Finite Group Databasebeta

← #28348447

Subgroups — #28348447

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#20293287C211yesmaximal, characteristic, complemented
2#20293277C211yesmaximal, characteristic, complemented
3#20292988C211yesmaximal, characteristic, complemented
4#20292960C211yesmaximal, characteristic, complemented
5#20292942C211yesmaximal, characteristic, complemented
6#20292921C211yesmaximal, characteristic, complemented
7#20292862C211yesmaximal, characteristic, complemented
8#20292844C211yesmaximal, characteristic, complemented
9#20292770C211yesmaximal, characteristic, complemented
10#20292764C211yesmaximal, characteristic, complemented
11#20292724C211yesmaximal, characteristic, complemented
12#20292696C211yesmaximal, characteristic, complemented
13#20292629C211yesmaximal, characteristic, complemented
14#103×D107C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C2×#13133817—21no—
18C2×Q8×D107C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22#16812108C2211yescharacteristic
23#16812098C2211yescharacteristic
24#16812095—21no—
25#16812091—21no—
26#16812090—21no—
27#16812087—21no—
28#16812086—21no—
29#16812084C2211yescharacteristic
30#16812082—21no—
31#16812080C2211yescharacteristic
32#16812074C2211yescharacteristic
33D4×Dic107C2211yescharacteristic
34#16812071—21no—
35C4×C22⋊D107C2211yescharacteristic
36C2×C4⋊Dic107C2211yescharacteristic
37C4⋊C4×D107C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812042C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812034C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51#16812019C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133818—22no—
60#13133818—41no—
61#13133817—44no—
62C2×Dic214—21no—
63C2×Dic214—21no—
64C2×Dic214C2311yescharacteristic
65Q8×D107—22no—
66Q8×D107—41no—
67C2×D428—21no—
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107D411yescharacteristic
72C22×Dic107—21no—
73C22×Dic107—21no—
74C2×C4×D107—21no—
75C2×C4×D107—21no—
76C2×C4×D107—21no—
77C2×C4×D107C2311yescharacteristic
78#13133806—22no—
79#13133803—41no—
80C2×C107⋊C8C2311yescharacteristic
81C2×C107⋊C8C2311yescharacteristic
82C8×D107—41no—
83C4⋊Dic107—21no—
84C4⋊Dic107C2311yescharacteristic
85C4⋊Dic107C2311yescharacteristic
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4D411yescharacteristic
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92C22⋊Dic107D411yescharacteristic
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4D411yescharacteristic
97C4×Dic107—21no—
98C4×Dic107C2311yescharacteristic
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103SD16×C107—41no—
104M16×C107—41no—
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107—21no—
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C22⋊C4×C107—21no—
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115C22×D107C2×D411yescharacteristic
116Q8×C107—21no—
117Q8×C107—21no—
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×C107C2×D411yescharacteristic
127C2×C4×C107C2411yescharacteristic
128C22⋊D107—22no—
129C22⋊D107—42no—
130C22⋊D107—42no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—42no—
135C2×Dic107—42no—
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—42no—
149Dic214—42no—
150Dic214#4112yescomplemented
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—21no—
169C4×C107—41no—
170C4×C107—41no—
171C4×C107—41no—
172C4×C107C2×#4111yescharacteristic
173C4×C107H(5,𝔽2)11yescharacteristic
174Dic107—22no—
175Dic107—21no—
176Dic107—21no—
177Dic107—21no—
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#468D10711yescharacteristic, complemented
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—1071no—
205C2×#41—2141no—
206C22×Q8—1071no—
207C2×SD16—1071no—
208C2×SD16C2×D10711yescharacteristic
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—2141no—
218C22⋊Q8—2141no—
219C22⋊Q8C2×D10711yescharacteristic
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#107C2×D10711yescharacteristic
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⋊C8—1071no—
246#97—1071no—
247#41—4281no—
248#41—4284no—
249#41—2142no—
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—1071no—
264C22×C4—2141no—
265C22×C4—2141no—
266C22×C4—2141no—
267C22×C4—2141no—
268C22×C4—2141no—
269C22×C4C22×D10711yescharacteristic
270SD16—41no—
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—21no—
290C22⋊C4—21no—
291C22⋊C4—1071no—
292C22⋊C4—1071no—
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—21no—
306Q8—1072no—
307Q8—4282no—
308Q8—4282no—
309Q8—2141no—
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—1071no—
325C2×C4—1071no—
326C2×C4—1071no—
327C2×C4—4282no—
328C2×C4—4282no—
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—41no—
358C4—41no—
359C4—41no—
360C4—4281no—
361C4—4281no—
362C4—2141no—
363C4—2141no—
364C4—2141no—
365C4—2142no—
366C4#41×D10711yescharacteristic
367C4#1681215411yescharacteristic
368C2—41no—
369C2—81no—
370C2—8561no—
371C2—2142no—
372C2#2029326611yescharacteristic
373C2AGL(1,ℤ/8)×D10711yescharacteristic
374C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
hypercenter#468D107
hypocenterC107#3308
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #20293287C2semidirect
2. #20293277C2semidirect
3. #20292988C2semidirect
4. #20292960C2semidirect
5. #20292942C2semidirect
6. #20292921C2semidirect
7. #20292862C2semidirect
8. #20292844C2semidirect
9. #20292770C2semidirect
10. #20292764C2semidirect
11. #20292724C2semidirect
12. #20292696C2semidirect
13. #20292629C2semidirect
14. #103×D107C2semidirect
15. #468×C107C2semidirect
150. Dic214#41semidirect
188. C107#3308semidirect
195. #468D107semidirect

Maximal subgroups

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

subgroupindex
#202932872
#202932772
#202929882
#202929602
#202929422
#202929212
#202928622
#202928442
#202927702
#202927642
#202927242
#202926962
#202926292
#103×D1072
#468×C1072
#3308107