Finite Group Databasebeta

← #28349239

Subgroups — #28349239

Order 13696.

340 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293352C211yesmaximal, characteristic, complemented
2#20293333C211yesmaximal, characteristic
3#20292994C211yesmaximal, characteristic, complemented
4#20292976C211yesmaximal, characteristic, complemented
5#20292936C211yesmaximal, characteristic, complemented
6#20292931C211yesmaximal, characteristic, complemented
7#20292920C211yesmaximal, characteristic, complemented
8#20292831C211yesmaximal, characteristic
9#20292831C211yesmaximal, characteristic
10#20292829C211yesmaximal, characteristic
11#20292755C211yesmaximal, characteristic, complemented
12#20292730C211yesmaximal, characteristic
13#20292729C211yesmaximal, characteristic
14#20292712C211yesmaximal, characteristic
15#491×C107C211yesmaximal, characteristic, complemented
16C2×#13133807C2211yescharacteristic
17C2×#13133805C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19#16812108C2211yescharacteristic
20#16812107C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812100—21no—
23#16812100C2211yescharacteristic
24#16812099—21no—
25#16812099—21no—
26#16812089—21no—
27#16812087—21no—
28#16812087—21no—
29Q8×Dic107C2211yescharacteristic
30Q8×Dic107C2211yescharacteristic
31C4×Dic214C2211yescharacteristic
32C4×Dic214C2211yescharacteristic
33#16812074—21no—
34C4×D428C2211yescharacteristic
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#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812032C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C4⋊Q8×C107C2211yescharacteristic
53#125×C107C2211yescharacteristic
54#103×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214D411yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×D428C2311yescharacteristic
63C2×C4×D107—21no—
64C2×C4×D107—21no—
65C2×C4×D107—21no—
66#13133807—22no—
67#13133805—22no—
68#13133805—22no—
69C2×C107⋊C8C2311yescharacteristic
70C2×C107⋊C8C2311yescharacteristic
71C4⋊Dic107—21no—
72C4⋊Dic107—21no—
73C4⋊Dic107—21no—
74C4⋊Dic107D411yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4D411yescharacteristic
85Dic107⋊C4D411yescharacteristic
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C4×Dic107C2311yescharacteristic
94C2×Q8×C107C2311yescharacteristic
95C2×Q8×C107C2311yescharacteristic
96C2×C8×C107C2311yescharacteristic
97C2×C8×C107C2311yescharacteristic
98C4⋊C4×C107—21no—
99C4⋊C4×C107—21no—
100C4⋊C4×C107—21no—
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C42×C107C2311yescharacteristic
105C22×D107—21no—
106Q8×C107—21no—
107Q8×C107—21no—
108Q8×C107—21no—
109Q8×C107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2411yescharacteristic
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107C2×D411yescharacteristic
121C2×Dic107C2×D411yescharacteristic
122C2×Dic107C2×D411yescharacteristic
123C2×Dic107C2×D411yescharacteristic
124D428—21no—
125D428#4112yes—
126C4×D107—42no—
127C4×D107—81no—
128C4×D107—81no—
129Dic214—22no—
130Dic214—21no—
131Dic214—21no—
132Dic214—81no—
133C8×C107—21no—
134C8×C107—21no—
135C107⋊C8—21no—
136C107⋊C8—21no—
137C22×C107C22×D411yescharacteristic
138C2×D107—22no—
139C2×D107—41no—
140C2×D107—41no—
141C4×C107—21no—
142C4×C107—21no—
143C4×C107—41no—
144C4×C107—41no—
145C4×C107—41no—
146C4×C107—41no—
147C4×C107C2×#4111yescharacteristic
148C4×C1072−1+411yescharacteristic
149Dic107—21no—
150Dic107—21no—
151Dic107—21no—
152Dic107—41no—
153Dic107—41no—
154Dic107—41no—
155Dic107#13712yes—
156C2×C107#54811yescharacteristic
157C2×C107C2×#13711yescharacteristic
158C2×C107#57711yescharacteristic
