Finite Group Databasebeta

← #28349209

Subgroups — #28349209

Order 13696.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293363C211yesmaximal, characteristic, complemented
2#20293333C211yesmaximal, characteristic, complemented
3#20292985C211yesmaximal, characteristic, complemented
4#20292980C211yesmaximal, characteristic, complemented
5#20292931C211yesmaximal, characteristic, complemented
6#20292925C211yesmaximal, characteristic, complemented
7#20292905C211yesmaximal, characteristic, complemented
8#20292841C211yesmaximal, characteristic, complemented
9#20292841C211yesmaximal, characteristic, complemented
10#20292819C211yesmaximal, characteristic
11#20292755C211yesmaximal, characteristic, complemented
12#20292736C211yesmaximal, characteristic, complemented
13#20292729C211yesmaximal, characteristic, complemented
14#20292707C211yesmaximal, characteristic, complemented
15#490×C107C211yesmaximal, characteristic, complemented
16C2×#13133807C2211yescharacteristic
17C2×#13133803C2211yescharacteristic
18C2×#13133803C2211yescharacteristic
19#16812108C2211yescharacteristic
20#16812100C2211yescharacteristic
21#16812096—21no—
22#16812096C2211yescharacteristic
23#16812096C2211yescharacteristic
24#16812090—21no—
25#16812090—21no—
26#16812089—21no—
27#16812087—21no—
28#16812087—21no—
29C4×Dic214C2211yescharacteristic
30C4×Dic214C2211yescharacteristic
31C4×D428C2211yescharacteristic
32D4×Dic107—21no—
33D4×Dic107C2211yescharacteristic
34D4×Dic107C2211yescharacteristic
35#16812071C2211yescharacteristic
36#16812071C2211yescharacteristic
37C4⋊C4×D107—21no—
38#16812043C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812040C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812030C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52#127×C107C2211yescharacteristic
53#125×C107C2211yescharacteristic
54#102×C107C2211yescharacteristic
55#102×C107C2211yescharacteristic
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214D411yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×D428C2311yescharacteristic
63C22×Dic107—21no—
64C22×Dic107—21no—
65C2×C4×D107—21no—
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68#13133807—22no—
69#13133803—22no—
70#13133803—22no—
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C4⋊Dic107—21no—
74C4⋊Dic107D411yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4D411yescharacteristic
81Dic107⋊C4D411yescharacteristic
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C4×Dic107C2311yescharacteristic
94C2×D4×C107—21no—
95C2×D4×C107—21no—
96C2×D4×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C4⋊C4×C107—21no—
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106C23×C107—21no—
107C22×D107—21no—
108D4×C107—21no—
109D4×C107—21no—
110D4×C107—21no—
111D4×C107—21no—
112D4×C107—42no—
113D4×C107—81no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2411yescharacteristic
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—81no—
123C2×Dic107—81no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128D428—21no—
129D428#4112yes—
130C4×D107—42no—
131C4×D107—81no—
132C4×D107—81no—
133Dic214—21no—
134Dic214—21no—
135Dic214—41no—
136Dic214—81no—
137C8×C107—21no—
138C8×C107—21no—
139C107⋊C8—21no—
140C107⋊C8—21no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107C22×D411yescharacteristic
148C2×D107—22no—
149C2×D107—41no—
150C2×D107—41no—
151C4×C107—21no—
152C4×C107—21no—
153C4×C107—41no—
154C4×C107—41no—
155C4×C107C2×#4111yescharacteristic
156C4×C1072−1+411yescharacteristic
157Dic107—21no—
158Dic107—21no—
159Dic107—21no—
160Dic107—21no—
161Dic107—41no—
162Dic107—41no—
163Dic107—41no—
164C2×C107—81no—
165C2×C107—81no—
166C2×C107#54811yescharacteristic
167C2×C107#57511yescharacteristic
168C2×C107#57611yescharacteristic
169D107—42no—
170#3370—1071nomaximal, Sylow
