Finite Group Databasebeta

← #28348258

Subgroups — #28348258

Order 13696.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293430C212yesmaximal, complemented
2#20293429C211yesmaximal, characteristic, complemented
3#20293002C211yesmaximal, characteristic, complemented
4#20292897C211yesmaximal, characteristic, complemented
5#20292895C212yesmaximal, complemented
6#20292894C211yesmaximal, characteristic, complemented
7#20292795C212yesmaximal, complemented
8#20292793C212yesmaximal, complemented
9#20292744C211yesmaximal, characteristic, complemented
10#20292742C211yesmaximal, characteristic, complemented
11#576×C107C211yesmaximal, characteristic, complemented
12#16812155C2212yescomplemented
13#16812136—22no—
14#16812135—22no—
15#16812135C2212yescomplemented
16#16812134—22no—
17#16812134C2212yescomplemented
18#16812134C2212yescomplemented
19#16812132—21no—
20#16812129—22no—
21#16812129C2211yescharacteristic
22C2×D856C2211yescharacteristic, complemented
23C2×#13133803—22no—
24C2×#13133803C2212yescomplemented
25D8×D107—21no—
26#16812116C2212yescomplemented
27#16812115C2211yescharacteristic, complemented
28#16812105C2212yescomplemented
29#16812045C2211yescharacteristic, complemented
30#16812044C2211yescharacteristic, complemented
31#16812036C2211yescharacteristic, complemented
32#16812036C2212yescomplemented
33#16812035C2211yescharacteristic, complemented
34#16812026C2212yescomplemented
35#16812025C2212yescomplemented
36C8×Dic107C2211yescharacteristic, complemented
37H(5,𝔽2)×C107C2212yescomplemented
38AGL(1,ℤ/8)×C107—22no—
39AGL(1,ℤ/8)×C107C2212yescomplemented
40#135×C107—21no—
41#135×C107C2211yescharacteristic
42C2×D8×C107—21no—
43C2×D8×C107C2211yescharacteristic, complemented
44#131×C107C2211yescharacteristic, complemented
45#13133819—22no—
46#13133818—22no—
47#13133818C2311yescharacteristic
48C2×D428—22no—
49C2×D428C2312yes—
50C2×C22⋊D107—42no—
51D4×D107—22no—
52D4×D107—42no—
53#13133808D411yescharacteristic, complemented
54#13133806—22no—
55#13133806D411yescharacteristic, complemented
56#13133805—22no—
57#13133805—22no—
58#13133805D411yescharacteristic, complemented
59D856—21no—
60D856—21no—
61#13133803—22no—
62#13133803—22no—
63#13133803—22no—
64#13133803—22no—
65#13133803—22no—
66#13133803D411yescharacteristic, complemented
67#13133802—22no—
68#13133802C2311yescharacteristic
69#13133802C2312yes—
70#13133801—21no—
71C2×C107⋊C8—22no—
72C2×C107⋊C8C2311yescharacteristic
73C8×D107—21no—
74C4×Dic107C2311yescharacteristic
75#41×C107—22no—
76#41×C107—22no—
77#41×C107C2312yes—
78#41×C107C2311yescharacteristic
79C2×D4×C107—22no—
80C2×D4×C107—22no—
81C2×D4×C107—42no—
82C2×D4×C107C2312yes—
83Q16×C107D411yescharacteristic, complemented
84SD16×C107—22no—
85SD16×C107D412yescomplemented
86D8×C107—22no—
87D8×C107—21no—
88D8×C107—21no—
89D8×C107D411yescharacteristic, complemented
90M16×C107—21no—
91M16×C107C2311yescharacteristic
92C2×C8×C107—21no—
93C2×C8×C107C2311yescharacteristic
94C23×C107—22no—
95C23×C107—42no—
96C22×D107—22no—
97C22×D107—42no—
98Q8×C107C2×D411yescharacteristic
99Q8×C107C2×D412yes—
100D4×C107—21no—
101D4×C107—22no—
102D4×C107—22no—
103D4×C107—22no—
104D4×C107—42no—
105D4×C107—42no—
106D4×C107C2×D411yescharacteristic
107D4×C107C2×D412yes—
108C2×C4×C107—21no—
109C2×C4×C107—22no—
110C2×C4×C107—42no—
111C2×C4×C107C2411yescharacteristic
112C22⋊D107—21no—
113C22⋊D107—42no—
114C22⋊D107—84no—
115C2×Dic107—21no—
116D428—22no—
117D428—22no—
118D428—42no—
119D428C2×D411yescharacteristic
120C4×D107—21no—
121C4×D107—42no—
122Dic214C2×D411yescharacteristic
123C8×C107—21no—
124C8×C107C2×D411yescharacteristic
125C8×C107C2×D411yescharacteristic
126C107⋊C8—21no—
127C107⋊C8—22no—
128C107⋊C8C2×D411yescharacteristic
129C107⋊C8C2×D411yescharacteristic
130C22×C107—22no—
131C22×C107—21no—
132C22×C107—42no—
133C22×C107—82no—
134C22×C107—162no—
135C22×C107H(5,𝔽2)11yescharacteristic
136C2×D107—21no—
137C2×D107—42no—
138C2×D107—82no—
139C2×D107—162no—
140C4×C107—22no—
141C4×C107—21no—
142C4×C107C22×D411yescharacteristic
143C4×C107C22×D411yescharacteristic
144Dic107—22no—
145Dic107—21no—
146C2×C107—21no—
147C2×C107—42no—
148C2×C107—41no—
149C2×C107—82no—
150C2×C107D4211yescharacteristic
151D107—41no—
152D107—82no—
153#3292—1071nomaximal, Sylow
154C107#329211yescharacteristic, complemented, Sylow
155#576—1072no—
156#576—1071no—
