Finite Group Databasebeta

← #28346939

Subgroups — #28346939

Order 13696.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16812140C211yesmaximal, characteristic, complemented
2C2×#16812134C211yesmaximal, characteristic, complemented
3C2×#16812129C211yesmaximal, characteristic, complemented
4C22×#13133805C211yesmaximal, characteristic, complemented
5#20293196C211yesmaximal, characteristic, complemented
6#20293019C212yesmaximal, complemented
7#20293017C212yesmaximal, complemented
8#20293009C212yesmaximal, complemented
9#20293001C212yesmaximal, complemented
10#20292583C211yesmaximal, characteristic, complemented
11#536×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14#16812140—24no—
15#16812134—24no—
16#16812129—24no—
17C2×#13133808C2211yescharacteristic, complemented
18C2×#13133808C2212yescomplemented
19C2×#13133806C2211yescharacteristic, complemented
20C2×#13133805—24no—
21C2×#13133805C2212yes—
22C2×#13133805C2212yes—
23C2×#13133805C2212yes—
24C2×#13133805C2211yescharacteristic, complemented
25C2×#13133805C2212yescomplemented
26C2×#13133803C2212yes—
27C2×#13133803C2211yescharacteristic, complemented
28C2×#13133802C2211yescharacteristic
29C22×C107⋊C8C2211yescharacteristic
30#16812106C2212yes—
31#16812097C2212yes—
32#16812084—22no—
33#16812079—22no—
34#16812020C2212yescomplemented
35C4×C107⋊C8C2212yescomplemented
36C2×#41×C107C2211yescharacteristic
37C22×Q8×C107C2211yescharacteristic
38C4⋊Q8×C107C2212yescomplemented
39#124×C107C2212yescomplemented
40#123×C107—22no—
41C22⋊Q8×C107—22no—
42#117×C107C2211yescharacteristic
43C23×D107—21no—
44#13133818—22no—
45#13133818—22no—
46C2×Dic214C2311yescharacteristic
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—21no—
50C2×D428—24no—
51C2×D428C2312yes—
52C2×D428C2311yescharacteristic
53C2×C22⋊D107—21no—
54C2×C4×D107—21no—
55#13133808—22no—
56#13133808—24no—
57#13133806—22no—
58#13133805—24no—
59#13133805—24no—
60#13133805—24no—
61#13133805—22no—
62#13133805—24no—
63#13133803—24no—
64#13133803—22no—
65#13133802D414yescomplemented
66C2×C107⋊C8D414yescomplemented
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C2×C107⋊C8C2312yes—
70(C2×D107)⋊C4—24no—
71#41×C107—22no—
72#41×C107—22no—
73C2×Q8×C107—21no—
74C2×Q8×C107—22no—
75C2×Q8×C107—22no—
76C2×Q8×C107C2312yes—
77C2×Q8×C107C2311yescharacteristic
78C2×D4×C107—21no—
79C2×D4×C107C2311yescharacteristic
80C22×C4×C107—21no—
81C22×C4×C107—21no—
82C22×C4×C107C2311yescharacteristic
83C4⋊C4×C107—24no—
84C4⋊C4×C107D412yes—
85C22⋊C4×C107—24no—
86C22⋊C4×C107D412yes—
87C42×C107C2312yes—
88C23×C107—21no—
89C23×C107C2×D411yescharacteristic
90C22×D107—22no—
91C22×D107—21no—
92C22×D107—42no—
93C22×D107—82no—
94Q8×C107—22no—
95Q8×C107—22no—
96Q8×C107—21no—
97Q8×C107—21no—
98Q8×C107—42no—
99D4×C107—21no—
100D4×C107—21no—
101D4×C107—42no—
102C2×C4×C107—22no—
103C2×C4×C107—21no—
104C2×C4×C107—42no—
105C2×C4×C107—42no—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2×D412yes—
108C2×C4×C107C2×D414yes—
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107C2411yescharacteristic
111C22⋊D107—42no—
112C2×Dic107—21no—
113D428—22no—
114D428—21no—
115D428—22no—
116D428—21no—
117D428—42no—
118D428—84no—
119C4×D107—42no—
120Dic214—21no—
121Dic214—21no—
122C107⋊C8C2×D412yes—
123C107⋊C8C2×D412yes—
124C107⋊C8C2×D414yes—
125C22×C107—22no—
126C22×C107—41no—
127C22×C107—42no—
128C22×C107#12712yes—
129C22×C107C22×D411yescharacteristic
130C2×D107—41no—
131C2×D107—42no—
132C2×D107—42no—
133C2×D107—44no—
134C2×D107—84no—
135C4×C107—24no—
136C4×C107—41no—
137C4×C107—42no—
138C4×C107#12712yes—
139C4×C107C22×D411yescharacteristic
140C4×C107C22×D411yescharacteristic
141Dic107—41no—
142C2×C107—22no—
143C2×C107—81no—
144C2×C107C2×#12711yescharacteristic
145C2×C107#57712yes—
146D107—81no—
147D107—82no—
148#3154—1071nomaximal, Sylow
