Finite Group Databasebeta

← #28346846

Subgroups — #28346846

Order 13696.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293182C211yesmaximal, characteristic, complemented
2#124×D107C211yesmaximal, characteristic
3#20292989C211yesmaximal, characteristic, complemented
4#20292987C211yesmaximal, characteristic, complemented
5#20292986C211yesmaximal, characteristic, complemented
6#20292983C212yesmaximal, complemented
7#20292980C212yesmaximal, complemented
8#20292634C212yesmaximal, complemented
9#20292630C212yesmaximal
10#20292579C211yesmaximal, characteristic, complemented
11#489×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2211yescharacteristic
13C2×D4×D107C2211yescharacteristic
14C2×#13133801C2212yes—
15#16812109C2211yescharacteristic, complemented
16#16812100C2212yescomplemented
17#16812098C2211yescharacteristic
18#16812097C2211yescharacteristic
19#16812096C2211yescharacteristic
20#16812095—21no—
21#16812095—21no—
22#16812089—21no—
23#16812085—21no—
24Q8×Dic107—21no—
25#16812074—21no—
26#16812067C2212yes—
27C22⋊C4×D107—21no—
28C22⋊C4×D107—21no—
29C42×D107C2211yescharacteristic
30#16812034C2212yescomplemented
31#16812033C2212yescomplemented
32#16812032C2212yescomplemented
33#16812031C2212yes—
34#16812030C2212yes—
35#16812029C2212yes—
36#16812020C2211yescharacteristic, complemented
37C8×Dic107C2212yescomplemented
38C4⋊Q8×C107C2211yescharacteristic, complemented
39#124×C107C2211yescharacteristic
40#103×C107C2212yescomplemented
41#102×C107C2212yes—
42#97×C107C2211yescharacteristic, complemented
43C23×D107—21no—
44C2×Dic214—21no—
45C2×Dic214—21no—
46C2×Dic214C2311yescharacteristic
47Q8×D107—24no—
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—21no—
50D4×D107—24no—
51C2×C4×D107—21no—
52C2×C4×D107D411yescharacteristic
53C2×C4×D107D411yescharacteristic
54C2×C4×D107C2311yescharacteristic
55#13133801—24no—
56C2×C107⋊C8C2312yes—
57C4⋊Dic107—21no—
58C4⋊Dic107—22no—
59C4⋊Dic107C2312yes—
60Dic107⋊C4—22no—
61C22⋊Dic107—21no—
62C22⋊Dic107—21no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4—21no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68C4×Dic107—22no—
69C4×Dic107D411yescharacteristic
70C4×Dic107D411yescharacteristic
71C4×Dic107C2311yescharacteristic
72C2×Q8×C107—21no—
73C2×Q8×C107C2311yescharacteristic
74C2×D4×C107C2311yescharacteristic
75C2×C8×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107C2312yes—
78C22⋊C4×C107—21no—
79C22⋊C4×C107—21no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C22×D107—21no—
83C22×D107—42no—
84C22×D107—42no—
85C22×D107—42no—
86C22×D107C2×D411yescharacteristic
87Q8×C107—22no—
88Q8×C107—42no—
89D4×C107—22no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—42no—
96C2×Dic107—22no—
97C2×Dic107—21no—
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101D428—22no—
102C4×D107—41no—
103C4×D107—41no—
104C4×D107—42no—
105C4×D107#4114yes—
106Dic214—22no—
107Dic214—42no—
108Dic214—42no—
109C8×C107—22no—
110C107⋊C8—22no—
111C22×C107—41no—
112C22×C107—41no—
113C22×C107—41no—
114C22×C107C22×D411yescharacteristic
115C2×D107—21no—
116C2×D107—21no—
117C2×D107—41no—
118C2×D107—41no—
119C2×D107—41no—
120C2×D107—84no—
121C2×D107#12412yes—
122C4×C107—21no—
123C4×C107—21no—
124C4×C107—41no—
125C4×C107—42no—
126C4×C107C2×#4112yes—
127Dic107—21no—
128Dic107—21no—
129Dic107—41no—
130Dic107—42no—
131Dic107#12412yes—
132C2×C107—81no—
133C2×C107C2×#12411yescharacteristic
134C2×C107#57511yescharacteristic
135C2×C107#57511yescharacteristic
136D107—22no—
137D107—81no—
138#3140—1071nomaximal, Sylow
