Finite Group Databasebeta

← #28346464

Subgroups — #28346464

Order 13696.

448 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 129 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#20293179C211yesmaximal, characteristic, complemented
2C2×D4×Dic107C211yesmaximal, characteristic
3#20292958C211yesmaximal, characteristic, complemented
4#20292955C211yesmaximal, characteristic, complemented
5#20292915C212yesmaximal, complemented
6#20292830C212yesmaximal, complemented
7#20292778C211yesmaximal, characteristic
8#20292718C212yesmaximal, complemented
9SD16×Dic107C212yesmaximal, complemented
10#20292677C211yesmaximal, characteristic, complemented
11#450×C107C211yesmaximal, characteristic
12C23×Dic107—21no—
13C2×#13133806C2212yes—
14#16812108C2211yescharacteristic, complemented
15#16812107C2211yescharacteristic, complemented
16#16812099—21no—
17#16812088C2211yescharacteristic
18#16812086—21no—
19Q8×Dic107C2211yescharacteristic, complemented
20#16812074—21no—
21#16812074C2211yescharacteristic, complemented
22D4×Dic107—21no—
23D4×Dic107—21no—
24D4×Dic107C2212yes—
25D4×Dic107C2212yes—
26#16812065—21no—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812041C2212yescomplemented
31#16812040C2212yescomplemented
32Dic107⋊C8C2212yescomplemented
33#16812034C2212yescomplemented
34#16812033C2212yescomplemented
35#16812030C2212yes—
36C22⋊(C107⋊C8)C2211yescharacteristic
37C8×Dic107C2212yescomplemented
38C22×D4×C107C2211yescharacteristic
39C2×SD16×C107C2212yes—
40C22⋊Q8×C107C2211yescharacteristic
41#102×C107C2212yes—
42C22⋊C8×C107C2211yescharacteristic
43C2×Dic214—21no—
44C2×Dic214C2311yescharacteristic
45C22×Dic107—21no—
46C22×Dic107—22no—
47C22×Dic107—22no—
48C22×Dic107—44no—
49C22×Dic107D411yescharacteristic
50C22×Dic107D411yescharacteristic
51#13133806—24no—
52C2×C107⋊C8C2312yes—
53C4⋊Dic107—21no—
54C4⋊Dic107—21no—
55C4⋊Dic107C2312yes—
56C4⋊Dic107C2311yescharacteristic
57Dic107⋊C4—21no—
58Dic107⋊C4—21no—
59Dic107⋊C4—21no—
60Dic107⋊C4—21no—
61Dic107⋊C4—21no—
62C22⋊Dic107—21no—
63C22⋊Dic107—22no—
64C22⋊Dic107—22no—
65C4×Dic107—21no—
66C4×Dic107—21no—
67C4×Dic107D411yescharacteristic
68C4×Dic107D411yescharacteristic
69C4×Dic107C2311yescharacteristic
70C4×Dic107C2311yescharacteristic
71C24×C107—21no—
72C2×Q8×C107C2311yescharacteristic
73C2×D4×C107—24no—
74C2×D4×C107—21no—
75C2×D4×C107C2312yes—
76C22×C4×C107C2311yescharacteristic
77SD16×C107—24no—
78C2×C8×C107C2312yes—
79C4⋊C4×C107—21no—
80C4⋊C4×C107C2311yescharacteristic
81C22⋊C4×C107—21no—
82C23×C107—24no—
83C23×C107—22no—
84C23×C107—42no—
85C23×C107—42no—
86C23×C107C2×D411yescharacteristic
87Q8×C107—22no—
88D4×C107—24no—
89D4×C107—22no—
90D4×C107#4114yes—
91C2×C4×C107—22no—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—22no—
99C2×Dic107—41no—
100C2×Dic107—42no—
101C2×Dic107—44no—
102C2×Dic107—44no—
103C2×Dic107SD1614yescomplemented
104C2×Dic107C2×D411yescharacteristic
105C2×Dic107C2×D411yescharacteristic
106C2×Dic107C2×D411yescharacteristic
107C2×Dic107C2×D411yescharacteristic
108Dic214—22no—
109Dic214—42no—
110C8×C107—22no—
111C107⋊C8—22no—
112C22×C107—21no—
113C22×C107—21no—
114C22×C107—24no—
115C22×C107—48no—
116C22×C107—42no—
117C22×C107—42no—
118C22×C107C2×SD1612yes—
119C22×C107C22×D411yescharacteristic
120C4×C107—21no—
121C4×C107—41no—
122C4×C107—41no—
123C4×C107C2×#4112yes—
124Dic107—21no—
125Dic107—21no—
126Dic107—41no—
127Dic107—42no—
128Dic107—41no—
129Dic107C2×SD1612yes—
130Dic107C2×SD1612yes—
131C2×C107—22no—
132C2×C107—44no—
133C2×C107#52511yescharacteristic