159D107—42no—
160#3372—1071nomaximal, Sylow
161C107#337211yescharacteristic, complemented, Sylow
162#557—1071no—
163#554—1071no—
164#491—1071no—
165#491D10711yescharacteristic, complemented
166#487—1071no—
167#479—1071no—
168#479—1071no—
169#477—1071no—
170#461—1071no—
171#461—1071no—
172#461—1071no—
173#446—1071no—
174#440—1071no—
175#440—1071no—
176C4×SD16—1071no—
177C2×SD16—1071no—
178C2×SD16—1071no—
179C2×SD16—1071no—
180C4⋊Q8—1071no—
181C4⋊Q8—1071no—
182C4⋊Q8—1071no—
183C4⋊Q8C2×D10711yescharacteristic
184#125—1071no—
185#125—2141no—
186#125—2141no—
187#125—2141no—
188#125C2×D10711yescharacteristic
189C22⋊Q8—2141no—
190C22⋊Q8—2141no—
191C22⋊Q8—2141no—
192C4×Q8—1071no—
193C4×Q8—1071no—
194C4×Q8—1071no—
195C4×Q8—1071no—
196C4×Q8—2141no—
197C4×D4—1071no—
198C4×D4—1071no—
199C4×D4—1071no—
200C2×C4⋊C4—2141no—
201#107—1071no—
202#107—1071no—
203#106—1071no—
204C4⋊C8—1071no—
205C4⋊C8—1071no—
206C4⋊C8—1071no—
207#103—1071no—
208#103—1071no—
209#103—1071no—
210#103—1071no—
211#103—1071no—
212#103C2×D10711yescharacteristic
213#103C2×D10711yescharacteristic
214#103C2×D10711yescharacteristic
215#103C2×D10711yescharacteristic
216#102—1071no—
217#97—1071no—
218#97C2×D10711yescharacteristic
219C4×C8—1071no—
220C2×Q8—1071no—
221C2×Q8—1071no—
222C2×Q8—2141no—
223C2×Q8C22×D10711yescharacteristic
224C2×Q8C22×D10711yescharacteristic
225C2×D4—1071no—
226C22×C4—2141no—
227C22×C4—2141no—
228C22×C4—2141no—
229SD16—2142no—
230SD16—2142no—
231SD16—2142no—
232C2×C8—1071no—
233C2×C8—1071no—
234C2×C8C22×D10711yescharacteristic
235C2×C8C22×D10711yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—1071no—
241C4⋊C4—1071no—
242C4⋊C4—1071no—
243C4⋊C4—1071no—
244C4⋊C4—1071no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4C22×D10711yescharacteristic
256C4⋊C4C22×D10711yescharacteristic
257C22⋊C4—2141no—
258C22⋊C4—2141no—
259C22⋊C4—2141no—
260C42—1071no—
261C42—1071no—
262C42—2141no—
263C42—2141no—
264C42—2141no—
265C42C22×D10711yescharacteristic
266C23—2141no—
267Q8—21no—
268Q8—21no—
269Q8—21no—
270Q8—21no—
271Q8—8561no—
272Q8—2142no—
273Q8—2141no—
274Q8—2141no—
275D4—1072no—
276D4—2141no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—8561no—
282C2×C4—8561no—
283C2×C4—1071no—
284C2×C4—1071no—
285C2×C4—1071no—
286C2×C4—1071no—
287C2×C4—4282no—
288C2×C4—2141no—
289C2×C4—2141no—
290C2×C4—2141no—
291C2×C4D4×D10711yescharacteristic
292C2×C4D4×D10711yescharacteristic
293C2×C4C23×D10711yescharacteristic
294C8—21no—
295C8—21no—
296C8—2141no—
297C8—2141no—
298C22—4281no—
299C22—4281no—
300C22—2142no—
301C22C2×D4×D10711yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—1072no—
309C4—4281no—
310C4—4281no—
311C4—4281no—
312C4—2141no—
313C4—2141no—
314C4—2141no—
315C4#41×D10711yescharacteristic
316C4#1681215611yescharacteristic
317C2—4282no—
318C2#2029328511yescharacteristic
319C2#137×D10711yescharacteristic
320C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#491×C107C2
socleC22×C107C22×D4
prime radicalC2×D428C23
hypercenter#491D107
hypocenterC107#3372
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #20293352C2semidirect
3. #20292994C2semidirect
4. #20292976C2semidirect
5. #20292936C2semidirect
6. #20292931C2semidirect
7. #20292920C2semidirect
11. #20292755C2semidirect
15. #491×C107C2semidirect
161. C107#3372semidirect
165. #491D107semidirect

Maximal subgroups

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

subgroupindex
#202933522
#202933332
#202929942
#202929762
#202929362
#202929312
#202929202
#202928312
#202928292
#202927552
#202927302
#202927292
#202927122
#491×C1072
#3372107