Finite Group Databasebeta

← #28345810

Subgroups — #28345810

Order 13696.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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
1#20293425C212yesmaximal, complemented
2#20293173C211yesmaximal, characteristic, complemented
3#20292810C212yesmaximal, complemented
4#20292805C212yesmaximal, complemented
5#20292800C212yesmaximal, complemented
6#20292795C212yesmaximal, complemented
7C2×#16812036C212yesmaximal, complemented
8#20292611C211yesmaximal, characteristic, complemented
9#585×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic, complemented
11#16812140—22no—
12#16812139C2214yescomplemented
13#16812135—22no—
14#16812134C2214yescomplemented
15#16812129—24no—
16C2×#13133802C2212yescomplemented
17#16812107C2211yescharacteristic, complemented
18#16812105C2211yescharacteristic, complemented
19#16812098C2211yescharacteristic, complemented
20#16812096C2211yescharacteristic, complemented
21#16812081—22no—
22C4×C22⋊D107—22no—
23C2×C4×Dic107C2211yescharacteristic, complemented
24#16812036C2214yescomplemented
25#16812036C2214yescomplemented
26#16812028C2212yescomplemented
27#16812026C2212yescomplemented
282−1+4×C107—22no—
292−1+4×C107C2212yes—
30H(5,𝔽2)×C107—22no—
31H(5,𝔽2)×C107—22no—
32H(5,𝔽2)×C107C2212yes—
33C2×#41×C107—24no—
34C2×#41×C107—22no—
35C2×#41×C107C2212yes—
36C2×#41×C107C2211yescharacteristic
37#13133818—22no—
38#13133818—22no—
39C2×Dic214C2311yescharacteristic
40C2×D428C2311yescharacteristic
41C2×C22⋊D107—22no—
42C2×C22⋊D107—21no—
43C22×Dic107—21no—
44C2×C4×D107—21no—
45#13133808—24no—
46#13133806—24no—
47#13133805—24no—
48#13133803—24no—
49#13133802—22no—
50#13133802C2314yes—
51C2×C107⋊C8—22no—
52Dic107⋊C4—22no—
53C22⋊Dic107—22no—
54(C2×D107)⋊C4—22no—
55C4×Dic107—21no—
56C4×Dic107C2311yescharacteristic
57C4×Dic107C2311yescharacteristic
58#41×C107—24no—
59#41×C107—28no—
60#41×C107—24no—
61#41×C107—24no—
62#41×C107—24no—
63#41×C107—22no—
64#41×C107—22no—
65#41×C107—24no—
66#41×C107—22no—
67#41×C107—22no—
68#41×C107D414yescomplemented
69#41×C107C2314yes—
70C2×Q8×C107—22no—
71C2×Q8×C107—24no—
72C2×Q8×C107D412yescomplemented
73C2×Q8×C107C2311yescharacteristic
74C2×D4×C107—28no—
75C2×D4×C107—22no—
76C2×D4×C107—24no—
77C2×D4×C107—22no—
78C2×D4×C107—24no—
79C2×D4×C107D412yes—
80C2×D4×C107D412yescomplemented
81C2×D4×C107C2311yescharacteristic
82C22×C4×C107—24no—
83C22×C4×C107—22no—
84C22×C4×C107D412yes—
85C22×C4×C107C2311yescharacteristic
86C23×C107—24no—
87C23×C107—22no—
88C23×C107C2×D412yes—
89C23×C107C2×D411yescharacteristic
90C22×D107—21no—
91Q8×C107—24no—
92Q8×C107—22no—
93Q8×C107—22no—
94Q8×C107C2×D414yes—
95D4×C107—28no—
96D4×C107—24no—
97D4×C107—24no—
98D4×C107—24no—
99D4×C107—24no—
100D4×C107—22no—
101D4×C107—22no—
102D4×C107C2×D414yes—
103C2×C4×C107—24no—
104C2×C4×C107—28no—
105C2×C4×C107—22no—
106C2×C4×C107—24no—
107C2×C4×C107—22no—
108C2×C4×C107—22no—
109C2×C4×C107—24no—
110C2×C4×C107C2×D412yes—
111C2×C4×C107C2×D411yescharacteristic
112C2×C4×C107C2×D412yes—
113C2×C4×C107C2×D412yes—
114C2×C4×C107C2411yescharacteristic
115C22⋊D107—44no—
116C22⋊D107—42no—
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—42no—
121D428—22no—
122C4×D107—41no—
123C4×D107—41no—
124Dic214—22no—
125C107⋊C8—24no—
126C22×C107—22no—
127C22×C107—24no—
128C22×C107—41no—
129C22×C107—42no—
130C22×C107—82no—
131C22×C107—84no—
132C22×C107#12012yes—
133C22×C107C22×D411yescharacteristic
134C2×D107—42no—
135C2×D107—41no—
136C4×C107—24no—
137C4×C107—22no—
138C4×C107#12011yescharacteristic
139C4×C107#12011yescharacteristic
140C4×C107C22×D412yes—
141Dic107—22no—
142Dic107—41no—
143Dic107—41no—
144C2×C107—21no—
145C2×C107—22no—
146C2×C107—42no—
147C2×C107—44no—
