Finite Group Databasebeta

← #28348366

Subgroups — #28348366

Order 13696.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293321C211yesmaximal, characteristic, complemented
2#20293287C211yesmaximal, characteristic, complemented
3#20292983C211yesmaximal, characteristic, complemented
4#20292947C211yesmaximal, characteristic, complemented
5#20292933C211yesmaximal, characteristic, complemented
6#20292864C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292845C211yesmaximal, characteristic, complemented
9#20292774C211yesmaximal, characteristic, complemented
10#20292767C211yesmaximal, characteristic, complemented
11#20292714C211yesmaximal, characteristic, complemented
12(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
13#20292621C211yesmaximal, characteristic, complemented
14#103×D107C211yesmaximal, characteristic, complemented
15#485×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C2×D856C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812103—21no—
24#16812100C2211yescharacteristic
25#16812098C2211yescharacteristic
26#16812095—21no—
27#16812091—21no—
28#16812087—21no—
29#16812084C2211yescharacteristic
30#16812084C2211yescharacteristic
31#16812083C2211yescharacteristic
32#16812082—21no—
33#16812082—21no—
34#16812074C2211yescharacteristic
35C4×D428—21no—
36#16812071—21no—
37#16812071C2211yescharacteristic
38C4×C22⋊D107C2211yescharacteristic
39C4⋊C4×D107C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C22⋊Q8×C107C2211yescharacteristic
53#117×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59#13133818—22no—
60#13133818—41no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63Q8×D107—22no—
64Q8×D107—41no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428C2311yescharacteristic
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107D411yescharacteristic
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107—21no—
75C2×C4×D107—21no—
76C2×C4×D107C2311yescharacteristic
77#13133807—22no—
78#13133805—41no—
79D856—41no—
80C2×C107⋊C8—41no—
81C2×C107⋊C8C2311yescharacteristic
82C2×C107⋊C8C2311yescharacteristic
83C8×D107—41no—
84C4⋊Dic107—21no—
85C4⋊Dic107C2311yescharacteristic
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4D411yescharacteristic
90C22⋊Dic107D411yescharacteristic
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97(C2×D107)⋊C4D411yescharacteristic
98C4×Dic107—21no—
99C4×Dic107C2311yescharacteristic
100C2×Q8×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C4⋊C4×C107—21no—
105C4⋊C4×C107C2311yescharacteristic
106C4⋊C4×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C22⋊C4×C107—21no—
109C22⋊C4×C107—21no—
110C42×C107—21no—
111C23×C107C2×D411yescharacteristic
112C22×D107—21no—
113C22×D107—21no—
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2411yescharacteristic
125C22⋊D107—22no—
126C22⋊D107—42no—
127C22⋊D107—81no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132C2×Dic107C2×D411yescharacteristic
133D428—21no—
134D428—21no—
135D428—21no—
136D428—21no—
137D428—42no—
138D428—42no—
139C4×D107—21no—
140C4×D107—21no—
141C4×D107—42no—
142C4×D107—42no—
143C4×D107—81no—
144C4×D107—81no—
145Dic214—21no—
146Dic214—42no—
147Dic214#4112yes—
148C8×C107—21no—
149C8×C107—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—41no—
162C2×D107—41no—
163C2×D107—41no—
164C2×D107#13512yes—
165C4×C107—21no—
166C4×C107—41no—
167C4×C107—41no—
168C4×C107—41no—
169C4×C107—41no—
170C4×C107C2×#4111yescharacteristic
171C4×C107H(5,𝔽2)11yescharacteristic
172Dic107—22no—
173Dic107—21no—
174Dic107—21no—
175Dic107—21no—
176Dic107—41no—
177C2×C107—41no—
178C2×C107#54611yescharacteristic
179C2×C107C2×#13511yescharacteristic
180C2×C107#57711yescharacteristic
181D107—22no—
182D107—81no—
183D107—81no—
184#3301—1071nomaximal, Sylow
185C107#330111yescharacteristic, complemented, Sylow
186#553—1071no—
187#548—1071no—
188#488—1071no—
189#485D10711yescharacteristic, complemented
190#481—1071no—
191#479—1071no—
192#466—1071no—
