Finite Group Databasebeta

← #28349176

Subgroups — #28349176

Order 13696.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293346C211yesmaximal, characteristic, complemented
2#20293292C211yesmaximal, characteristic, complemented
3#20292985C211yesmaximal, characteristic, complemented
4#20292976C211yesmaximal, characteristic, complemented
5#20292920C211yesmaximal, characteristic, complemented
6#20292919C211yesmaximal, characteristic, complemented
7#20292905C211yesmaximal, characteristic, complemented
8#20292841C211yesmaximal, characteristic
9#20292839C211yesmaximal, characteristic
10#20292831C211yesmaximal, characteristic, complemented
11#20292755C211yesmaximal, characteristic, complemented
12#20292730C211yesmaximal, characteristic, complemented
13#20292729C211yesmaximal, characteristic, complemented
14#20292707C211yesmaximal, characteristic
15#489×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107—21no—
17C2×#13133807C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812107C2211yescharacteristic
23#16812102—21no—
24#16812102—21no—
25#16812096C2211yescharacteristic
26#16812087—21no—
27#16812087—21no—
28#16812086—21no—
29#16812085—21no—
30Q8×Dic107C2211yescharacteristic
31C4×Dic214C2211yescharacteristic
32C4×Dic214C2211yescharacteristic
33C4×D428C2211yescharacteristic
34D4×Dic107C2211yescharacteristic
35#16812071C2211yescharacteristic
36#16812071C2211yescharacteristic
37#16812065—21no—
38#16812043C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812030C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C4⋊Q8×C107C2211yescharacteristic
53#124×C107C2211yescharacteristic
54#103×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×Dic214D411yescharacteristic
62C2×Dic214C2311yescharacteristic
63Q8×D107—44no—
64C2×D428C2311yescharacteristic
65C22×Dic107—21no—
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69#13133807—22no—
70#13133805—22no—
71#13133803—22no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C4⋊Dic107—21no—
75C4⋊Dic107D411yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84Dic107⋊C4D411yescharacteristic
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×Dic107—21no—
93C4×Dic107C2311yescharacteristic
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107—21no—
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C4⋊C4×C107—21no—
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C42×C107C2311yescharacteristic
106C23×C107—21no—
107C22×D107—21no—
108Q8×C107—21no—
109Q8×C107—21no—
110Q8×C107—42no—
111D4×C107—21no—
112D4×C107—21no—
113C2×C4×C107—21no—
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—42no—
123C2×Dic107C2×D411yescharacteristic
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127D428—21no—
128D428#4112yes—
129C4×D107—42no—
130C4×D107—42no—
131C4×D107—42no—
132Dic214—22no—
133Dic214—21no—
134Dic214—21no—
135Dic214—42no—
136Dic214—42no—
137C8×C107—21no—
138C8×C107—21no—
139C107⋊C8—21no—
140C107⋊C8—21no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107C22×D411yescharacteristic
145C2×D107—22no—
146C2×D107—41no—
147C2×D107—41no—
148C4×C107—21no—
149C4×C107—21no—
150C4×C107—41no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107C2×#4111yescharacteristic
154C4×C1072−1+411yescharacteristic
155Dic107—21no—
156Dic107—21no—
157Dic107—21no—
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161Dic107AGL(1,ℤ/8)12yes—
162C2×C107—81no—
163C2×C107#54811yescharacteristic
164C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
165C2×C107#57811yescharacteristic
166D107—42no—
167#3367—1071nomaximal, Sylow
168C107#336711yescharacteristic, complemented, Sylow
169#556—1071no—
170D4×Q8—1071no—
171#489—1071no—
