Finite Group Databasebeta

← #28345680

Subgroups — #28345680

Order 13696.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133817C211yesmaximal, characteristic, complemented
2C2×#16812130C212yesmaximal, complemented
3#20293140C211yesmaximal, characteristic, complemented
4#20292786C212yesmaximal, complemented
5#20292782C211yesmaximal, characteristic, complemented
6#20292773C212yesmaximal, complemented
7#20292768C211yesmaximal, characteristic, complemented
8#20292761C211yesmaximal, characteristic, complemented
9#20292620C212yesmaximal, complemented
10C22⋊C8×D107C211yesmaximal, characteristic, complemented
11#450×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic, complemented
13C2×#13133817—28no—
14C2×#13133817—21no—
15C2×#13133817—21no—
16C2×#13133817C2212yes—
17C2×#13133817C2212yes—
18C22×Dic214C2211yescharacteristic
19C22×C22⋊D107—21no—
20C23×Dic107—21no—
21C22×C4×D107C2211yescharacteristic
22#16812130—28no—
23C2×#13133808C2212yescomplemented
24C2×#13133807C2212yescomplemented
25C2×#13133803C2212yescomplemented
26C2×C8×D107C2212yescomplemented
27#16812094—21no—
28#16812085C2211yescharacteristic, complemented
29#16812084C2211yescharacteristic, complemented
30#16812083C2211yescharacteristic
31#16812079—21no—
32#16812071—21no—
33(C22⋊D107)⋊C4—21no—
34#16812067C2211yescharacteristic, complemented
35#16812034C2212yescomplemented
36#16812032C2212yescomplemented
37#16812030C2212yescomplemented
38C22⋊(C107⋊C8)C2211yescharacteristic
39(C2×D107)⋊C8C2212yescomplemented
40C22×D4×C107C2211yescharacteristic
41C2×SD16×C107C2212yescomplemented
42C22⋊Q8×C107C2211yescharacteristic
43#102×C107C2212yescomplemented
44C22⋊C8×C107C2211yescharacteristic
45C23×D107D411yescharacteristic
46#13133819—24no—
47#13133817—28no—
48#13133817—28no—
49#13133817—24no—
50#13133817—24no—
51#13133817—24no—
52#13133817D418yescomplemented
53C2×Dic214—21no—
54C2×Dic214—24no—
55C2×Dic214C2312yes—
56C2×D428—21no—
57C2×D428—21no—
58C2×D428C2311yescharacteristic
59C2×C22⋊D107—28no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—22no—
62C2×C22⋊D107—21no—
63C22×Dic107—28no—
64C22×Dic107—21no—
65C22×Dic107—22no—
66C22×Dic107—22no—
67C22×Dic107D411yescharacteristic
68C2×C4×D107—24no—
69C2×C4×D107—21no—
70C2×C4×D107D411yescharacteristic
71C2×C4×D107D411yescharacteristic
72C2×C4×D107C2311yescharacteristic
73C2×C4×D107C2311yescharacteristic
74#13133808—24no—
75#13133807—24no—
76#13133803—24no—
77C2×C107⋊C8C2312yes—
78C8×D107—24no—
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85C4×Dic107—21no—
86C24×C107—21no—
87C2×Q8×C107C2311yescharacteristic
88C2×D4×C107—24no—
89C2×D4×C107—21no—
90C2×D4×C107C2312yes—
91C22×C4×C107C2311yescharacteristic
92SD16×C107—24no—
93C2×C8×C107C2312yes—
94C4⋊C4×C107—21no—
95C4⋊C4×C107C2311yescharacteristic
96C22⋊C4×C107—21no—
97C23×C107—24no—
98C23×C107—22no—
99C23×C107—42no—
100C23×C107—42no—
101C23×C107C2×D411yescharacteristic
102C22×D107—22no—
103C22×D107—22no—
104C22×D107—21no—
105C22×D107—41no—
106C22×D107C2×D411yescharacteristic
107C22×D107C2×D411yescharacteristic
108Q8×C107—22no—
109D4×C107—24no—
110D4×C107—22no—
111D4×C107C2×D414yes—
112C2×C4×C107—22no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2411yescharacteristic
117C22⋊D107—28no—
118C22⋊D107—28no—
119C22⋊D107—42no—
120C2×Dic107—24no—
121C2×Dic107—28no—
122C2×Dic107—22no—
123C2×Dic107—22no—
124C2×Dic107—28no—
125C2×Dic107—21no—
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128D428—22no—
129D428—42no—
130D428—42no—
131C4×D107—22no—
132C4×D107—22no—
133C4×D107—22no—
134C4×D107—42no—
135C4×D107C2×D414yes—
136Dic214—24no—
137Dic214—22no—
138Dic214C2×D414yes—
139C8×C107—22no—
140C107⋊C8—22no—
141C22×C107—21no—
142C22×C107—21no—
143C22×C107—24no—
144C22×C107—48no—
145C22×C107—42no—
146C22×C107—42no—
147C22×C107#13512yescomplemented
148C22×C107C22×D411yescharacteristic
149C2×D107—21no—
150C2×D107—21no—
151C2×D107—21no—
152C2×D107—21no—
153C2×D107—21no—
154C2×D107—44no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C2×D107#12012yes—
159C4×C107—21no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107C22×D412yes—
163Dic107—24no—
164Dic107—21no—
165Dic107—41no—
166Dic107#12012yes—
167C2×C107—22no—
168C2×C107—44no—
169C2×C107C2×#12011yescharacteristic
170C2×C107C2×#13511yescharacteristic
171C2×C107#57511yescharacteristic
