Finite Group Databasebeta

← #28346813

Subgroups — #28346813

Order 13696.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20293258C211yesmaximal, characteristic, complemented
2#20293194C211yesmaximal, characteristic
3#20292990C212yesmaximal, complemented
4#20292985C212yesmaximal, complemented
5#20292980C212yesmaximal, complemented
6#20292972C212yesmaximal, complemented
7#20292632C211yesmaximal, characteristic, complemented
8#20292626C211yesmaximal, characteristic, complemented
9#20292624C211yesmaximal, characteristic, complemented
10#20292582C211yesmaximal, characteristic, complemented
11C2×#102×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14#16812108—21no—
15#16812108C2212yescomplemented
16#16812105C2212yes—
17#16812100—21no—
18#16812100C2212yescomplemented
19#16812096C2212yes—
20#16812088—22no—
21#16812080—21no—
22#16812080—21no—
23#16812078—22no—
24#16812068C2211yescharacteristic
25#16812068C2211yescharacteristic
26#16812034C2211yescharacteristic, complemented
27#16812034C2211yescharacteristic, complemented
28#16812032C2212yescomplemented
29#16812031C2211yescharacteristic, complemented
30#16812031C2211yescharacteristic, complemented
31#16812030C2212yes—
32#16812030C2212yes—
33#16812029C2212yescomplemented
34#16812019C2212yescomplemented
35C8×Dic107C2212yescomplemented
36C22×D4×C107C2211yescharacteristic
37C22×C8×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#102×C107C2212yes—
40#102×C107C2212yes—
41#13133818—22no—
42#13133818—22no—
43C2×Dic214—22no—
44C2×Dic214C2311yescharacteristic
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—21no—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C2×C4×D107—21no—
50#13133802—22no—
51C2×C107⋊C8C2312yes—
52C4⋊Dic107—22no—
53C4⋊Dic107D411yescharacteristic
54C4⋊Dic107D411yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59C22⋊Dic107—21no—
60C22⋊Dic107—22no—
61C22⋊Dic107—22no—
62C22⋊Dic107D412yes—
63C4×Dic107—21no—
64C4×Dic107C2312yes—
65C24×C107—21no—
66C2×D4×C107—21no—
67C2×D4×C107—22no—
68C2×D4×C107—22no—
69C2×D4×C107C2312yes—
70C22×C4×C107—21no—
71C22×C4×C107C2311yescharacteristic
72C2×C8×C107—22no—
73C2×C8×C107C2311yescharacteristic
74C2×C8×C107C2311yescharacteristic
75C4⋊C4×C107—21no—
76C4⋊C4×C107C2312yes—
77C23×C107—22no—
78C23×C107—42no—
79C23×C107—81no—
80C23×C107—81no—
81C23×C107C2×D411yescharacteristic
82C22×D107—21no—
83D4×C107—22no—
84D4×C107—22no—
85D4×C107—42no—
86C2×C4×C107—22no—
87C2×C4×C107—81no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107#4112yes—
90C2×C4×C107#4112yes—
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—42no—
93C22⋊D107—82no—
94C22⋊D107—82no—
95C2×Dic107—21no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107C2×D412yes—
99C2×Dic107C2×D412yes—
100D428—21no—
101D428—21no—
102C4×D107—42no—
103Dic214—21no—
104Dic214—21no—
105Dic214—82no—
106C8×C107—21no—
107C8×C107—21no—
108C107⋊C8—22no—
109C22×C107—21no—
110C22×C107—21no—
111C22×C107—42no—
112C22×C107—42no—
113C22×C107—42no—
114C22×C107—84no—
115C22×C107#12412yes—
116C22×C107C22×D411yescharacteristic
117C2×D107—41no—
118C2×D107—41no—
119C2×D107—41no—
120C4×C107—42no—
121C4×C107#12412yes—
122C4×C107C2×#4111yescharacteristic
123C4×C107C2×#4111yescharacteristic
124Dic107—22no—
125Dic107—22no—
126Dic107—41no—
127Dic107—41no—
128Dic107—41no—
129C2×C107—22no—
130C2×C107—82no—
131C2×C107C2×#12411yescharacteristic
132C2×C107#57611yescharacteristic
133C2×C107#57711yescharacteristic
134D107—81no—
135#3136—1071nomaximal, Sylow
