Finite Group Databasebeta

← #28345923

Subgroups — #28345923

Order 13696.

548 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812136C211yesmaximal, characteristic, complemented
2C2×#16812133C211yesmaximal, characteristic, complemented
3#20293176C211yesmaximal, characteristic, complemented
4C4×D4×D107C211yesmaximal, characteristic, complemented
5#20292843C211yesmaximal, characteristic, complemented
6#20292842C211yesmaximal, characteristic, complemented
7#20292839C211yesmaximal, characteristic, complemented
8#20292831C211yesmaximal, characteristic, complemented
9#20292830C211yesmaximal, characteristic, complemented
10#20292821C211yesmaximal, characteristic, complemented
11#20292820C211yesmaximal, characteristic, complemented
12#20292649C211yesmaximal, characteristic, complemented
13#20292627C211yesmaximal, characteristic, complemented
14#20292625C211yesmaximal, characteristic, complemented
15#463×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19C22×C4×D107—21no—
20#16812136—24no—
21#16812133—24no—
22C2×#13133807C2211yescharacteristic
23C2×#13133806C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25C2×D856C2211yescharacteristic
26C2×#13133803C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133801C2211yescharacteristic
29C2×#13133801C2211yescharacteristic
30#16812107C2211yescharacteristic, complemented
31#16812105C2211yescharacteristic, complemented
32#16812097C2211yescharacteristic, complemented
33#16812082—21no—
34#16812082—21no—
35C4×D428C2211yescharacteristic
36D4×Dic107C2211yescharacteristic
37#16812071C2211yescharacteristic
38C4×C22⋊D107—21no—
39(C22⋊D107)⋊C4—21no—
40(C22⋊D107)⋊C4—21no—
41(C22⋊D107)⋊C4—21no—
42C4⋊C4×D107C2211yescharacteristic
43C22⋊C4×D107—21no—
44C42×D107C2211yescharacteristic
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46Dic107⋊C8C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812033C2211yescharacteristic, complemented
50#16812032C2211yescharacteristic, complemented
51#16812031C2211yescharacteristic, complemented
52#16812030C2211yescharacteristic, complemented
53#16812029C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C2×D8×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133819—22no—
63#13133819—22no—
64#13133817—22no—
65#13133817—22no—
66C2×Dic214C2311yescharacteristic
67C2×D428—21no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74D4×D107—22no—
75D4×D107D414yescomplemented
76C22×Dic107—21no—
77C22×Dic107—21no—
78C2×C4×D107—21no—
79C2×C4×D107—21no—
80C2×C4×D107—21no—
81C2×C4×D107—21no—
82C2×C4×D107—44no—
83C2×C4×D107D411yescharacteristic
84C2×C4×D107D411yescharacteristic
85C2×C4×D107C2311yescharacteristic
86#13133807—22no—
87#13133806—22no—
88#13133805—22no—
89D856—22no—
90#13133803—22no—
91#13133803—22no—
92#13133801—22no—
93#13133801—22no—
94C2×C107⋊C8C2311yescharacteristic
95C2×C107⋊C8C2311yescharacteristic
96C4⋊Dic107C2311yescharacteristic
97Dic107⋊C4—21no—
98Dic107⋊C4—21no—
99Dic107⋊C4—21no—
100C22⋊Dic107—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105C4×Dic107D411yescharacteristic
106C4×Dic107D411yescharacteristic
107C4×Dic107C2311yescharacteristic
108C2×Q8×C107C2311yescharacteristic
109C2×D4×C107C2311yescharacteristic
110C2×D4×C107C2311yescharacteristic
111C22×C4×C107—21no—
112SD16×C107—22no—
113D8×C107—22no—
114C2×C8×C107C2311yescharacteristic
115C2×C8×C107C2311yescharacteristic
116C4⋊C4×C107C2311yescharacteristic
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107—21no—