148C2×C107C2×#12011yescharacteristic
149D107—81no—
150#3018—1071nomaximal, Sylow
151C107#301811yescharacteristic, complemented, Sylow
152#585D10711yescharacteristic, complemented
153#575—1072no—
154#532—1071no—
155#456—1072no—
156#455—1072no—
157#454—1072no—
158#453—1072no—
159C2×#104—1072no—
160#413—1071no—
1612−1+4—22no—
1622−1+4C2×D10712yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D10712yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—1071no—
169C2×#41C2×D10712yes—
170C2×#41C2×D10711yescharacteristic
171#137—1074no—
172#137—2142no—
173AGL(1,ℤ/8)—1074no—
174AGL(1,ℤ/8)—2142no—
175#135—2144no—
176C2×M16—1072no—
177C4⋊Q8—1071no—
178#127—1071no—
179#124—1071no—
180#124—1071no—
181#121—2142no—
182C4×D4—2142no—
183C2×C42—1071no—
184#104—1074no—
185#104—1074no—
186#101—1072no—
187#100—1072no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—2142no—
199#41—2142no—
200#41C22⋊D10714yescomplemented
201#41C22×D10714yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—1071no—
205C2×Q8C22⋊D10712yescomplemented
206C2×Q8C22×D10711yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—1071no—
213C2×D4—2142no—
214C2×D4—2141no—
215C2×D4C22⋊D10712yes—
216C2×D4C22⋊D10712yescomplemented
217C2×D4C22×D10711yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—2141no—
221C22×C4—2141no—
222C22×C4C22⋊D10712yes—
223C22×C4C22×D10711yescharacteristic
224Q16—2144no—
225SD16—2144no—
226SD16—2144no—
227D8—2144no—
228M16—1074no—
229M16—2142no—
230C2×C8—2142no—
231C4⋊C4—2142no—
232C22⋊C4—2142no—
233C22⋊C4—2142no—
234C42—1071no—
235C42—1071no—
236C42—2141no—
237C23—24no—
238C23—22no—
239C23—2141no—
240C23C2×C22⋊D10712yes—
241C23C2×C22⋊D10711yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—2142no—
246Q8C2×C22⋊D10714yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—4282no—
255D4—4284no—
256D4—2142no—
257D4C2×C22⋊D10714yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—4282no—
266C2×C4—4281no—
267C2×C4—4281no—
268C2×C4—2141no—
269C2×C4—2141no—
270C2×C4—2141no—
271C2×C4C2×C22⋊D10712yes—
272C2×C4C2×C22⋊D10711yescharacteristic
273C2×C4C2×C22⋊D10712yes—
274C2×C4C2×C22⋊D10712yes—
275C2×C4C23×D10711yescharacteristic
276C8—2144no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—4282no—
284C22—4281no—
285C22#1681207812yes—
286C22C22×C22⋊D10711yescharacteristic
287C4—24no—
288C4—22no—
289C4—4281no—
290C4—4281no—
291C4—2142no—
292C4#1681207811yescharacteristic
293C4#1681207811yescharacteristic
294C4C22×C22⋊D10712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—8561no—
300C2C2×#1681207811yescharacteristic
subgroupG/H
centerC4#16812078
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#585×C107C2
socleC2×C107C2×#120
hypercenter#585D107
hypocenterC107#3018
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20293425C2semidirect
2. #20293173C2semidirect
3. #20292810C2semidirect
4. #20292805C2semidirect
5. #20292800C2semidirect
6. #20292795C2semidirect
7. C2×#16812036C2semidirect
8. #20292611C2semidirect
9. #585×C107C2semidirect
10. C2×#13133818C22semidirect
12. #16812139C22semidirect
14. #16812134C22semidirect
16. C2×#13133802C22semidirect
17. #16812107C22semidirect
18. #16812105C22semidirect
19. #16812098C22semidirect
20. #16812096C22semidirect
23. C2×C4×Dic107C22semidirect
24. #16812036C22semidirect
25. #16812036C22semidirect
26. #16812028C22semidirect
27. #16812026C22semidirect
68. #41×C107D4semidirect
72. C2×Q8×C107D4semidirect
80. C2×D4×C107D4semidirect
151. C107#3018semidirect
152. #585D107semidirect
200. #41C22⋊D107semidirect
205. C2×Q8C22⋊D107semidirect
216. C2×D4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
#202931732
#202928102
#202928052
#202928002
#202927952
C2×#168120362
#202926112
#585×C1072
#3018107