Finite Group Databasebeta

← #28346619

Subgroups — #28346619

Order 13696.

386 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#20293181C211yesmaximal, characteristic, complemented
2C4×#13133817C211yesmaximal, characteristic
3#20292970C211yesmaximal, characteristic, complemented
4#20292966C211yesmaximal, characteristic, complemented
5#20292921C211yesmaximal, characteristic, complemented
6#20292915C211yesmaximal, characteristic, complemented
7#20292836C211yesmaximal, characteristic, complemented
8#20292830C211yesmaximal, characteristic, complemented
9#20292790C211yesmaximal, characteristic
10#20292786C211yesmaximal, characteristic
11#20292735C211yesmaximal, characteristic, complemented
12#20292732C211yesmaximal, characteristic, complemented
13#20292727C211yesmaximal, characteristic, complemented
14#20292682C211yesmaximal, characteristic, complemented
15#440×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812099—21no—
23#16812089C2211yescharacteristic
24#16812087—21no—
25#16812085C2211yescharacteristic
26C4×Dic214—21no—
27C4×Dic214C2211yescharacteristic
28C4×Dic214C2211yescharacteristic
29#16812074C2211yescharacteristic
30D4×Dic107C2211yescharacteristic
31C4×C22⋊D107—21no—
32(C22⋊D107)⋊C4—21no—
33#16812067C2211yescharacteristic
34#16812066—21no—
35#16812065—21no—
36C2×C4×Dic107—21no—
37C42×D107C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812030C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51#16812020C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic
54C4×D4×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59#13133817—22no—
60#13133817—22no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63C2×Dic214C2311yescharacteristic
64C2×C22⋊D107—21no—
65C22×Dic107—21no—
66C22×Dic107—21no—
67C2×C4×D107D411yescharacteristic
68C2×C4×D107D411yescharacteristic
69C2×C4×D107C2311yescharacteristic
70Dic428—22no—
71#13133808—22no—
72#13133806—22no—
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C4⋊Dic107—21no—
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85C22⋊Dic107—21no—
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×Dic107—21no—
93C4×Dic107D411yescharacteristic
94C4×Dic107D411yescharacteristic
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107—21no—
99SD16×C107—22no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C22×D107C2×D411yescharacteristic
110Q8×C107—21no—
111Q8×C107—21no—
112D4×C107—21no—
113D4×C107#4112yes—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—42no—
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2411yescharacteristic
121C22⋊D107—24no—
122C2×Dic107—24no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—42no—
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C4×D107—21no—
132C4×D107—21no—
133C4×D107—22no—
134C4×D107—41no—
135Dic214—21no—
136Dic214—21no—
137Dic214—21no—
138Dic214—42no—
139Dic214#4112yes—
140C8×C107—21no—
141C8×C107—21no—
142C107⋊C8—21no—
143C107⋊C8—21no—
144C22×C107—22no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107C22×D411yescharacteristic
148C2×D107—21no—
149C2×D107—21no—
150C2×D107—21no—
151C4×C107—21no—
152C4×C107—21no—
153C4×C107—41no—
154C4×C107—41no—
155C4×C107—41no—
156C4×C107C2×#4111yescharacteristic
157C4×C107C2×#4111yescharacteristic
158Dic107—22no—
159Dic107—21no—
160Dic107—21no—
161Dic107—41no—
162Dic107—41no—
163Dic107—41no—
164Dic107#13712yes—
165C2×C107—42no—
166C2×C107#52511yescharacteristic
167C2×C107C2×#13711yescharacteristic
168C2×C107#57511yescharacteristic
169D107—41no—
170#3116—1071nomaximal, Sylow
171C107#311611yescharacteristic, complemented, Sylow
172#533—1071no—
173C4×#41—1071no—
174#485—1071no—
175#485—1071no—
176#477—1071no—
177#476—1071no—
178#462—1071no—