193#465—1071no—
194#463—1071no—
195#449—1071no—
196#448—1071no—
197C4×SD16—1071no—
198C8×D4—1071no—
199#416—1071no—
200C2×#103—1071no—
201C2×#41—1071no—
202C22×Q8—1071no—
203C2×SD16—1071no—
204C2×SD16—1071no—
205C2×D8—1071no—
206C22×C8—1071no—
207C22×C8—1071no—
208#126—2141no—
209#125—1071no—
210#124—1071no—
211#124—2141no—
212#123—2141no—
213C22⋊Q8—2141no—
214C22⋊Q8C2×D10711yescharacteristic
215#121—1071no—
216#121—1071no—
217#121—1071no—
218#121—2141no—
219#121—2141no—
220C4×Q8—1071no—
221C4×D4—1071no—
222C4×D4—1071no—
223C4×D4—2141no—
224C4×D4—2141no—
225#117C2×D10711yescharacteristic
226C2×C4⋊C4—1071no—
227#107C2×D10711yescharacteristic
228#106—1071no—
229#106C2×D10711yescharacteristic
230C4⋊C8—1071no—
231#103—1071no—
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103C2×D10711yescharacteristic
236#103C2×D10711yescharacteristic
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102—1071no—
241C22⋊C8—1071no—
242C22⋊C8C2×D10711yescharacteristic
243C4×C8—1071no—
244#41—4281no—
245#41—2142no—
246C2×Q8—1071no—
247C2×Q8—4281no—
248C2×Q8—2141no—
249C2×Q8—2142no—
250C2×Q8C22×D10711yescharacteristic
251C2×D4—1071no—
252C2×D4—1071no—
253C2×D4—1071no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2141no—
257C2×D4—2141no—
258C22×C4—1071no—
259C22×C4—2141no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4—2141no—
263C22×C4C22×D10711yescharacteristic
264SD16—4281no—
265SD16—2142no—
266D8—4281no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—4281no—
270C2×C8—4281no—
271C2×C8C22×D10711yescharacteristic
272C2×C8C22×D10711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—2141no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4C22×D10711yescharacteristic
281C4⋊C4C22×D10711yescharacteristic
282C4⋊C4C22×D10711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1071no—
286C22⋊C4—1071no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C42—21no—
294C42—1071no—
295C42—2141no—
296C23—1071no—
297C23—2141no—
298C23—2141no—
299C23D4×D10711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—1072no—
303Q8—4282no—
304Q8—2141no—
305D4—8561no—
306D4—4282no—
307D4—4282no—
308D4—4282no—
309D4—2142no—
310D4—2141no—
311D4—2141no—
312D4—2141no—
313D4—2141no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—8561no—
321C2×C4—8561no—
322C2×C4—1071no—
323C2×C4—1071no—
324C2×C4—1071no—
325C2×C4—4282no—
326C2×C4—4282no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4D4×D10711yescharacteristic
332C2×C4C23×D10711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2141no—
336C8—2141no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—1072no—
341C22—4281no—
342C22—4281no—
343C22—4281no—
344C22—4281no—
345C22—4281no—
346C22—4281no—
347C22—2141no—
348C22—2141no—
349C22C2×D4×D10711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—4281no—
356C4—2141no—
357C4—2141no—
358C4—2141no—
359C4—2142no—
360C4#41×D10711yescharacteristic
361C4#1681215511yescharacteristic
362C2—41no—
363C2—8561no—
364C2—8561no—
365C2—2142no—
366C2#2029326911yescharacteristic
367C2#135×D10711yescharacteristic
368C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#485×C107C2
socleC22×C107C22×D4
hypercenter#485D107
hypocenterC107#3301
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #20293321C2semidirect
2. #20293287C2semidirect
3. #20292983C2semidirect
4. #20292947C2semidirect
5. #20292933C2semidirect
6. #20292864C2semidirect
7. #20292858C2semidirect
8. #20292845C2semidirect
9. #20292774C2semidirect
10. #20292767C2semidirect
11. #20292714C2semidirect
12. (C22⋊D107)⋊C8C2semidirect
13. #20292621C2semidirect
14. #103×D107C2semidirect
15. #485×C107C2semidirect
185. C107#3301semidirect
189. #485D107semidirect

Maximal subgroups

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

subgroupindex
#202933212
#202932872
#202929832
#202929472
#202929332
#202928642
#202928582
#202928452
#202927742
#202927672
#202927142
(C22⋊D107)⋊C82
#202926212
#103×D1072
#485×C1072
#3301107