Finite Group Databasebeta

← #28346376

Subgroups — #28346376

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
2C2×#16812068C211yesmaximal, characteristic, complemented
3#20292969C212yesmaximal, complemented
4#20292963C211yesmaximal, characteristic, complemented
5#20292958C211yesmaximal, characteristic, complemented
6#20292952C212yesmaximal, complemented
7#20292946C211yesmaximal, characteristic, complemented
8#20292941C211yesmaximal, characteristic, complemented
9#20292665C212yesmaximal, complemented
10#20292624C212yesmaximal, complemented
11C2×C22⋊C8×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812092—21no—
14#16812088C2211yescharacteristic
15#16812088C2211yescharacteristic
16#16812083C2211yescharacteristic
17#16812083C2211yescharacteristic
18#16812078—21no—
19C4×C22⋊D107—21no—
20C4×C22⋊D107—21no—
21#16812068—21no—
22#16812068—21no—
23#16812068C2212yes—
24#16812068C2211yescharacteristic
25#16812068C2212yes—
26C2×C4⋊Dic107C2211yescharacteristic
27C2×C4⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107—21no—
29C2×C4×Dic107—21no—
30#16812043C2212yescomplemented
31#16812043C2212yescomplemented
32#16812041C2212yescomplemented
33#16812041C2212yescomplemented
34#16812034C2212yescomplemented
35#16812034C2212yescomplemented
36#16812031C2212yescomplemented
37#16812031C2212yescomplemented
38C23×C4×C107C2211yescharacteristic
39C22×C8×C107C2212yes—
40C22⋊C8×C107C2211yescharacteristic
41C22⋊C8×C107C2211yescharacteristic
42C22⋊C8×C107C2212yes—
43#13133818—24no—
44C2×Dic214C2311yescharacteristic
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—21no—
47C2×C22⋊D107—21no—
48C2×C22⋊D107—21no—
49C22×Dic107—21no—
50C22×Dic107—21no—
51C2×C4×D107—21no—
52C4⋊Dic107—21no—
53C4⋊Dic107—21no—
54C4⋊Dic107C2312yes—
55C4⋊Dic107C2312yes—
56C4⋊Dic107C2311yescharacteristic
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—21no—
59Dic107⋊C4—21no—
60C22⋊Dic107—22no—
61C22⋊Dic107—22no—
62C22⋊Dic107—21no—
63C22⋊Dic107—21no—
64C22⋊Dic107—21no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—21no—
67C4×Dic107—22no—
68C4×Dic107—22no—
69C4×Dic107—21no—
70C24×C107D411yescharacteristic
71C22×C4×C107—24no—
72C22×C4×C107D411yescharacteristic
73C22×C4×C107D411yescharacteristic
74C22×C4×C107D411yescharacteristic
75C22×C4×C107C2311yescharacteristic
76C22×C4×C107C2311yescharacteristic
77C22×C4×C107C2311yescharacteristic
78C2×C8×C107—24no—
79C2×C8×C107C2312yes—
80C2×C8×C107C2312yes—
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×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107#4114yes—
97C2×C4×C107C2411yescharacteristic
98C22⋊D107—42no—
99C22⋊D107—42no—
100C22⋊D107—42no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105C2×Dic107—21no—
106C2×Dic107—42no—
107C2×Dic107—42no—
108D428—22no—
109C4×D107—42no—
110Dic214—22no—
111C8×C107—22no—
112C8×C107—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#12312yes—
130C4×C107C2×#4112yes—
131Dic107—41no—
132Dic107—42no—
133Dic107—41no—
134Dic107—42no—
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×#117—1071no—
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⋊C8D10711yescharacteristic, complemented
156C2×#41—1071no—
157C23×C4C2×D10711yescharacteristic
