Finite Group Databasebeta

← #28348287

Subgroups — #28348287

Order 13696.

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

normal subgroups only

This group has 462 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#20293337C211yesmaximal, characteristic, complemented
2D4×D428C211yesmaximal, characteristic
3#20292974C211yesmaximal, characteristic, complemented
4#20292947C212yesmaximal, complemented
5#20292917C211yesmaximal, characteristic, complemented
6#20292859C212yesmaximal, complemented
7#20292827C211yesmaximal, characteristic, complemented
8#20292776C212yesmaximal, complemented
9C8×D428C211yesmaximal, characteristic, complemented
10#102×D107C212yesmaximal, complemented
11C4×SD16×C107C211yesmaximal, characteristic, complemented
12C22×D428—21no—
13C2×D4×D107C2212yes—
14C2×#13133805C2212yescomplemented
15C2×C8×D107C2212yescomplemented
16#16812109C2211yescharacteristic, complemented
17#16812106C2211yescharacteristic
18#16812101—21no—
19#16812087—22no—
20#16812085C2212yescomplemented
21#16812084C2212yes—
22#16812083—21no—
23#16812079—22no—
24C4×D428—21no—
25C4×D428C2211yescharacteristic
26C4⋊C4×D107C2212yescomplemented
27#16812040C2212yescomplemented
28C4⋊(C107⋊C8)C2211yescharacteristic, complemented
29#16812031C2212yescomplemented
30#16812031C2212yescomplemented
31#16812030C2212yes—
32#16812029C2212yescomplemented
33(C2×D107)⋊C8C2212yescomplemented
34C2×SD16×C107C2211yescharacteristic
35C4×Q8×C107C2211yescharacteristic, complemented
36C4×D4×C107C2211yescharacteristic
37#106×C107C2211yescharacteristic, complemented
38#103×C107C2211yescharacteristic, complemented
39#102×C107C2211yescharacteristic
40C4×C8×C107C2211yescharacteristic, complemented
41C23×D107—22no—
42C2×Dic214—21no—
43C2×D428—22no—
44C2×D428—21no—
45C2×D428—21no—
46C2×D428—44no—
47C2×D428D411yescharacteristic
48C2×D428C2312yes—
49C2×C22⋊D107—22no—
50D4×D107—24no—
51D4×D107—42no—
52C2×C4×D107—22no—
53C2×C4×D107C2312yes—
54#13133805—42no—
55C2×C107⋊C8C2312yes—
56C8×D107—42no—
57C4⋊Dic107—21no—
58C4⋊Dic107D411yescharacteristic
59C4⋊Dic107C2312yes—
60Dic107⋊C4—22no—
61Dic107⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4D412yes—
65C2×Q8×C107C2311yescharacteristic
66C2×D4×C107C2311yescharacteristic
67C22×C4×C107—21no—
68SD16×C107—22no—
69C2×C8×C107C2311yescharacteristic
70C2×C8×C107C2311yescharacteristic
71C4⋊C4×C107—21no—
72C4⋊C4×C107C2311yescharacteristic
73C4⋊C4×C107C2311yescharacteristic
74C22⋊C4×C107—21no—
75C42×C107—21no—
76C42×C107C2311yescharacteristic
77C23×C107—21no—
78C22×D107—22no—
79C22×D107—44no—
80C22×D107—44no—
81C22×D107—82no—
82C22×D107C2×D412yes—
83Q8×C107—21no—
84Q8×C107—21no—
85D4×C107—21no—
86D4×C107#4112yes—
87C2×C4×C107—21no—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—42no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—44no—
95C2×Dic107—22no—
96C2×Dic107C2×D412yes—
97D428—22no—
98D428—22no—
99D428—44no—
100D428—42no—
101D428—81no—
102D428SD1614yescomplemented
103C4×D107—22no—
104C4×D107—22no—
105C4×D107—82no—
106Dic214—81no—
107C8×C107—21no—
108C8×C107—21no—
109C107⋊C8—22no—
110C22×C107—22no—
111C22×C107—41no—
112C22×C107—41no—
113C22×C107C22×D411yescharacteristic
114C2×D107—22no—
115C2×D107—22no—
116C2×D107—42no—
117C2×D107—42no—
118C2×D107—42no—
119C2×D107—88no—
120C2×D107C2×SD1614yes—
121C4×C107—21no—
122C4×C107—41no—
123C4×C107—41no—
124C4×C107—41no—
125C4×C107C2×SD1612yes—
126C4×C107C2×#4111yescharacteristic
127C4×C107H(5,𝔽2)11yescharacteristic
128Dic107—22no—
129Dic107—42no—
130C2×C107—42no—
131C2×C107#54611yescharacteristic
132C2×C107C22×SD1611yescharacteristic
133C2×C107#57611yescharacteristic
134D107—24no—
135D107—82no—
136#3295—1071nomaximal, Sylow
137C107#329511yescharacteristic, complemented, Sylow
138#554—1071no—
139D42—1071no—
140#486—1071no—
