Finite Group Databasebeta

← #28346374

Subgroups — #28346374

Order 13696.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#20293138C211yesmaximal, characteristic
2#20292968C212yesmaximal, complemented
3#20292962C211yesmaximal, characteristic, complemented
4#20292957C211yesmaximal, characteristic, complemented
5#20292951C212yesmaximal, complemented
6#20292945C211yesmaximal, characteristic, complemented
7#20292940C211yesmaximal, characteristic, complemented
8#20292663C212yesmaximal, complemented
9#20292621C212yesmaximal, complemented
10C2×C22⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
11C2×#117×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×C107⋊C8C2212yes—
14#16812092—21no—
15#16812088C2211yescharacteristic
16#16812088C2211yescharacteristic
17#16812083C2211yescharacteristic
18#16812083C2211yescharacteristic
19#16812078—21no—
20C4×C22⋊D107—21no—
21C4×C22⋊D107—21no—
22#16812068C2211yescharacteristic
23#16812042C2212yescomplemented
24#16812042C2212yescomplemented
25#16812040C2212yescomplemented
26#16812040C2212yescomplemented
27#16812032C2212yescomplemented
28#16812032C2212yescomplemented
29#16812029C2212yescomplemented
30#16812029C2212yescomplemented
31C22⋊(C107⋊C8)C2211yescharacteristic
32C22⋊(C107⋊C8)C2211yescharacteristic
33C22⋊(C107⋊C8)C2212yes—
34C23×C4×C107C2211yescharacteristic
35#117×C107—21no—
36#117×C107—21no—
37#117×C107C2212yes—
38#117×C107C2212yes—
39C2×C4⋊C4×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41C2×C22⋊C4×C107—21no—
42C2×C42×C107—21no—
43#13133818—24no—
44C2×Dic214C2311yescharacteristic
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—21no—
47C2×C22⋊D107—21no—
48C2×C22⋊D107—21no—
49C2×C4×D107—21no—
50C2×C107⋊C8—24no—
51C2×C107⋊C8C2312yes—
52C2×C107⋊C8C2312yes—
53C4⋊Dic107C2311yescharacteristic
54C4⋊Dic107C2311yescharacteristic
55Dic107⋊C4—21no—
56Dic107⋊C4—21no—
57C22⋊Dic107—21no—
58C22⋊Dic107—21no—
59C22⋊Dic107—21no—
60(C2×D107)⋊C4—21no—
61(C2×D107)⋊C4—21no—
62C4×Dic107—21no—
63C24×C107D411yescharacteristic
64C22×C4×C107—24no—
65C22×C4×C107—21no—
66C22×C4×C107—21no—
67C22×C4×C107D411yescharacteristic
68C22×C4×C107D411yescharacteristic
69C22×C4×C107D411yescharacteristic
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2311yescharacteristic
73C4⋊C4×C107—21no—
74C4⋊C4×C107—21no—
75C4⋊C4×C107C2312yes—
76C4⋊C4×C107C2312yes—
77C22⋊C4×C107—22no—
78C22⋊C4×C107—22no—
79C42×C107—22no—
80C42×C107—22no—
81C23×C107—22no—
82C23×C107—22no—
83C23×C107—41no—
84C23×C107C2×D411yescharacteristic
85C23×C107C2×D411yescharacteristic
86C23×C107C2×D411yescharacteristic
87C22×D107—21no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—42no—
96C2×C4×C107—42no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107#4114yes—
101C2×C4×C107C2411yescharacteristic
102C22⋊D107—42no—
103C22⋊D107—42no—
104C22⋊D107—42no—
105C2×Dic107—21no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108D428—22no—
109C4×D107—42no—
110Dic214—22no—
111C107⋊C8—22no—
112C107⋊C8—22no—
113C22×C107—21no—
114C22×C107—21no—
115C22×C107—21no—
116C22×C107—21no—
117C22×C107—21no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107—44no—
121C22×C107#12312yes—
122C22×C107#13512yes—
123C22×C107C22×D411yescharacteristic
124C2×D107—41no—
125C2×D107—41no—
126C2×D107—41no—
127C4×C107—21no—
128C4×C107—21no—
129C4×C107—42no—
130C4×C107—42no—
131C4×C107#12312yes—
132C4×C107C2×#4112yes—
133Dic107—41no—
134Dic107—41no—
135Dic107—41no—
136C2×C107—22no—
137C2×C107—22no—
138C2×C107—41no—
139C2×C107C2×#12311yescharacteristic
140C2×C107C2×#13511yescharacteristic
141C2×C107#57511yescharacteristic
142D107—81no—
143#3091—1071nomaximal, Sylow
