Finite Group Databasebeta

← #28346257

Subgroups — #28346257

Order 13696.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#20293180C211yesmaximal, characteristic, complemented
2C2×#16812071C211yesmaximal, characteristic
3#20292930C212yesmaximal, complemented
4#20292926C212yesmaximal, complemented
5#20292916C212yesmaximal, complemented
6#20292906C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic, complemented
8#20292650C211yesmaximal, characteristic, complemented
9#20292633C211yesmaximal, characteristic
10#20292631C211yesmaximal, characteristic
11#428×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C4×D107—21no—
14C2×#13133802C2211yescharacteristic
15#16812108C2211yescharacteristic, complemented
16#16812108C2211yescharacteristic, complemented
17#16812100C2211yescharacteristic, complemented
18#16812100C2211yescharacteristic, complemented
19#16812086—21no—
20#16812086—21no—
21C4×Dic214—21no—
22C4×Dic214—21no—
23#16812071—21no—
24#16812071—21no—
25#16812071C2212yes—
26#16812071C2212yes—
27#16812068C2211yescharacteristic
28C2×(C2×D107)⋊C4—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812042C2212yescomplemented
31#16812040C2212yescomplemented
32Dic107⋊C8C2212yescomplemented
33Dic107⋊C8C2212yescomplemented
34#16812031C2212yes—
35#16812031C2212yes—
36#16812029C2212yes—
37#16812029C2212yes—
38C2×M16×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#107×C107C2212yescomplemented
42#106×C107C2212yescomplemented
43C23×D107—21no—
44C2×Dic214—21no—
45C2×Dic214—21no—
46C2×D428—21no—
47C2×D428—22no—
48C2×D428Q814yescomplemented
49C2×D428C2312yes—
50C22×Dic107D411yescharacteristic
51C22×Dic107D411yescharacteristic
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107—44no—
55#13133802—22no—
56C2×C107⋊C8C2312yes—
57C4⋊Dic107—21no—
58C4⋊Dic107—21no—
59C4⋊Dic107C2311yescharacteristic
60C4⋊Dic107C2311yescharacteristic
61Dic107⋊C4—21no—
62Dic107⋊C4—21no—
63Dic107⋊C4—21no—
64Dic107⋊C4—21no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68C4×Dic107—21no—
69C4×Dic107D411yescharacteristic
70C4×Dic107D411yescharacteristic
71C4×Dic107C2311yescharacteristic
72C4×Dic107C2311yescharacteristic
73C22×C4×C107—21no—
74C22×C4×C107C2311yescharacteristic
75M16×C107—22no—
76C2×C8×C107C2312yes—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C4⋊C4×C107C2311yescharacteristic
81C22⋊C4×C107—21no—
82C42×C107—21no—
83C23×C107C2×D411yescharacteristic
84C22×D107—24no—
85C22×D107—22no—
86C22×D107—42no—
87C22×D107—42no—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107—42no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×Q812yes—
94C2×C4×C107#4112yes—
95C2×C4×C107C2411yescharacteristic
96C2×Dic107—22no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—41no—
100C2×Dic107C2×D411yescharacteristic
101C2×Dic107C2×D411yescharacteristic
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D411yescharacteristic
104D428—24no—
105D428—22no—
106D428C2×Q814yes—
107C4×D107—44no—
108C4×D107—44no—
109Dic214—42no—
110Dic214—42no—
111C8×C107—22no—
112C107⋊C8—22no—
113C22×C107—21no—
114C22×C107—21no—
115C22×C107C22⋊Q812yes—
116C22×C107C22×D411yescharacteristic
117C2×D107—24no—
118C2×D107—48no—
119C2×D107—42no—
120C2×D107—42no—
121C4×C107—41no—
122C4×C107—42no—
123C4×C107—41no—
124C4×C107C22⋊Q812yes—
125C4×C107C22×Q811yescharacteristic
126C4×C107C2×#4111yescharacteristic
127Dic107—21no—
128Dic107—21no—
129Dic107—21no—
130Dic107—41no—
131Dic107—41no—
132Dic107AGL(1,ℤ/8)12yes—
133C2×C107—22no—
134C2×C107C2×C22⋊Q811yescharacteristic
135C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C107#57511yescharacteristic