179#461—1071no—
180#452—1071no—
181#452—1071no—
182#441—1071no—
183#441—1071no—
184#440—1071no—
185#440D10711yescharacteristic, complemented
186#432—1071no—
187C2×#41—1071no—
188C2×Q16—1071no—
189C2×Q16—1071no—
190C2×SD16—1071no—
191C2×SD16C2×D10711yescharacteristic
192C4⋊Q8—1071no—
193C4⋊Q8—1071no—
194#125—2141no—
195C22⋊Q8—1071no—
196C22⋊Q8—1071no—
197C22⋊Q8—2141no—
198C4×Q8—1071no—
199C4×Q8—1071no—
200C4×Q8—1071no—
201C4×Q8—2141no—
202C4×Q8C2×D10711yescharacteristic
203C4×D4—1071no—
204C4×D4—2141no—
205C4×D4—2141no—
206C4×D4C2×D10711yescharacteristic
207#117—1071no—
208#117—2141no—
209#117—2141no—
210C2×C42—1071no—
211C2×C42—2141no—
212#107—1071no—
213#107C2×D10711yescharacteristic
214#106—1071no—
215#106—1071no—
216C4⋊C8—1071no—
217C4⋊C8—1071no—
218#103—1071no—
219#103—1071no—
220#103—1071no—
221#103—1071no—
222#103—1071no—
223#103C2×D10711yescharacteristic
224#102—1071no—
225#102C2×D10711yescharacteristic
226C22⋊C8—1071no—
227C22⋊C8—1071no—
228#97—1071no—
229#97C2×D10711yescharacteristic
230#41—2142no—
231#41—2142no—
232C2×Q8—1071no—
233C2×Q8—1071no—
234C2×Q8—2141no—
235C2×Q8C22×D10711yescharacteristic
236C2×D4—2141no—
237C2×D4C22×D10711yescharacteristic
238C22×C4—21no—
239C22×C4—1071no—
240C22×C4—1071no—
241C22×C4—1071no—
242C22×C4—2141no—
243C22×C4—2141no—
244Q16—2142no—
245Q16—2142no—
246SD16—22no—
247SD16—2142no—
248C2×C8—1071no—
249C2×C8—1071no—
250C2×C8C22×D10711yescharacteristic
251C2×C8C22×D10711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—1071no—
254C4⋊C4—1071no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4C22×D10711yescharacteristic
264C4⋊C4C22×D10711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C42—21no—
271C42—1071no—
272C42—1071no—
273C42—1071no—
274C42—2141no—
275C42—2141no—
276C42—2141no—
277C42—2141no—
278C42C22×D10711yescharacteristic
279C23—21no—
280C23—1071no—
281Q8—21no—
282Q8—21no—
283Q8—1072no—
284Q8—4282no—
285Q8—2141no—
286Q8—2141no—
287Q8—2141no—
288D4—21no—
289D4—2144no—
290D4#1313381812yes—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—1071no—
296C2×C4—1071no—
297C2×C4—1071no—
298C2×C4—4281no—
299C2×C4—4282no—
300C2×C4—2142no—
301C2×C4—2141no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4—2144no—
305C2×C4—2141no—
306C2×C4—2141no—
307C2×C4—2141no—
308C2×C4D4×D10711yescharacteristic
309C2×C4D4×D10711yescharacteristic
310C2×C4C23×D10711yescharacteristic
311C8—21no—
312C8—21no—
313C8—2141no—
314C8—2141no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—2141no—
319C22—2141no—
320C22—2141no—
321C22C2×D4×D10711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—1072no—
328C4—4281no—
329C4—4281no—
330C4—4281no—
331C4—2141no—
332C4—2142no—
333C4—2141no—
334C4#41×D10711yescharacteristic
335C4C2×#1313381811yescharacteristic
336C2—42no—
337C2—4281no—
338C2#2029313611yescharacteristic
339C2#137×D10711yescharacteristic
340C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#440×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133817C22
hypercenter#440D107
hypocenterC107#3116
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #20293181C2semidirect
3. #20292970C2semidirect
4. #20292966C2semidirect
5. #20292921C2semidirect
6. #20292915C2semidirect
7. #20292836C2semidirect
8. #20292830C2semidirect
11. #20292735C2semidirect
12. #20292732C2semidirect
13. #20292727C2semidirect
14. #20292682C2semidirect
15. #440×C107C2semidirect
171. C107#3116semidirect
185. #440D107semidirect

Maximal subgroups

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

subgroupindex
#202931812
C4×#131338172
#202929702
#202929662
#202929212
#202929152
#202928362
#202928302
#202927902
#202927862
#202927352
#202927322
#202927272
#202926822
#440×C1072
#3116107