Finite Group Databasebeta

← #28348285

Subgroups — #28348285

Order 13696.

462 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 141 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#20293334C211yesmaximal, characteristic, complemented
2#20292973C211yesmaximal, characteristic, complemented
3#20292943C212yesmaximal, complemented
4#20292917C211yesmaximal, characteristic, complemented
5#20292854C212yesmaximal, complemented
6#20292824C211yesmaximal, characteristic, complemented
7#20292778C212yesmaximal, complemented
8C4×#13133806C211yesmaximal, characteristic, complemented
9D4×C107⋊C8C211yesmaximal, characteristic, complemented
10C2×#16812030C212yesmaximal, complemented
11D42×C107C211yesmaximal, characteristic
12C2×#13133806C2211yescharacteristic
13C2×#13133806C2212yescomplemented
14C22×C107⋊C8C2212yescomplemented
15#16812109C2211yescharacteristic, complemented
16#16812100—21no—
17#16812088C2212yescomplemented
18#16812086—22no—
19C4×Dic214C2211yescharacteristic, complemented
20D4×Dic107—21no—
21C2×C4⋊Dic107C2212yescomplemented
22#16812040C2212yescomplemented
23#16812040C2211yescharacteristic, complemented
24C4⋊(C107⋊C8)C2211yescharacteristic, complemented
25#16812032C2211yescharacteristic, complemented
26#16812030C2211yescharacteristic
27#16812030C2212yes—
28#16812030C2212yescomplemented
29#16812030C2212yescomplemented
30#16812030C2212yescomplemented
31C22⋊(C107⋊C8)C2212yescomplemented
32C4×C107⋊C8C2211yescharacteristic, complemented
33C22×D4×C107—21no—
34C22×D4×C107C2212yes—
35#127×C107C2211yescharacteristic
36#121×C107—21no—
37#121×C107C2212yes—
38#120×C107—22no—
39C4×D4×C107C2211yescharacteristic
40C4×D4×C107C2211yescharacteristic
41C2×Dic214—21no—
42C2×Dic214C2311yescharacteristic
43C22×Dic107—22no—
44#13133806—22no—
45#13133806—42no—
46C2×C107⋊C8—42no—
47C2×C107⋊C8C2311yescharacteristic
48C2×C107⋊C8C2312yes—
49C2×C107⋊C8C2311yescharacteristic
50C4⋊Dic107—22no—
51C4⋊Dic107—21no—
52C4⋊Dic107C2311yescharacteristic
53C4⋊Dic107C2312yes—
54Dic107⋊C4—21no—
55Dic107⋊C4—22no—
56C22⋊Dic107—22no—
57C4×Dic107—21no—
58C24×C107—22no—
59C2×D4×C107—24no—
60C2×D4×C107—22no—
61C2×D4×C107—22no—
62C2×D4×C107—21no—
63C2×D4×C107—21no—
64C2×D4×C107—42no—
65C2×D4×C107—44no—
66C2×D4×C107D411yescharacteristic
67C2×D4×C107C2312yes—
68C2×D4×C107C2311yescharacteristic
69C22×C4×C107—21no—
70C22×C4×C107C2312yes—
71C4⋊C4×C107D411yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C22⋊C4×C107—22no—
75C22⋊C4×C107D412yes—
76C42×C107C2311yescharacteristic
77C23×C107—21no—
78C23×C107—22no—
79C23×C107—44no—
80C23×C107—44no—
81C23×C107—82no—
82C23×C107C2×D412yes—
83D4×C107—22no—
84D4×C107—21no—
85D4×C107—22no—
86D4×C107—44no—
87D4×C107—44no—
88D4×C107—42no—
89D4×C107—81no—
90D4×C107SD1614yescomplemented
91D4×C107#4112yes—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—21no—
95C2×C4×C107—42no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2411yescharacteristic
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—82no—
104Dic214—21no—
105Dic214—21no—
106Dic214—81no—
107C107⋊C8—21no—
108C107⋊C8—21no—
109C107⋊C8—22no—
110C22×C107—22no—
111C22×C107—22no—
112C22×C107—22no—
113C22×C107—41no—
114C22×C107—41no—
115C22×C107—42no—
116C22×C107—42no—
117C22×C107—42no—
118C22×C107—88no—
119C22×C107C2×SD1614yes—
120C22×C107C22×D411yescharacteristic
121C4×C107—22no—
122C4×C107—21no—
123C4×C107—41no—
124C4×C107C2×SD1612yes—
125C4×C107C2×#4111yescharacteristic
126C4×C107H(5,𝔽2)11yescharacteristic
127Dic107—41no—
128Dic107—42no—
129Dic107—41no—
130C2×C107—24no—
131C2×C107—42no—
132C2×C107—82no—
133C2×C107#54611yescharacteristic
134C2×C107C22×SD1611yescharacteristic
