Finite Group Databasebeta

← #28346135

Subgroups — #28346135

Order 13696.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812117C211yesmaximal, characteristic, complemented
2#20293205C211yesmaximal, characteristic, complemented
3#20293194C211yesmaximal, characteristic, complemented
4#20292877C212yesmaximal, complemented
5#20292870C212yesmaximal, complemented
6#20292860C212yesmaximal, complemented
7#20292853C212yesmaximal, complemented
8#20292669C211yesmaximal, characteristic, complemented
9#20292623C211yesmaximal, characteristic, complemented
10C2×#16812030C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812117—24no—
14C2×#13133801C2212yescomplemented
15C22×C107⋊C8C2211yescharacteristic
16C2×C8×D107C2212yescomplemented
17#16812105—21no—
18#16812096—21no—
19#16812088—21no—
20#16812085C2212yescomplemented
21#16812083—21no—
22#16812081—22no—
23#16812080C2212yes—
24#16812080C2212yes—
25#16812080C2212yescomplemented
26#16812078—22no—
27#16812068C2211yescharacteristic
28C2×C4⋊Dic107C2211yescharacteristic
29#16812043C2212yescomplemented
30#16812041C2212yescomplemented
31#16812033C2211yescharacteristic, complemented
32#16812033C2211yescharacteristic, complemented
33#16812030C2212yes—
34#16812030C2212yes—
35#16812030C2211yescharacteristic, complemented
36#16812030C2211yescharacteristic, complemented
37C2×#41×C107C2211yescharacteristic
38C22×D4×C107C2211yescharacteristic
39AGL(1,ℤ/8)×C107—24no—
40C2×SD16×C107C2212yes—
41C2×D8×C107C2212yes—
42C2×M16×C107C2211yescharacteristic
43#13133818—22no—
44#13133818—22no—
45C2×Dic214D411yescharacteristic
46C2×D428D411yescharacteristic
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107D412yes—
51C22×Dic107—21no—
52C2×C4×D107C2312yes—
53#13133801—24no—
54C2×C107⋊C8—22no—
55C2×C107⋊C8C2311yescharacteristic
56C2×C107⋊C8C2311yescharacteristic
57C8×D107—24no—
58C4⋊Dic107—21no—
59C4⋊Dic107C2311yescharacteristic
60C4⋊Dic107C2311yescharacteristic
61C4⋊Dic107C2312yes—
62Dic107⋊C4—22no—
63C22⋊Dic107—21no—
64C22⋊Dic107—22no—
65C22⋊Dic107—22no—
66C22⋊Dic107—22no—
67(C2×D107)⋊C4—22no—
68C4×Dic107—21no—
69C24×C107—21no—
70#41×C107—22no—
71#41×C107—22no—
72C2×Q8×C107C2311yescharacteristic
73C2×D4×C107—21no—
74C2×D4×C107—22no—
75C2×D4×C107—21no—
76C2×D4×C107—22no—
77C2×D4×C107C2312yes—
78C2×D4×C107C2311yescharacteristic
79C22×C4×C107—21no—
80C22×C4×C107C2311yescharacteristic
81SD16×C107—24no—
82D8×C107—24no—
83M16×C107D414yescomplemented
84C2×C8×C107C2312yes—
85C23×C107—21no—
86C23×C107—22no—
87C23×C107—42no—
88C23×C107—81no—
89C23×C107—81no—
90C23×C107C2×D411yescharacteristic
91C22×D107C2×D412yes—
92Q8×C107—21no—
93Q8×C107—21no—
94D4×C107—21no—
95D4×C107—22no—
96D4×C107—22no—
97D4×C107—21no—
98D4×C107—42no—
99D4×C107—42no—
100C2×C4×C107—21no—
101C2×C4×C107—42no—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2×D412yes—
104C2×C4×C107#4112yes—
105C2×C4×C107C2411yescharacteristic
106C22⋊D107—24no—
107C22⋊D107—82no—
108C22⋊D107—82no—
109C22⋊D107—82no—
110C2×Dic107—21no—
111C2×Dic107—21no—
112C2×Dic107—22no—
113C2×Dic107—81no—
114C2×Dic107C2×D412yes—
115D428—22no—
116C4×D107—22no—
117C4×D107—22no—
118Dic214—22no—
119C8×C107C2×D414yes—
120C107⋊C8—21no—
121C107⋊C8—21no—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107—41no—
125C22×C107—42no—
126C22×C107—41no—
127C22×C107—42no—
128C22×C107—42no—
129C22×C107—41no—
130C22×C107—84no—
131C22×C107#12112yes—
132C22×C107C22×D411yescharacteristic
133C2×D107—22no—
134C2×D107—22no—
135C2×D107—22no—
136C4×C107—41no—
137C4×C107#12112yes—
