Finite Group Databasebeta

← #28348452

Subgroups — #28348452

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293281C211yesmaximal, characteristic, complemented
2#20292986C211yesmaximal, characteristic, complemented
3#20292960C211yesmaximal, characteristic, complemented
4#20292938C211yesmaximal, characteristic, complemented
5#20292922C211yesmaximal, characteristic, complemented
6#20292875C211yesmaximal, characteristic, complemented
7#20292861C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292772C211yesmaximal, characteristic, complemented
10#20292766C211yesmaximal, characteristic, complemented
11#20292725C211yesmaximal, characteristic, complemented
12#20292699C211yesmaximal, characteristic, complemented
13#20292628C211yesmaximal, characteristic, complemented
14C2×#16812033C211yesmaximal, characteristic, complemented
15#548×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C2×#13133805C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×#13133802C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22#16812109C2211yescharacteristic
23#16812093—21no—
24#16812093—21no—
25#16812089—21no—
26#16812083C2211yescharacteristic
27#16812083C2211yescharacteristic
28C4×D428C2211yescharacteristic
29C2×C4⋊Dic107C2211yescharacteristic
30C2×C4⋊Dic107C2211yescharacteristic
31#16812042C2211yescharacteristic
32#16812042C2211yescharacteristic
33#16812042C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35#16812033C2211yescharacteristic
36#16812033C2211yescharacteristic
37#16812033C2211yescharacteristic
38#16812033C2211yescharacteristic
39#16812033C2211yescharacteristic
40#16812033C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812030C2211yescharacteristic
43#16812030C2211yescharacteristic
44#16812029C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic
47#16812020C2211yescharacteristic
48C2×#41×C107C2211yescharacteristic
49C22×Q8×C107C2211yescharacteristic
50#124×C107—21no—
51#124×C107C2211yescharacteristic
52C22⋊Q8×C107—21no—
53C22⋊Q8×C107C2211yescharacteristic
54#121×C107—21no—
55#121×C107C2211yescharacteristic
56C4×Q8×C107C2211yescharacteristic
57C4×D4×C107—21no—
58C4×D4×C107C2211yescharacteristic
59#13133817—44no—
60C2×Dic214—21no—
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C22×Dic107—21no—
65C22×Dic107—21no—
66C2×C4×D107—21no—
67#13133805—22no—
68#13133805—41no—
69#13133803—41no—
70#13133802—41no—
71C2×C107⋊C8—41no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107—21no—
79C4⋊Dic107C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85#41×C107—22no—
86#41×C107—41no—
87C2×Q8×C107—21no—
88C2×Q8×C107—22no—
89C2×Q8×C107—41no—
90C2×Q8×C107C2311yescharacteristic
91C2×Q8×C107C2311yescharacteristic
92C2×D4×C107—21no—
93C2×D4×C107—21no—
94C2×D4×C107D411yescharacteristic
95C2×D4×C107C2311yescharacteristic
96C22×C4×C107—21no—
97C22×C4×C107—21no—
98C22×C4×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C4⋊C4×C107—21no—
101C4⋊C4×C107—21no—
102C4⋊C4×C107D411yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C22⋊C4×C107—21no—
108C22⋊C4×C107D411yescharacteristic
109C22⋊C4×C107D411yescharacteristic
110C42×C107—21no—
111C42×C107C2311yescharacteristic
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C23×C107C2×D411yescharacteristic
115C22×D107—21no—
116Q8×C107—21no—
117Q8×C107—21no—
118Q8×C107—21no—
119Q8×C107—42no—
120Q8×C107#4112yescomplemented
121D4×C107—22no—
122D4×C107—21no—
123D4×C107—21no—
124D4×C107—42no—
125D4×C107—42no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107—42no—
134C2×C4×C107—42no—
135C2×C4×C107C2×D411yescharacteristic
136C2×C4×C107C2×D411yescharacteristic
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2411yescharacteristic
140C22⋊D107—42no—
141C22⋊D107—42no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—42no—
146C2×Dic107—42no—
147D428—21no—
148D428—21no—
149C4×D107—42no—
150Dic214—42no—
151C107⋊C8—21no—
152C107⋊C8—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107—21no—
158C22×C107—21no—
159C22×C107—21no—
160C22×C107—41no—
161C22×C107—41no—
162C22×C107—41no—
163C22×C107AGL(1,ℤ/8)12yes—
164C22×C107C22×D411yescharacteristic
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—41no—
168C4×C107—21no—
169C4×C107—21no—
170C4×C107—22no—
171C4×C107—21no—
172C4×C107—21no—
173C4×C107—41no—
174C4×C107—41no—
175C4×C107C2×#4111yescharacteristic
176C4×C107H(5,𝔽2)11yescharacteristic
177Dic107—41no—
178Dic107—41no—
179Dic107—41no—
