Finite Group Databasebeta

← #28347050

Subgroups — #28347050

Order 13696.

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

normal subgroups only

This group has 408 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#20293161C211yesmaximal, characteristic, complemented
2#20293033C211yesmaximal, characteristic, complemented
3#20293014C211yesmaximal, characteristic, complemented
4#20292991C211yesmaximal, characteristic, complemented
5#20292989C211yesmaximal, characteristic, complemented
6#20292878C212yesmaximal, complemented
7#20292871C212yesmaximal, complemented
8C4×#13133806C212yesmaximal, complemented
9C4×#13133803C212yesmaximal, complemented
10C4×#13133802C211yesmaximal, characteristic, complemented
11#532×C107C211yesmaximal, characteristic, complemented
12C2×#13133806C2212yes—
13C2×#13133803C2212yes—
14C2×#13133802C2212yes—
15#16812103—21no—
16#16812101C2211yescharacteristic
17#16812097C2211yescharacteristic
18#16812092—21no—
19#16812088C2211yescharacteristic
20#16812083C2211yescharacteristic
21C4×Dic214C2211yescharacteristic
22C4×D428C2211yescharacteristic
23C4×C22⋊D107—21no—
24#16812066—21no—
25#16812042C2212yes—
26#16812040C2212yes—
27#16812032C2212yes—
28#16812030C2212yes—
29#16812030C2212yes—
30#16812029C2212yes—
31#16812020C2211yescharacteristic
32#16812020C2211yescharacteristic
33C4×C107⋊C8C2212yes—
34C2×#41×C107—21no—
35C4⋊Q8×C107C2211yescharacteristic
36#127×C107C2211yescharacteristic
37#124×C107—21no—
38#121×C107—21no—
39#121×C107C2212yes—
40C4×D4×C107—21no—
41C4×D4×C107C2212yes—
42C2×C42×C107C2211yescharacteristic
43C2×Dic214C2311yescharacteristic
44C2×D428C2311yescharacteristic
45C2×C22⋊D107—21no—
46C2×C4×D107—21no—
47#13133806—24no—
48#13133803—24no—
49#13133802—24no—
50C2×C107⋊C8C2312yes—
51C2×C107⋊C8C2312yes—
52C4⋊Dic107C2311yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54Dic107⋊C4—21no—
55Dic107⋊C4—21no—
56Dic107⋊C4—21no—
57Dic107⋊C4—21no—
58C22⋊Dic107—21no—
59(C2×D107)⋊C4—21no—
60(C2×D107)⋊C4—21no—
61(C2×D107)⋊C4—21no—
62(C2×D107)⋊C4—21no—
63C4×Dic107—21no—
64#41×C107—44no—
65C2×Q8×C107—21no—
66C2×D4×C107—22no—
67C2×D4×C107—21no—
68C2×D4×C107—21no—
69C2×D4×C107C2312yes—
70C22×C4×C107—22no—
71C22×C4×C107D411yescharacteristic
72C22×C4×C107D411yescharacteristic
73C22×C4×C107C2311yescharacteristic
74C4⋊C4×C107—21no—
75C4⋊C4×C107C2312yes—
76C22⋊C4×C107—22no—
77C22⋊C4×C107—22no—
78C42×C107D411yescharacteristic
79C42×C107D411yescharacteristic
80C42×C107C2311yescharacteristic
81C42×C107C2311yescharacteristic
82C23×C107—22no—
83C23×C107C2×D411yescharacteristic
84C22×D107—21no—
85Q8×C107—42no—
86D4×C107—22no—
87D4×C107—22no—
88D4×C107—42no—
89D4×C107—42no—
90D4×C107—44no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—41no—
95C2×C4×C107—44no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107C2411yescharacteristic
102C22⋊D107—42no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105C2×Dic107—21no—
106D428—22no—
107C4×D107—42no—
108Dic214—22no—
109C107⋊C8—22no—
110C107⋊C8#4114yescomplemented
111C22×C107—21no—
112C22×C107—21no—
113C22×C107—21no—
114C22×C107—42no—
115C22×C107—42no—
116C22×C107—42no—
117C22×C107C22×D411yescharacteristic
118C2×D107—41no—
119C2×D107—41no—
120C2×D107—41no—
121C4×C107—21no—
122C4×C107—21no—
123C4×C107—21no—
124C4×C107—21no—
125C4×C107—42no—
126C4×C107AGL(1,ℤ/8)12yes—
127C4×C107C2×#4112yes—
128Dic107—41no—
129Dic107—41no—
130Dic107—41no—
131C2×C107—41no—
132C2×C107—82no—