172D107—22no—
173D107—41no—
174D107—81no—
175#3002—1071nomaximal, Sylow
176C107#300211yescharacteristic, complemented, Sylow
177C22×#41—1071no—
178C2×#135—1072no—
179#525—1071no—
180#452—1072no—
181#451—1071no—
182#450D10711yescharacteristic, complemented
183#449—1072no—
184#448—1071no—
185#447—1071no—
186#416—1072no—
187C2×C22⋊C8—1071no—
188C2×#41—1071no—
189C2×#41—1072no—
190C2×#41—1072no—
191C2×#41—2141no—
192C2×#41—2141no—
193C2×#41—2148no—
194C22×Q8—1071no—
195C22×D4—2141no—
196C22×D4C2×D10711yescharacteristic
197C23×C4—1071no—
198C23×C4—2141no—
199#135—2148no—
200C2×Q16—1072no—
201C2×SD16—1072no—
202C2×SD16C2×D10712yescomplemented
203C2×D8—1072no—
204C22×C8—1072no—
205#123—2141no—
206C22⋊Q8—1071no—
207C22⋊Q8C2×D10711yescharacteristic
208#121—1071no—
209#121—1071no—
210#120—2141no—
211C4×D4—2141no—
212C4×D4—2141no—
213#117—1071no—
214#103—1072no—
215#103—1072no—
216#102—1072no—
217#102C2×D10712yescomplemented
218C22⋊C8—1071no—
219C22⋊C8—1072no—
220C22⋊C8C2×D10711yescharacteristic
221C24—21no—
222C24—1071no—
223#41—1078no—
224#41—2144no—
225#41—2144no—
226#41—2144no—
227#41—2148no—
228#41—2148no—
229#41—2144no—
230C2×Q8—1072no—
231C2×Q8—2144no—
232C2×Q8—2141no—
233C2×Q8C22×D10711yescharacteristic
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—1071no—
237C2×D4—2141no—
238C2×D4—2141no—
239C2×D4—2141no—
240C2×D4—2142no—
241C2×D4—2142no—
242C2×D4—2148no—
243C2×D4C22×D10712yes—
244C22×C4—1071no—
245C22×C4—1071no—
246C22×C4—1071no—
247C22×C4—1071no—
248C22×C4—1071no—
249C22×C4—2141no—
250C22×C4—2144no—
251C22×C4—2142no—
252C22×C4—2142no—
253C22×C4—2148no—
254C22×C4—2141no—
255C22×C4C22×D10711yescharacteristic
256Q16—2144no—
257SD16—24no—
258SD16—2144no—
259D8—2144no—
260C2×C8—1072no—
261C2×C8—2144no—
262C2×C8C22×D10712yes—
263C4⋊C4—21no—
264C4⋊C4—1071no—
265C4⋊C4—2141no—
266C4⋊C4C22×D10711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C42—2141no—
273C23—24no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—1071no—
278C23—1071no—
279C23—4281no—
280C23—2142no—
281C23—2141no—
282C23—2142no—
283C23D4×D10711yescharacteristic
284Q8—22no—
285Q8—1074no—
286Q8—2144no—
287Q8—2142no—
288D4—24no—
289D4—22no—
290D4—4282no—
291D4—4282no—
292D4—4282no—
293D4—2148no—
294D4—2148no—
295D4—2142no—
296D4D4×D10714yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—1071no—
301C2×C4—1074no—
302C2×C4—1071no—
303C2×C4—4282no—
304C2×C4—2142no—
305C2×C4—2142no—
306C2×C4—2142no—
307C2×C4—2148no—
308C2×C4—2141no—
309C2×C4—2144no—
310C2×C4—2148no—
311C2×C4—2142no—
312C2×C4—2142no—
313C2×C4D4×D10711yescharacteristic
314C2×C4C23×D10711yescharacteristic
315C8—22no—
316C8—2142no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—1072no—
324C22—4284no—
325C22—4281no—
326C22—4281no—
327C22—4281no—
328C22—2141no—
329C22—2141no—
330C22—2141no—
331C22—2141no—
332C22—2141no—
333C22#1681213012yescomplemented
334C22C2×D4×D10711yescharacteristic
335C4—21no—
336C4—41no—
337C4—41no—
338C4—1072no—
339C4—4281no—
340C4—2141no—
341C4—2144no—
342C4C2×D4×D10712yes—
343C2—22no—
344C2—44no—
345C2—8561no—
346C2—4281no—
347C2—2142no—
348C2#120×D10711yescharacteristic
349C2C2×#1681213011yescharacteristic
350C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
hypercenter#450D107
hypocenterC107#3002
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C2×#16812130C2semidirect
3. #20293140C2semidirect
4. #20292786C2semidirect
5. #20292782C2semidirect
6. #20292773C2semidirect
7. #20292768C2semidirect
8. #20292761C2semidirect
9. #20292620C2semidirect
10. C22⋊C8×D107C2semidirect
11. #450×C107C2semidirect
12. C2×#13133819C22semidirect
23. C2×#13133808C22semidirect
24. C2×#13133807C22semidirect
25. C2×#13133803C22semidirect
26. C2×C8×D107C22semidirect
28. #16812085C22semidirect
29. #16812084C22semidirect
34. #16812067C22semidirect
35. #16812034C22semidirect
36. #16812032C22semidirect
37. #16812030C22semidirect
39. (C2×D107)⋊C8C22semidirect
41. C2×SD16×C107C22semidirect
43. #102×C107C22semidirect
52. #13133817D4semidirect
147. C22×C107#135semidirect
176. C107#3002semidirect
182. #450D107semidirect
202. C2×SD16C2×D107semidirect
217. #102C2×D107semidirect
333. C22#16812130semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168121302
#202931402
#202927862
#202927822
#202927732
#202927682
#202927612
#202926202
C22⋊C8×D1072
#450×C1072
#3002107