Finite Group Databasebeta

← #28372316

Subgroups — #28372316

Order 13952.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812746C211yesmaximal, characteristic, complemented
2C2×#16812725C211yesmaximal, characteristic, complemented
3C22×#13182017C211yesmaximal, characteristic, complemented
4#20295169C214yesmaximal, complemented
5#20295164C214yesmaximal, complemented
6#20295034C211yesmaximal, characteristic, complemented
7#20295020C211yesmaximal, characteristic, complemented
8#20294993C211yesmaximal, characteristic, complemented
9#535×C109C211yesmaximal, characteristic, complemented
10#16812746—24no—
11#16812746—24no—
12C2×#13182029C2211yescharacteristic
13C2×#13182029C2212yescomplemented
14C2×#13182028—22no—
15C2×#13182028—21no—
16C22×Dic218C2211yescharacteristic
17C22×Dic218C2212yescomplemented
18C2×Q8×D109—21no—
19#16812725—24no—
20C2×#13182017—24no—
21C2×#13182017C2214yes—
22C2×#13182017C2212yes—
23C2×#13182014C2212yes—
24C2×#13182013C2211yescharacteristic
25C22×#11724784C2211yescharacteristic
26#16812622C2214yescomplemented
27#16812622C2212yescomplemented
28#16812619C2212yescomplemented
29C22⋊#11724784C2214yescomplemented
30C2×#41×C109C2211yescharacteristic
31C22×D4×C109C2211yescharacteristic
32#127×C109—21no—
33#124×C109—21no—
34#121×C109C2214yes—
35#120×C109—22no—
36#117×C109C2211yescharacteristic
37#13182029—22no—
38#13182029—24no—
39#13182029—24no—
40#13182029D414yescomplemented
41#13182028—216no—
42#13182028—28no—
43C2×Dic218—21no—
44C2×Dic218—22no—
45C2×Dic218—22no—
46C2×Dic218—24no—
47C2×Dic218—24no—
48C2×Dic218D414yescomplemented
49C2×Dic218C2312yes—
50C2×Dic218C2311yescharacteristic
51Q8×D109—28no—
52C2×D436C2311yescharacteristic
53C2×C22⋊D109—21no—
54C2×C22⋊D109—22no—
55C22×Dic109—21no—
56C22×Dic109—22no—
57C2×C4×D109—21no—
58C2×C4×D109—22no—
59#13182017—28no—
60#13182017—24no—
61#13182014—24no—
62#13182013—22no—
63C2×#11724784—22no—
64C2×#11724784C2312yes—
65C2×#11724784C2312yes—
66C24×C109—21no—
67#41×C109—22no—
68#41×C109—22no—
69C2×Q8×C109D411yescharacteristic
70C2×D4×C109—24no—
71C2×D4×C109—22no—
72C2×D4×C109—22no—
73C2×D4×C109—21no—
74C2×D4×C109D412yes—
75C2×D4×C109D411yescharacteristic
76C2×D4×C109C2312yes—
77C22×C4×C109C2312yes—
78C22×C4×C109C2311yescharacteristic
79C4⋊C4×C109C2312yes—
80C22⋊C4×C109—21no—
81C22⋊C4×C109—24no—
82C42×C109—21no—
83C23×C109—22no—
84C23×C109—42no—
85C23×C109—82no—
86C23×C109C2×D412yes—
87C23×C109C2×D411yescharacteristic
88C22×D109—21no—
89Q8×C109—22no—
90D4×C109—24no—
91D4×C109—22no—
92D4×C109—22no—
93D4×C109—22no—
94D4×C109—42no—
95D4×C109—84no—
96C2×C4×C109—22no—
97C2×C4×C109—22no—
98C2×C4×C109—22no—
99C2×C4×C109C2×D412yes—
100C2×C4×C109C2×D412yes—
101C2×C4×C109C2×D412yes—
102C2×C4×C109C2×D411yescharacteristic
103C2×C4×C109C2411yescharacteristic
104C22⋊D109—28no—
105C22⋊D109—24no—
106C2×Dic109—28no—
107C2×Dic109—24no—
108C2×Dic109—21no—
109C2×Dic109—22no—
110D436—21no—
111D436C2×D412yes—
112C4×D109—28no—
113C4×D109—24no—
114Dic218—28no—
115Dic218—24no—
116Dic218—22no—
117Dic218—21no—
118Dic218C2×D412yes—
119Dic218C2×D414yes—
120#11724784—22no—
121#11724784—22no—
122C22×C109—22no—
123C22×C109—22no—
124C22×C109—24no—
125C22×C109—42no—
126C22×C109—44no—
127C22×C109—84no—
128C22×C109#12012yes—
