Finite Group Databasebeta

← #28345999

Subgroups — #28345999

Order 13696.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293191C211yesmaximal, characteristic, complemented
4#20293108C211yesmaximal, characteristic, complemented
5#20292852C212yesmaximal, complemented
6#20292845C212yesmaximal, complemented
7#20292826C212yesmaximal, complemented
8#20292821C212yesmaximal, complemented
9#20292655C211yesmaximal, characteristic, complemented
10C2×#16812029C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14#16812137—24no—
15#16812129—24no—
16C2×#13133808C2211yescharacteristic, complemented
17C2×#13133807C2212yes—
18C2×#13133806C2211yescharacteristic, complemented
19C2×#13133805C2211yescharacteristic, complemented
20C2×D856C2212yes—
21C2×#13133803C2211yescharacteristic, complemented
22C22×C107⋊C8C2211yescharacteristic
23#16812105C2212yes—
24#16812098C2212yes—
25#16812083—21no—
26#16812083—21no—
27#16812077—22no—
28#16812074C2212yescomplemented
29#16812071C2212yescomplemented
30C4×C22⋊D107—22no—
31#16812068C2211yescharacteristic
32#16812067—21no—
33C4⋊C4×D107—21no—
34Dic107⋊C8C2212yescomplemented
35Dic107⋊C8C2212yescomplemented
36#16812029C2212yes—
37#16812029C2212yes—
38C2×#41×C107C2211yescharacteristic
39C2×M16×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#103×C107C2212yescomplemented
42#102×C107C2212yescomplemented
43C23×D107—21no—
44#13133818—22no—
45#13133818D414yescomplemented
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×C4×D107—21no—
56C2×C4×D107—22no—
57#13133808—22no—
58#13133807—24no—
59#13133806—22no—
60#13133805—22no—
61D856—24no—
62#13133803—22no—
63C2×C107⋊C8—22no—
64C2×C107⋊C8C2311yescharacteristic
65C2×C107⋊C8C2311yescharacteristic
66C4⋊Dic107D411yescharacteristic
67C4⋊Dic107D411yescharacteristic
68Dic107⋊C4—22no—
69C22⋊Dic107D412yes—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—22no—
72(C2×D107)⋊C4—22no—
73C4×Dic107—22no—
74C4×Dic107C2312yes—
75#41×C107—22no—
76#41×C107—22no—
77C2×Q8×C107C2311yescharacteristic
78C2×D4×C107—21no—
79C2×D4×C107C2311yescharacteristic
80C22×C4×C107—21no—
81C22×C4×C107—21no—
82C22×C4×C107C2311yescharacteristic
83M16×C107—22no—
84C2×C8×C107C2312yes—
85C4⋊C4×C107—21no—
86C4⋊C4×C107C2312yes—
87C23×C107—21no—
88C23×C107C2×D411yescharacteristic
89C22×D107—21no—
90C22×D107—22no—
91C22×D107—42no—
92C22×D107—81no—
93C22×D107—81no—
94Q8×C107—21no—
95Q8×C107—21no—
96D4×C107—21no—
97D4×C107—21no—
98D4×C107—42no—
99C2×C4×C107—22no—
100C2×C4×C107—21no—
101C2×C4×C107—42no—
102C2×C4×C107—81no—
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D412yes—
105C2×C4×C107#4112yes—
106C2×C4×C107C2411yescharacteristic
107C22⋊D107—24no—
108C22⋊D107—82no—
109C22⋊D107—82no—
110C2×Dic107—21no—
111C2×Dic107C2×D412yes—
112C2×Dic107C2×D412yes—
113D428—22no—
114D428—21no—
115D428—22no—
116D428—42no—
117D428C2×D412yes—
118C4×D107—24no—
119C4×D107—82no—
120Dic214—21no—
121Dic214C2×D412yes—
122C8×C107—22no—
123C107⋊C8—21no—
124C107⋊C8—21no—
125C22×C107—21no—
126C22×C107—21no—
127C22×C107—41no—
128C22×C107—41no—
129C22×C107—41no—
130C22×C107#12112yes—
131C22×C107C22×D411yescharacteristic
132C2×D107—22no—
133C2×D107—41no—
134C2×D107—41no—
135C2×D107—42no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107—84no—
139C4×C107—41no—
140C4×C107—42no—
141C4×C107#12112yes—
142C4×C107C22×D411yescharacteristic
143C4×C107C2×#4111yescharacteristic
144Dic107—22no—
145Dic107—22no—
146Dic107—22no—
147C2×C107—22no—
148C2×C107—81no—
149C2×C107C2×#12111yescharacteristic
150C2×C107#57611yescharacteristic
151C2×C107#57711yescharacteristic
