Finite Group Databasebeta

← #28347058

Subgroups — #28347058

Order 13696.

408 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 119 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#20293178C211yesmaximal, characteristic, complemented
2#20293162C211yesmaximal, characteristic, complemented
3#20293016C211yesmaximal, characteristic, complemented
4#20292993C211yesmaximal, characteristic, complemented
5#20292989C211yesmaximal, characteristic, complemented
6#20292881C212yesmaximal, complemented
7#20292874C212yesmaximal, complemented
8#20292714C212yesmaximal, complemented
9#20292709C212yesmaximal, complemented
10#20292579C211yesmaximal, characteristic, complemented
11#501×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—21no—
13C2×#13133807C2212yes—
14C2×D856C2212yes—
15C2×#13133801C2212yes—
16#16812106C2211yescharacteristic
17#16812104—21no—
18#16812098—21no—
19#16812097C2211yescharacteristic
20#16812094—21no—
21#16812089C2211yescharacteristic
22#16812084—21no—
23#16812084C2212yes—
24#16812084C2211yescharacteristic
25#16812074C2211yescharacteristic
26#16812071—21no—
27#16812071—21no—
28#16812071C2211yescharacteristic
29#16812071C2212yes—
30#16812067—21no—
31C42×D107C2211yescharacteristic
32#16812032C2212yes—
33#16812029C2212yes—
34#16812029C2212yes—
35#16812029C2212yes—
36#16812020C2211yescharacteristic
37C8×Dic107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39#125×C107C2211yescharacteristic
40#107×C107C2212yes—
41#106×C107C2212yes—
42#97×C107C2211yescharacteristic
43#13133819—44no—
44C2×Dic214C2311yescharacteristic
45C2×D428—21no—
46C2×D428—22no—
47C2×D428—21no—
48C2×D428—21no—
49C2×D428C2311yescharacteristic
50C2×D428C2312yes—
51C2×C4×D107—21no—
52C2×C4×D107—22no—
53C2×C4×D107D411yescharacteristic
54C2×C4×D107D411yescharacteristic
55C2×C4×D107C2311yescharacteristic
56#13133807—24no—
57D856—24no—
58#13133801—24no—
59C2×C107⋊C8C2312yes—
60C4⋊Dic107—21no—
61Dic107⋊C4—21no—
62(C2×D107)⋊C4—21no—
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4—21no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—21no—
69C4×Dic107—21no—
70C4×Dic107D411yescharacteristic
71C4×Dic107D411yescharacteristic
72C4×Dic107C2311yescharacteristic
73C2×Q8×C107—21no—
74C2×C8×C107C2312yes—
75C4⋊C4×C107—21no—
76C4⋊C4×C107—21no—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C4⋊C4×C107C2311yescharacteristic
81C42×C107C2311yescharacteristic
82C22×D107—21no—
83C22×D107—22no—
84C22×D107C2×D411yescharacteristic
85Q8×C107—42no—
86C2×C4×C107—21no—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C2×Dic107—21no—
93C2×Dic107C2×D411yescharacteristic
94C2×Dic107C2×D411yescharacteristic
95C2×Dic107C2×D411yescharacteristic
96D428—22no—
97D428—22no—
98D428—22no—
99D428—42no—
100D428—42no—
101D428—42no—
102D428—44no—
103C4×D107—22no—
104C4×D107—22no—
105C4×D107—41no—
106C4×D107—42no—
107C4×D107—44no—
108Dic214—22no—
109C8×C107#4114yescomplemented
110C107⋊C8—22no—
111C22×C107C22×D411yescharacteristic
112C2×D107—21no—
113C2×D107—21no—
114C2×D107—21no—
115C2×D107—41no—
116C2×D107—42no—
117C2×D107—42no—
118C2×D107—41no—
119C2×D107—42no—
120C2×D107—41no—
121C4×C107—21no—
122C4×C107—21no—
123C4×C107—42no—
124C4×C107—41no—
125C4×C107—41no—
126C4×C107C2×#4112yes—
127Dic107—21no—
128Dic107—21no—
129Dic107—41no—
130Dic107AGL(1,ℤ/8)12yes—
131C2×C107#53211yescharacteristic
132C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
