Finite Group Databasebeta

← #28348275

Subgroups — #28348275

Order 13696.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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#20293281C211yesmaximal, characteristic, complemented
2D4×D428C211yesmaximal, characteristic, complemented
3#20292974C211yesmaximal, characteristic, complemented
4#20292941C212yesmaximal, complemented
5#20292908C211yesmaximal, characteristic, complemented
6#20292863C212yesmaximal, complemented
7#20292823C211yesmaximal, characteristic, complemented
8#20292763C212yesmaximal, complemented
9C4×D856C211yesmaximal, characteristic, complemented
10C2×#16812031C212yesmaximal, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—21no—
13C22×D428C2212yes—
14C2×D4×D107—21no—
15C2×D856C2211yescharacteristic
16C2×D856C2212yescomplemented
17#16812109C2211yescharacteristic, complemented
18#16812106C2211yescharacteristic
19#16812093—22no—
20#16812089—21no—
21#16812084—21no—
22#16812083C2212yes—
23#16812083C2212yescomplemented
24#16812079—22no—
25C4×D428C2211yescharacteristic
26C4×D428C2211yescharacteristic, complemented
27C2×C4⋊Dic107C2212yescomplemented
28#16812043C2212yescomplemented
29#16812043C2211yescharacteristic, complemented
30#16812031C2211yescharacteristic
31#16812031C2212yes—
32#16812031C2212yescomplemented
33#16812031C2212yescomplemented
34#16812031C2212yescomplemented
35#16812031C2211yescharacteristic, complemented
36C22×C8×C107C2212yescomplemented
37C4×D4×C107C2211yescharacteristic
38C4⋊C8×C107C2211yescharacteristic, complemented
39C22⋊C8×C107C2212yescomplemented
40C4×C8×C107C2211yescharacteristic, complemented
41C23×D107—22no—
42#13133817—44no—
43C2×Dic214—21no—
44C2×D428—24no—
45C2×D428—22no—
46C2×D428—21no—
47C2×D428—21no—
48C2×D428—42no—
49C2×D428C2311yescharacteristic
50C2×D428C2312yes—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—22no—
53C2×C22⋊D107—22no—
54D4×D107—44no—
55C22×Dic107—22no—
56C2×C4×D107—21no—
57C2×C4×D107—21no—
58D856—22no—
59D856—42no—
60C4⋊Dic107—22no—
61C4⋊Dic107—21no—
62C4⋊Dic107C2311yescharacteristic
63C4⋊Dic107C2311yescharacteristic
64C4⋊Dic107C2312yes—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69C2×D4×C107D411yescharacteristic
70C22×C4×C107C2312yes—
71C2×C8×C107—42no—
72C2×C8×C107C2311yescharacteristic
73C2×C8×C107C2312yes—
74C2×C8×C107C2311yescharacteristic
75C4⋊C4×C107D411yescharacteristic
76C22⋊C4×C107D412yes—
77C42×C107C2311yescharacteristic
78C23×C107C2×D412yes—
79C22×D107—21no—
80C22×D107—21no—
81C22×D107—22no—
82C22×D107—44no—
83C22×D107—44no—
84C22×D107—82no—
85D4×C107D814yescomplemented
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—44no—
93C22⋊D107—44no—
94C2×Dic107—21no—
95C2×Dic107—21no—
96C2×Dic107—22no—
97C2×Dic107—44no—
98D428—21no—
99D428—22no—
100D428—21no—
101D428—21no—
102D428—22no—
103D428—44no—
104D428—42no—
105D428—81no—
106D428#4112yescomplemented
107C4×D107—42no—
108C4×D107—42no—
109Dic214—42no—
110C8×C107—21no—
111C8×C107—21no—
112C8×C107—22no—
113C22×C107—22no—
114C22×C107—22no—
115C22×C107C2×D814yes—
116C22×C107C22×D411yescharacteristic
117C2×D107—22no—
118C2×D107—41no—
119C2×D107—41no—
120C2×D107—42no—
121C2×D107—41no—
122C2×D107—41no—
123C2×D107—42no—
124C2×D107—42no—
125C2×D107—41no—
126C2×D107—88no—
127C4×C107—22no—
128C4×C107—21no—
129C4×C107C2×D812yes—
130C4×C107C2×#4111yescharacteristic
131C4×C107H(5,𝔽2)11yescharacteristic
132Dic107—42no—
133Dic107—41no—
134Dic107—41no—
135C2×C107—24no—
136C2×C107#54611yescharacteristic
137C2×C107C22×D811yescharacteristic
138C2×C107#57711yescharacteristic
139D107—42no—
140D107—81no—
141D107—82no—
142#3294—1071nomaximal, Sylow
143C107#329411yescharacteristic, complemented, Sylow
144#547—1071no—
145D42—1071no—
146#486—1071no—
147#480—1072no—
148#474—1071no—
149#466—1072no—
150#459—1071no—
