Finite Group Databasebeta

← #28345809

Subgroups — #28345809

Order 13696.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#20293463C211yesmaximal, characteristic, complemented
2#20293425C212yesmaximal, complemented
3#20292809C212yesmaximal, complemented
4#20292804C212yesmaximal, complemented
5#20292799C212yesmaximal, complemented
6#20292794C212yesmaximal, complemented
7C2×#16812037C212yesmaximal, complemented
8#20292611C211yesmaximal, characteristic, complemented
9#532×C107C211yesmaximal, characteristic, complemented
10#16812157C2212yes—
11#16812156—22no—
12#16812155—22no—
13#16812154—22no—
14#16812154C2212yes—
15C2×#13133818—22no—
16C2×#13133818C2212yes—
17#41×D107—24no—
18#16812140C2214yescomplemented
19#16812139—22no—
20#16812135C2214yescomplemented
21#16812134—22no—
22#16812129—24no—
23C2×#13133802C2212yescomplemented
24#16812037C2214yescomplemented
25#16812037C2214yescomplemented
26#16812028C2212yescomplemented
27#16812026C2212yescomplemented
28C2×#41×C107C2211yescharacteristic
29C2×#41×C107C2211yescharacteristic, complemented
30C4⋊Q8×C107C2211yescharacteristic, complemented
31#127×C107C2211yescharacteristic, complemented
32#124×C107C2211yescharacteristic, complemented
33#124×C107C2211yescharacteristic, complemented
34#121×C107—22no—
35C4×D4×C107—22no—
36C2×C42×C107C2211yescharacteristic, complemented
37#13133819—24no—
38#13133818—24no—
39#13133818—24no—
40#13133818—22no—
41#13133818—24no—
42#13133818—22no—
43#13133818D414yescomplemented
44#13133818C2314yes—
45#13133817—28no—
46#13133817—24no—
47C2×Dic214—22no—
48C2×Dic214D412yescomplemented
49Q8×D107—24no—
50C2×D428—22no—
51C2×D428D412yescomplemented
52C2×C22⋊D107—24no—
53C2×C22⋊D107—22no—
54D4×D107—28no—
55D4×D107—24no—
56C2×C4×D107—24no—
57C2×C4×D107—22no—
58#13133808—24no—
59#13133806—24no—
60#13133805—24no—
61#13133803—24no—
62#13133802—22no—
63#13133802C2314yes—
64C2×C107⋊C8—22no—
65#41×C107—22no—
66#41×C107—22no—
67#41×C107—22no—
68#41×C107—22no—
69C2×Q8×C107C2311yescharacteristic
70C2×Q8×C107C2311yescharacteristic
71C2×D4×C107—22no—
72C2×D4×C107—21no—
73C2×D4×C107D412yes—
74C2×D4×C107C2311yescharacteristic
75C2×D4×C107C2311yescharacteristic
76C22×C4×C107—21no—
77C22×C4×C107—21no—
78C22×C4×C107D412yes—
79C22×C4×C107C2311yescharacteristic
80C4⋊C4×C107—22no—
81C22⋊C4×C107—22no—
82C22⋊C4×C107—22no—
83C42×C107—21no—
84C42×C107C2311yescharacteristic
85C42×C107C2311yescharacteristic
86C23×C107—21no—
87C23×C107C2×D412yes—
88C23×C107C2×D411yescharacteristic
89C22×D107—24no—
90C22×D107—22no—
91Q8×C107—22no—
92Q8×C107—22no—
93D4×C107—24no—
94D4×C107—22no—
95D4×C107—22no—
96D4×C107—44no—
97D4×C107—42no—
98C2×C4×C107—21no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—21no—
102C2×C4×C107—21no—
103C2×C4×C107—42no—
104C2×C4×C107—41no—
105C2×C4×C107—41no—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2×D412yes—
108C2×C4×C107C2×D412yes—
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107C2411yescharacteristic
111C22⋊D107—28no—
112C22⋊D107—24no—
113C22⋊D107—24no—
114C2×Dic107—24no—
115C2×Dic107—22no—
116D428—24no—
117D428—22no—
118D428C2×D414yes—
119C4×D107—28no—
120C4×D107—24no—
121C4×D107—24no—
122Dic214—24no—
123Dic214—22no—
124Dic214C2×D414yes—
125C107⋊C8—24no—
126C22×C107—22no—
127C22×C107—41no—
128C22×C107—42no—
129C22×C107—42no—
130C22×C107—41no—
131C22×C107#12012yes—
132C22×C107C22×D411yescharacteristic
133C2×D107—24no—
134C2×D107—82no—
135C2×D107—84no—
136C4×C107—22no—
137C4×C107—22no—
138C4×C107—41no—
139C4×C107—41no—
140C4×C107#12011yescharacteristic
141C4×C107#12011yescharacteristic
142C4×C107C22×D412yes—
143Dic107—24no—
144C2×C107—21no—
145C2×C107—22no—
146C2×C107—42no—
147C2×C107—81no—
148C2×C107C2×#12011yescharacteristic
149D107—44no—
150#3018—1071nomaximal, Sylow
151C107#301811yescharacteristic, complemented, Sylow
152#585—1071no—
153#575—1072no—
154#532D10711yescharacteristic, complemented