134C2×C107C22×SD1611yescharacteristic
135C2×C107#57511yescharacteristic
136#3101—1071nomaximal, Sylow
137C107#310111yescharacteristic, complemented, Sylow
138#533—1071no—
139C2×C4×D4—1071no—
140#483—1071no—
141#483—1071no—
142#476—1072no—
143#461—1072no—
144#450—1071no—
145#450D10711yescharacteristic
146C4×SD16—1072no—
147C4×SD16—1072no—
148#431—1071no—
149C22×D4C2×D10711yescharacteristic
150C23×C4—2141no—
151C2×SD16—1072no—
152C2×SD16C2×D10712yes—
153C4⋊Q8—1071no—
154C4⋊Q8—1071no—
155#125—2141no—
156C22⋊Q8—1071no—
157C22⋊Q8—2141no—
158C22⋊Q8C2×D10711yescharacteristic
159C4×Q8—1071no—
160C4×Q8—1071no—
161C4×Q8—2141no—
162C4×D4—1072no—
163C4×D4—1072no—
164C4×D4—2141no—
165C4×D4—2141no—
166#117—2141no—
167C2×C4⋊C4—1071no—
168C2×C22⋊C4—2141no—
169C2×C42—1071no—
170#106—1072no—
171#106—1072no—
172C4⋊C8—1072no—
173#103—1072no—
174#103—1072no—
175#102—1072no—
176#102C2×D10712yes—
177C22⋊C8—1071no—
178C22⋊C8C2×D10711yescharacteristic
179C4×C8—1072no—
180C24—21no—
181C2×Q8—1071no—
182C2×Q8—2141no—
183C2×Q8C22×D10711yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D10712yes—
187C22×C4—1071no—
188C22×C4—1071no—
189C22×C4—4284no—
190C22×C4—2142no—
191C22×C4—2142no—
192C22×C4—2141no—
193C22×C4C22×D10711yescharacteristic
194SD16—24no—
195SD16—2144no—
196C2×C8—1072no—
197C2×C8C22×D10712yes—
198C4⋊C4—21no—
199C4⋊C4—1071no—
200C4⋊C4—1072no—
201C4⋊C4—2141no—
202C4⋊C4—2141no—
203C4⋊C4—2141no—
204C4⋊C4—2141no—
205C4⋊C4—2141no—
206C4⋊C4—2141no—
207C4⋊C4—2141no—
208C4⋊C4C22×D10711yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—2142no—
211C22⋊C4—2142no—
212C22⋊C4—2141no—
213C42—1071no—
214C42—1071no—
215C42—1071no—
216C42—1071no—
217C42—2141no—
218C42—2141no—
219C23—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D10711yescharacteristic
224Q8—22no—
225Q8—4282no—
226Q8—2142no—
227D4—24no—
228D4—22no—
229D4#1313381714yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—1074no—
234C2×C4—1071no—
235C2×C4—1071no—
236C2×C4—1071no—
237C2×C4—1071no—
238C2×C4—4284no—
239C2×C4—4284no—
240C2×C4—4282no—
241C2×C4—4281no—
242C2×C4—2141no—
243C2×C4—2141no—
244C2×C4—2142no—
245C2×C4D4×D10711yescharacteristic
246C2×C4C23×D10711yescharacteristic
247C8—22no—
248C8—2142no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22SD16×D10712yes—
256C22C2×D4×D10711yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—1072no—
261C4—1072no—
262C4—4281no—
263C4—4282no—
264C4—4281no—
265C4—2141no—
266C4—2141no—
267C4C2×#1313381712yes—
268C2—22no—
269C2—44no—
270C2#2029313311yescharacteristic
271C2C2×SD16×D10711yescharacteristic
272C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
prime radicalC22×D4×C107C22
hypercenter#450D107
hypocenterC107#3101
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20293179C2semidirect
3. #20292958C2semidirect
4. #20292955C2semidirect
5. #20292915C2semidirect
6. #20292830C2semidirect
8. #20292718C2semidirect
9. SD16×Dic107C2semidirect
10. #20292677C2semidirect
14. #16812108C22semidirect
15. #16812107C22semidirect
19. Q8×Dic107C22semidirect
21. #16812074C22semidirect
30. #16812041C22semidirect
31. #16812040C22semidirect
32. Dic107⋊C8C22semidirect
33. #16812034C22semidirect
34. #16812033C22semidirect
37. C8×Dic107C22semidirect
103. C2×Dic107SD16semidirect
137. C107#3101semidirect

Maximal subgroups

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

subgroupindex
#202931792
C2×D4×Dic1072
#202929582
#202929552
#202929152
#202928302
#202927782
#202927182
SD16×Dic1072
#202926772
#450×C1072
#3101107