138C4×C107C22×D411yescharacteristic
139C4×C107C2×#4111yescharacteristic
140Dic107—22no—
141Dic107—42no—
142Dic107—41no—
143Dic107—41no—
144C2×C107—22no—
145C2×C107—81no—
146C2×C107—82no—
147C2×C107C2×#12111yescharacteristic
148C2×C107#57611yescharacteristic
149C2×C107#57711yescharacteristic
150D107—42no—
151#3062—1071nomaximal, Sylow
152C107#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
154C2×#131—1071no—
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×D4C2×D10711yescharacteristic
167AGL(1,ℤ/8)—24no—
168C2×SD16C2×D10712yes—
169C2×D8C2×D10712yes—
170#131—2144no—
171C2×M16—1072no—
172C2×M16C2×D10711yescharacteristic
173C22×C8—1071no—
174C22×C8—1072no—
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—21no—
196#41—22no—
197#41—22no—
198#41—2142no—
199#41—2142no—
200C2×Q8—1071no—
201C2×Q8C22×D10711yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—1072no—
207C2×D4—1071no—
208C2×D4—2142no—
209C2×D4—2142no—
210C2×D4—2142no—
211C2×D4C22×D10711yescharacteristic
212C2×D4C22×D10712yes—
213C22×C4—21no—
214C22×C4—1072no—
215C22×C4—2141no—
216C22×C4C22×D10711yescharacteristic
217SD16—24no—
218D8—24no—
219M16—2144no—
220M16C22⋊D10714yescomplemented
221C2×C8—1071no—
222C2×C8—1071no—
223C2×C8—2142no—
224C2×C8—2144no—
225C2×C8C22×D10712yes—
226C4⋊C4—1072no—
227C4⋊C4—1071no—
228C4⋊C4—1071no—
229C4⋊C4—2142no—
230C4⋊C4—2141no—
231C22⋊C4—2142no—
232C22⋊C4—2142no—
233C22⋊C4—2142no—
234C22⋊C4—2142no—
235C22⋊C4—2141no—
236C42—2141no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—81no—
241C23—81no—
242C23—1072no—
243C23D4×D10711yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—2142no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—21no—
251D4—42no—
252D4—42no—
253D4—8562no—
254D4—8562no—
255D4—8562no—
256D4—2144no—
257D4—2142no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—8561no—
261C2×C4—1072no—
262C2×C4—2141no—
263C2×C4—2141no—
264C2×C4—2142no—
265C2×C4—2142no—
266C2×C4—2142no—
267C2×C4D4×D10711yescharacteristic
268C2×C4C2×C22⋊D10712yes—
269C2×C4#1313381712yes—
270C2×C4C23×D10711yescharacteristic
271C8—2141no—
272C8—2141no—
273C8C2×C22⋊D10714yes—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—84no—
283C22—2142no—
284C22—2142no—
285C22—2142no—
286C22#1681208012yes—
287C22C2×D4×D10711yescharacteristic
288C4—41no—
289C4—4282no—
290C4—4281no—
291C4—4281no—
292C4—2142no—
293C4#1681208012yes—
294C4C22×C22⋊D10711yescharacteristic
295C4C2×#1313381711yescharacteristic
296C2—22no—
297C2—82no—
298C2—81no—
299C2—4282no—
300C2C2×#1681208011yescharacteristic
301C2#2029343111yescharacteristic
302C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3062
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812117C2semidirect
2. #20293205C2semidirect
3. #20293194C2semidirect
4. #20292877C2semidirect
5. #20292870C2semidirect
6. #20292860C2semidirect
7. #20292853C2semidirect
8. #20292669C2semidirect
9. #20292623C2semidirect
10. C2×#16812030C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
14. C2×#13133801C22semidirect
16. C2×C8×D107C22semidirect
20. #16812085C22semidirect
25. #16812080C22semidirect
29. #16812043C22semidirect
30. #16812041C22semidirect
31. #16812033C22semidirect
32. #16812033C22semidirect
35. #16812030C22semidirect
36. #16812030C22semidirect
83. M16×C107D4semidirect
152. C107#3062semidirect
153. C2×AGL(1,ℤ/8)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×#168121172
#202932052
#202931942
#202928772
#202928702
#202928602
#202928532
#202926692
#202926232
C2×#168120302
C2×AGL(1,ℤ/8)×C1072
#3062107