Finite Group Databasebeta

← #28348284

Subgroups — #28348284

Order 13696.

462 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 123 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#20293338C211yesmaximal, characteristic, complemented
2#20293262C211yesmaximal, characteristic
3#20292944C212yesmaximal, complemented
4#20292914C211yesmaximal, characteristic, complemented
5#20292855C212yesmaximal, complemented
6#20292824C211yesmaximal, characteristic, complemented
7#20292779C212yesmaximal, complemented
8C4×#13133807C211yesmaximal, characteristic, complemented
9C8×D428C211yesmaximal, characteristic, complemented
10#102×D107C212yesmaximal, complemented
11#486×C107C211yesmaximal, characteristic, complemented
12C2×D4×D107—21no—
13C2×D4×D107C2212yes—
14C2×#13133807C2211yescharacteristic
15C2×#13133806C2212yescomplemented
16C2×C8×D107C2212yescomplemented
17#16812100—21no—
18#16812089C2212yescomplemented
19#16812085—22no—
20#16812080—21no—
21#16812080C2212yes—
22#16812077—22no—
23C4×Dic214C2211yescharacteristic, complemented
24C4×D428C2211yescharacteristic
25C4×D428C2211yescharacteristic
26#16812071—21no—
27C4⋊C4×D107C2212yescomplemented
28#16812041C2211yescharacteristic, complemented
29#16812040C2212yescomplemented
30C4⋊(C107⋊C8)C2211yescharacteristic, complemented
31#16812034C2211yescharacteristic, complemented
32#16812031C2211yescharacteristic
33#16812030C2212yescomplemented
34#16812029C2212yes—
35(C2×D107)⋊C8C2212yescomplemented
36C4⋊Q8×C107C2211yescharacteristic, complemented
37#127×C107C2211yescharacteristic
38#102×C107C2212yescomplemented
39#102×C107C2212yescomplemented
40C4×C8×C107C2211yescharacteristic, complemented
41C23×D107—22no—
42C2×Dic214C2311yescharacteristic
43C2×D428D411yescharacteristic
44C2×D428C2311yescharacteristic
45C2×C22⋊D107—22no—
46C2×C22⋊D107—22no—
47D4×D107—24no—
48D4×D107—42no—
49D4×D107—44no—
50C2×C4×D107—21no—
51C2×C4×D107—22no—
52C2×C4×D107C2312yes—
53#13133807—22no—
54#13133806—42no—
55C2×C107⋊C8C2312yes—
56C8×D107—42no—
57C4⋊Dic107—22no—
58C4⋊Dic107D411yescharacteristic
59C4⋊Dic107C2311yescharacteristic
60C4⋊Dic107C2311yescharacteristic
61Dic107⋊C4—21no—
62Dic107⋊C4—22no—
63C22⋊Dic107—22no—
64(C2×D107)⋊C4—22no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4D412yes—
67C4×Dic107—21no—
68C2×Q8×C107—21no—
69C2×D4×C107—21no—
70C2×D4×C107—21no—
71C2×D4×C107C2312yes—
72C2×C8×C107C2311yescharacteristic
73C2×C8×C107C2311yescharacteristic
74C4⋊C4×C107—21no—
75C4⋊C4×C107C2312yes—
76C42×C107C2311yescharacteristic
77C23×C107—22no—
78C22×D107—21no—
79C22×D107—44no—
80C22×D107—44no—
81C22×D107—82no—
82C22×D107C2×D412yes—
83Q8×C107—81no—
84D4×C107—22no—
85D4×C107—22no—
86D4×C107—42no—
87D4×C107—81no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—44no—
93C22⋊D107—44no—
94C2×Dic107—21no—
95C2×Dic107—21no—
96C2×Dic107—21no—
97C2×Dic107C2×D412yes—
98D428—21no—
99D428SD1614yescomplemented
100D428#4112yes—
101C4×D107—22no—
102C4×D107—22no—
103C4×D107—42no—
104C4×D107—82no—
105Dic214—21no—
106Dic214—21no—
107C8×C107—21no—
108C8×C107—21no—
109C107⋊C8—22no—
110C22×C107—42no—
111C22×C107—42no—
112C22×C107—42no—
113C22×C107C22×D411yescharacteristic
114C2×D107—22no—
115C2×D107—22no—
116C2×D107—22no—
117C2×D107—41no—
118C2×D107—41no—
119C2×D107—88no—
120C2×D107C2×SD1614yes—
121C4×C107—21no—
122C4×C107—42no—
123C4×C107C2×SD1612yes—
124C4×C107C2×#4111yescharacteristic
125C4×C107H(5,𝔽2)11yescharacteristic
126Dic107—22no—
127Dic107—41no—
128Dic107—41no—
129Dic107—41no—
130C2×C107—82no—
131C2×C107#54611yescharacteristic
132C2×C107C22×SD1611yescharacteristic
133C2×C107#57611yescharacteristic
134D107—24no—
135D107—42no—
136#3295—1071nomaximal, Sylow
