Finite Group Databasebeta

← #28346004

Subgroups — #28346004

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×#16812129C211yesmaximal, characteristic, complemented
2#20293194C211yesmaximal, characteristic, complemented
3#20293110C211yesmaximal, characteristic, complemented
4#20292847C212yesmaximal, complemented
5#20292840C212yesmaximal, complemented
6#20292825C212yesmaximal, complemented
7#20292820C212yesmaximal, complemented
8#20292655C211yesmaximal, characteristic, complemented
9#20292629C211yesmaximal, characteristic, complemented
10C2×#16812030C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812129—24no—
14C2×#13133808C2211yescharacteristic, complemented
15C2×#13133806C2211yescharacteristic, complemented
16C2×#13133805C2211yescharacteristic, complemented
17C2×#13133803C2211yescharacteristic, complemented
18C22×C107⋊C8C2211yescharacteristic
19#16812105C2212yes—
20#16812096C2212yes—
21#16812080—21no—
22#16812080—21no—
23#16812078—22no—
24Q8×Dic107C2212yescomplemented
25D4×Dic107—22no—
26D4×Dic107C2212yescomplemented
27#16812068—21no—
28#16812068C2211yescharacteristic
29C2×C4⋊Dic107—21no—
30C2×C4⋊Dic107C2211yescharacteristic
31Dic107⋊C8C2212yescomplemented
32Dic107⋊C8C2212yescomplemented
33#16812034C2212yescomplemented
34#16812031C2212yescomplemented
35#16812030C2212yes—
36#16812030C2212yes—
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×Dic214C2311yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—21no—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C22×Dic107—21no—
51C22×Dic107—22no—
52C2×C4×D107—21no—
53#13133808—22no—
54#13133806—22no—
55#13133805—22no—
56#13133803—22no—
57C2×C107⋊C8—22no—
58C2×C107⋊C8C2311yescharacteristic
59C2×C107⋊C8C2311yescharacteristic
60C4⋊Dic107—21no—
61C4⋊Dic107—22no—
62C4⋊Dic107D411yescharacteristic
63C4⋊Dic107D411yescharacteristic
64C4⋊Dic107C2312yes—
65C22⋊Dic107—22no—
66C22⋊Dic107—22no—
67C22⋊Dic107—22no—
68C22⋊Dic107D412yes—
69C4×Dic107—22no—
70C4×Dic107C2312yes—
71C24×C107—21no—
72#41×C107—22no—
73#41×C107D414yescomplemented
74C2×Q8×C107C2311yescharacteristic
75C2×D4×C107—21no—
76C2×D4×C107—22no—
77C2×D4×C107—21no—
78C2×D4×C107—22no—
79C2×D4×C107C2312yes—
80C2×D4×C107C2311yescharacteristic
81C22×C4×C107—21no—
82C22×C4×C107C2311yescharacteristic
83SD16×C107—24no—
84D8×C107—24no—
85M16×C107—22no—
86C2×C8×C107C2312yes—
87C23×C107—21no—
88C23×C107—22no—
89C23×C107—42no—
90C23×C107—81no—
91C23×C107—81no—
92C23×C107C2×D411yescharacteristic
93C22×D107—21no—
94Q8×C107—21no—
95Q8×C107C2×D412yes—
96D4×C107—24no—
97D4×C107—22no—
98D4×C107—21no—
99D4×C107—22no—
100D4×C107—42no—
101D4×C107C2×D412yes—
102C2×C4×C107—21no—
103C2×C4×C107—24no—
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107C2×D412yes—
106C2×C4×C107#4112yes—
107C2×C4×C107C2411yescharacteristic
108C22⋊D107—42no—
109C22⋊D107—82no—
110C22⋊D107—82no—
111C2×Dic107—22no—
112C2×Dic107—21no—
113C2×Dic107—82no—
114C2×Dic107—81no—
115C2×Dic107C2×D412yes—
116C2×Dic107C2×D412yes—
117D428—21no—
118D428—21no—
119C4×D107—42no—
120Dic214—21no—
121Dic214—21no—
122C8×C107—22no—
123C107⋊C8—21no—
124C107⋊C8—21no—
125C22×C107—21no—
126C22×C107—21no—
127C22×C107—22no—
128C22×C107—41no—
129C22×C107—41no—
130C22×C107—42no—
131C22×C107—42no—
132C22×C107—42no—
133C22×C107—84no—
134C22×C107#12112yes—
135C22×C107C22×D411yescharacteristic
136C2×D107—41no—
137C2×D107—41no—
138C2×D107—41no—
139C4×C107—22no—
140C4×C107#12112yes—
141C4×C107C22×D411yescharacteristic
142C4×C107C2×#4111yescharacteristic
143Dic107—22no—
144Dic107—22no—
145Dic107—41no—
146Dic107—42no—
147C2×C107—22no—
148C2×C107—42no—
149C2×C107—82no—
150C2×C107C2×#12111yescharacteristic
151C2×C107#57611yescharacteristic
152C2×C107#57711yescharacteristic
