Finite Group Databasebeta

← #28346141

Subgroups — #28346141

Order 13696.

504 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16812137C211yesmaximal, characteristic, complemented
2#20293205C211yesmaximal, characteristic, complemented
3#20293191C211yesmaximal, characteristic, complemented
4#20292880C212yesmaximal, complemented
5#20292873C212yesmaximal, complemented
6#20292863C212yesmaximal, complemented
7#20292856C212yesmaximal, complemented
8#20292669C211yesmaximal, characteristic, complemented
9#20292624C211yesmaximal, characteristic, complemented
10C2×#16812031C211yesmaximal, characteristic, complemented
11C2×#131×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14#16812137—24no—
15C2×#13133807C2212yes—
16C2×D856C2212yes—
17#16812105—21no—
18#16812098—21no—
19#16812088C2212yescomplemented
20#16812085—21no—
21#16812083C2212yes—
22#16812083C2212yes—
23#16812083C2212yescomplemented
24#16812081—22no—
25#16812080—21no—
26#16812077—22no—
27#16812068C2211yescharacteristic
28C2×C4⋊Dic107C2211yescharacteristic
29#16812043C2212yescomplemented
30#16812041C2212yescomplemented
31#16812034C2211yescharacteristic, complemented
32#16812034C2211yescharacteristic, complemented
33#16812031C2212yes—
34#16812031C2212yes—
35#16812031C2211yescharacteristic, complemented
36#16812031C2211yescharacteristic, complemented
37C2×#41×C107C2211yescharacteristic
38#131×C107—24no—
39C2×M16×C107C2211yescharacteristic
40C2×M16×C107C2212yescomplemented
41C22×C8×C107C2211yescharacteristic
42C22×C8×C107C2212yescomplemented
43C23×D107—21no—
44#13133818—22no—
45#13133818—22no—
46C2×Dic214C2311yescharacteristic
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—21no—
50C2×D428C2312yes—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107—22no—
55C2×C22⋊D107—22no—
56C22×Dic107—21no—
57C2×C4×D107—21no—
58#13133807—24no—
59D856—24no—
60C4⋊Dic107—21no—
61C4⋊Dic107C2311yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63C4⋊Dic107C2312yes—
64Dic107⋊C4—22no—
65C22⋊Dic107—21no—
66C22⋊Dic107—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70C4×Dic107—21no—
71#41×C107—22no—
72#41×C107—22no—
73C2×Q8×C107D411yescharacteristic
74C2×D4×C107D412yes—
75C2×D4×C107D411yescharacteristic
76C22×C4×C107C2312yes—
77C22×C4×C107C2311yescharacteristic
78M16×C107—24no—
79M16×C107D414yescomplemented
80C2×C8×C107—22no—
81C2×C8×C107—24no—
82C2×C8×C107C2311yescharacteristic
83C2×C8×C107C2311yescharacteristic
84C2×C8×C107C2312yes—
85C23×C107C2×D412yes—
86C23×C107C2×D411yescharacteristic
87C22×D107—21no—
88C22×D107—22no—
89C22×D107—42no—
90C22×D107—81no—
91C22×D107—81no—
92Q8×C107—22no—
93D4×C107—24no—
94D4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107#4112yes—
101C2×C4×C107C2411yescharacteristic
102C22⋊D107—42no—
103C22⋊D107—82no—
104C22⋊D107—82no—
105C22⋊D107—82no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107—22no—
109C2×Dic107—21no—
110C2×Dic107—81no—
111D428—21no—
112D428—22no—
113D428—22no—
114D428—21no—
115D428—42no—
116C4×D107—42no—
117Dic214—21no—
118Dic214—21no—
119C8×C107—21no—
120C8×C107—21no—
121C8×C107C2×D414yes—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107—22no—
125C22×C107—22no—
126C22×C107—22no—
127C22×C107#12112yes—
128C22×C107C22×D411yescharacteristic
129C2×D107—41no—
130C2×D107—42no—
131C2×D107—41no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—41no—
135C2×D107—84no—
136C4×C107—22no—
137C4×C107#12112yes—
138C4×C107C22×D411yescharacteristic
139C4×C107C2×#4111yescharacteristic
140Dic107—41no—
141Dic107—42no—
142Dic107—41no—
143Dic107—41no—
144C2×C107—22no—
145C2×C107—42no—
146C2×C107C2×#12111yescharacteristic
147C2×C107#57611yescharacteristic
148C2×C107#57711yescharacteristic
149D107—81no—
150D107—82no—
151#3062—1071nomaximal, Sylow
152C107#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1071no—