155#456—1072no—
156#455—1072no—
157#454—1072no—
158#453—1072no—
159C2×#104—1072no—
160#413—1071no—
1612−1+4—1072no—
1622−1+4—2142no—
163H(5,𝔽2)—1072no—
164H(5,𝔽2)—2142no—
165H(5,𝔽2)—2142no—
166C2×#41—1072no—
167C2×#41—2144no—
168C2×#41—2142no—
169C2×#41C2×D10711yescharacteristic
170C2×#41C2×D10711yescharacteristic, complemented
171#137—1074no—
172#137—2142no—
173AGL(1,ℤ/8)—1074no—
174AGL(1,ℤ/8)—2142no—
175#135—2144no—
176C2×M16—1072no—
177C4⋊Q8C2×D10711yescharacteristic, complemented
178#127C2×D10711yescharacteristic, complemented
179#124C2×D10711yescharacteristic, complemented
180#124C2×D10711yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D10711yescharacteristic, complemented
184#104—1074no—
185#104—1074no—
186#101—1072no—
187#100—1072no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—1074no—
193#41—1074no—
194#41—2144no—
195#41—2144no—
196#41—2144no—
197#41—2148no—
198#41—2144no—
199#41—2142no—
200#41—2142no—
201#41—2144no—
202C2×Q8—1072no—
203C2×Q8—2142no—
204C2×Q8—2144no—
205C2×Q8C22×D10711yescharacteristic
206C2×Q8C22×D10711yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—1072no—
210C2×D4—2142no—
211C2×D4—2144no—
212C2×D4—2148no—
213C2×D4—2142no—
214C2×D4—2144no—
215C2×D4C22⋊D10712yes—
216C2×D4C22×D10711yescharacteristic
217C2×D4C22×D10711yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—2144no—
221C22×C4—2142no—
222C22×C4C22⋊D10712yes—
223C22×C4C22×D10711yescharacteristic
224Q16—2144no—
225SD16—2144no—
226SD16—2144no—
227D8—2144no—
228M16—1074no—
229M16—2142no—
230C2×C8—2142no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D10711yescharacteristic
236C42C22×D10711yescharacteristic
237C23—21no—
238C23—2144no—
239C23—2142no—
240C23C2×C22⋊D10712yes—
241C23C2×C22⋊D10711yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—1074no—
245Q8—2144no—
246Q8—2142no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—1074no—
253D4—2144no—
254D4—2148no—
255D4—2144no—
256D4—2144no—
257D4—2142no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—2148no—
267C2×C4—2144no—
268C2×C4—2142no—
269C2×C4—2144no—
270C2×C4—2144no—
271C2×C4D4×D10712yes—
272C2×C4D4×D10712yes—
273C2×C4C2×C22⋊D10712yes—
274C2×C4C2×C22⋊D10711yescharacteristic
275C2×C4C23×D10711yescharacteristic
276C8—2144no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—8562no—
283C22—8564no—
284C22—2144no—
285C22#1681207712yes—
286C22C22×C22⋊D10711yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—2144no—
292C4#1681207711yescharacteristic
293C4#1681207711yescharacteristic
294C4C2×D4×D10712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—4284no—
300C2C2×#1681207711yescharacteristic
subgroupG/H
centerC4#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#532×C107C2
socleC2×C107C2×#120
hypercenter#532D107
hypocenterC107#3018
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20293463C2semidirect
2. #20293425C2semidirect
3. #20292809C2semidirect
4. #20292804C2semidirect
5. #20292799C2semidirect
6. #20292794C2semidirect
7. C2×#16812037C2semidirect
8. #20292611C2semidirect
9. #532×C107C2semidirect
18. #16812140C22semidirect
20. #16812135C22semidirect
23. C2×#13133802C22semidirect
24. #16812037C22semidirect
25. #16812037C22semidirect
26. #16812028C22semidirect
27. #16812026C22semidirect
29. C2×#41×C107C22semidirect
30. C4⋊Q8×C107C22semidirect
31. #127×C107C22semidirect
32. #124×C107C22semidirect
33. #124×C107C22semidirect
36. C2×C42×C107C22semidirect
43. #13133818D4semidirect
48. C2×Dic214D4semidirect
51. C2×D428D4semidirect
151. C107#3018semidirect
154. #532D107semidirect
170. C2×#41C2×D107semidirect
177. C4⋊Q8C2×D107semidirect
178. #127C2×D107semidirect
179. #124C2×D107semidirect
180. #124C2×D107semidirect
183. C2×C42C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934632
#202934252
#202928092
#202928042
#202927992
#202927942
C2×#168120372
#202926112
#532×C1072
#3018107