Finite Group Databasebeta

← #28348446

Subgroups — #28348446

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#20293286C211yesmaximal, characteristic, complemented
2#20292989C211yesmaximal, characteristic, complemented
3#20292962C211yesmaximal, characteristic, complemented
4#20292940C211yesmaximal, characteristic, complemented
5#20292919C211yesmaximal, characteristic, complemented
6#20292871C211yesmaximal, characteristic, complemented
7#20292861C211yesmaximal, characteristic, complemented
8#20292842C211yesmaximal, characteristic, complemented
9#20292770C211yesmaximal, characteristic, complemented
10#20292765C211yesmaximal, characteristic, complemented
11#20292727C211yesmaximal, characteristic, complemented
12#20292699C211yesmaximal, characteristic, complemented
13#20292626C211yesmaximal, characteristic, complemented
14C2×#16812032C211yesmaximal, characteristic, complemented
15#547×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21C2×#13133802C2211yescharacteristic
22C22×C107⋊C8C2211yescharacteristic
23#16812097C2211yescharacteristic
24#16812095—21no—
25#16812088C2211yescharacteristic
26#16812087—21no—
27#16812083C2211yescharacteristic
28#16812081—21no—
29C4×Dic214C2211yescharacteristic
30(C22⋊D107)⋊C4—21no—
31#16812042C2211yescharacteristic
32#16812042C2211yescharacteristic
33#16812042C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35#16812032C2211yescharacteristic
36#16812032C2211yescharacteristic
37#16812032C2211yescharacteristic
38#16812032C2211yescharacteristic
39#16812032C2211yescharacteristic
40#16812032C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812029C2211yescharacteristic
43#16812029C2211yescharacteristic
44#16812029C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic
47#16812020C2211yescharacteristic
48C2×#41×C107—21no—
49C4⋊Q8×C107C2211yescharacteristic
50#123×C107—21no—
51#123×C107—21no—
52C22⋊Q8×C107—21no—
53#121×C107C2211yescharacteristic
54#121×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56C4×D4×C107C2211yescharacteristic
57C2×C4⋊C4×C107C2211yescharacteristic
58C2×C4⋊C4×C107C2211yescharacteristic
59#13133818—22no—
60#13133818—41no—
61C2×Dic214—22no—
62C2×Dic214—21no—
63C2×Dic214—41no—
64C2×Dic214C2311yescharacteristic
65C2×Dic214C2311yescharacteristic
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C22×Dic107—21no—
70C2×C4×D107—21no—
71#13133806—22no—
72#13133806—41no—
73#13133803—41no—
74#13133802—41no—
75C2×C107⋊C8—41no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87C4×Dic107—21no—
88#41×C107—44no—
89C2×Q8×C107—21no—
90C2×D4×C107—21no—
91C2×D4×C107—21no—
92C2×D4×C107D411yescharacteristic
93C2×D4×C107C2311yescharacteristic
94C22×C4×C107—21no—
95C22×C4×C107—21no—
96C22×C4×C107—21no—
97C22×C4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C4⋊C4×C107—21no—
100C4⋊C4×C107—21no—
101C4⋊C4×C107—21no—
102C4⋊C4×C107D411yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C22⋊C4×C107—21no—
107C22⋊C4×C107—21no—
108C22⋊C4×C107—21no—
109C22⋊C4×C107D411yescharacteristic
110C22⋊C4×C107D411yescharacteristic
111C42×C107C2311yescharacteristic
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C23×C107C2×D411yescharacteristic
115C22×D107—21no—
116Q8×C107—42no—
117D4×C107—22no—
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—42no—
121D4×C107—42no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—42no—
130C2×C4×C107—42no—
131C2×C4×C107—42no—
132C2×C4×C107C2×D411yescharacteristic
133C2×C4×C107C2×D411yescharacteristic
134C2×C4×C107C2×D411yescharacteristic
135C2×C4×C107C2×D411yescharacteristic
136C2×C4×C107C2411yescharacteristic
137C22⋊D107—42no—
138C22⋊D107—42no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—42no—
143D428—21no—
144D428—21no—
145C4×D107—42no—
146Dic214—21no—
147Dic214—21no—
148Dic214—21no—
149Dic214—42no—
150Dic214#4112yescomplemented
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—21no—
171C4×C107—21no—
172C4×C107—41no—
173C4×C107—41no—
174C4×C107—41no—
175C4×C107C2×#4111yescharacteristic
176C4×C107H(5,𝔽2)11yescharacteristic
177Dic107—22no—
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#548—1071no—
190#547D10711yescharacteristic, complemented
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—21no—
205C2×#41—1071no—
206C22×Q8—1071no—
207C2×SD16—1071no—
208C2×SD16—1071no—
209C2×D8—1071no—
210C2×M16—1071no—
211C22×C8—1071no—
212C4⋊Q8C2×D10711yescharacteristic
213#124—1071no—
214#124—2141no—
215#123—21no—
216#123—21no—
217C22⋊Q8—21no—
218C22⋊Q8—1071no—
219C22⋊Q8—2141no—
220#121—1071no—
221#121—2141no—
222#121C2×D10711yescharacteristic
223#121C2×D10711yescharacteristic
224C4×Q8—1071no—
225C4×D4—2141no—
226C4×D4C2×D10711yescharacteristic
227C4×D4C2×D10711yescharacteristic
228C2×C4⋊C4C2×D10711yescharacteristic
229C2×C4⋊C4C2×D10711yescharacteristic
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—44no—
248#41—4281no—
249#41—2142no—
250C2×Q8—21no—
251C2×Q8—1071no—
252C2×Q8—1071no—
253C2×Q8—4281no—
254C2×Q8—2142no—
255C2×Q8—2141no—
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—21no—
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—21no—
282C4⋊C4—1071no—
283C4⋊C4—2141no—
284C4⋊C4—2141no—
285C4⋊C4C22⋊D10711yescharacteristic
286C4⋊C4C22×D10711yescharacteristic
287C4⋊C4C22×D10711yescharacteristic
288C4⋊C4C22×D10711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4C22⋊D10711yescharacteristic
297C22⋊C4C22⋊D10711yescharacteristic
298C42—2141no—
299C42C22×D10711yescharacteristic
300C23—21no—
301C23—2141no—
302C23C2×C22⋊D10711yescharacteristic
303C23C2×C22⋊D10711yescharacteristic
304Q8—42no—
305Q8—1072no—
306Q8—4282no—
307Q8—2141no—
308Q8—2141no—
309Q8—2141no—
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—42no—
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#1681213512yes—
355C22C22×C22⋊D10711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—4281no—
364C4—4281no—
365C4—2142no—
366C4#41×D10711yescharacteristic
367C4#1681215511yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—8561no—
372C2#2029326511yescharacteristic
373C2C2×#1681213511yescharacteristic
374C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3308
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20293286C2semidirect
2. #20292989C2semidirect
3. #20292962C2semidirect
4. #20292940C2semidirect
5. #20292919C2semidirect
6. #20292871C2semidirect
7. #20292861C2semidirect
8. #20292842C2semidirect
9. #20292770C2semidirect
10. #20292765C2semidirect
11. #20292727C2semidirect
12. #20292699C2semidirect
13. #20292626C2semidirect
14. C2×#16812032C2semidirect
15. #547×C107C2semidirect
150. Dic214#41semidirect
188. C107#3308semidirect
190. #547D107semidirect

Maximal subgroups

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

subgroupindex
#202932862
#202929892
#202929622
#202929402
#202929192
#202928712
#202928612
#202928422
#202927702
#202927652
#202927272
#202926992
#202926262
C2×#168120322
#547×C1072
#3308107