Finite Group Databasebeta

← #28348453

Subgroups — #28348453

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#20293284C211yesmaximal, characteristic, complemented
2#20293278C211yesmaximal, characteristic, complemented
3#20292985C211yesmaximal, characteristic, complemented
4#20292962C211yesmaximal, characteristic, complemented
5#20292939C211yesmaximal, characteristic, complemented
6#20292920C211yesmaximal, characteristic, complemented
7#20292873C211yesmaximal, characteristic, complemented
8#20292862C211yesmaximal, characteristic, complemented
9#20292841C211yesmaximal, characteristic, complemented
10#20292772C211yesmaximal, characteristic, complemented
11#20292767C211yesmaximal, characteristic, complemented
12#20292730C211yesmaximal, characteristic, complemented
13#20292696C211yesmaximal, characteristic, complemented
14#103×D107C211yesmaximal, characteristic, complemented
15#417×C107C211yesmaximal, characteristic, complemented
16C2×#13133818—21no—
17C2×Q8×D107C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22#16812108C2211yescharacteristic
23#16812096—21no—
24#16812096C2211yescharacteristic
25#16812094—21no—
26#16812092—21no—
27#16812088—21no—
28#16812088C2211yescharacteristic
29#16812087—21no—
30#16812084C2211yescharacteristic
31#16812083C2211yescharacteristic
32#16812080—21no—
33#16812080C2211yescharacteristic
34Q8×Dic107C2211yescharacteristic
35#16812071C2211yescharacteristic
36C4×C22⋊D107—21no—
37C4×C22⋊D107C2211yescharacteristic
38C4⋊C4×D107C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812042C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51#16812019C2211yescharacteristic
52C2×#41×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54C2×C4⋊C4×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133818—44no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63Q8×D107—22no—
64Q8×D107—41no—
65C2×D428—21no—
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107D411yescharacteristic
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74#13133807—41no—
75#13133805—22no—
76#13133803—41no—
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C8×D107—41no—
80C4⋊Dic107—21no—
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4D411yescharacteristic
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90C22⋊Dic107—21no—
91C22⋊Dic107D411yescharacteristic
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4D411yescharacteristic
96C4×Dic107—21no—
97C4×Dic107C2311yescharacteristic
98#41×C107—22no—
99#41×C107—41no—
100C2×Q8×C107C2311yescharacteristic
101C2×D4×C107—21no—
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C22×C4×C107—21no—
105C22×C4×C107C2311yescharacteristic
106M16×C107—41no—
107C2×C8×C107C2311yescharacteristic
108C2×C8×C107C2311yescharacteristic
109C4⋊C4×C107—21no—
110C4⋊C4×C107C2311yescharacteristic
111C4⋊C4×C107C2311yescharacteristic
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115C22×D107C2×D411yescharacteristic
116Q8×C107—21no—
117Q8×C107#4112yescomplemented
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—42no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—42no—
127C2×C4×C107—42no—
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—22no—
131C22⋊D107—42no—
132C22⋊D107—42no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138C2×Dic107C2×D411yescharacteristic
139D428—21no—
140D428—21no—
141D428—42no—
142C4×D107—21no—
143C4×D107—21no—
144C4×D107—42no—
145C4×D107—42no—
146C4×D107—42no—
147Dic214—21no—
148Dic214—21no—
149Dic214—42no—
150Dic214—42no—
151C8×C107—21no—
152C8×C107—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107—21no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—41no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107AGL(1,ℤ/8)12yes—
168C4×C107—22no—
169C4×C107—21no—
170C4×C107—41no—
171C4×C107—41no—
172C4×C107C2×#4111yescharacteristic
173C4×C107H(5,𝔽2)11yescharacteristic
174Dic107—21no—
175Dic107—21no—
176Dic107—21no—
177Dic107—41no—
178Dic107—41no—
179Dic107—41no—
180C2×C107—41no—
181C2×C107—81no—
182C2×C107#54611yescharacteristic