133C2×C107#53211yescharacteristic
134C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C107#57511yescharacteristic
136D107—81no—
137#3171—1071nomaximal, Sylow
138C107#317111yescharacteristic, complemented, Sylow
139#532D10711yescharacteristic, complemented
140#529—1071no—
141#501—1071no—
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—21no—
151C2×SD16—1072no—
152C2×D8—1072no—
153C2×M16—1072no—
154C4⋊Q8C2×D10711yescharacteristic
155#127C2×D10711yescharacteristic
156#126—2141no—
157#125—1071no—
158#124—21no—
159#124—1071no—
160#123—2141no—
161C22⋊Q8—1071no—
162#121—21no—
163#121—1071no—
164#121C2×D10712yes—
165C4×Q8—1071no—
166C4×D4—21no—
167C4×D4—1071no—
168C4×D4—2141no—
169C4×D4C2×D10712yes—
170#117—2141no—
171C2×C42C2×D10711yescharacteristic
172#107—1072no—
173#106—1072no—
174#103—1072no—
175#102—1072no—
176#102—1072no—
177#102—1072no—
178#97—1071no—
179#97—1071no—
180C4×C8—1072no—
181#41—44no—
182C2×Q8—21no—
183C2×Q8—1071no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—1071no—
188C2×D4—2141no—
189C2×D4C22×D10712yes—
190C22×C4—22no—
191C22×C4—2141no—
192C22×C4C22⋊D10711yescharacteristic
193C22×C4C22⋊D10711yescharacteristic
194C22×C4C22×D10711yescharacteristic
195SD16—2144no—
196D8—2144no—
197M16—2144no—
198C2×C8—1072no—
199C2×C8—1072no—
200C4⋊C4—21no—
201C4⋊C4—1071no—
202C4⋊C4—1071no—
203C4⋊C4—2141no—
204C4⋊C4—2141no—
205C4⋊C4—2141no—
206C4⋊C4—2141no—
207C4⋊C4C22×D10712yes—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—2141no—
211C22⋊C4—2141no—
212C22⋊C4—2141no—
213C22⋊C4—2141no—
214C22⋊C4—2141no—
215C42—2141no—
216C42C22⋊D10711yescharacteristic
217C42C22⋊D10711yescharacteristic
218C42C22×D10711yescharacteristic
219C42C22×D10711yescharacteristic
220C23—22no—
221C23—2141no—
222C23C2×C22⋊D10711yescharacteristic
223Q8—42no—
224Q8—2142no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—42no—
229D4—44no—
230D4—4282no—
231D4—2142no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—44no—
237C2×C4—4282no—
238C2×C4—2141no—
239C2×C4—2141no—
240C2×C4—2141no—
241C2×C4C2×C22⋊D10711yescharacteristic
242C2×C4C2×C22⋊D10711yescharacteristic
243C2×C4C2×C22⋊D10711yescharacteristic
244C2×C4C2×C22⋊D10711yescharacteristic
245C2×C4C2×C22⋊D10711yescharacteristic
246C2×C4C23×D10711yescharacteristic
247C8—1074no—
248C8—2142no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—4281no—
256C22—4281no—
257C22—4281no—
258C22C22×C22⋊D10711yescharacteristic
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—4281no—
265C4—4281no—
266C4—4281no—
267C4#1681213512yes—
268C4#41×D10712yes—
269C2—41no—
270C2—82no—
271C2—8561no—
272C2#2029317311yescharacteristic
273C2C2×#1681213511yescharacteristic
274C2#2029342511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#532×C107C2
socleC22×C107C22×D4
hypercenter#532D107
hypocenterC107#3171
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20293161C2semidirect
2. #20293033C2semidirect
3. #20293014C2semidirect
4. #20292991C2semidirect
5. #20292989C2semidirect
6. #20292878C2semidirect
7. #20292871C2semidirect
8. C4×#13133806C2semidirect
9. C4×#13133803C2semidirect
10. C4×#13133802C2semidirect
11. #532×C107C2semidirect
110. C107⋊C8#41semidirect
138. C107#3171semidirect
139. #532D107semidirect

Maximal subgroups

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

subgroupindex
#202931612
#202930332
#202930142
#202929912
#202929892
#202928782
#202928712
C4×#131338062
C4×#131338032
C4×#131338022
#532×C1072
#3171107