149C107#315411yescharacteristic, complemented, Sylow
150C2×#137—1071no—
151C2×AGL(1,ℤ/8)—1071no—
152C2×#135—1071no—
153C22×SD16—1071no—
154#536D10711yescharacteristic, complemented
155#535—1071no—
156#497—1072no—
157#496—1072no—
158#495—1072no—
159#492—1072no—
160#405—1071no—
161C2×#41—1071no—
162C2×#41C2×D10711yescharacteristic
163C22×Q8C2×D10711yescharacteristic
164C22×D4—1071no—
165#137—2144no—
166AGL(1,ℤ/8)—2144no—
167#135—2144no—
168C2×Q16—1071no—
169C2×Q16—1072no—
170C2×SD16—1072no—
171C2×SD16—1072no—
172C2×SD16—1071no—
173C2×SD16—1072no—
174C2×SD16—1072no—
175C2×SD16—1071no—
176C2×SD16—2144no—
177C2×D8—1071no—
178C2×D8—1072no—
179C2×M16—1071no—
180C22×C8—1071no—
181C4⋊Q8C2×D10712yescomplemented
182#127—1072no—
183#124—1072no—
184#124C2×D10712yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—2142no—
188#120—2142no—
189#117C2×D10711yescharacteristic
190#97—1072no—
191C4×C8—1072no—
192C24—2141no—
193#41—22no—
194#41—22no—
195#41—2142no—
196#41—2142no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1071no—
201C2×Q8C22×D10712yes—
202C2×Q8C22×D10711yescharacteristic
203C2×D4—21no—
204C2×D4—1072no—
205C2×D4—1071no—
206C2×D4—2141no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4—2144no—
210C2×D4—2141no—
211C2×D4C22×D10711yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—2141no—
215C22×C4C22×D10711yescharacteristic
216Q16—2142no—
217Q16—2144no—
218SD16—2142no—
219SD16—2144no—
220SD16—2144no—
221SD16—2142no—
222SD16—2144no—
223SD16—2144no—
224D8—2142no—
225D8—2144no—
226M16—1074no—
227C2×C8—1074no—
228C2×C8—1071no—
229C2×C8—1072no—
230C2×C8—1071no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D10712yes—
233C22⋊C4—24no—
234C22⋊C4—2144no—
235C22⋊C4C22⋊D10712yes—
236C42C22×D10712yes—
237C23—21no—
238C23—8562no—
239C23—4282no—
240C23—2142no—
241C23—2141no—
242C23C2×C22⋊D10711yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—2141no—
249Q8—2141no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—8564no—
254D4—4282no—
255D4—4282no—
256D4—2142no—
257D4—2141no—
258D4—2142no—
259D4—2141no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—4282no—
265C2×C4—2141no—
266C2×C4D4×D10712yes—
267C2×C4D4×D10712yes—
268C2×C4C2×C22⋊D10714yes—
269C2×C4C2×C22⋊D10711yescharacteristic
270C2×C4C23×D10711yescharacteristic
271C8—1072no—
272C8—1072no—
273C8—1074no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—8564no—
278C22—4281no—
279C22—4282no—
280C22—4282no—
281C22—4284no—
282C22#1681210512yes—
283C22C22×C22⋊D10711yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—4281no—
288C4#1681210512yes—
289C4C2×D4×D10711yescharacteristic
290C4C2×D4×D10711yescharacteristic
291C2—22no—
292C2—81no—
293C2—8562no—
294C2—8561no—
295C2C2×#1681210511yescharacteristic
296C2#2029343512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#536×C107C2
socleC22×C107C22×D4
hypercenter#536D107
hypocenterC107#3154
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C2×#16812134C2semidirect
3. C2×#16812129C2semidirect
4. C22×#13133805C2semidirect
5. #20293196C2semidirect
6. #20293019C2semidirect
7. #20293017C2semidirect
8. #20293009C2semidirect
9. #20293001C2semidirect
10. #20292583C2semidirect
11. #536×C107C2semidirect
17. C2×#13133808C22semidirect
18. C2×#13133808C22semidirect
19. C2×#13133806C22semidirect
24. C2×#13133805C22semidirect
25. C2×#13133805C22semidirect
27. C2×#13133803C22semidirect
34. #16812020C22semidirect
35. C4×C107⋊C8C22semidirect
38. C4⋊Q8×C107C22semidirect
39. #124×C107C22semidirect
65. #13133802D4semidirect
66. C2×C107⋊C8D4semidirect
149. C107#3154semidirect
154. #536D107semidirect
181. C4⋊Q8C2×D107semidirect
184. #124C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C2×#168121342
C2×#168121292
C22×#131338052
#202931962
#202930192
#202930172
#202930092
#202930012
#202925832
#536×C1072
#3154107