Finite Group Databasebeta

← #28348445

Subgroups — #28348445

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#20293277C211yesmaximal, characteristic, complemented
3#20292985C211yesmaximal, characteristic, complemented
4#20292963C211yesmaximal, characteristic, complemented
5#20292939C211yesmaximal, characteristic, complemented
6#20292920C211yesmaximal, characteristic, complemented
7#20292871C211yesmaximal, characteristic, complemented
8#20292860C211yesmaximal, characteristic, complemented
9#20292840C211yesmaximal, characteristic, complemented
10#20292764C211yesmaximal, characteristic, complemented
11#20292728C211yesmaximal, characteristic, complemented
12#20292701C211yesmaximal, characteristic, complemented
13#20292628C211yesmaximal, characteristic, complemented
14#103×D107C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C2×Q8×D107C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133802C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812096—21no—
23#16812096C2211yescharacteristic
24#16812091—21no—
25#16812090—21no—
26#16812088—21no—
27#16812088C2211yescharacteristic
28#16812086—21no—
29#16812080—21no—
30#16812080C2211yescharacteristic
31#16812080C2211yescharacteristic
32Q8×Dic107C2211yescharacteristic
33D4×Dic107C2211yescharacteristic
34C4×C22⋊D107—21no—
35C4×C22⋊D107C2211yescharacteristic
36C2×C4⋊Dic107C2211yescharacteristic
37C4⋊C4×D107C2211yescharacteristic
38#16812043C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812030C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51#16812019C2211yescharacteristic
52C2×#41×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C2×D8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59#13133817—44no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63Q8×D107—22no—
64Q8×D107—41no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67C2×C22⋊D107D411yescharacteristic
68C22×Dic107—21no—
69C22×Dic107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107C2311yescharacteristic
73#13133806—41no—
74#13133802—41no—
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C8×D107—41no—
78C4⋊Dic107—21no—
79C4⋊Dic107—21no—
80C4⋊Dic107C2311yescharacteristic
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⋊Dic107—21no—
92C22⋊Dic107—21no—
93C22⋊Dic107D411yescharacteristic
94(C2×D107)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97#41×C107—22no—
98#41×C107—41no—
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—21no—
101C2×D4×C107—21no—
102C2×D4×C107C2311yescharacteristic
103C2×D4×C107C2311yescharacteristic
104C22×C4×C107—21no—
105C22×C4×C107C2311yescharacteristic
106SD16×C107—22no—
107D8×C107—41no—
108C2×C8×C107C2311yescharacteristic
109C2×C8×C107C2311yescharacteristic
110C4⋊C4×C107C2311yescharacteristic
111C22⋊C4×C107—21no—
112C23×C107—21no—
113C23×C107—21no—
114C23×C107C2×D411yescharacteristic
115C22×D107C2×D411yescharacteristic
116Q8×C107—21no—
117Q8×C107#4112yescomplemented
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—21no—
121D4×C107—21no—
122D4×C107—42no—
123D4×C107—42no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—42no—
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—22no—
132C22⋊D107—42no—
133C22⋊D107—42no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—42no—
139C2×Dic107—42no—
140C2×Dic107C2×D411yescharacteristic
141C2×Dic107C2×D411yescharacteristic
142C2×Dic107C2×D411yescharacteristic
143C4×D107—21no—
144C4×D107—21no—
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×C107—41no—
162C22×C107—41no—
163C22×C107—41no—
164C22×C107C22×D411yescharacteristic
165C2×D107—21no—
166C2×D107—21no—
167C2×D107AGL(1,ℤ/8)12yes—
168C4×C107—22no—
169C4×C107—21no—
170C4×C107—41no—
171C4×C107C2×#4111yescharacteristic