157#576D10711yescharacteristic, complemented
158#493—1071no—
159#472—1072no—
160#472—1071no—
161#472—1071no—
162#453—1072no—
163#453—1072no—
164#443—1071no—
165#443—1071no—
166H(5,𝔽2)—1072no—
167H(5,𝔽2)C2×D10712yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—1072no—
170AGL(1,ℤ/8)—1072no—
171AGL(1,ℤ/8)—1072no—
172AGL(1,ℤ/8)—2142no—
173AGL(1,ℤ/8)—2142no—
174AGL(1,ℤ/8)—2142no—
175AGL(1,ℤ/8)C2×D10712yescomplemented
176#135—21no—
177#135—1071no—
178#135—2142no—
179#135—2141no—
180#135C2×D10711yescharacteristic
181C2×D8—21no—
182C2×D8—1072no—
183C2×D8—1071no—
184C2×D8—2142no—
185C2×D8—2141no—
186C2×D8C2×D10711yescharacteristic, complemented
187#131—1072no—
188#131—1071no—
189#131C2×D10711yescharacteristic, complemented
190#127—1072no—
191#108—1071no—
192#108—1071no—
193#104—1071no—
194#104—1072no—
195#104—1071no—
196#100—1072no—
197#100—1072no—
198C4×C8—1071no—
199#41—22no—
200#41—22no—
201#41—1071no—
202#41—2142no—
203#41—2142no—
204#41C22×D10711yescharacteristic
205#41C22×D10712yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—1072no—
210C2×D4—4282no—
211C2×D4—4282no—
212C2×D4—2142no—
213C2×D4—2142no—
214C2×D4C22×D10712yes—
215Q16—1071no—
216Q16C22⋊D10711yescharacteristic, complemented
217SD16—22no—
218SD16—1071no—
219SD16—1071no—
220SD16—2142no—
221SD16—2142no—
222SD16—2142no—
223SD16C22⋊D10712yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—1071no—
228D8—2142no—
229D8—2142no—
230D8—2142no—
231D8—2142no—
232D8—2142no—
233D8—2141no—
234D8—2141no—
235D8C22⋊D10711yescharacteristic, complemented
236M16—21no—
237M16—1072no—
238M16—1071no—
239M16—2142no—
240M16—2141no—
241M16C22×D10711yescharacteristic
242C2×C8—21no—
243C2×C8—1071no—
244C2×C8—2142no—
245C2×C8—2141no—
246C2×C8C22×D10711yescharacteristic
247C42—1071no—
248C23—22no—
249C23—42no—
250C23—4282no—
251C23—2142no—
252Q8—1071no—
253Q8D4×D10711yescharacteristic
254Q8C2×C22⋊D10712yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—8564no—
262D4—1071no—
263D4—4282no—
264D4—4282no—
265D4—2141no—
266D4—2142no—
267D4—2142no—
268D4D4×D10711yescharacteristic
269D4C2×C22⋊D10712yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—4282no—
274C2×C4—2141no—
275C2×C4—2141no—
276C2×C4C23×D10711yescharacteristic
277C8—21no—
278C8—1071no—
279C8—1071no—
280C8—2142no—
281C8—2141no—
282C8C2×C22⋊D10711yescharacteristic
283C8C2×C22⋊D10711yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—8562no—
290C22—4282no—
291C22—17122no—
292C22—2141no—
293C22#1681215411yescharacteristic
294C4—21no—
295C4—22no—
296C4—2142no—
297C4—2141no—
298C4C2×D4×D10711yescharacteristic
299C4C22×C22⋊D10711yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—8562no—
305C2—4281no—
306C2D4×C22⋊D10711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#576×C107C2
socleC2×C107D42
hypercenter#576D107
hypocenterC107#3292
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20293430C2semidirect
2. #20293429C2semidirect
3. #20293002C2semidirect
4. #20292897C2semidirect
5. #20292895C2semidirect
6. #20292894C2semidirect
7. #20292795C2semidirect
8. #20292793C2semidirect
9. #20292744C2semidirect
10. #20292742C2semidirect
11. #576×C107C2semidirect
12. #16812155C22semidirect
15. #16812135C22semidirect
17. #16812134C22semidirect
18. #16812134C22semidirect
22. C2×D856C22semidirect
24. C2×#13133803C22semidirect
26. #16812116C22semidirect
27. #16812115C22semidirect
28. #16812105C22semidirect
29. #16812045C22semidirect
30. #16812044C22semidirect
31. #16812036C22semidirect
32. #16812036C22semidirect
33. #16812035C22semidirect
34. #16812026C22semidirect
35. #16812025C22semidirect
36. C8×Dic107C22semidirect
37. H(5,𝔽2)×C107C22semidirect
39. AGL(1,ℤ/8)×C107C22semidirect
43. C2×D8×C107C22semidirect
44. #131×C107C22semidirect
53. #13133808D4semidirect
55. #13133806D4semidirect
58. #13133805D4semidirect
66. #13133803D4semidirect
83. Q16×C107D4semidirect
85. SD16×C107D4semidirect
89. D8×C107D4semidirect
154. C107#3292semidirect
157. #576D107semidirect
167. H(5,𝔽2)C2×D107semidirect
175. AGL(1,ℤ/8)C2×D107semidirect
186. C2×D8C2×D107semidirect
189. #131C2×D107semidirect
216. Q16C22⋊D107semidirect
223. SD16C22⋊D107semidirect
235. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202934302
#202934292
#202930022
#202928972
#202928952
#202928942
#202927952
#202927932
#202927442
#202927422
#576×C1072
#3292107