135C2×C107#57611yescharacteristic
136#3295—1071nomaximal, Sylow
137C107#329511yescharacteristic, complemented, Sylow
138#554—1071no—
139D42D10711yescharacteristic
140#486—1071no—
141#481—1072no—
142#476—1071no—
143#465—1072no—
144#460—1071no—
145#450—1072no—
146C4×SD16—1071no—
147C8×D4—1071no—
148C2×#102—1072no—
149C22×D4—21no—
150C22×D4C2×D10712yes—
151C2×SD16—1072no—
152C2×SD16—1071no—
153C22×C8—1072no—
154C4⋊Q8—1071no—
155#127C2×D10711yescharacteristic
156#125—2141no—
157C22⋊Q8—1072no—
158C22⋊Q8—2142no—
159#121—21no—
160#121C2×D10712yes—
161#120—22no—
162C4×Q8—1071no—
163C4×D4—2141no—
164C4×D4C2×D10711yescharacteristic
165C4×D4C2×D10711yescharacteristic
166C2×C4⋊C4—1072no—
167#106—1071no—
168#106—1072no—
169C4⋊C8—1071no—
170#103—1071no—
171#102—1071no—
172#102—1072no—
173#102—1072no—
174#102—1072no—
175#102—1072no—
176C22⋊C8—1072no—
177C4×C8—1071no—
178C24—22no—
179C2×Q8—1071no—
180C2×Q8—2141no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4C22⋊D10711yescharacteristic
189C2×D4C22×D10712yes—
190C2×D4C22×D10711yescharacteristic
191C22×C4—21no—
192C22×C4—2142no—
193C22×C4C22×D10712yes—
194SD16—4282no—
195SD16—2142no—
196C2×C8—1071no—
197C2×C8—1071no—
198C2×C8—1072no—
199C2×C8—4282no—
200C4⋊C4—1071no—
201C4⋊C4—1072no—
202C4⋊C4—2142no—
203C4⋊C4—2141no—
204C4⋊C4—2141no—
205C4⋊C4—2142no—
206C4⋊C4C22⋊D10711yescharacteristic
207C4⋊C4C22×D10711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—2142no—
211C22⋊C4C22⋊D10712yes—
212C42—2141no—
213C42C22×D10711yescharacteristic
214C23—21no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—82no—
219C23C2×C22⋊D10712yes—
220Q8—8561no—
221Q8—2141no—
222Q8—2141no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1313380614yescomplemented
231D4#1313381712yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—8562no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4—2142no—
240C2×C4C2×C22⋊D10711yescharacteristic
241C2×C4C2×C22⋊D10712yes—
242C2×C4C2×C22⋊D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—2141no—
245C8—2141no—
246C8—2142no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—88no—
256C22C2×#1313380614yes—
257C22C22×C22⋊D10711yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—4281no—
262C4—4282no—
263C4—4281no—
264C4C2×#1313380612yes—
265C4C2×#1313381711yescharacteristic
266C4#1681215411yescharacteristic
267C2—24no—
268C2—42no—
269C2—82no—
270C2#2029327111yescharacteristic
271C2C22×#1313380611yescharacteristic
272C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalD42×C107C2
socleC22×C107C22×D4
hypercenterD42D107
hypocenterC107#3295
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20293334C2semidirect
2. #20292973C2semidirect
3. #20292943C2semidirect
4. #20292917C2semidirect
5. #20292854C2semidirect
6. #20292824C2semidirect
7. #20292778C2semidirect
8. C4×#13133806C2semidirect
9. D4×C107⋊C8C2semidirect
10. C2×#16812030C2semidirect
13. C2×#13133806C22semidirect
14. C22×C107⋊C8C22semidirect
15. #16812109C22semidirect
17. #16812088C22semidirect
19. C4×Dic214C22semidirect
21. C2×C4⋊Dic107C22semidirect
22. #16812040C22semidirect
23. #16812040C22semidirect
24. C4⋊(C107⋊C8)C22semidirect
25. #16812032C22semidirect
28. #16812030C22semidirect
29. #16812030C22semidirect
30. #16812030C22semidirect
31. C22⋊(C107⋊C8)C22semidirect
32. C4×C107⋊C8C22semidirect
90. D4×C107SD16semidirect
137. C107#3295semidirect
230. D4#13133806semidirect

Maximal subgroups

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

subgroupindex
#202933342
#202929732
#202929432
#202929172
#202928542
#202928242
#202927782
C4×#131338062
D4×C107⋊C82
C2×#168120302
D42×C1072
#3295107