Finite Group Databasebeta

← #28346139

Subgroups — #28346139

Order 13696.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812135C211yesmaximal, characteristic, complemented
2C2×#16812115C211yesmaximal, characteristic, complemented
3#20293208C211yesmaximal, characteristic, complemented
4#20293194C211yesmaximal, characteristic, complemented
5#20292879C212yesmaximal, complemented
6#20292872C212yesmaximal, complemented
7#20292862C212yesmaximal, complemented
8#20292855C212yesmaximal, complemented
9#20292667C211yesmaximal, characteristic, complemented
10#20292624C211yesmaximal, characteristic, complemented
11C2×#102×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133818C2211yescharacteristic
14#16812135—24no—
15C2×#13133806C2212yes—
16C2×#13133803C2212yes—
17#16812115—24no—
18C2×#13133802C2211yescharacteristic
19C2×#13133801C2212yescomplemented
20C2×C8×D107C2212yescomplemented
21#16812105—21no—
22#16812096—21no—
23#16812089—21no—
24#16812089C2212yescomplemented
25#16812084—21no—
26#16812084C2212yescomplemented
27#16812083—22no—
28#16812080C2212yes—
29#16812080C2212yes—
30#16812078—22no—
31#16812068C2211yescharacteristic
32#16812042C2212yescomplemented
33#16812040C2212yescomplemented
34#16812034C2211yescharacteristic, complemented
35#16812034C2211yescharacteristic, complemented
36#16812031C2211yescharacteristic, complemented
37#16812031C2211yescharacteristic, complemented
38C22×D4×C107C2211yescharacteristic
39C22×C8×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#102×C107C2212yes—
42#102×C107C2212yes—
43#13133818—22no—
44#13133818—22no—
45#13133818—22no—
46#13133818—22no—
47C2×Dic214D411yescharacteristic
48C2×Dic214C2311yescharacteristic
49C2×D428—22no—
50C2×D428D411yescharacteristic
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107—22no—
55C2×C22⋊D107D412yes—
56C2×C4×D107—21no—
57C2×C4×D107C2312yes—
58#13133806—24no—
59#13133803—24no—
60#13133802D414yescomplemented
61#13133801—24no—
62C2×C107⋊C8C2312yes—
63C8×D107—24no—
64C4⋊Dic107—22no—
65C4⋊Dic107C2311yescharacteristic
66C4⋊Dic107C2311yescharacteristic
67C22⋊Dic107—21no—
68C22⋊Dic107—22no—
69C22⋊Dic107—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—22no—
72C4×Dic107—21no—
73C24×C107—21no—
74C2×D4×C107—22no—
75C2×D4×C107—21no—
76C2×D4×C107—22no—
77C2×D4×C107C2312yes—
78C22×C4×C107—21no—
79C22×C4×C107C2311yescharacteristic
80C2×C8×C107—22no—
81C2×C8×C107C2311yescharacteristic
82C2×C8×C107C2311yescharacteristic
83C4⋊C4×C107—21no—
84C4⋊C4×C107C2312yes—
85C23×C107—22no—
86C23×C107—42no—
87C23×C107—81no—
88C23×C107—81no—
89C23×C107C2×D411yescharacteristic
90C22×D107—21no—
91C22×D107C2×D412yes—
92D4×C107—22no—
93D4×C107—22no—
94D4×C107—42no—
95C2×C4×C107—22no—
96C2×C4×C107—81no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D412yes—
99C2×C4×C107#4112yes—
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—24no—
102C22⋊D107—42no—
103C22⋊D107—82no—
104C22⋊D107—82no—
105C2×Dic107—21no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107C2×D412yes—
109D428—22no—
110D428—21no—
111D428—21no—
112D428—82no—
113C4×D107—22no—
114C4×D107—22no—
115C4×D107—42no—
116Dic214—22no—
117Dic214—21no—
118Dic214—21no—
119C8×C107—21no—
120C8×C107—21no—
121C107⋊C8C2×D414yes—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107—84no—
128C22×C107#12112yes—
129C22×C107C22×D411yescharacteristic
130C2×D107—22no—
131C2×D107—22no—
132C2×D107—22no—
133C2×D107—41no—
134C2×D107—41no—
135C2×D107—41no—
136C4×C107—42no—
137C4×C107#12112yes—
138C4×C107C22×D411yescharacteristic
139C4×C107C2×#4111yescharacteristic
140Dic107—22no—
141Dic107—41no—
142Dic107—41no—
143Dic107—41no—
144C2×C107—22no—
145C2×C107—82no—
146C2×C107C2×#12111yescharacteristic
147C2×C107#57611yescharacteristic
148C2×C107#57711yescharacteristic
149D107—42no—
150D107—81no—