180C2×C107—22no—
181C2×C107—41no—
182C2×C107—81no—
183C2×C107#54611yescharacteristic
184C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C107#57811yescharacteristic
186D107—81no—
187#3308—1071nomaximal, Sylow
188C107#330811yescharacteristic, complemented, Sylow
189#548D10711yescharacteristic, complemented
190#547—1071no—
191#489—1071no—
192#484—1071no—
193#480—1071no—
194#477—1071no—
195#468—1071no—
196#466—1071no—
197#463—1071no—
198#449—1071no—
199#448—1071no—
200#440—1071no—
201#435—1071no—
202#417—1071no—
203C2×#103—1071no—
204C2×#41—2141no—
205C2×#41C2×D10711yescharacteristic
206C22×Q8C2×D10711yescharacteristic
207C2×SD16—1071no—
208C2×SD16—1071no—
209C2×D8—1071no—
210C2×M16—1071no—
211C22×C8—1071no—
212C4⋊Q8—1071no—
213#124—21no—
214#124C2×D10711yescharacteristic
215#123—2141no—
216#123—2141no—
217C22⋊Q8—21no—
218C22⋊Q8—2141no—
219C22⋊Q8C2×D10711yescharacteristic
220#121—21no—
221#121—1071no—
222#121—1071no—
223#121C2×D10711yescharacteristic
224C4×Q8C2×D10711yescharacteristic
225C4×D4—21no—
226C4×D4—1071no—
227C4×D4C2×D10711yescharacteristic
228C2×C4⋊C4—1071no—
229C2×C4⋊C4—1071no—
230#107—1071no—
231#107—1071no—
232#107—1071no—
233C4⋊C8—1071no—
234#103—1071no—
235#103—1071no—
236#103—1071no—
237#103—1071no—
238#103—1071no—
239#103—1071no—
240#102—1071no—
241#102—1071no—
242#102—1071no—
243#102—1071no—
244C22⋊C8—1071no—
245C22⋊C8—1071no—
246#97—1071no—
247#41—22no—
248#41—41no—
249#41—4284no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—2141no—
254C2×Q8C22×D10711yescharacteristic
255C2×Q8C22×D10711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—1071no—
259C2×D4—2141no—
260C2×D4—2141no—
261C2×D4C22⋊D10711yescharacteristic
262C2×D4C22×D10711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—2141no—
266C22×C4—2141no—
267C22×C4—2141no—
268C22×C4C22×D10711yescharacteristic
269C22×C4C22×D10711yescharacteristic
270SD16—4281no—
271SD16—2142no—
272D8—4281no—
273M16—4281no—
274C2×C8—1071no—
275C2×C8—1071no—
276C2×C8—1071no—
277C2×C8—1071no—
278C2×C8—4281no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—1071no—
282C4⋊C4—1071no—
283C4⋊C4—1071no—
284C4⋊C4—2141no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4C22⋊D10711yescharacteristic
288C4⋊C4C22×D10711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4C22⋊D10711yescharacteristic
297C22⋊C4C22⋊D10711yescharacteristic
298C42—21no—
299C42C22×D10711yescharacteristic
300C23—21no—
301C23—2141no—
302C23C2×C22⋊D10711yescharacteristic
303C23C2×C22⋊D10711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—4282no—
309Q8#1313381712yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—4282no—
316D4—4282no—
317D4—2141no—
318D4—2141no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—4282no—
329C2×C4—4282no—
330C2×C4—4282no—
331C2×C4—2141no—
332C2×C4—2141no—
333C2×C4—2141no—
334C2×C4C2×C22⋊D10711yescharacteristic
335C2×C4C2×C22⋊D10711yescharacteristic
336C2×C4C2×C22⋊D10711yescharacteristic
337C2×C4C2×C22⋊D10711yescharacteristic
338C2×C4C23×D10711yescharacteristic
339C8—2141no—
340C8—2141no—
341C8—2141no—
342C8—2141no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—4281no—
352C22—4281no—
353C22—4281no—
354C22#1681213412yes—
355C22C22×C22⋊D10711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—4281no—
364C4—4281no—
365C4—4281no—
366C4C2×#1313381711yescharacteristic
367C4#1681215411yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—8561no—
372C2#2029327111yescharacteristic
373C2C2×#1681213411yescharacteristic
374C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#548×C107C2
socleC22×C107C22×D4
hypercenter#548D107
hypocenterC107#3308
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20293281C2semidirect
2. #20292986C2semidirect
3. #20292960C2semidirect
4. #20292938C2semidirect
5. #20292922C2semidirect
6. #20292875C2semidirect
7. #20292861C2semidirect
8. #20292839C2semidirect
9. #20292772C2semidirect
10. #20292766C2semidirect
11. #20292725C2semidirect
12. #20292699C2semidirect
13. #20292628C2semidirect
14. C2×#16812033C2semidirect
15. #548×C107C2semidirect
120. Q8×C107#41semidirect
188. C107#3308semidirect
189. #548D107semidirect
309. Q8#13133817semidirect

Maximal subgroups

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

subgroupindex
#202932812
#202929862
#202929602
#202929382
#202929222
#202928752
#202928612
#202928392
#202927722
#202927662
#202927252
#202926992
#202926282
C2×#168120332
#548×C1072
#3308107