Finite Group Databasebeta

← #28346104

Subgroups — #28346104

Order 13696.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D107C211yesmaximal, characteristic, complemented
2C2×#16812115C211yesmaximal, characteristic, complemented
3#20293193C212yesmaximal, complemented
4#20292879C212yesmaximal, complemented
5#20292878C212yesmaximal, complemented
6#20292864C211yesmaximal, characteristic, complemented
7#20292863C211yesmaximal, characteristic, complemented
8#20292860C211yesmaximal, characteristic, complemented
9#20292664C211yesmaximal, characteristic, complemented
10#20292630C212yesmaximal, complemented
11#466×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×D4×D107C2212yes—
14C2×D856C2211yescharacteristic
15C2×#13133803C2212yes—
16D8×D107—24no—
17#16812115—24no—
18C2×#13133802C2211yescharacteristic, complemented
19C2×#13133801C2211yescharacteristic
20C2×#13133801C2211yescharacteristic, complemented
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic, complemented
23#16812105—22no—
24#16812095—22no—
25#16812084C2212yescomplemented
26#16812083C2212yescomplemented
27#16812080C2212yescomplemented
28#16812079—22no—
29#16812077—22no—
30#16812067C2212yes—
31#16812043C2211yescharacteristic, complemented
32#16812040C2212yescomplemented
33#16812031C2212yescomplemented
34#16812030C2212yescomplemented
35#16812029C2212yescomplemented
36C2×D8×C107C2211yescharacteristic
37C22×C8×C107C2211yescharacteristic, complemented
38#121×C107C2212yescomplemented
39#107×C107C2211yescharacteristic, complemented
40#102×C107C2212yescomplemented
41C23×D107—22no—
42#13133818—22no—
43#13133818—22no—
44C2×Dic214D411yescharacteristic
45C2×D428—22no—
46C2×D428D411yescharacteristic
47C2×D428C2312yes—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107D411yescharacteristic
52C2×C22⋊D107D411yescharacteristic
53D4×D107—24no—
54D4×D107—24no—
55C2×C4×D107C2311yescharacteristic
56C2×C4×D107C2311yescharacteristic
57D856—22no—
58#13133803—24no—
59#13133802—22no—
60#13133801—22no—
61#13133801—22no—
62C2×C107⋊C8C2311yescharacteristic
63C2×C107⋊C8C2311yescharacteristic
64C8×D107—22no—
65C8×D107D414yescomplemented
66C4⋊Dic107C2312yes—
67C22⋊Dic107—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C4×Dic107—22no—
72C2×D4×C107—22no—
73C2×D4×C107C2312yes—
74C22×C4×C107C2311yescharacteristic
75D8×C107—22no—
76C2×C8×C107—22no—
77C2×C8×C107C2311yescharacteristic
78C2×C8×C107C2311yescharacteristic
79C4⋊C4×C107C2312yes—
80C22⋊C4×C107—22no—
81C23×C107—22no—
82C23×C107C2×D411yescharacteristic
83C22×D107—22no—
84C22×D107—44no—
85C22×D107—82no—
86C22×D107—82no—
87C22×D107C2×D411yescharacteristic
88C22×D107C2×D411yescharacteristic
89D4×C107—22no—
90D4×C107—22no—
91D4×C107—82no—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—22no—
98C22⋊D107—22no—
99C22⋊D107—44no—
100C22⋊D107—82no—
101C22⋊D107—82no—
102C2×Dic107—22no—
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105D428—22no—
106D428—22no—
107D428—22no—
108D428—82no—
109C4×D107—21no—
110C4×D107—21no—
111C4×D107C2×D412yes—
112C4×D107#4112yes—
113Dic214—22no—
114C8×C107—21no—
115C8×C107C2×D412yes—
116C107⋊C8—21no—
117C107⋊C8C2×D412yes—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107—21no—
121C22×C107—42no—
122C22×C107—42no—
123C22×C107—42no—
124C22×C107C22×D411yescharacteristic
125C2×D107—21no—
126C2×D107—21no—
127C2×D107—21no—
128C2×D107—21no—
129C2×D107—21no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—42no—
133C2×D107—88no—
134C2×D107#12112yes—
135C4×C107—21no—
136C4×C107—42no—
137C4×C107C22×D411yescharacteristic
138C4×C107C2×#4111yescharacteristic
139Dic107—21no—
140Dic107—42no—
141Dic107#12112yes—
142C2×C107—41no—
143C2×C107—82no—
144C2×C107C2×#12111yescharacteristic
145C2×C107#57611yescharacteristic
146C2×C107#57611yescharacteristic
147D107—22no—
148D107—41no—
149D107—82no—
150#3058—1071nomaximal, Sylow
151C107#305811yescharacteristic, complemented, Sylow