136C107#313611yescharacteristic, complemented, Sylow
137#544—1071no—
138#535—1071no—
139#490—1072no—
140#489—1072no—
141#488—1072no—
142#486—1072no—
143#418—1071no—
144#417—1071no—
145#416—1071no—
146C2×#102D10711yescharacteristic, complemented
147#405—1071no—
148C2×#41—1071no—
149C22×D4C2×D10711yescharacteristic
150C2×M16—1071no—
151C22×C8C2×D10711yescharacteristic
152C4⋊Q8—1072no—
153C4⋊Q8—2141no—
154#127—1072no—
155#125—1072no—
156#125—2141no—
157#124—1072no—
158C22⋊Q8—2142no—
159#121—2141no—
160#121—2141no—
161#120—2142no—
162#117—1071no—
163#117—1071no—
164C2×C4⋊C4C2×D10711yescharacteristic
165#103—1071no—
166#103—1071no—
167#103—1072no—
168#102—1072no—
169#102—1072no—
170#102—1071no—
171#102—1071no—
172#102—1072no—
173#102C2×D10712yes—
174#102C2×D10712yes—
175#97—1072no—
176C4×C8—1072no—
177C24—21no—
178#41—2142no—
179#41—2142no—
180C2×Q8—1071no—
181C2×Q8—2142no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—1071no—
186C2×D4—2142no—
187C2×D4—2142no—
188C2×D4—2141no—
189C2×D4C22×D10712yes—
190C22×C4—21no—
191C22×C4—2141no—
192C22×C4C22×D10711yescharacteristic
193M16—2142no—
194C2×C8—22no—
195C2×C8—1072no—
196C2×C8C22×D10711yescharacteristic
197C2×C8C22×D10711yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—1071no—
200C4⋊C4—1071no—
201C4⋊C4—1071no—
202C4⋊C4—1071no—
203C4⋊C4—2142no—
204C4⋊C4—2142no—
205C4⋊C4—2142no—
206C4⋊C4C22×D10712yes—
207C22⋊C4—1072no—
208C22⋊C4—2142no—
209C22⋊C4—2142no—
210C22⋊C4—2141no—
211C42—1072no—
212C42—2141no—
213C23—22no—
214C23—42no—
215C23—81no—
216C23—81no—
217C23—2141no—
218C23D4×D10711yescharacteristic
219Q8—8562no—
220Q8—2141no—
221Q8—2141no—
222D4—22no—
223D4—22no—
224D4—42no—
225D4—8562no—
226D4—8562no—
227D4—4282no—
228D4—2141no—
229D4—2141no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—1072no—
233C2×C4—1072no—
234C2×C4—4282no—
235C2×C4—2141no—
236C2×C4—2141no—
237C2×C4—2141no—
238C2×C4D4×D10711yescharacteristic
239C2×C4#1313381712yes—
240C2×C4#1313381812yes—
241C2×C4C23×D10711yescharacteristic
242C8—21no—
243C8—21no—
244C8—2142no—
245C22—21no—
246C22—21no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—84no—
251C22—4281no—
252C22—4281no—
253C22—4281no—
254C22#1681209512yes—
255C22C2×D4×D10711yescharacteristic
256C4—42no—
257C4—4281no—
258C4—4281no—
259C4—4281no—
260C4—2142no—
261C4—2142no—
262C4#1681209512yes—
263C4C2×#1313381711yescharacteristic
264C4C2×#1313381811yescharacteristic
265C2—22no—
266C2—82no—
267C2—8561no—
268C2C2×#1681209511yescharacteristic
269C2#2029342811yescharacteristic
270C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#102×C107C2
socleC22×C107C22×D4
prime radical#20293194C2
hypercenterC2×#102D107
hypocenterC107#3136
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #20293258C2semidirect
3. #20292990C2semidirect
4. #20292985C2semidirect
5. #20292980C2semidirect
6. #20292972C2semidirect
7. #20292632C2semidirect
8. #20292626C2semidirect
9. #20292624C2semidirect
10. #20292582C2semidirect
11. C2×#102×C107C2semidirect
15. #16812108C22semidirect
18. #16812100C22semidirect
26. #16812034C22semidirect
27. #16812034C22semidirect
28. #16812032C22semidirect
29. #16812031C22semidirect
30. #16812031C22semidirect
33. #16812029C22semidirect
34. #16812019C22semidirect
35. C8×Dic107C22semidirect
136. C107#3136semidirect
146. C2×#102D107semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202931942
#202929902
#202929852
#202929802
#202929722
#202926322
#202926262
#202926242
#202925822
C2×#102×C1072
#3136107