137D107—44no—
138#3077—1071nomaximal, Sylow
139C107#307711yescharacteristic, complemented, Sylow
140#533—1071no—
141C2×C4×D4—1071no—
142#478—1072no—
143#478—1072no—
144#476—1072no—
145#474—1072no—
146#428—1071no—
147#428D10711yescharacteristic, complemented
148#423—1071no—
149#418—1071no—
150#418—1071no—
151C22×D4—1071no—
152C23×C4—2141no—
153C2×M16—1071no—
154C2×M16C2×D10711yescharacteristic
155C4⋊Q8—1071no—
156C4⋊Q8—1071no—
157#125—1071no—
158#125—1071no—
159C22⋊Q8—2141no—
160C22⋊Q8—2141no—
161C4×Q8—2141no—
162C4×Q8—2141no—
163C4×D4—1072no—
164C4×D4—1072no—
165C4×D4—2141no—
166C4×D4—2141no—
167#117—1071no—
168#117C2×D10711yescharacteristic
169C2×C4⋊C4C2×D10711yescharacteristic
170C2×C22⋊C4—2141no—
171C2×C42—1071no—
172#107—1072no—
173#107C2×D10712yescomplemented
174#106—1072no—
175#106C2×D10712yescomplemented
176C4⋊C8—1072no—
177C4⋊C8—1072no—
178#102—1072no—
179#102—1072no—
180#102—1072no—
181#102—1072no—
182C24—2141no—
183C2×Q8—2141no—
184C2×Q8—2141no—
185C2×D4—1074no—
186C2×D4—1072no—
187C2×D4—2141no—
188C2×D4—2142no—
189C22×C4—21no—
190C22×C4—1071no—
191C22×C4—1071no—
192C22×C4—4284no—
193C22×C4—2142no—
194C22×C4—2142no—
195C22×C4C22×D10711yescharacteristic
196M16—22no—
197M16—2142no—
198C2×C8—1072no—
199C2×C8C22×D10712yes—
200C4⋊C4—21no—
201C4⋊C4—1071no—
202C4⋊C4—1071no—
203C4⋊C4—2141no—
204C4⋊C4—2141no—
205C4⋊C4—2141no—
206C4⋊C4—2141no—
207C4⋊C4—2141no—
208C4⋊C4—2141no—
209C4⋊C4C22×D10711yescharacteristic
210C4⋊C4C22×D10711yescharacteristic
211C4⋊C4C22×D10712yes—
212C22⋊C4—21no—
213C22⋊C4—2142no—
214C22⋊C4—2142no—
215C22⋊C4—2141no—
216C42—21no—
217C42—1071no—
218C42—1071no—
219C42—1071no—
220C42—1071no—
221C42—2141no—
222C23—4282no—
223C23—4282no—
224C23—2144no—
225C23—2142no—
226C23D4×D10711yescharacteristic
227Q8—4282no—
228Q8—4282no—
229D4—1074no—
230D4—2144no—
231D4—2142no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—1071no—
237C2×C4—1071no—
238C2×C4—1071no—
239C2×C4—1071no—
240C2×C4—4284no—
241C2×C4—4284no—
242C2×C4—4281no—
243C2×C4—2142no—
244C2×C4—2141no—
245C2×C4—2141no—
246C2×C4D4×D10711yescharacteristic
247C2×C4Q8×D10712yes—
248C2×C4#1313381812yes—
249C2×C4C23×D10711yescharacteristic
250C8—22no—
251C8—2142no—
252C22—21no—
253C22—21no—
254C22—4288no—
255C22—4282no—
256C22—4282no—
257C22—2144no—
258C22#1681208712yes—
259C22C2×D4×D10711yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—1072no—
264C4—4281no—
265C4—4281no—
266C4—2141no—
267C4—2141no—
268C4—2141no—
269C4#1681208712yes—
270C4C2×Q8×D10711yescharacteristic
271C4C2×#1313381811yescharacteristic
272C2—22no—
273C2—4284no—
274C2C2×#1681208711yescharacteristic
275C2AGL(1,ℤ/8)×D10711yescharacteristic
276C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC22×D428C22
hypercenter#428D107
hypocenterC107#3077
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #20293180C2semidirect
3. #20292930C2semidirect
4. #20292926C2semidirect
5. #20292916C2semidirect
6. #20292906C2semidirect
7. #20292667C2semidirect
8. #20292650C2semidirect
11. #428×C107C2semidirect
15. #16812108C22semidirect
16. #16812108C22semidirect
17. #16812100C22semidirect
18. #16812100C22semidirect
30. #16812042C22semidirect
31. #16812040C22semidirect
32. Dic107⋊C8C22semidirect
33. Dic107⋊C8C22semidirect
41. #107×C107C22semidirect
42. #106×C107C22semidirect
48. C2×D428Q8semidirect
139. C107#3077semidirect
147. #428D107semidirect
173. #107C2×D107semidirect
175. #106C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931802
C2×#168120712
#202929302
#202929262
#202929162
#202929062
#202926672
#202926502
#202926332
#202926312
#428×C1072
#3077107