151#3062—1071nomaximal, Sylow
152C107#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1071no—
154C2×#131—1071no—
155#537—1071no—
156#535—1071no—
157#469—1072no—
158#468—1072no—
159#466—1072no—
160#465—1072no—
161#428—1071no—
162#416—1071no—
163C2×#102D10711yescharacteristic, complemented
164C2×#41—1071no—
165C2×#41—1071no—
166C22×D4C2×D10711yescharacteristic
167AGL(1,ℤ/8)—2144no—
168C2×SD16—1072no—
169C2×D8—1072no—
170#131—2144no—
171C2×M16—1071no—
172C2×M16—1072no—
173C22×C8—1072no—
174C22×C8C2×D10711yescharacteristic
175#127—2141no—
176#124—2141no—
177C22⋊Q8—1072no—
178C22⋊Q8—2141no—
179#121—1072no—
180#121—1072no—
181#121—1072no—
182#121—2141no—
183#121—2142no—
184#120—2142no—
185#117—1071no—
186C2×C4⋊C4C2×D10711yescharacteristic
187#107—1072no—
188#106—1072no—
189#103—1071no—
190#103—1071no—
191#102—1071no—
192#102—1071no—
193#102C2×D10712yes—
194#102C2×D10712yes—
195C24—21no—
196#41—2142no—
197#41—2142no—
198#41—2142no—
199#41—2142no—
200C2×Q8—1071no—
201C2×Q8—1071no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—1072no—
206C2×D4—1071no—
207C2×D4—1071no—
208C2×D4—2142no—
209C2×D4—2142no—
210C2×D4—2142no—
211C2×D4—2141no—
212C2×D4C22×D10712yes—
213C22×C4—21no—
214C22×C4—1072no—
215C22×C4—2141no—
216C22×C4C22×D10711yescharacteristic
217SD16—2144no—
218D8—2144no—
219M16—1074no—
220M16—2144no—
221C2×C8—22no—
222C2×C8—1072no—
223C2×C8—2144no—
224C2×C8C22×D10711yescharacteristic
225C2×C8C22×D10711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—1071no—
228C4⋊C4—1071no—
229C4⋊C4—2142no—
230C4⋊C4C22×D10712yes—
231C22⋊C4—2142no—
232C22⋊C4—2142no—
233C22⋊C4—2142no—
234C22⋊C4—2142no—
235C22⋊C4—2141no—
236C42—2141no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—1072no—
242C23—2141no—
243C23D4×D10711yescharacteristic
244Q8—2141no—
245Q8—2141no—
246Q8—2142no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—8562no—
251D4—8562no—
252D4—8562no—
253D4—4282no—
254D4—2144no—
255D4—2142no—
256D4—2141no—
257D4—2141no—
258C2×C4—22no—
259C2×C4—81no—
260C2×C4—1072no—
261C2×C4—4282no—
262C2×C4—2141no—
263C2×C4—2141no—
264C2×C4—2141no—
265C2×C4—2142no—
266C2×C4—2142no—
267C2×C4D4×D10711yescharacteristic
268C2×C4D4×D10712yes—
269C2×C4#1313381812yes—
270C2×C4C23×D10711yescharacteristic
271C8—21no—
272C8—21no—
273C8—1074no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—84no—
280C22—4281no—
281C22—4281no—
282C22—4281no—
283C22—2142no—
284C22—2142no—
285C22—2142no—
286C22#1681208212yes—
287C22C2×D4×D10711yescharacteristic
288C4—42no—
289C4—4281no—
290C4—4281no—
291C4—4281no—
292C4—2142no—
293C4#1681208212yes—
294C4C2×D4×D10711yescharacteristic
295C4C2×#1313381811yescharacteristic
296C2—22no—
297C2—82no—
298C2—8561no—
299C2—4282no—
300C2C2×#1681208211yescharacteristic
301C2#2029342811yescharacteristic
302C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#102×C107C2
socleC22×C107C22×D4
hypercenterC2×#102D107
hypocenterC107#3062
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. C2×#16812135C2semidirect
2. C2×#16812115C2semidirect
3. #20293208C2semidirect
4. #20293194C2semidirect
5. #20292879C2semidirect
6. #20292872C2semidirect
7. #20292862C2semidirect
8. #20292855C2semidirect
9. #20292667C2semidirect
10. #20292624C2semidirect
11. C2×#102×C107C2semidirect
19. C2×#13133801C22semidirect
20. C2×C8×D107C22semidirect
24. #16812089C22semidirect
26. #16812084C22semidirect
32. #16812042C22semidirect
33. #16812040C22semidirect
34. #16812034C22semidirect
35. #16812034C22semidirect
36. #16812031C22semidirect
37. #16812031C22semidirect
60. #13133802D4semidirect
152. C107#3062semidirect
163. C2×#102D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C2×#168121152
#202932082
#202931942
#202928792
#202928722
#202928622
#202928552
#202926672
#202926242
C2×#102×C1072
#3062107