172C4×C107H(5,𝔽2)11yescharacteristic
173Dic107—21no—
174Dic107—21no—
175Dic107—21no—
176Dic107—41no—
177Dic107—41no—
178Dic107—41no—
179Dic107—41no—
180C2×C107—41no—
181C2×C107—81no—
182C2×C107—81no—
183C2×C107#54611yescharacteristic
184C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C107#57811yescharacteristic
186D107—22no—
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#449D10711yescharacteristic, complemented
199#448—1071no—
200#440—1071no—
201#435—1071no—
202#417—1071no—
203C2×#103—1071no—
204C2×#41—2141no—
205C2×#41C2×D10711yescharacteristic
206C22×Q8—1071no—
207C2×SD16—1071no—
208C2×SD16C2×D10711yescharacteristic
209C2×D8C2×D10711yescharacteristic
210C2×M16—1071no—
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—2141no—
223#121C2×D10711yescharacteristic
224C4×Q8—1071no—
225C4×D4—1071no—
226C4×D4—1071no—
227C4×D4—2141no—
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#103C2×D10711yescharacteristic
240#102—1071no—
241#102—1071no—
242#102—1071no—
243#102C2×D10711yescharacteristic
244C22⋊C8—1071no—
245C22⋊C8C2×D10711yescharacteristic
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—21no—
258C2×D4—1071no—
259C2×D4—2141no—
260C2×D4—2141no—
261C2×D4C22×D10711yescharacteristic
262C2×D4C22×D10711yescharacteristic
263C22×C4—21no—
264C22×C4—1071no—
265C22×C4—2141no—
266C22×C4—2141no—
267C22×C4—2141no—
268C22×C4—2141no—
269C22×C4C22×D10711yescharacteristic
270SD16—22no—
271SD16—4281no—
272D8—41no—
273M16—4281no—
274C2×C8—1071no—
275C2×C8—1071no—
276C2×C8—4281no—
277C2×C8C22×D10711yescharacteristic
278C2×C8C22×D10711yescharacteristic
279C4⋊C4—1071no—
280C4⋊C4—1071no—
281C4⋊C4—1071no—
282C4⋊C4—1071no—
283C4⋊C4—2141no—
284C4⋊C4—2141no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4—2141no—
288C4⋊C4C22×D10711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—1071no—
291C22⋊C4—1071no—
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—21no—
302C23—1071no—
303C23D4×D10711yescharacteristic
304Q8—21no—
305Q8—4282no—
306Q8—4282no—
307Q8—2141no—
308Q8—2141no—
309Q8#1313381712yescomplemented
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—42no—
316D4—4282no—
317D4—4282no—
318D4—2142no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—1071no—
325C2×C4—1071no—
326C2×C4—1071no—
327C2×C4—4282no—
328C2×C4—4282no—
329C2×C4—4282no—
330C2×C4—4282no—
331C2×C4—2141no—
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—41no—
350C22—41no—
351C22—41no—
352C22—1072no—
353C22—2141no—
354C22—2141no—
355C22C2×D4×D10711yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—4281no—
360C4—4281no—
361C4—4281no—
362C4—4281no—
363C4—2141no—
364C4—2141no—
365C4—2141no—
366C4C2×#1313381711yescharacteristic
367C4#1681215411yescharacteristic
368C2—41no—
369C2—81no—
370C2—81no—
371C2—2142no—
372C2#2029326411yescharacteristic
373C2AGL(1,ℤ/8)×D10711yescharacteristic
374C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107#3308
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #20293284C2semidirect
2. #20293277C2semidirect
3. #20292985C2semidirect
4. #20292963C2semidirect
5. #20292939C2semidirect
6. #20292920C2semidirect
7. #20292871C2semidirect
8. #20292860C2semidirect
9. #20292840C2semidirect
10. #20292764C2semidirect
11. #20292728C2semidirect
12. #20292701C2semidirect
13. #20292628C2semidirect
14. #103×D107C2semidirect
15. #449×C107C2semidirect
117. Q8×C107#41semidirect
188. C107#3308semidirect
198. #449D107semidirect
309. Q8#13133817semidirect

Maximal subgroups

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

subgroupindex
#202932842
#202932772
#202929852
#202929632
#202929392
#202929202
#202928712
#202928602
#202928402
#202927642
#202927282
#202927012
#202926282
#103×D1072
#449×C1072
#3308107