139C107#314011yescharacteristic, complemented, Sylow
140#533—1071no—
141C2×#124—1071no—
142#489—1071no—
143#489—1071no—
144#489—1071no—
145#489D10711yescharacteristic, complemented
146#488—1072no—
147#488—1072no—
148#419—1072no—
149#418—1072no—
150C4×M16—1071no—
151C22×Q8—1071no—
152C22×D4—1071no—
153C2×M16—1072no—
154C4⋊Q8—1071no—
155C4⋊Q8C2×D10711yescharacteristic, complemented
156#125—1072no—
157#124—1071no—
158#124—1071no—
159#124—1071no—
160#124—2141no—
161#124—2141no—
162#124C2×D10711yescharacteristic
163C22⋊Q8—2141no—
164C22⋊Q8—2141no—
165C4×Q8—2141no—
166C4×Q8—2141no—
167#117—1072no—
168C2×C22⋊C4—2141no—
169C2×C22⋊C4—2141no—
170C2×C42—1071no—
171#103—1072no—
172#103—1072no—
173#103—1072no—
174#103C2×D10712yescomplemented
175#102—1072no—
176#102—1072no—
177#102—1072no—
178#102C2×D10712yes—
179#97—1071no—
180#97C2×D10711yescharacteristic, complemented
181C4×C8—1072no—
182C24—2141no—
183C2×Q8—21no—
184C2×Q8—1071no—
185C2×Q8—2141no—
186C2×Q8—2144no—
187C2×Q8—2141no—
188C2×Q8C22×D10711yescharacteristic
189C2×D4—1071no—
190C2×D4—2144no—
191C2×D4—2141no—
192C2×D4C22×D10711yescharacteristic
193C22×C4—1071no—
194C22×C4—1071no—
195C22×C4—1071no—
196C22×C4—2141no—
197M16—2144no—
198C2×C8—1072no—
199C2×C8C22×D10712yes—
200C4⋊C4—21no—
201C4⋊C4—1072no—
202C4⋊C4—2142no—
203C4⋊C4—2142no—
204C4⋊C4—2141no—
205C4⋊C4C22×D10712yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—2141no—
209C22⋊C4—2141no—
210C22⋊C4—2142no—
211C22⋊C4—2141no—
212C22⋊C4—2141no—
213C22⋊C4—2141no—
214C22⋊C4—2141no—
215C42—1071no—
216C42—1071no—
217C42—1071no—
218C42—2142no—
219C42C22×D10711yescharacteristic
220C23—21no—
221C23—1071no—
222C23—4282no—
223C23—4282no—
224C23—4282no—
225C23—2141no—
226Q8—22no—
227Q8—42no—
228Q8—4282no—
229Q8—4282no—
230Q8—2142no—
231D4—22no—
232D4—4282no—
233D4—2142no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—1071no—
237C2×C4—1074no—
238C2×C4—1071no—
239C2×C4—1071no—
240C2×C4—4281no—
241C2×C4—4281no—
242C2×C4—4282no—
243C2×C4—2142no—
244C2×C4—2141no—
245C2×C4D4×D10711yescharacteristic
246C2×C4D4×D10711yescharacteristic
247C2×C4C23×D10711yescharacteristic
248C8—22no—
249C8—2142no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—8564no—
254C22—1072no—
255C22—4281no—
256C22—4281no—
257C22—4281no—
258C22—2141no—
259C22—2141no—
260C22C2×D4×D10711yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—1072no—
266C4—4281no—
267C4—4282no—
268C4—2141no—
269C4—2141no—
270C4#41×D10712yes—
271C2—81no—
272C2—8561no—
273C2—2142no—
274C2#124×D10711yescharacteristic
275C2#2029342311yescharacteristic
276C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#489×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×D107C22
hypercenter#489D107
hypocenterC107#3140
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #20293182C2semidirect
3. #20292989C2semidirect
4. #20292987C2semidirect
5. #20292986C2semidirect
6. #20292983C2semidirect
7. #20292980C2semidirect
8. #20292634C2semidirect
10. #20292579C2semidirect
11. #489×C107C2semidirect
15. #16812109C22semidirect
16. #16812100C22semidirect
30. #16812034C22semidirect
31. #16812033C22semidirect
32. #16812032C22semidirect
36. #16812020C22semidirect
37. C8×Dic107C22semidirect
38. C4⋊Q8×C107C22semidirect
40. #103×C107C22semidirect
42. #97×C107C22semidirect
139. C107#3140semidirect
145. #489D107semidirect
155. C4⋊Q8C2×D107semidirect
174. #103C2×D107semidirect
180. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931822
#124×D1072
#202929892
#202929872
#202929862
#202929832
#202929802
#202926342
#202926302
#202925792
#489×C1072
#3140107