Finite Group Databasebeta

← #28346924

Subgroups — #28346924

Order 13696.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 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
1C2×#16812141C212yesmaximal, complemented
2C2×#16812136C212yesmaximal, complemented
3#20293178C211yesmaximal, characteristic, complemented
4#124×D107C211yesmaximal, characteristic, complemented
5#20293019C211yesmaximal, characteristic, complemented
6#20293017C211yesmaximal, characteristic, complemented
7#20293016C211yesmaximal, characteristic, complemented
8#20293012C212yesmaximal, complemented
9#20293008C212yesmaximal, complemented
10#20292579C211yesmaximal, characteristic, complemented
11#497×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×Q8×D107C2211yescharacteristic
14C2×D4×D107C2211yescharacteristic
15#16812141—28no—
16#16812136—28no—
17C2×#13133808C2212yescomplemented
18C2×#13133807C2212yescomplemented
19C2×#13133805C2212yes—
20C2×#13133805C2212yescomplemented
21C2×D856C2212yes—
22C2×#13133803C2212yes—
23C2×#13133801C2212yes—
24#16812106C2211yescharacteristic, complemented
25#16812098C2211yescharacteristic
26#16812098C2212yescomplemented
27#16812097C2211yescharacteristic
28#16812096C2211yescharacteristic
29#16812095—22no—
30#16812084—22no—
31#16812071—22no—
32C22⋊C4×D107—22no—
33C42×D107C2211yescharacteristic
34#16812020C2211yescharacteristic, complemented
35C8×Dic107C2212yescomplemented
36C2×Q16×C107C2212yescomplemented
37C2×SD16×C107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic, complemented
39#124×C107C2211yescharacteristic
40#97×C107C2211yescharacteristic, complemented
41C23×D107—21no—
42#13133819—24no—
43#13133819—24no—
44C2×Dic214—21no—
45C2×Dic214C2311yescharacteristic
46Q8×D107—24no—
47C2×D428—22no—
48C2×D428—22no—
49C2×D428C2311yescharacteristic
50C2×D428C2312yes—
51C2×C22⋊D107—21no—
52D4×D107—24no—
53C2×C4×D107—21no—
54C2×C4×D107—22no—
55C2×C4×D107D412yes—
56C2×C4×D107C2311yescharacteristic
57#13133808—24no—
58#13133807—24no—
59#13133805—24no—
60#13133805—24no—
61D856—24no—
62#13133803—24no—
63#13133801D418yescomplemented
64C2×C107⋊C8C2312yes—
65C22⋊Dic107—22no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—24no—
69C4×Dic107D412yes—
70C4×Dic107C2311yescharacteristic
71C2×Q8×C107C2311yescharacteristic
72C2×Q8×C107C2312yes—
73C2×D4×C107C2311yescharacteristic
74Q16×C107—24no—
75SD16×C107—24no—
76C2×C8×C107C2312yes—
77C4⋊C4×C107—22no—
78C22⋊C4×C107—22no—
79C42×C107C2311yescharacteristic
80C23×C107—21no—
81C22×D107—21no—
82C22×D107—22no—
83C22×D107—44no—
84C22×D107—42no—
85C22×D107C2×D411yescharacteristic
86Q8×C107—22no—
87Q8×C107—22no—
88Q8×C107—22no—
89D4×C107—22no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107C2×D412yes—
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—42no—
95C2×Dic107—21no—
96C2×Dic107C2×D412yes—
97C2×Dic107C2×D411yescharacteristic
98D428—22no—
99D428—22no—
100D428—22no—
101D428—44no—
102D428—44no—
103C4×D107—42no—
104C4×D107—42no—
105C4×D107—44no—
106C4×D107C2×D414yes—
107Dic214—22no—
108Dic214—42no—
109C8×C107C2×D414yes—
110C107⋊C8C2×D414yes—
111C22×C107—41no—
112C22×C107—42no—
113C22×C107C22×D411yescharacteristic
114C2×D107—22no—
115C2×D107—42no—
116C2×D107—41no—
117C2×D107—44no—
118C2×D107—42no—
119C2×D107—84no—
120C2×D107#12712yes—
121C4×C107—22no—
122C4×C107—41no—
123C4×C107—42no—
124C4×C107C22×D412yes—
125Dic107—22no—
126Dic107—41no—
127Dic107#12712yes—
128C2×C107—81no—
129C2×C107C2×#12711yescharacteristic
130C2×C107#57512yes—
131D107—22no—
132D107—82no—
133D107—81no—
134#3153—1071nomaximal, Sylow