158C22×C8C2×D10712yes—
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—1072no—
168#117—1072no—
169#117—1071no—
170#117—2141no—
171#117—2141no—
172C2×C4⋊C4—1071no—
173C2×C4⋊C4—1071no—
174C2×C22⋊C4—2141no—
175C2×C42—2141no—
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⋊C8C2×D10711yescharacteristic
185C22⋊C8C2×D10711yescharacteristic
186C22⋊C8C2×D10712yes—
187C24D42811yescharacteristic
188#41—2144no—
189C2×Q8—1071no—
190C2×D4—1071no—
191C2×D4—2141no—
192C2×D4—2141no—
193C2×D4—2141no—
194C22×C4—24no—
195C22×C4—2141no—
196C22×C4—2141no—
197C22×C4—2141no—
198C22×C4D42811yescharacteristic
199C22×C4D42811yescharacteristic
200C22×C4D42811yescharacteristic
201C22×C4C22×D10711yescharacteristic
202C22×C4C22×D10711yescharacteristic
203C22×C4C22×D10711yescharacteristic
204C2×C8—24no—
205C2×C8C22×D10712yes—
206C2×C8C22×D10712yes—
207C4⋊C4—1072no—
208C4⋊C4—1072no—
209C4⋊C4—1071no—
210C4⋊C4—1071no—
211C4⋊C4—2141no—
212C4⋊C4—2141no—
213C4⋊C4—2141no—
214C4⋊C4—2141no—
215C22⋊C4—2142no—
216C22⋊C4—2142no—
217C22⋊C4—2141no—
218C22⋊C4—2141no—
219C22⋊C4—2141no—
220C22⋊C4—2141no—
221C22⋊C4—2141no—
222C42—2142no—
223C42—2142no—
224C42—2141no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—2141no—
229C23C2×D42811yescharacteristic
230C23C2×D42811yescharacteristic
231C23C2×D42811yescharacteristic
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—4282no—
243C2×C4—4282no—
244C2×C4—4282no—
245C2×C4—2142no—
246C2×C4—2142no—
247C2×C4—2141no—
248C2×C4—2141no—
249C2×C4—2141no—
250C2×C4C2×D42811yescharacteristic
251C2×C4C2×D42811yescharacteristic
252C2×C4C2×D42811yescharacteristic
253C2×C4#1313381714yes—
254C2×C4C23×D10711yescharacteristic
255C8—22no—
256C8—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—4281no—
266C22—4281no—
267C22—4281no—
268C22#1681209312yes—
269C22#1681213112yes—
270C22C22×D42811yescharacteristic
271C4—21no—
272C4—21no—
273C4—4281no—
274C4—4282no—
275C4—4281no—
276C4—4282no—
277C4—4281no—
278C4#1681209312yes—
279C4C2×#1313381712yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—8561no—
284C2C2×#1681209311yescharacteristic
285C2C2×#1681213111yescharacteristic
286C2#2029342711yescharacteristic
subgroupG/H
centerC2×C4C2×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C22⋊C8×C107C2
socleC22×C107C22×D4
prime radical#20293138C2
hypercenterC2×C22⋊C8D107
hypocenterC107#3091
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16812068C2semidirect
3. #20292969C2semidirect
4. #20292963C2semidirect
5. #20292958C2semidirect
6. #20292952C2semidirect
7. #20292946C2semidirect
8. #20292941C2semidirect
9. #20292665C2semidirect
10. #20292624C2semidirect
11. C2×C22⋊C8×C107C2semidirect
30. #16812043C22semidirect
31. #16812043C22semidirect
32. #16812041C22semidirect
33. #16812041C22semidirect
34. #16812034C22semidirect
35. #16812034C22semidirect
36. #16812031C22semidirect
37. #16812031C22semidirect
144. C107#3091semidirect
155. C2×C22⋊C8D107semidirect

Maximal subgroups

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

subgroupindex
#202931382
C2×#168120682
#202929692
#202929632
#202929582
#202929522
#202929462
#202929412
#202926652
#202926242
C2×C22⋊C8×C1072
#3091107