Finite Group Databasebeta

← #28345723

Subgroups — #28345723

Order 13696.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#13133818C211yesmaximal, characteristic
2C2×#16812142C212yesmaximal, complemented
3C2×#16812088C211yesmaximal, characteristic, complemented
4#20292791C214yesmaximal, complemented
5#20292785C212yesmaximal, complemented
6#20292780C212yesmaximal, complemented
7#20292629C212yesmaximal, complemented
8#407×C107C211yesmaximal, characteristic, complemented
9C2×#13133818—28no—
10C2×#13133818—21no—
11C2×#13133818—21no—
12C2×#13133818C2212yes—
13C2×#13133818C2212yes—
14C22×Dic214C2211yescharacteristic
15C22×Dic214C2211yescharacteristic, complemented
16C22×D428C2211yescharacteristic
17C22×C22⋊D107—21no—
18C22×C4×D107—21no—
19#16812142—28no—
20C2×Dic428C2214yescomplemented
21C2×#13133807C2214yescomplemented
22#16812088—21no—
23#16812088—21no—
24#16812088C2212yes—
25#16812088C2212yescomplemented
26C2×C4⋊Dic107C2211yescharacteristic, complemented
27C2×Dic107⋊C4—21no—
28C2×C22⋊Dic107—21no—
29#16812034C2214yescomplemented
30#16812031C2214yescomplemented
31C23×C4×C107C2211yescharacteristic
32C2×M16×C107C2212yescomplemented
33C22⋊C8×C107C2212yes—
34C22⋊C8×C107C2212yescomplemented
35C23×D107—21no—
36#13133818—28no—
37#13133818—28no—
38#13133818—24no—
39#13133818—24no—
40#13133818—24no—
41#13133818D418yes—
42C2×Dic214—21no—
43C2×Dic214—24no—
44C2×Dic214—24no—
45C2×Dic214—21no—
46C2×Dic214C2312yes—
47C2×Dic214C2312yes—
48C2×D428—21no—
49C2×D428—24no—
50C2×D428C2312yes—
51C2×C22⋊D107—28no—
52C2×C22⋊D107—22no—
53C2×C22⋊D107—22no—
54C22×Dic107—21no—
55C22×Dic107—21no—
56C22×Dic107—21no—
57C2×C4×D107—28no—
58C2×C4×D107—22no—
59C2×C4×D107—22no—
60Dic428—28no—
61#13133807—28no—
62C4⋊Dic107—21no—
63C4⋊Dic107C2312yes—
64Dic107⋊C4—22no—
65Dic107⋊C4—22no—
66C22⋊Dic107—22no—
67C22⋊Dic107—22no—
68C24×C107D411yescharacteristic
69C22×C4×C107—24no—
70C22×C4×C107D411yescharacteristic
71C22×C4×C107D411yescharacteristic
72C22×C4×C107D411yescharacteristic
73C22×C4×C107C2311yescharacteristic
74C22×C4×C107C2311yescharacteristic
75C22×C4×C107C2311yescharacteristic
76M16×C107—24no—
77C2×C8×C107C2314yes—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—41no—
81C23×C107C2×D411yescharacteristic
82C23×C107C2×D411yescharacteristic
83C23×C107C2×D411yescharacteristic
84C22×D107—24no—
85C22×D107—22no—
86C22×D107—42no—
87C22×D107—42no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107C2×D414yes—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2411yescharacteristic
98C22⋊D107—28no—
99C22⋊D107—28no—
100C2×Dic107—24no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—42no—
105C2×Dic107—42no—
106D428—24no—
107D428—22no—
108D428C2×D414yes—
109C4×D107—28no—
110C4×D107—28no—
111Dic214—24no—
112Dic214—22no—
113Dic214—24no—
114Dic214—42no—
115Dic214C2×D414yes—
116C8×C107—24no—
117C22×C107—21no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107—21no—
121C22×C107—21no—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107—44no—
125C22×C107#12012yes—
126C22×C107#13712yescomplemented
127C22×C107C22×D411yescharacteristic
128C2×D107—24no—
129C2×D107—48no—
130C2×D107—42no—
131C2×D107—42no—
132C4×C107—21no—
133C4×C107—21no—
134C4×C107#12012yes—
135C4×C107C22×D412yes—
136Dic107—24no—
137Dic107—42no—
138Dic107—42no—
139C2×C107—22no—
140C2×C107—22no—
141C2×C107—41no—
142C2×C107C2×#12011yescharacteristic
143C2×C107C2×#13711yescharacteristic
144C2×C107#57511yescharacteristic
145D107—44no—
146#3007—1071nomaximal, Sylow
147C107#300711yescharacteristic, complemented, Sylow
148C22×#41—1071no—
149C2×#137—1072no—