183C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C107#57811yescharacteristic
185D107—22no—
186D107—81no—
187#3308—1071nomaximal, Sylow
188C107#330811yescharacteristic, complemented, Sylow
189#548—1071no—
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#417D10711yescharacteristic, complemented
203C2×#103—1071no—
204C2×#41—2141no—
205C2×#41C2×D10711yescharacteristic
206C22×Q8—1071no—
207C2×SD16—1071no—
208C2×SD16—1071no—
209C2×D8—1071no—
210C2×M16C2×D10711yescharacteristic
211C22×C8—1071no—
212C4⋊Q8—1071no—
213#124—1071no—
214#124—2141no—
215#123—2141no—
216#123—2141no—
217C22⋊Q8—1071no—
218C22⋊Q8—2141no—
219C22⋊Q8—2141no—
220#121—1071no—
221#121—1071no—
222#121—1071no—
223#121—2141no—
224C4×Q8—1071no—
225C4×D4—1071no—
226C4×D4—1071no—
227C4×D4—2141no—
228C2×C4⋊C4—1071no—
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#103C2×D10711yescharacteristic
239#103C2×D10711yescharacteristic
240#102—1071no—
241#102—1071no—
242#102C2×D10711yescharacteristic
243#102C2×D10711yescharacteristic
244C22⋊C8—1071no—
245C22⋊C8—1071no—
246#97—1071no—
247#41—22no—
248#41—41no—
249#41—4284no—
250C2×Q8—1071no—
251C2×Q8—4281no—
252C2×Q8—2141no—
253C2×Q8—2141no—
254C2×Q8—2142no—
255C2×Q8C22×D10711yescharacteristic
256C2×D4—21no—
257C2×D4—1071no—
258C2×D4—1071no—
259C2×D4—2141no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4C22×D10711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1071no—
266C22×C4—2141no—
267C22×C4—2141no—
268C22×C4—2141no—
269C22×C4C22×D10711yescharacteristic
270SD16—4281no—
271SD16—2142no—
272D8—4281no—
273M16—41no—
274C2×C8—1071no—
275C2×C8—1071no—
276C2×C8—4281no—
277C2×C8C22×D10711yescharacteristic
278C2×C8C22×D10711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—1071no—
281C4⋊C4—1071no—
282C4⋊C4—1071no—
283C4⋊C4—2141no—
284C4⋊C4—2141no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4C22×D10711yescharacteristic
288C4⋊C4C22×D10711yescharacteristic
289C22⋊C4—1071no—
290C22⋊C4—1071no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C42—1071no—
299C42—2141no—
300C23—21no—
301C23—1071no—
302C23—2141no—
303C23D4×D10711yescharacteristic
304Q8—21no—
305Q8—4282no—
306Q8—4282no—
307Q8—2141no—
308Q8—2141no—
309Q8#1313381812yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—4282no—
314D4—4282no—
315D4—4282no—
316D4—2142no—
317D4—2141no—
318D4—2141no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—1071no—
327C2×C4—1071no—
328C2×C4—1071no—
329C2×C4—4282no—
330C2×C4—4282no—
331C2×C4—4282no—
332C2×C4—2141no—
333C2×C4—2141no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4D4×D10711yescharacteristic
338C2×C4C23×D10711yescharacteristic
339C8—21no—
340C8—21no—
341C8—2141no—
342C8—2141no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—1072no—
350C22—4281no—
351C22—4281no—
352C22—4281no—
353C22—2141no—
354C22—2141no—
355C22C2×D4×D10711yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—4281no—
361C4—4281no—
362C4—4281no—
363C4—2141no—
364C4—2141no—
365C4—2141no—
366C4C2×#1313381811yescharacteristic
367C4#1681215411yescharacteristic
368C2—41no—
369C2—81no—
370C2—8561no—
371C2—2142no—
372C2#2029326711yescharacteristic
373C2AGL(1,ℤ/8)×D10711yescharacteristic
374C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
hypercenter#417D107
hypocenterC107#3308
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #20293284C2semidirect
2. #20293278C2semidirect
3. #20292985C2semidirect
4. #20292962C2semidirect
5. #20292939C2semidirect
6. #20292920C2semidirect
7. #20292873C2semidirect
8. #20292862C2semidirect
9. #20292841C2semidirect
10. #20292772C2semidirect
11. #20292767C2semidirect
12. #20292730C2semidirect
13. #20292696C2semidirect
14. #103×D107C2semidirect
15. #417×C107C2semidirect
117. Q8×C107#41semidirect
188. C107#3308semidirect
202. #417D107semidirect
309. Q8#13133818semidirect

Maximal subgroups

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

subgroupindex
#202932842
#202932782
#202929852
#202929622
#202929392
#202929202
#202928732
#202928622
#202928412
#202927722
#202927672
#202927302
#202926962
#103×D1072
#417×C1072
#3308107