152C22×D8—1071no—
153C2×#131—1071no—
154#535—1072no—
155#469—1072no—
156#469—1072no—
157#466—1071no—
158#466—1071no—
159#466—1071no—
160#466D10711yescharacteristic, complemented
161#427—1071no—
162#418—1072no—
163C2×#41—1071no—
164C22×D4—1072no—
165C2×D8—1072no—
166C2×D8—1071no—
167C2×D8—2144no—
168C2×D8C2×D10711yescharacteristic
169#131—2144no—
170C2×M16—1071no—
171C2×M16—1071no—
172C2×M16—1071no—
173C22×C8—1071no—
174C22×C8—1071no—
175C22×C8C2×D10711yescharacteristic, complemented
176#127—2142no—
177#124—2142no—
178#121—1072no—
179#121—1072no—
180#121—1072no—
181#121C2×D10712yescomplemented
182#120—2142no—
183#120—2142no—
184#117—1072no—
185#107—1071no—
186#107C2×D10711yescharacteristic, complemented
187#106—1072no—
188#102—1072no—
189#102—1072no—
190#102—1072no—
191#102C2×D10712yescomplemented
192C24—2142no—
193#41—2142no—
194#41—2142no—
195C2×Q8—1071no—
196C2×D4—22no—
197C2×D4—1071no—
198C2×D4—1071no—
199C2×D4—1071no—
200C2×D4—1072no—
201C2×D4—2142no—
202C2×D4—2144no—
203C2×D4—2144no—
204C2×D4—2142no—
205C2×D4—2142no—
206C2×D4—2142no—
207C2×D4C22×D10712yes—
208C22×C4—1071no—
209C22×C4—1071no—
210C22×C4C22×D10711yescharacteristic
211D8—22no—
212D8—2144no—
213D8—2142no—
214M16—2142no—
215M16—2142no—
216M16—2142no—
217C2×C8—22no—
218C2×C8—1071no—
219C2×C8—1074no—
220C2×C8—1071no—
221C2×C8—2142no—
222C2×C8C22×D10711yescharacteristic
223C2×C8C22×D10711yescharacteristic
224C4⋊C4—1072no—
225C4⋊C4C22×D10712yes—
226C22⋊C4—22no—
227C22⋊C4—2142no—
228C22⋊C4—2142no—
229C22⋊C4—2142no—
230C22⋊C4—2142no—
231C42—2142no—
232C23—22no—
233C23—8562no—
234C23—8562no—
235C23—1071no—
236C23—1071no—
237C23—4284no—
238C23—2142no—
239C23D4×D10711yescharacteristic
240Q8—2142no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—8562no—
245D4—8562no—
246D4—8562no—
247D4—4284no—
248D4—2142no—
249D4—2142no—
250D4—2142no—
251D4—2142no—
252D4—2142no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—1071no—
257C2×C4—1072no—
258C2×C4—1072no—
259C2×C4—1071no—
260C2×C4—2142no—
261C2×C4—2141no—
262C2×C4—2141no—
263C2×C4D4×D10711yescharacteristic
264C2×C4C23×D10711yescharacteristic
265C8—21no—
266C8—1072no—
267C8—2141no—
268C8D4×D10712yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—8568no—
276C22—1072no—
277C22—4282no—
278C22—4282no—
279C22—4282no—
280C22—2141no—
281C22—2141no—
282C22—2141no—
283C22—2141no—
284C22—2141no—
285C22C2×D4×D10711yescharacteristic
286C4—21no—
287C4—42no—
288C4—1072no—
289C4—4282no—
290C4—2141no—
291C4C2×D4×D10711yescharacteristic
292C4#41×D10711yescharacteristic
293C2—41no—
294C2—82no—
295C2—8562no—
296C2—4281no—
297C2—2142no—
298C2#121×D10711yescharacteristic
299C2#2029342811yescharacteristic
300C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#466×C107C2
socleC22×C107C22×D4
hypercenter#466D107
hypocenterC107#3058
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D8×D107C2semidirect
2. C2×#16812115C2semidirect
3. #20293193C2semidirect
4. #20292879C2semidirect
5. #20292878C2semidirect
6. #20292864C2semidirect
7. #20292863C2semidirect
8. #20292860C2semidirect
9. #20292664C2semidirect
10. #20292630C2semidirect
11. #466×C107C2semidirect
18. C2×#13133802C22semidirect
20. C2×#13133801C22semidirect
22. C2×C8×D107C22semidirect
25. #16812084C22semidirect
26. #16812083C22semidirect
27. #16812080C22semidirect
31. #16812043C22semidirect
32. #16812040C22semidirect
33. #16812031C22semidirect
34. #16812030C22semidirect
35. #16812029C22semidirect
37. C22×C8×C107C22semidirect
38. #121×C107C22semidirect
39. #107×C107C22semidirect
40. #102×C107C22semidirect
65. C8×D107D4semidirect
151. C107#3058semidirect
160. #466D107semidirect
175. C22×C8C2×D107semidirect
181. #121C2×D107semidirect
186. #107C2×D107semidirect
191. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1072
C2×#168121152
#202931932
#202928792
#202928782
#202928642
#202928632
#202928602
#202926642
#202926302
#466×C1072
#3058107