135C107#315311yescharacteristic, complemented, Sylow
136C2×#137—1072no—
137C2×AGL(1,ℤ/8)—1072no—
138#532—1071no—
139C2×#124—1071no—
140#497—1071no—
141#497D10711yescharacteristic, complemented
142#496—1071no—
143#496—1071no—
144#495—1072no—
145#495—1072no—
146C4×M16—1071no—
147C2×#41—1072no—
148C22×Q8—1071no—
149C22×D4—1071no—
150#137—2148no—
151AGL(1,ℤ/8)—2148no—
152C2×Q16—1072no—
153C2×Q16C2×D10712yescomplemented
154C2×SD16—1072no—
155C2×SD16—1072no—
156C2×SD16—1072no—
157C2×SD16C2×D10712yes—
158C2×D8—1072no—
159C2×D8—1072no—
160C2×M16—1072no—
161C4⋊Q8C2×D10711yescharacteristic, complemented
162#127—1071no—
163#124—1071no—
164#124—1072no—
165#124—1071no—
166#124—1071no—
167#124—2142no—
168#124C2×D10711yescharacteristic
169#121—2142no—
170C4×D4—2142no—
171C2×C22⋊C4—2142no—
172C2×C42—1071no—
173#97—1071no—
174#97C2×D10711yescharacteristic, complemented
175C4×C8—1072no—
176C24—2141no—
177#41—2144no—
178#41—2144no—
179C2×Q8—1071no—
180C2×Q8—2141no—
181C2×Q8—2144no—
182C2×Q8C22×D10711yescharacteristic
183C2×Q8C22×D10712yes—
184C2×D4—1071no—
185C2×D4—1072no—
186C2×D4—2142no—
187C2×D4—2144no—
188C2×D4—2141no—
189C2×D4—2142no—
190C2×D4C22×D10711yescharacteristic
191C22×C4—1071no—
192C22×C4—1072no—
193C22×C4—2142no—
194C22×C4—2141no—
195Q16—24no—
196Q16—2144no—
197SD16—24no—
198SD16—2144no—
199SD16—2144no—
200SD16—2144no—
201D8—2144no—
202D8—2144no—
203M16—1078no—
204C2×C8—1072no—
205C2×C8C22×D10712yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—2142no—
209C22⋊C4—2144no—
210C22⋊C4—2142no—
211C22⋊C4—2142no—
212C42—1072no—
213C42—1071no—
214C42C22×D10711yescharacteristic
215C23—21no—
216C23—1071no—
217C23—4282no—
218C23—4284no—
219C23—2141no—
220C23—2142no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—4282no—
225Q8—2142no—
226D4—22no—
227D4—4284no—
228D4—4284no—
229D4—4282no—
230D4—2142no—
231D4—2142no—
232D4—2142no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—1071no—
236C2×C4—1074no—
237C2×C4—1072no—
238C2×C4—4282no—
239C2×C4—4284no—
240C2×C4—4282no—
241C2×C4—2141no—
242C2×C4D4×D10712yes—
243C2×C4C23×D10711yescharacteristic
244C8—1074no—
245C8D4×D10714yes—
246C22—41no—
247C22—42no—
248C22—8564no—
249C22—1072no—
250C22—4282no—
251C22—4281no—
252C22—4284no—
253C22—4282no—
254C22—2142no—
255C22C2×D4×D10711yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—1072no—
260C4—4281no—
261C4—2142no—
262C4C2×D4×D10712yes—
263C2—81no—
264C2—8561no—
265C2—8562no—
266C2—2142no—
267C2#127×D10711yescharacteristic
268C2#2029342612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#497×C107C2
socleC22×C107C22×D4
hypercenter#497D107
hypocenterC107#3153
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
2. C2×#16812136C2semidirect
3. #20293178C2semidirect
4. #124×D107C2semidirect
5. #20293019C2semidirect
6. #20293017C2semidirect
7. #20293016C2semidirect
8. #20293012C2semidirect
9. #20293008C2semidirect
10. #20292579C2semidirect
11. #497×C107C2semidirect
17. C2×#13133808C22semidirect
18. C2×#13133807C22semidirect
20. C2×#13133805C22semidirect
24. #16812106C22semidirect
26. #16812098C22semidirect
34. #16812020C22semidirect
35. C8×Dic107C22semidirect
36. C2×Q16×C107C22semidirect
38. C4⋊Q8×C107C22semidirect
40. #97×C107C22semidirect
63. #13133801D4semidirect
135. C107#3153semidirect
141. #497D107semidirect
153. C2×Q16C2×D107semidirect
161. C4⋊Q8C2×D107semidirect
174. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
C2×#168121362
#202931782
#124×D1072
#202930192
#202930172
#202930162
#202930122
#202930082
#202925792
#497×C1072
#3153107