154C2×#131D10711yescharacteristic, complemented
155#537—1071no—
156#535—1071no—
157#469—1072no—
158#468—1072no—
159#466—1072no—
160#465—1072no—
161#428—1071no—
162#416—1071no—
163C2×#102—1071no—
164C2×#41—1071no—
165C2×#41C2×D10711yescharacteristic
166C22×D4—1071no—
167AGL(1,ℤ/8)—2144no—
168C2×SD16—1072no—
169C2×D8—1072no—
170#131—24no—
171C2×M16C2×D10711yescharacteristic
172C2×M16C2×D10712yescomplemented
173C22×C8C2×D10711yescharacteristic
174C22×C8C2×D10712yescomplemented
175#127—2141no—
176#124—2141no—
177C22⋊Q8—1072no—
178C22⋊Q8—2141no—
179#121—1072no—
180#121—1072no—
181#121—1072no—
182#121—2141no—
183#121—2142no—
184#120—2142no—
185#117—1071no—
186C2×C4⋊C4—1071no—
187#107—1072no—
188#106—1072no—
189#103—1071no—
190#103—1071no—
191#102—1072no—
192#102—1072no—
193#102—1071no—
194#102—1071no—
195C24—2141no—
196#41—22no—
197#41—22no—
198#41—2142no—
199#41—2142no—
200C2×Q8—1071no—
201C2×Q8D42811yescharacteristic
202C2×D4—1071no—
203C2×D4—1072no—
204C2×D4—2142no—
205C2×D4—2142no—
206C2×D4—2142no—
207C2×D4—2141no—
208C2×D4—2142no—
209C2×D4—2141no—
210C2×D4—2142no—
211C2×D4D42812yes—
212C2×D4D42811yescharacteristic
213C22×C4—2141no—
214C22×C4—2141no—
215C22×C4C22×D10712yes—
216C22×C4C22×D10711yescharacteristic
217SD16—2144no—
218D8—2144no—
219M16—24no—
220M16C22⋊D10714yescomplemented
221C2×C8—22no—
222C2×C8—24no—
223C2×C8C22×D10711yescharacteristic
224C2×C8C22×D10711yescharacteristic
225C2×C8C22×D10712yes—
226C4⋊C4—1072no—
227C4⋊C4—1071no—
228C4⋊C4—1071no—
229C4⋊C4—2141no—
230C4⋊C4—2142no—
231C22⋊C4—2141no—
232C22⋊C4—2142no—
233C22⋊C4—2142no—
234C22⋊C4—2142no—
235C22⋊C4—2142no—
236C42—2141no—
237C23—8561no—
238C23—8561no—
239C23—4282no—
240C23—2141no—
241C23—2142no—
242C23C2×D42812yes—
243C23C2×D42811yescharacteristic
244Q8—22no—
245Q8—2141no—
246Q8—2141no—
247D4—24no—
248D4—22no—
249D4—8562no—
250D4—8562no—
251D4—8562no—
252D4—4282no—
253D4—4282no—
254D4—2141no—
255D4—2142no—
256D4—2142no—
257D4—2141no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—8561no—
261C2×C4—4282no—
262C2×C4—2141no—
263C2×C4—2141no—
264C2×C4—2142no—
265C2×C4—2141no—
266C2×C4C2×C22⋊D10712yes—
267C2×C4C2×D42812yes—
268C2×C4C2×D42811yescharacteristic
269C2×C4#1313381812yes—
270C2×C4C23×D10711yescharacteristic
271C8—21no—
272C8—21no—
273C8C2×C22⋊D10714yes—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—8564no—
280C22—4281no—
281C22—4282no—
282C22—4282no—
283C22—4281no—
284C22—4281no—
285C22—4282no—
286C22#1681208312yes—
287C22C22×D42811yescharacteristic
288C4—22no—
289C4—4282no—
290C4—4281no—
291C4—4281no—
292C4—4281no—
293C4#1681208312yes—
294C4C22×C22⋊D10711yescharacteristic
295C4C2×#1313381811yescharacteristic
296C2—22no—
297C2—42no—
298C2—8562no—
299C2—8561no—
300C2C2×#1681208311yescharacteristic
301C2#2029343211yescharacteristic
302C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#131×C107C2
socleC22×C107C22×D4
hypercenterC2×#131D107
hypocenterC107#3062
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. #20293205C2semidirect
3. #20293191C2semidirect
4. #20292880C2semidirect
5. #20292873C2semidirect
6. #20292863C2semidirect
7. #20292856C2semidirect
8. #20292669C2semidirect
9. #20292624C2semidirect
10. C2×#16812031C2semidirect
11. C2×#131×C107C2semidirect
19. #16812088C22semidirect
23. #16812083C22semidirect
29. #16812043C22semidirect
30. #16812041C22semidirect
31. #16812034C22semidirect
32. #16812034C22semidirect
35. #16812031C22semidirect
36. #16812031C22semidirect
40. C2×M16×C107C22semidirect
42. C22×C8×C107C22semidirect
79. M16×C107D4semidirect
152. C107#3062semidirect
154. C2×#131D107semidirect
172. C2×M16C2×D107semidirect
174. C22×C8C2×D107semidirect
220. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
#202932052
#202931912
#202928802
#202928732
#202928632
#202928562
#202926692
#202926242
C2×#168120312
C2×#131×C1072
#3062107