151#447—1072no—
152C4×D8—1071no—
153C8×D4D10711yescharacteristic, complemented
154C2×#102—1072no—
155C2×#41—2141no—
156C22×D4—1072no—
157C22×D4—2141no—
158C2×D8—1072no—
159C2×D8—1071no—
160C22×C8C2×D10712yescomplemented
161C4⋊Q8—1071no—
162#127—1071no—
163#123—2142no—
164C22⋊Q8—2141no—
165#121—1072no—
166#121—1072no—
167#121—2141no—
168#120—2142no—
169C4×D4—1071no—
170C4×D4—1071no—
171C4×D4C2×D10711yescharacteristic
172C2×C4⋊C4—1072no—
173#107—1071no—
174#107—1072no—
175C4⋊C8C2×D10711yescharacteristic, complemented
176#102—1071no—
177#102—1072no—
178#102—1072no—
179#102—1072no—
180#102—1071no—
181#102—1072no—
182C22⋊C8C2×D10712yescomplemented
183C4×C8C2×D10711yescharacteristic, complemented
184C24—2142no—
185#41—4284no—
186C2×Q8—2141no—
187C2×D4—1071no—
188C2×D4—1072no—
189C2×D4—1071no—
190C2×D4—4282no—
191C2×D4—4284no—
192C2×D4—2144no—
193C2×D4—2142no—
194C2×D4—2142no—
195C2×D4—2141no—
196C2×D4—2142no—
197C2×D4—2141no—
198C2×D4D42811yescharacteristic
199C22×C4—2142no—
200C22×C4—2141no—
201C22×C4—2141no—
202C22×C4C22×D10712yes—
203D8—4282no—
204D8—2142no—
205C2×C8—42no—
206C2×C8C22×D10711yescharacteristic
207C2×C8C22×D10711yescharacteristic
208C2×C8C22×D10712yes—
209C4⋊C4—1071no—
210C4⋊C4—1072no—
211C4⋊C4—1071no—
212C4⋊C4—2142no—
213C4⋊C4—2141no—
214C4⋊C4D42811yescharacteristic
215C22⋊C4—2141no—
216C22⋊C4—2141no—
217C22⋊C4—2142no—
218C22⋊C4—2142no—
219C22⋊C4D42812yes—
220C42C22×D10711yescharacteristic
221C23—8562no—
222C23—4284no—
223C23—4284no—
224C23—2141no—
225C23—2142no—
226C23—2141no—
227C23C2×D42812yes—
228Q8—4282no—
229D4—8561no—
230D4—1072no—
231D4—4284no—
232D4—4284no—
233D4—4284no—
234D4—4282no—
235D4—2141no—
236D4—2142no—
237D4—2141no—
238D4—2142no—
239D4—2141no—
240D4D85614yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—4284no—
244C2×C4—4282no—
245C2×C4—4282no—
246C2×C4—2141no—
247C2×C4—2141no—
248C2×C4—2142no—
249C2×C4C2×D42811yescharacteristic
250C2×C4C2×D42812yes—
251C2×C4C2×D42811yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—8568no—
259C22—4281no—
260C22—4282no—
261C22—4282no—
262C22—4281no—
263C22—4282no—
264C22—4281no—
265C22—4281no—
266C22—4281no—
267C22—2142no—
268C22C2×D85614yes—
269C22C22×D42811yescharacteristic
270C4—21no—
271C4—22no—
272C4—4281no—
273C4—4281no—
274C4—4282no—
275C4C2×D85612yes—
276C4#41×D10711yescharacteristic
277C4#1681215411yescharacteristic
278C2—24no—
279C2—8562no—
280C2—8561no—
281C2—4282no—
282C2#2029327211yescharacteristic
283C2C22×D85611yescharacteristic
284C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
hypercenterC8×D4D107
hypocenterC107#3294
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20293281C2semidirect
2. D4×D428C2semidirect
3. #20292974C2semidirect
4. #20292941C2semidirect
5. #20292908C2semidirect
6. #20292863C2semidirect
7. #20292823C2semidirect
8. #20292763C2semidirect
9. C4×D856C2semidirect
10. C2×#16812031C2semidirect
11. C8×D4×C107C2semidirect
16. C2×D856C22semidirect
17. #16812109C22semidirect
23. #16812083C22semidirect
26. C4×D428C22semidirect
27. C2×C4⋊Dic107C22semidirect
28. #16812043C22semidirect
29. #16812043C22semidirect
32. #16812031C22semidirect
33. #16812031C22semidirect
34. #16812031C22semidirect
35. #16812031C22semidirect
36. C22×C8×C107C22semidirect
38. C4⋊C8×C107C22semidirect
39. C22⋊C8×C107C22semidirect
40. C4×C8×C107C22semidirect
85. D4×C107D8semidirect
106. D428#41semidirect
143. C107#3294semidirect
153. C8×D4D107semidirect
160. C22×C8C2×D107semidirect
175. C4⋊C8C2×D107semidirect
182. C22⋊C8C2×D107semidirect
183. C4×C8C2×D107semidirect
240. D4D856semidirect

Maximal subgroups

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

subgroupindex
#202932812
D4×D4282
#202929742
#202929412
#202929082
#202928632
#202928232
#202927632
C4×D8562
C2×#168120312
C8×D4×C1072
#3294107