153D107—81no—
154#3045—1071nomaximal, Sylow
155C107#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
157C2×#135—1071no—
158#535—1071no—
159#520—1071no—
160#464—1072no—
161#463—1072no—
162#460—1072no—
163#459—1072no—
164#424—1071no—
165#417—1071no—
166C2×#102—1071no—
167C2×#41—1071no—
168C2×#41C2×D10711yescharacteristic
169C22×D4C2×D10711yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—2144no—
172C2×Q16—1071no—
173C2×SD16—1071no—
174C2×SD16—1071no—
175C2×SD16C2×D10712yes—
176C2×D8—1071no—
177C2×D8C2×D10712yes—
178C2×M16C2×D10711yescharacteristic
179C22×C8—1071no—
180#127—1072no—
181#124—1072no—
182#121—2141no—
183#121—2141no—
184#120—2142no—
185C4×Q8—1072no—
186C4×D4—1072no—
187C4×D4—2142no—
188#117—1071no—
189#117—2141no—
190C2×C4⋊C4—1071no—
191C2×C4⋊C4—2141no—
192C4⋊C8—1072no—
193C4⋊C8—1072no—
194#103—1072no—
195#102—1072no—
196#102—1072no—
197#102—1072no—
198C24—21no—
199#41—22no—
200#41—2142no—
201#41—2142no—
202#41C22⋊D10714yescomplemented
203C2×Q8—1071no—
204C2×Q8C22×D10711yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—1071no—
210C2×D4—2142no—
211C2×D4—2142no—
212C2×D4—2141no—
213C2×D4C22×D10712yes—
214C2×D4C22×D10711yescharacteristic
215C22×C4—21no—
216C22×C4—2142no—
217C22×C4—2141no—
218C22×C4—2141no—
219C22×C4C22×D10711yescharacteristic
220Q16—2142no—
221SD16—24no—
222SD16—2142no—
223SD16—2142no—
224D8—24no—
225D8—2142no—
226M16—22no—
227C2×C8—1071no—
228C2×C8—1071no—
229C2×C8—2142no—
230C2×C8C22×D10712yes—
231C4⋊C4—1071no—
232C4⋊C4—1071no—
233C4⋊C4—1072no—
234C4⋊C4—2142no—
235C4⋊C4—2141no—
236C22⋊C4—1072no—
237C22⋊C4—2142no—
238C22⋊C4—2142no—
239C22⋊C4—2142no—
240C42—1072no—
241C42—2142no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2141no—
248C23D4×D10711yescharacteristic
249Q8—21no—
250Q8—2141no—
251Q8—2141no—
252Q8C2×C22⋊D10712yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—8562no—
259D4—8562no—
260D4—4282no—
261D4—2141no—
262D4—2141no—
263D4C2×C22⋊D10712yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—8561no—
267C2×C4—8562no—
268C2×C4—1072no—
269C2×C4—1072no—
270C2×C4—4282no—
271C2×C4—2142no—
272C2×C4—2141no—
273C2×C4D4×D10711yescharacteristic
274C2×C4C2×C22⋊D10712yes—
275C2×C4#1313381712yes—
276C2×C4C23×D10711yescharacteristic
277C8—22no—
278C8—2141no—
279C8—2141no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—4281no—
290C22—4281no—
291C22—4281no—
292C22#1681208112yes—
293C22C2×D4×D10711yescharacteristic
294C4—22no—
295C4—4282no—
296C4—4281no—
297C4—2142no—
298C4—2142no—
299C4#1681208112yes—
300C4C22×C22⋊D10711yescharacteristic
301C4C2×#1313381711yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—8561no—
306C2C2×#1681208111yescharacteristic
307C2#2029343111yescharacteristic
308C2#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#3045
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×#16812129C2semidirect
2. #20293194C2semidirect
3. #20293110C2semidirect
4. #20292847C2semidirect
5. #20292840C2semidirect
6. #20292825C2semidirect
7. #20292820C2semidirect
8. #20292655C2semidirect
9. #20292629C2semidirect
10. C2×#16812030C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
14. C2×#13133808C22semidirect
15. C2×#13133806C22semidirect
16. C2×#13133805C22semidirect
17. C2×#13133803C22semidirect
24. Q8×Dic107C22semidirect
26. D4×Dic107C22semidirect
31. Dic107⋊C8C22semidirect
32. Dic107⋊C8C22semidirect
33. #16812034C22semidirect
34. #16812031C22semidirect
73. #41×C107D4semidirect
155. C107#3045semidirect
156. C2×AGL(1,ℤ/8)D107semidirect
202. #41C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
#202931942
#202931102
#202928472
#202928402
#202928252
#202928202
#202926552
#202926292
C2×#168120302
C2×AGL(1,ℤ/8)×C1072
#3045107