144C107#309111yescharacteristic, complemented, Sylow
145#525—1071no—
146C2×#117D10711yescharacteristic, complemented
147#485—1072no—
148#484—1071no—
149#483—1071no—
150#482—1072no—
151#481—1071no—
152#480—1071no—
153#427—1072no—
154#416—1072no—
155C2×C22⋊C8—1071no—
156C2×#41—1071no—
157C23×C4C2×D10711yescharacteristic
158C22×C8—1072no—
159#123—2141no—
160C22⋊Q8—1071no—
161C22⋊Q8—1071no—
162#121—1071no—
163#121—1071no—
164#120—2141no—
165C4×D4—2141no—
166C4×D4—2141no—
167#117—21no—
168#117—21no—
169#117—1071no—
170#117C2×D10712yes—
171#117C2×D10712yes—
172C2×C4⋊C4C2×D10711yescharacteristic
173C2×C4⋊C4C2×D10711yescharacteristic
174C2×C22⋊C4—21no—
175C2×C42—21no—
176#107—1072no—
177#107—1072no—
178#106—1072no—
179#106—1072no—
180#103—1072no—
181#103—1072no—
182#102—1072no—
183#102—1072no—
184C22⋊C8—1071no—
185C22⋊C8—1071no—
186C22⋊C8—1072no—
187C24C22⋊D10711yescharacteristic
188#41—2144no—
189C2×Q8—1071no—
190C2×D4—1071no—
191C2×D4—2141no—
192C2×D4—2141no—
193C2×D4—2141no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—2141no—
198C22×C4C22⋊D10711yescharacteristic
199C22×C4C22⋊D10711yescharacteristic
200C22×C4C22⋊D10711yescharacteristic
201C22×C4C22×D10711yescharacteristic
202C22×C4C22×D10711yescharacteristic
203C22×C4C22×D10711yescharacteristic
204C2×C8—1072no—
205C2×C8—1072no—
206C2×C8—2144no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—1071no—
210C4⋊C4—1071no—
211C4⋊C4—2141no—
212C4⋊C4—2141no—
213C4⋊C4C22×D10712yes—
214C4⋊C4C22×D10712yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—2141no—
218C22⋊C4—2141no—
219C22⋊C4—2141no—
220C22⋊C4—2141no—
221C22⋊C4—2141no—
222C42—22no—
223C42—22no—
224C42—2141no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—2141no—
229C23C2×C22⋊D10711yescharacteristic
230C23C2×C22⋊D10711yescharacteristic
231C23C2×C22⋊D10711yescharacteristic
232Q8—2142no—
233D4—4282no—
234D4—4282no—
235D4—4282no—
236D4—2142no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—4282no—
247C2×C4—2141no—
248C2×C4—2141no—
249C2×C4—2141no—
250C2×C4C2×C22⋊D10711yescharacteristic
251C2×C4C2×C22⋊D10711yescharacteristic
252C2×C4C2×C22⋊D10711yescharacteristic
253C2×C4#1313381814yes—
254C2×C4C23×D10711yescharacteristic
255C8—2142no—
256C8—2142no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—4281no—
266C22—4281no—
267C22—4281no—
268C22#1681209212yes—
269C22#1681212912yes—
270C22C22×C22⋊D10711yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—4281no—
276C4—4281no—
277C4—4281no—
278C4#1681209212yes—
279C4C2×#1313381812yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—8561no—
284C2C2×#1681209211yescharacteristic
285C2C2×#1681212911yescharacteristic
286C2#2029342511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#117×C107C2
socleC22×C107C22×D4
prime radical#20293138C2
hypercenterC2×#117D107
hypocenterC107#3091
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20292968C2semidirect
3. #20292962C2semidirect
4. #20292957C2semidirect
5. #20292951C2semidirect
6. #20292945C2semidirect
7. #20292940C2semidirect
8. #20292663C2semidirect
9. #20292621C2semidirect
10. C2×C22⋊(C107⋊C8)C2semidirect
11. C2×#117×C107C2semidirect
23. #16812042C22semidirect
24. #16812042C22semidirect
25. #16812040C22semidirect
26. #16812040C22semidirect
27. #16812032C22semidirect
28. #16812032C22semidirect
29. #16812029C22semidirect
30. #16812029C22semidirect
144. C107#3091semidirect
146. C2×#117D107semidirect

Maximal subgroups

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

subgroupindex
#202931382
#202929682
#202929622
#202929572
#202929512
#202929452
#202929402
#202926632
#202926212
C2×C22⋊(C107⋊C8)2
C2×#117×C1072
#3091107