152D107—42no—
153D107—82no—
154#3045—1071nomaximal, Sylow
155C107#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)—1071no—
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#417D10711yescharacteristic, complemented
166C2×#102—1071no—
167C2×#41—1071no—
168C2×#41C2×D10711yescharacteristic
169C22×D4—1071no—
170AGL(1,ℤ/8)—2144no—
171#135—2144no—
172C2×Q16—1071no—
173C2×SD16—1071no—
174C2×SD16—1071no—
175C2×SD16—1072no—
176C2×D8—1071no—
177C2×D8—1072no—
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—2141no—
191C2×C4⋊C4C2×D10711yescharacteristic
192C4⋊C8—1072no—
193C4⋊C8—1072no—
194#103C2×D10712yescomplemented
195#102—1072no—
196#102—1072no—
197#102C2×D10712yescomplemented
198C24—2141no—
199#41—22no—
200#41—22no—
201#41—1074no—
202#41—2142no—
203C2×Q8—1071no—
204C2×Q8C22×D10711yescharacteristic
205C2×D4—21no—
206C2×D4—1071no—
207C2×D4—1072no—
208C2×D4—2142no—
209C2×D4—2142no—
210C2×D4—2141no—
211C2×D4—2141no—
212C2×D4—2142no—
213C2×D4—2142no—
214C2×D4C22×D10711yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—2142no—
218C22×C4—2141no—
219C22×C4C22×D10711yescharacteristic
220Q16—2142no—
221SD16—2144no—
222SD16—2142no—
223SD16—2142no—
224D8—2144no—
225D8—2142no—
226M16—22no—
227C2×C8—1071no—
228C2×C8—1071no—
229C2×C8—2142no—
230C2×C8C22×D10712yes—
231C4⋊C4—21no—
232C4⋊C4—1071no—
233C4⋊C4—1071no—
234C4⋊C4—2142no—
235C4⋊C4C22×D10712yes—
236C22⋊C4—1072no—
237C22⋊C4—2142no—
238C22⋊C4—2142no—
239C22⋊C4—2142no—
240C42—1072no—
241C42—2142no—
242C23—21no—
243C23—8561no—
244C23—8561no—
245C23—4282no—
246C23—2142no—
247C23—2141no—
248C23D4×D10711yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—1072no—
252Q8—2141no—
253D4—21no—
254D4—21no—
255D4—42no—
256D4—8562no—
257D4—8562no—
258D4—1072no—
259D4—4282no—
260D4—2144no—
261D4—2142no—
262D4—2142no—
263D4—2141no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—8562no—
269C2×C4—1072no—
270C2×C4—1072no—
271C2×C4—2141no—
272C2×C4—2144no—
273C2×C4D4×D10711yescharacteristic
274C2×C4D4×D10712yes—
275C2×C4#1313381812yes—
276C2×C4C23×D10711yescharacteristic
277C8—22no—
278C8—2141no—
279C8—2141no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—8564no—
286C22—4282no—
287C22—4282no—
288C22—4282no—
289C22—4281no—
290C22—4281no—
291C22—2142no—
292C22#1681208212yes—
293C22C2×D4×D10711yescharacteristic
294C4—42no—
295C4—41no—
296C4—2142no—
297C4—2142no—
298C4—2142no—
299C4#1681208212yes—
300C4C2×D4×D10711yescharacteristic
301C4C2×#1313381811yescharacteristic
302C2—22no—
303C2—81no—
304C2—8562no—
305C2—4282no—
306C2C2×#1681208211yescharacteristic
307C2#2029343111yescharacteristic
308C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
hypercenter#417D107
hypocenterC107#3045
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812129C2semidirect
3. #20293191C2semidirect
4. #20293108C2semidirect
5. #20292852C2semidirect
6. #20292845C2semidirect
7. #20292826C2semidirect
8. #20292821C2semidirect
9. #20292655C2semidirect
10. C2×#16812029C2semidirect
11. #417×C107C2semidirect
16. C2×#13133808C22semidirect
18. C2×#13133806C22semidirect
19. C2×#13133805C22semidirect
21. C2×#13133803C22semidirect
28. #16812074C22semidirect
29. #16812071C22semidirect
34. Dic107⋊C8C22semidirect
35. Dic107⋊C8C22semidirect
41. #103×C107C22semidirect
42. #102×C107C22semidirect
45. #13133818D4semidirect
155. C107#3045semidirect
165. #417D107semidirect
194. #103C2×D107semidirect
197. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121292
#202931912
#202931082
#202928522
#202928452
#202928262
#202928212
#202926552
C2×#168120292
#417×C1072
#3045107