120C42×C107C2311yescharacteristic
121C23×C107—21no—
122C23×C107—21no—
123C22×D107—21no—
124C22×D107—24no—
125C22×D107—21no—
126C22×D107—42no—
127C22×D107—42no—
128C22×D107C2×D411yescharacteristic
129Q8×C107—21no—
130Q8×C107—21no—
131D4×C107—21no—
132D4×C107—21no—
133D4×C107—21no—
134D4×C107C2×D412yes—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—42no—
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2411yescharacteristic
141C22⋊D107—24no—
142C22⋊D107—42no—
143C22⋊D107—42no—
144C22⋊D107—42no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—42no—
148C2×Dic107—42no—
149C2×Dic107C2×D411yescharacteristic
150C2×Dic107C2×D411yescharacteristic
151C2×Dic107C2×D411yescharacteristic
152D428—21no—
153D428—21no—
154D428—21no—
155D428—42no—
156D428C2×D412yes—
157C4×D107—22no—
158C4×D107—41no—
159C4×D107—42no—
160C4×D107—42no—
161C4×D107—42no—
162C4×D107—42no—
163C4×D107C2×D412yes—
164C4×D107#4112yes—
165Dic214—21no—
166Dic214—21no—
167C8×C107—21no—
168C8×C107—21no—
169C107⋊C8—21no—
170C107⋊C8—21no—
171C22×C107—22no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107C22×D411yescharacteristic
178C2×D107—21no—
179C2×D107—21no—
180C2×D107—22no—
181C2×D107—48no—
182C2×D107—41no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107#12112yes—
188C4×C107—21no—
189C4×C107—21no—
190C4×C107—41no—
191C4×C107—41no—
192C4×C107C22×D411yescharacteristic
193C4×C107C2×#4111yescharacteristic
194Dic107—21no—
195Dic107—41no—
196Dic107—41no—
197Dic107#12112yes—
198Dic107AGL(1,ℤ/8)12yescomplemented
199C2×C107—42no—
200C2×C107—81no—
201C2×C107C2×#12111yescharacteristic
202C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C107#57511yescharacteristic
204D107—22no—
205D107—42no—
206D107—81no—
207#3036—1071nomaximal, Sylow
208C107#303611yescharacteristic, complemented, Sylow
209C2×AGL(1,ℤ/8)—1071no—
210C2×AGL(1,ℤ/8)—1071no—
211#532—1071no—
212C2×C4×D4—1071no—
213#463—1071no—
214#463—1071no—
215#463—1071no—
216#463D10711yescharacteristic, complemented
217#461—1071no—
218#461—1071no—
219#459—1071no—
220#459—1071no—
221#423—1071no—
222#417—1071no—
223#417—1071no—
224C2×#41—1071no—
225C2×#41—1071no—
226C22×D4—1071no—
227C23×C4—2141no—
228AGL(1,ℤ/8)—2144no—
229AGL(1,ℤ/8)—2144no—
230C2×SD16—1071no—
231C2×SD16—1071no—
232C2×SD16—1071no—
233C2×SD16C2×D10711yescharacteristic
234C2×D8—1071no—
235C2×D8—1071no—
236C2×D8—1071no—
237C2×D8C2×D10711yescharacteristic
238C2×M16—1071no—
239C2×M16—1071no—
240C4⋊Q8—1071no—
241#127—1071no—
242#124—1071no—
243#124C2×D10711yescharacteristic, complemented
244#121—2141no—
245#121—2141no—
246C4×D4—1071no—
247C4×D4—1071no—
248C4×D4—1071no—
249C4×D4—2141no—
250C4×D4—2141no—
251C4×D4—2141no—
252C4×D4—2141no—
253C4×D4C2×D10711yescharacteristic
254C2×C4⋊C4—1071no—
255C2×C22⋊C4—2141no—
256C2×C42—1071no—
257C4⋊C8—1071no—
258C4⋊C8—1071no—
259C4⋊C8—1071no—
260C4⋊C8C2×D10711yescharacteristic, complemented
261#103—1071no—
262#103—1071no—
263#103—1071no—
264#103C2×D10711yescharacteristic, complemented
265#102—1071no—
266#102—1071no—
267#102—1071no—
268#102C2×D10711yescharacteristic, complemented
269C24—2141no—
270#41—2142no—
271#41—2142no—
272#41—2142no—
273#41—2142no—
274C2×Q8—1071no—
275C2×Q8C22×D10711yescharacteristic
276C2×D4—1074no—
277C2×D4—1071no—
278C2×D4—1071no—
279C2×D4—2141no—
280C2×D4—2142no—
281C2×D4—2141no—
282C2×D4—2141no—
283C2×D4—2141no—
284C2×D4—2141no—
285C2×D4C22×D10711yescharacteristic
286C2×D4C22×D10711yescharacteristic