172#489D10711yescharacteristic, complemented
173#487—1071no—
174#477—1071no—
175#477—1071no—
176#474—1071no—
177#463—1071no—
178#463—1071no—
179#461—1071no—
180#446—1071no—
181#440—1071no—
182#440—1071no—
183C4×D8—1071no—
184C22×Q8—2141no—
185C2×SD16—1071no—
186C2×SD16—1071no—
187C2×D8—1071no—
188C4⋊Q8—1071no—
189C4⋊Q8—1071no—
190C4⋊Q8—1071no—
191C4⋊Q8C2×D10711yescharacteristic
192#126—2141no—
193#126—2141no—
194#124—1071no—
195#124C2×D10711yescharacteristic
196C22⋊Q8—2141no—
197C22⋊Q8—2141no—
198C22⋊Q8—2141no—
199C22⋊Q8—2141no—
200C4×Q8—1071no—
201C4×Q8—1071no—
202C4×Q8—1071no—
203C4×D4—1071no—
204C4×D4—1071no—
205C4×D4—1071no—
206C4×D4—1071no—
207#117—2141no—
208#107—1071no—
209#107—1071no—
210#107—1071no—
211C4⋊C8—1071no—
212C4⋊C8—1071no—
213C4⋊C8—1071no—
214#103—1071no—
215#103—1071no—
216#103—1071no—
217#103—1071no—
218#103C2×D10711yescharacteristic
219#103C2×D10711yescharacteristic
220#102—1071no—
221#102—1071no—
222#102C2×D10711yescharacteristic
223#102C2×D10711yescharacteristic
224#97—1071no—
225#97C2×D10711yescharacteristic
226C4×C8—1071no—
227C2×Q8—21no—
228C2×Q8—1071no—
229C2×Q8—1071no—
230C2×Q8—4284no—
231C2×Q8—2141no—
232C2×Q8—2141no—
233C2×Q8C22×D10711yescharacteristic
234C2×D4—1071no—
235C2×D4C22×D10711yescharacteristic
236C22×C4—2141no—
237C22×C4—2141no—
238C22×C4—2141no—
239C22×C4—2141no—
240SD16—2142no—
241SD16—2142no—
242D8—2142no—
243C2×C8—1071no—
244C2×C8—1071no—
245C2×C8C22×D10711yescharacteristic
246C2×C8C22×D10711yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—1071no—
249C4⋊C4—1071no—
250C4⋊C4—1071no—
251C4⋊C4—1071no—
252C4⋊C4—1071no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4C22×D10711yescharacteristic
260C4⋊C4C22×D10711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C42—1071no—
270C42—1071no—
271C42—2141no—
272C42—2141no—
273C42C22×D10711yescharacteristic
274C23—21no—
275C23—2141no—
276Q8—21no—
277Q8—21no—
278Q8—42no—
279Q8—4282no—
280Q8—4282no—
281Q8—2142no—
282Q8—2141no—
283Q8—2141no—
284D4—21no—
285D4—21no—
286D4—1072no—
287D4—2141no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—1071no—
292C2×C4—1071no—
293C2×C4—1071no—
294C2×C4—1071no—
295C2×C4—4282no—
296C2×C4—4282no—
297C2×C4—4282no—
298C2×C4—4282no—
299C2×C4—2141no—
300C2×C4—2141no—
301C2×C4—2141no—
302C2×C4D4×D10711yescharacteristic
303C2×C4D4×D10711yescharacteristic
304C2×C4C23×D10711yescharacteristic
305C8—21no—
306C8—21no—
307C8—2141no—
308C8—2141no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—4281no—
313C22—4281no—
314C22—2142no—
315C22C2×D4×D10711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—1072no—
322C4—4281no—
323C4—4281no—
324C4—4281no—
325C4—2141no—
326C4—2141no—
327C4—2141no—
328C4#41×D10711yescharacteristic
329C4#1681215611yescharacteristic
330C2—81no—
331C2—4282no—
332C2#2029328511yescharacteristic
333C2AGL(1,ℤ/8)×D10711yescharacteristic
334C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#489×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133803C22
hypercenter#489D107
hypocenterC107#3367
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #20293346C2semidirect
2. #20293292C2semidirect
3. #20292985C2semidirect
4. #20292976C2semidirect
5. #20292920C2semidirect
6. #20292919C2semidirect
7. #20292905C2semidirect
10. #20292831C2semidirect
11. #20292755C2semidirect
12. #20292730C2semidirect
13. #20292729C2semidirect
15. #489×C107C2semidirect
168. C107#3367semidirect
172. #489D107semidirect

Maximal subgroups

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

subgroupindex
#202933462
#202932922
#202929852
#202929762
#202929202
#202929192
#202929052
#202928412
#202928392
#202928312
#202927552
#202927302
#202927292
#202927072
#489×C1072
#3367107