141#481—1072no—
142#476—1071no—
143#465—1072no—
144#460—1071no—
145#450—1072no—
146C4×SD16D10711yescharacteristic, complemented
147C8×D4—1071no—
148C2×#102—1072no—
149C22×D4—1072no—
150C22×D4—2141no—
151C2×SD16—1072no—
152C2×SD16C2×D10711yescharacteristic
153C22×C8—1072no—
154C4⋊Q8—1071no—
155#127—1071no—
156#125—2141no—
157C22⋊Q8—1072no—
158C22⋊Q8—2142no—
159#121—1072no—
160#121—2141no—
161#120—2142no—
162C4×Q8C2×D10711yescharacteristic, complemented
163C4×D4—1071no—
164C4×D4—2141no—
165C4×D4C2×D10711yescharacteristic
166C2×C4⋊C4—1072no—
167#106—1072no—
168#106C2×D10711yescharacteristic, complemented
169C4⋊C8—1071no—
170#103C2×D10711yescharacteristic, complemented
171#102—1072no—
172#102—1072no—
173#102—1072no—
174#102—1072no—
175#102C2×D10711yescharacteristic
176C22⋊C8—1072no—
177C4×C8C2×D10711yescharacteristic, complemented
178C24—2142no—
179C2×Q8—2141no—
180C2×Q8C22×D10711yescharacteristic
181C2×D4—1071no—
182C2×D4—1072no—
183C2×D4—4284no—
184C2×D4—4282no—
185C2×D4—2142no—
186C2×D4—2142no—
187C2×D4—2144no—
188C2×D4—2141no—
189C2×D4—2141no—
190C2×D4C22×D10711yescharacteristic
191C22×C4—21no—
192C22×C4—1072no—
193C22×C4—2142no—
194SD16—22no—
195SD16—4282no—
196C2×C8—1072no—
197C2×C8—4282no—
198C2×C8C22×D10711yescharacteristic
199C2×C8C22×D10711yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—1071no—
202C4⋊C4—1072no—
203C4⋊C4—2142no—
204C4⋊C4—2142no—
205C4⋊C4—2141no—
206C4⋊C4C22×D10711yescharacteristic
207C4⋊C4C22×D10711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—1072no—
210C22⋊C4—2142no—
211C22⋊C4—2142no—
212C42—21no—
213C42C22×D10711yescharacteristic
214C23—21no—
215C23—8562no—
216C23—1072no—
217C23—4284no—
218C23—4284no—
219C23—2142no—
220Q8—21no—
221Q8—21no—
222Q8—8561no—
223D4—21no—
224D4—8561no—
225D4—1074no—
226D4—4284no—
227D4—4284no—
228D4—4282no—
229D4—2142no—
230D4—2142no—
231D4#1313381812yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—8562no—
237C2×C4—1072no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4D4×D10711yescharacteristic
242C2×C4D4×D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—21no—
245C8—21no—
246C8—2142no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—8568no—
251C22—1074no—
252C22—4282no—
253C22—4282no—
254C22—4282no—
255C22—2142no—
256C22—2142no—
257C22C2×D4×D10711yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—4282no—
263C4—2142no—
264C4SD16×D10712yes—
265C4C2×#1313381811yescharacteristic
266C4#1681215511yescharacteristic
267C2—42no—
268C2—8562no—
269C2—2144no—
270C2#2029327311yescharacteristic
271C2C2×SD16×D10711yescharacteristic
272C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radicalD4×D428C2
hypercenterC4×SD16D107
hypocenterC107#3295
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #20293337C2semidirect
3. #20292974C2semidirect
4. #20292947C2semidirect
5. #20292917C2semidirect
6. #20292859C2semidirect
7. #20292827C2semidirect
8. #20292776C2semidirect
9. C8×D428C2semidirect
10. #102×D107C2semidirect
11. C4×SD16×C107C2semidirect
14. C2×#13133805C22semidirect
15. C2×C8×D107C22semidirect
16. #16812109C22semidirect
20. #16812085C22semidirect
26. C4⋊C4×D107C22semidirect
27. #16812040C22semidirect
28. C4⋊(C107⋊C8)C22semidirect
29. #16812031C22semidirect
30. #16812031C22semidirect
32. #16812029C22semidirect
33. (C2×D107)⋊C8C22semidirect
35. C4×Q8×C107C22semidirect
37. #106×C107C22semidirect
38. #103×C107C22semidirect
40. C4×C8×C107C22semidirect
102. D428SD16semidirect
137. C107#3295semidirect
146. C4×SD16D107semidirect
162. C4×Q8C2×D107semidirect
168. #106C2×D107semidirect
170. #103C2×D107semidirect
177. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933372
D4×D4282
#202929742
#202929472
#202929172
#202928592
#202928272
#202927762
C8×D4282
#102×D1072
C4×SD16×C1072
#3295107