Finite Group Databasebeta

← #28346621

Subgroups — #28346621

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×#13133819C211yesmaximal, characteristic
3#20292971C211yesmaximal, characteristic, complemented
4#20292967C211yesmaximal, characteristic, complemented
5#20292920C211yesmaximal, characteristic, complemented
6#20292916C211yesmaximal, characteristic, complemented
7#20292835C211yesmaximal, characteristic, complemented
8#20292831C211yesmaximal, characteristic, complemented
9#20292792C211yesmaximal, characteristic
10#20292788C211yesmaximal, characteristic
11#20292732C211yesmaximal, characteristic, complemented
12#20292729C211yesmaximal, characteristic, complemented
13#20292725C211yesmaximal, characteristic, complemented
14#20292682C211yesmaximal, characteristic, complemented
15#441×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133808C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812099—21no—
23#16812089C2211yescharacteristic
24#16812087—21no—
25#16812085C2211yescharacteristic
26Q8×Dic107C2211yescharacteristic
27C4×Dic214—21no—
28C4×Dic214C2211yescharacteristic
29C4×D428—21no—
30C4×D428C2211yescharacteristic
31#16812071—21no—
32#16812071C2211yescharacteristic
33#16812067—21no—
34#16812067C2211yescharacteristic
35#16812066—21no—
36C42×D107—21no—
37C42×D107C2211yescharacteristic
38#16812043C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812029C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51#16812020C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59#13133819—22no—
60#13133819—22no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×C4×D107—21no—
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107D411yescharacteristic
69C2×C4×D107D411yescharacteristic
70C2×C4×D107C2311yescharacteristic
71#13133808—22no—
72#13133807—22no—
73#13133805—22no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107D411yescharacteristic
94C4×Dic107D411yescharacteristic
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×Q8×C107C2311yescharacteristic
98Q16×C107—22no—
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C42×C107—21no—
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C22×D107—21no—
109C22×D107C2×D411yescharacteristic
110Q8×C107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113Q8×C107#4112yes—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2411yescharacteristic
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127D428—24no—
128D428—21no—
129D428#4112yes—
130C4×D107—24no—
131C4×D107—21no—
132C4×D107—21no—
133C4×D107—22no—
134C4×D107—41no—
135C4×D107—42no—
136C4×D107—42no—
137Dic214—21no—
138Dic214—21no—
139Dic214—42no—
140C8×C107—21no—
141C8×C107—21no—
142C107⋊C8—21no—
143C107⋊C8—21no—
144C22×C107C22×D411yescharacteristic
145C2×D107—21no—
146C2×D107—22no—
147C2×D107—21no—
148C2×D107—21no—
149C2×D107—41no—
150C2×D107—41no—
151C4×C107—22no—
152C4×C107—21no—
153C4×C107—21no—
154C4×C107—41no—
155C4×C107—41no—
156C4×C107—41no—
157C4×C107C2×#4111yescharacteristic
158C4×C107C2×#4111yescharacteristic
159Dic107—21no—
160Dic107—21no—
161Dic107—41no—
162Dic107—41no—
163Dic107—41no—
164Dic107#13712yes—
165C2×C107#52511yescharacteristic
166C2×C107C2×#13711yescharacteristic
167C2×C107#57511yescharacteristic
168D107—41no—
169D107—42no—
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#441D10711yescharacteristic, complemented
184#440—1071no—
185#440—1071no—
186#432—1071no—
187C2×#41—1071no—
188C2×Q16—1071no—
189C2×Q16C2×D10711yescharacteristic
190C2×SD16—1071no—
191C2×SD16—1071no—
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—2141no—
201C4×Q8C2×D10711yescharacteristic
202C4×Q8C2×D10711yescharacteristic
203C4×D4—1071no—
204C4×D4—1071no—
205C4×D4—2141no—
206C4×D4—2141no—
207#117—1071no—
208#117—2141no—
209#117—2141no—
210C2×C42—1071no—
211C2×C42—2141no—
212#107—1071no—
213#107—1071no—
214#106—1071no—
215#106C2×D10711yescharacteristic
216C4⋊C8—1071no—
217C4⋊C8—1071no—
218#103—1071no—
219#103—1071no—
220#103—1071no—
221#103—1071no—
222#103C2×D10711yescharacteristic
223#103C2×D10711yescharacteristic
224#102—1071no—
225#102—1071no—
226C22⋊C8—1071no—
227C22⋊C8—1071no—
228#97—1071no—
229#97C2×D10711yescharacteristic
230#41—2142no—
231#41—2142no—
232C2×Q8—1071no—
233C2×Q8—2141no—
234C2×Q8C22×D10711yescharacteristic
235C2×Q8C22×D10711yescharacteristic
236C2×D4—1071no—
237C2×D4—2141no—
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—1071no—
241C22×C4—2141no—
242C22×C4—2141no—
243C22×C4—2141no—
244Q16—22no—
245Q16—2142no—
246SD16—2142no—
247SD16—2142no—
248C2×C8—1071no—
249C2×C8—1071no—
250C2×C8C22×D10711yescharacteristic
251C2×C8C22×D10711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—1071no—
255C4⋊C4—1071no—
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—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C42—21no—
271C42—21no—
272C42—1071no—
273C42—1071no—
274C42—1071no—
275C42—2141no—
276C42—2141no—
277C42—2141no—
278C42C22×D10711yescharacteristic
279C23—1071no—
280C23—2141no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—4282no—
285Q8—2141no—
286Q8—2141no—
287Q8#1313381812yes—
288D4—1072no—
289D4—2144no—
290D4—2141no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—1071no—
296C2×C4—1071no—
297C2×C4—1071no—
298C2×C4—4282no—
299C2×C4—4281no—
300C2×C4—4282no—
301C2×C4—2142no—
302C2×C4—2144no—
303C2×C4—2141no—
304C2×C4—2141no—
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—4281no—
316C22—4281no—
317C22—2142no—
318C22—2141no—
319C22—2141no—
320C22—2141no—
321C22C2×D4×D10711yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—1072no—
329C4—4281no—
330C4—4281no—
331C4—4281no—
332C4—2141no—
333C4—2141no—
334C4#41×D10711yescharacteristic
335C4C2×#1313381811yescharacteristic
336C2—4282no—
337C2—4281no—
338C2#2029313611yescharacteristic
339C2#137×D10711yescharacteristic
340C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#441×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133819C22
hypercenter#441D107
hypocenterC107#3116
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #20293181C2semidirect
3. #20292971C2semidirect
4. #20292967C2semidirect
5. #20292920C2semidirect
6. #20292916C2semidirect
7. #20292835C2semidirect
8. #20292831C2semidirect
11. #20292732C2semidirect
12. #20292729C2semidirect
13. #20292725C2semidirect
14. #20292682C2semidirect
15. #441×C107C2semidirect
171. C107#3116semidirect
183. #441D107semidirect

Maximal subgroups

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

subgroupindex
#202931812
C4×#131338192
#202929712
#202929672
#202929202
#202929162
#202928352
#202928312
#202927922
#202927882
#202927322
#202927292
#202927252
#202926822
#441×C1072
#3116107