150C2×C22⋊Q8—1071no—
151#452—1074no—
152#451—1072no—
153#450—1072no—
154#417—1072no—
155#407D10711yescharacteristic, complemented
156C2×#41—1072no—
157C2×#41—1072no—
158C2×#41—2148no—
159C2×#41—2141no—
160C2×#41—2141no—
161C22×Q8—1071no—
162C22×Q8—1071no—
163C22×D4—1071no—
164C22×D4—2141no—
165C23×C4—2141no—
166C23×C4C2×D10711yescharacteristic
167#137—2148no—
168C2×Q16—1074no—
169C2×SD16—1074no—
170C2×M16C2×D10712yescomplemented
171C22⋊Q8—1072no—
172C22⋊Q8—1072no—
173C22⋊Q8—2141no—
174C22⋊Q8—2141no—
175C2×C4⋊C4—1071no—
176C2×C4⋊C4—2141no—
177C2×C22⋊C4—2141no—
178#103—1074no—
179#102—1074no—
180C22⋊C8C2×D10712yes—
181C22⋊C8C2×D10712yescomplemented
182C24—2141no—
183C24D42811yescharacteristic
184#41—1078no—
185#41—2144no—
186#41—2144no—
187#41—2148no—
188#41—2148no—
189#41—2144no—
190C2×Q8—1072no—
191C2×Q8—1072no—
192C2×Q8—2144no—
193C2×Q8—2144no—
194C2×Q8—2141no—
195C2×Q8—2141no—
196C2×D4—1072no—
197C2×D4—2144no—
198C2×D4—2148no—
199C2×D4—2142no—
200C2×D4—2142no—
201C2×D4—2141no—
202C22×C4—24no—
203C22×C4—2148no—
204C22×C4—2141no—
205C22×C4—2141no—
206C22×C4—2141no—
207C22×C4—2142no—
208C22×C4—2142no—
209C22×C4D42811yescharacteristic
210C22×C4D42811yescharacteristic
211C22×C4D42811yescharacteristic
212C22×C4C22×D10711yescharacteristic
213C22×C4C22×D10711yescharacteristic
214C22×C4C22×D10711yescharacteristic
215Q16—2148no—
216SD16—2148no—
217M16—24no—
218C2×C8C22×D10714yes—
219C4⋊C4—1072no—
220C4⋊C4—2141no—
221C4⋊C4—2142no—
222C4⋊C4—2142no—
223C22⋊C4—2142no—
224C22⋊C4—2142no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—4282no—
229C23—4282no—
230C23—2144no—
231C23—2142no—
232C23C2×D42811yescharacteristic
233C23C2×D42811yescharacteristic
234C23C2×D42811yescharacteristic
235Q8—1074no—
236Q8—4282no—
237Q8—2144no—
238Q8—2144no—
239Q8—2142no—
240D4—1074no—
241D4—2144no—
242D4—2148no—
243D4—2148no—
244D4—2142no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—4282no—
251C2×C4—4282no—
252C2×C4—2148no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2144no—
256C2×C4—2142no—
257C2×C4—2148no—
258C2×C4D4×D10714yes—
259C2×C4C2×D42811yescharacteristic
260C2×C4C2×D42811yescharacteristic
261C2×C4C2×D42811yescharacteristic
262C2×C4C23×D10711yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—4288no—
273C22—4282no—
274C22—4282no—
275C22—2144no—
276C22#1681207912yes—
277C22#1681214212yescomplemented
278C22C22×D42811yescharacteristic
279C4—21no—
280C4—21no—
281C4—4282no—
282C4—4282no—
283C4—2144no—
284C4#1681207912yes—
285C4C2×D4×D10712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—4284no—
290C2C2×#1681207911yescharacteristic
291C2C2×#1681214211yescharacteristic
292C2#2029342711yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#407×C107C2
socleC22×C107C22×D4
prime radicalC22×#13133818C2
hypercenter#407D107
hypocenterC107#3007
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16812142C2semidirect
3. C2×#16812088C2semidirect
4. #20292791C2semidirect
5. #20292785C2semidirect
6. #20292780C2semidirect
7. #20292629C2semidirect
8. #407×C107C2semidirect
15. C22×Dic214C22semidirect
20. C2×Dic428C22semidirect
21. C2×#13133807C22semidirect
25. #16812088C22semidirect
26. C2×C4⋊Dic107C22semidirect
29. #16812034C22semidirect
30. #16812031C22semidirect
32. C2×M16×C107C22semidirect
34. C22⋊C8×C107C22semidirect
126. C22×C107#137semidirect
147. C107#3007semidirect
155. #407D107semidirect
170. C2×M16C2×D107semidirect
181. C22⋊C8C2×D107semidirect
277. C22#16812142semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121422
C2×#168120882
#202927912
#202927852
#202927802
#202926292
#407×C1072
#3007107