171C107#337011yescharacteristic, complemented, Sylow
172#559—1071no—
173#554—1071no—
174#490D10711yescharacteristic, complemented
175#489—1071no—
176#488—1071no—
177#479—1071no—
178#478—1071no—
179#474—1071no—
180#463—1071no—
181#463—1071no—
182#459—1071no—
183#446—1071no—
184#442—1071no—
185#440—1071no—
186C4×D8—1071no—
187C2×SD16—1071no—
188C2×D8—1071no—
189C2×D8—1071no—
190C4⋊Q8—1071no—
191#127C2×D10711yescharacteristic
192#125—1071no—
193#125C2×D10711yescharacteristic
194#124—1071no—
195#124—1071no—
196#124—2141no—
197#123—2141no—
198#123—2141no—
199C22⋊Q8—2141no—
200C22⋊Q8—2141no—
201C22⋊Q8—2141no—
202C4×Q8—1071no—
203C4×Q8—1071no—
204C4×D4—1071no—
205C4×D4—1071no—
206C4×D4—1071no—
207C4×D4—1071no—
208C4×D4—1071no—
209C4×D4—2141no—
210C2×C4⋊C4—2141no—
211#107—1071no—
212#107—1071no—
213#106—1071no—
214C4⋊C8—1071no—
215C4⋊C8—1071no—
216C4⋊C8—1071no—
217#103—1071no—
218#103—1071no—
219#103—1071no—
220#102—1071no—
221#102—1071no—
222#102—1071no—
223#102C2×D10711yescharacteristic
224#102C2×D10711yescharacteristic
225#102C2×D10711yescharacteristic
226#102C2×D10711yescharacteristic
227#97—1071no—
228#97C2×D10711yescharacteristic
229C4×C8—1071no—
230C2×Q8—1071no—
231C2×Q8—1071no—
232C2×Q8—2141no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—1071no—
236C2×D4C22×D10711yescharacteristic
237C2×D4C22×D10711yescharacteristic
238C22×C4—2141no—
239C22×C4—2141no—
240C22×C4—2141no—
241C22×C4—2141no—
242C22×C4—2141no—
243SD16—2142no—
244D8—2142no—
245D8—2142no—
246C2×C8—1071no—
247C2×C8—1071no—
248C2×C8C22×D10711yescharacteristic
249C2×C8C22×D10711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—1071no—
253C4⋊C4—1071no—
254C4⋊C4—1071no—
255C4⋊C4—1071no—
256C4⋊C4—1071no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4C22×D10711yescharacteristic
262C4⋊C4C22×D10711yescharacteristic
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C42—1071no—
273C42—1071no—
274C42—2141no—
275C42C22×D10711yescharacteristic
276C23—21no—
277C23—21no—
278C23—2141no—
279Q8—8561no—
280Q8—4281no—
281Q8—2141no—
282Q8—2141no—
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—42no—
288D4—81no—
289D4—1072no—
290D4—2141no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—8561no—
294C2×C4—8561no—
295C2×C4—8561no—
296C2×C4—8561no—
297C2×C4—1071no—
298C2×C4—1071no—
299C2×C4—1071no—
300C2×C4—1071no—
301C2×C4—4282no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4D4×D10711yescharacteristic
306C2×C4D4×D10711yescharacteristic
307C2×C4C23×D10711yescharacteristic
308C8—21no—
309C8—21no—
310C8—2141no—
311C8—2141no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—4281no—
319C22—4281no—
320C22—2142no—
321C22C2×D4×D10711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—4281no—
327C4—4281no—
328C4—4281no—
329C4—2141no—
330C4—2141no—
331C4—2141no—
332C4—2141no—
333C4#41×D10711yescharacteristic
334C4#1681215611yescharacteristic
335C2—81no—
336C2—81no—
337C2—4282no—
338C2#2029328511yescharacteristic
339C2#2029342311yescharacteristic
340C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#490×C107C2
socleC22×C107C22×D4
prime radical#20292819C2
hypercenter#490D107
hypocenterC107#3370
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #20293363C2semidirect
2. #20293333C2semidirect
3. #20292985C2semidirect
4. #20292980C2semidirect
5. #20292931C2semidirect
6. #20292925C2semidirect
7. #20292905C2semidirect
8. #20292841C2semidirect
9. #20292841C2semidirect
11. #20292755C2semidirect
12. #20292736C2semidirect
13. #20292729C2semidirect
14. #20292707C2semidirect
15. #490×C107C2semidirect
171. C107#3370semidirect
174. #490D107semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202933332
#202929852
#202929802
#202929312
#202929252
#202929052
#202928412
#202928192
#202927552
#202927362
#202927292
#202927072
#490×C1072
#3370107