137C107#329511yescharacteristic, complemented, Sylow
138#554—1071no—
139D42—1071no—
140#486D10711yescharacteristic, complemented
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—1072no—
150C22×D4—2141no—
151C2×SD16—1072no—
152C2×SD16—1071no—
153C22×C8—1072no—
154C4⋊Q8C2×D10711yescharacteristic, complemented
155#127C2×D10711yescharacteristic
156#125—2141no—
157C22⋊Q8—1072no—
158C22⋊Q8—2142no—
159#121—1072no—
160#121—2141no—
161#120—2142no—
162C4×Q8—1071no—
163C4×D4—1071no—
164C4×D4—1071no—
165C4×D4—2141no—
166C2×C4⋊C4—1072no—
167#106—1071no—
168#106—1072no—
169C4⋊C8—1071no—
170#103—1071no—
171#102—1072no—
172#102—1071no—
173#102—1072no—
174#102C2×D10712yescomplemented
175#102C2×D10712yescomplemented
176C22⋊C8—1072no—
177C4×C8C2×D10711yescharacteristic, complemented
178C24—2142no—
179C2×Q8—21no—
180C2×Q8—1071no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—1071no—
184C2×D4—1071no—
185C2×D4—4284no—
186C2×D4—4282no—
187C2×D4—2142no—
188C2×D4—2144no—
189C2×D4—2142no—
190C2×D4C22×D10712yes—
191C22×C4—1072no—
192C22×C4—2142no—
193C22×C4—2141no—
194SD16—4282no—
195SD16—2142no—
196C2×C8—1072no—
197C2×C8—4282no—
198C2×C8C22×D10711yescharacteristic
199C2×C8C22×D10711yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—1071no—
202C4⋊C4—1071no—
203C4⋊C4—1071no—
204C4⋊C4—2142no—
205C4⋊C4—2142no—
206C4⋊C4—2141no—
207C4⋊C4C22×D10712yes—
208C22⋊C4—1072no—
209C22⋊C4—2142no—
210C22⋊C4—2142no—
211C22⋊C4—2141no—
212C42—2141no—
213C42C22×D10711yescharacteristic
214C23—22no—
215C23—8562no—
216C23—1072no—
217C23—4284no—
218C23—4284no—
219C23—2141no—
220Q8—81no—
221Q8—2141no—
222Q8—2141no—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—81no—
227D4—1074no—
228D4—1072no—
229D4—4284no—
230D4—4284no—
231D4—2141no—
232C2×C4—22no—
233C2×C4—8562no—
234C2×C4—1072no—
235C2×C4—4282no—
236C2×C4—2141no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4D4×D10711yescharacteristic
242C2×C4D4×D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—21no—
245C8—21no—
246C8—2142no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—8568no—
251C22—1074no—
252C22—4281no—
253C22—4281no—
254C22—2142no—
255C22—2142no—
256C22—2142no—
257C22C2×D4×D10711yescharacteristic
258C4—21no—
259C4—42no—
260C4—4281no—
261C4—4281no—
262C4—4281no—
263C4—2142no—
264C4SD16×D10712yes—
265C4#41×D10711yescharacteristic
266C4#1681215511yescharacteristic
267C2—82no—
268C2—4282no—
269C2—2144no—
270C2#2029327011yescharacteristic
271C2C2×SD16×D10711yescharacteristic
272C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#486×C107C2
socleC22×C107C22×D4
prime radical#20293262C2
hypercenter#486D107
hypocenterC107#3295
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #20293338C2semidirect
3. #20292944C2semidirect
4. #20292914C2semidirect
5. #20292855C2semidirect
6. #20292824C2semidirect
7. #20292779C2semidirect
8. C4×#13133807C2semidirect
9. C8×D428C2semidirect
10. #102×D107C2semidirect
11. #486×C107C2semidirect
15. C2×#13133806C22semidirect
16. C2×C8×D107C22semidirect
18. #16812089C22semidirect
23. C4×Dic214C22semidirect
27. C4⋊C4×D107C22semidirect
28. #16812041C22semidirect
29. #16812040C22semidirect
30. C4⋊(C107⋊C8)C22semidirect
31. #16812034C22semidirect
33. #16812030C22semidirect
35. (C2×D107)⋊C8C22semidirect
36. C4⋊Q8×C107C22semidirect
38. #102×C107C22semidirect
39. #102×C107C22semidirect
40. C4×C8×C107C22semidirect
99. D428SD16semidirect
137. C107#3295semidirect
140. #486D107semidirect
154. C4⋊Q8C2×D107semidirect
174. #102C2×D107semidirect
175. #102C2×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
#202933382
#202932622
#202929442
#202929142
#202928552
#202928242
#202927792
C4×#131338072
C8×D4282
#102×D1072
#486×C1072
#3295107