×#41×C107C22semidirect
56. C2×Q16×C107C22semidirect
58. C2×SD16×C107C22semidirect
60. C22×C8×C107C22semidirect
119. D8×C107D4semidirect
209. C107#3267semidirect
211. C2×#135D107semidirect
227. C2×#41C2×D107semidirect
232. C2×Q16C2×D107semidirect
237. C2×SD16C2×D107semidirect
241. C22×C8C2×D107semidirect
306. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202932762
#202932652
#202930202
#202928822
#202928562
#202928442
#202928302
#202927892
#202927862
#202927762
#202927722
#202927392
#202926982
C2×#135×C1072
#3267107