287C22×C4—21no—
288C22×C4—1071no—
289C22×C4—1071no—
290C22×C4—1071no—
291C22×C4—4284no—
292C22×C4—2141no—
293C22×C4—2141no—
294C22×C4—2141no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4—2141no—
298SD16—22no—
299SD16—2142no—
300SD16—2142no—
301SD16—2142no—
302D8—22no—
303D8—2142no—
304D8—2142no—
305D8—2142no—
306M16—2142no—
307M16—2142no—
308C2×C8—1071no—
309C2×C8—1071no—
310C2×C8C22×D10711yescharacteristic
311C2×C8C22×D10711yescharacteristic
312C4⋊C4—1071no—
313C4⋊C4—2141no—
314C4⋊C4—2141no—
315C4⋊C4—2141no—
316C4⋊C4C22×D10711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—2141no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C42—1071no—
326C42—1071no—
327C42—1071no—
328C42C22×D10711yescharacteristic
329C23—21no—
330C23—21no—
331C23—1071no—
332C23—4282no—
333C23—4282no—
334C23—2144no—
335C23—2141no—
336C23—2141no—
337Q8—21no—
338Q8—21no—
339Q8—2141no—
340Q8—2141no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—1072no—
345D4—4282no—
346D4—4282no—
347D4—4282no—
348D4—4282no—
349D4—2144no—
350D4—2141no—
351D4—2141no—
352D4—2141no—
353D4D4×D10712yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—1071no—
358C2×C4—1071no—
359C2×C4—1072no—
360C2×C4—1072no—
361C2×C4—1071no—
362C2×C4—4282no—
363C2×C4—4281no—
364C2×C4—4282no—
365C2×C4—4282no—
366C2×C4—4282no—
367C2×C4—4282no—
368C2×C4—4282no—
369C2×C4—2142no—
370C2×C4—2141no—
371C2×C4—2141no—
372C2×C4D4×D10711yescharacteristic
373C2×C4D4×D10711yescharacteristic
374C2×C4C23×D10711yescharacteristic
375C8—21no—
376C8—21no—
377C8—2141no—
378C8—2141no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—1072no—
386C22—4288no—
387C22—4281no—
388C22—4281no—
389C22—4281no—
390C22—4281no—
391C22—4281no—
392C22—2142no—
393C22—2141no—
394C22—2141no—
395C22C2×D4×D10711yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—1072no—
401C4—1072no—
402C4—4281no—
403C4—4281no—
404C4—2141no—
405C4C2×D4×D10711yescharacteristic
406C4#41×D10711yescharacteristic
407C2—42no—
408C2—81no—
409C2—8561no—
410C2—4282no—
411C2—2142no—
412C2#121×D10711yescharacteristic
413C2AGL(1,ℤ/8)×D10711yescharacteristic
414C2#2029342411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#463×C107C2
socleC22×C107C22×D4
hypercenter#463D107
hypocenterC107#3036
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812133C2semidirect
3. #20293176C2semidirect
4. C4×D4×D107C2semidirect
5. #20292843C2semidirect
6. #20292842C2semidirect
7. #20292839C2semidirect
8. #20292831C2semidirect
9. #20292830C2semidirect
10. #20292821C2semidirect
11. #20292820C2semidirect
12. #20292649C2semidirect
13. #20292627C2semidirect
14. #20292625C2semidirect
15. #463×C107C2semidirect
30. #16812107C22semidirect
31. #16812105C22semidirect
32. #16812097C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. Dic107⋊C8C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812033C22semidirect
50. #16812032C22semidirect
51. #16812031C22semidirect
52. #16812030C22semidirect
53. #16812029C22semidirect
56. #124×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
75. D4×D107D4semidirect
198. Dic107AGL(1,ℤ/8)semidirect
208. C107#3036semidirect
216. #463D107semidirect
243. #124C2×D107semidirect
260. C4⋊C8C2×D107semidirect
264. #103C2×D107semidirect
268. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121332
#202931762
C4×D4×D1072
#202928432
#202928422
#202928392
#202928312
#202928302
#202928212
#202928202
#202926492
#202926272
#202926252
#463×C1072
#3036107