133C2×C107#57511yescharacteristic
134D107—41no—
135D107—82no—
136D107—81no—
137#3171—1071nomaximal, Sylow
138C107#317111yescharacteristic, complemented, Sylow
139#532—1071no—
140#529—1071no—
141#501D10711yescharacteristic, complemented
142#496—1071no—
143#490—1071no—
144#489—1071no—
145#469—1072no—
146#468—1072no—
147C4×SD16—1072no—
148C4×D8—1072no—
149C4×M16—1071no—
150C2×#41—2141no—
151C2×SD16—1072no—
152C2×D8—1072no—
153C2×M16—1072no—
154C4⋊Q8C2×D10711yescharacteristic
155#127—1071no—
156#126—2141no—
157#125C2×D10711yescharacteristic
158#124—1071no—
159#124—2141no—
160#123—2141no—
161C22⋊Q8—1071no—
162#121—1071no—
163#121—1072no—
164#121—2141no—
165C4×Q8—1071no—
166C4×D4—1071no—
167C4×D4—1072no—
168C4×D4—2141no—
169C4×D4—2141no—
170#117—2141no—
171C2×C42—1071no—
172#107C2×D10712yes—
173#106C2×D10712yes—
174#103—1072no—
175#102—1072no—
176#102—1072no—
177#102—1072no—
178#97—1071no—
179#97C2×D10711yescharacteristic
180C4×C8—1072no—
181#41—4284no—
182C2×Q8—21no—
183C2×Q8—1071no—
184C2×D4—1071no—
185C2×D4—1072no—
186C2×D4—2141no—
187C2×D4—2142no—
188C2×D4—2141no—
189C2×D4—2141no—
190C22×C4—1071no—
191C22×C4—1071no—
192C22×C4—1071no—
193C22×C4—2141no—
194C22×C4—2142no—
195SD16—2144no—
196D8—2144no—
197M16—2144no—
198C2×C8—1072no—
199C2×C8C22×D10712yes—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—2141no—
204C4⋊C4—2141no—
205C4⋊C4C22×D10711yescharacteristic
206C4⋊C4C22×D10711yescharacteristic
207C4⋊C4C22×D10712yes—
208C22⋊C4—2141no—
209C22⋊C4—2142no—
210C22⋊C4—2141no—
211C22⋊C4—2142no—
212C22⋊C4—2141no—
213C22⋊C4—2141no—
214C22⋊C4—2141no—
215C42—1071no—
216C42—1071no—
217C42—1071no—
218C42—2141no—
219C42C22×D10711yescharacteristic
220C23—1071no—
221C23—2141no—
222C23—2142no—
223Q8—42no—
224Q8—2142no—
225D4—4282no—
226D4—4282no—
227D4—4284no—
228D4—4282no—
229D4—2142no—
230D4—2142no—
231D4—2142no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—1071no—
236C2×C4—1071no—
237C2×C4—1071no—
238C2×C4—4281no—
239C2×C4—4284no—
240C2×C4—4282no—
241C2×C4—2142no—
242C2×C4—2141no—
243C2×C4—2142no—
244C2×C4D4×D10711yescharacteristic
245C2×C4D4×D10711yescharacteristic
246C2×C4C23×D10711yescharacteristic
247C8—2142no—
248C8#1313381914yescomplemented
249C22—4282no—
250C22—4281no—
251C22—4282no—
252C22—4281no—
253C22—4281no—
254C22—4282no—
255C22—2141no—
256C22—2141no—
257C22—2141no—
258C22C2×D4×D10711yescharacteristic
259C4—21no—
260C4—21no—
261C4—42no—
262C4—41no—
263C4—41no—
264C4—1072no—
265C4—4281no—
266C4—2141no—
267C4—2141no—
268C4C2×#1313381912yes—
269C2—8562no—
270C2—8561no—
271C2—4281no—
272C2#2029317811yescharacteristic
273C2AGL(1,ℤ/8)×D10711yescharacteristic
274C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#501×C107C2
socleC22×C107C22×D4
hypercenter#501D107
hypocenterC107#3171
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #20293178C2semidirect
2. #20293162C2semidirect
3. #20293016C2semidirect
4. #20292993C2semidirect
5. #20292989C2semidirect
6. #20292881C2semidirect
7. #20292874C2semidirect
8. #20292714C2semidirect
9. #20292709C2semidirect
10. #20292579C2semidirect
11. #501×C107C2semidirect
109. C8×C107#41semidirect
138. C107#3171semidirect
141. #501D107semidirect
248. C8#13133819semidirect

Maximal subgroups

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

subgroupindex
#202931782
#202931622
#202930162
#202929932
#202929892
#202928812
#202928742
#202927142
#202927092
#202925792
#501×C1072
#3171107