129C22×C109C22×D411yescharacteristic
130C2×D109—22no—
131C2×D109—42no—
132C4×C109—22no—
133C4×C109—42no—
134C4×C109#12012yes—
135C4×C109C22×D411yescharacteristic
136C4×C109C22×D411yescharacteristic
137Dic109—22no—
138Dic109—24no—
139C2×C109—22no—
140C2×C109—42no—
141C2×C109—82no—
142C2×C109C2×#12011yescharacteristic
143C2×C109#57712yes—
144D109—42no—
145#3010—1091nomaximal, Sylow
146C109#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1091no—
148C2×AGL(1,ℤ/8)—1091no—
149C22×SD16—1091no—
150#535D10911yescharacteristic, complemented
151#449—1094no—
152#448—1094no—
153#419—1091no—
154#416—1091no—
155#408—1091no—
1562−1+4—2184no—
1572−1+4—2184no—
158C2×#41—1091no—
159C2×#41—1092no—
160C2×#41—2182no—
161C2×#41—2181no—
162C2×#41C2×D10911yescharacteristic
163C22×Q8—1091no—
164C22×Q8—1092no—
165C22×Q8—2181no—
166C22×D4C2×D10911yescharacteristic
167AGL(1,ℤ/8)—2184no—
168C2×SD16—1094no—
169C2×SD16—1092no—
170C2×SD16—2184no—
171C2×D8—1092no—
172C2×M16—1091no—
173C22×C8—1091no—
174#127—21no—
175#124—21no—
176#121C2×D10914yes—
177#120—22no—
178#117C2×D10911yescharacteristic
179#103—1092no—
180#103—1094no—
181#102—1092no—
182C22⋊C8—1094no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—1094no—
187#41—2184no—
188#41—2184no—
189#41—21816no—
190#41—2188no—
191#41—2182no—
192C2×Q8—1094no—
193C2×Q8—1092no—
194C2×Q8—1091no—
195C2×Q8—2184no—
196C2×Q8—2184no—
197C2×Q8—2188no—
198C2×Q8—2182no—
199C2×Q8—2182no—
200C2×Q8—2181no—
201C2×Q8C22⋊D10911yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—1091no—
207C2×D4—2182no—
208C2×D4—2181no—
209C2×D4C22⋊D10912yes—
210C2×D4C22⋊D10911yescharacteristic
211C2×D4C22×D10912yes—
212C22×C4—2182no—
213C22×C4—2181no—
214C22×C4—2182no—
215C22×C4—2181no—
216C22×C4C22×D10912yes—
217C22×C4C22×D10911yescharacteristic
218SD16—2188no—
219SD16—2184no—
220D8—2184no—
221M16—2182no—
222C2×C8—1092no—
223C2×C8—1092no—
224C2×C8—2182no—
225C4⋊C4C22×D10912yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—2181no—
233C23C2×C22⋊D10912yes—
234C23C2×C22⋊D10911yescharacteristic
235Q8—22no—
236Q8—1094no—
237Q8—1092no—
238Q8—2188no—
239Q8—2184no—
240Q8—2182no—
241Q8—2181no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—1092no—
249D4—2188no—
250D4—2184no—
251D4—2181no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—2188no—
256C2×C4—2188no—
257C2×C4—2184no—
258C2×C4—2182no—
259C2×C4—2181no—
260C2×C4—2184no—
261C2×C4D4×D10912yes—
262C2×C4D4×D10912yes—
263C2×C4C2×C22⋊D10912yes—
264C2×C4C2×C22⋊D10911yescharacteristic
265C2×C4C23×D10911yescharacteristic
266C8—2182no—
267C8—2182no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—42no—
272C22—44no—
273C22—84no—
274C22—4362no—
275C22—2182no—
276C22#1681266712yes—
277C22C22×C22⋊D10911yescharacteristic
278C4—22no—
279C4—42no—
280C4—2184no—
281C4—2182no—
282C4#1681266712yes—
283C4C2×D4×D10911yescharacteristic
284C4C2×D4×D10911yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—4362no—
289C2C2×#1681266711yescharacteristic
290C2#2029583712yes—
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#535×C109C2
socleC22×C109C22×D4
hypercenter#535D109
hypocenterC109#3010
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16812746C2semidirect
2. C2×#16812725C2semidirect
3. C22×#13182017C2semidirect
4. #20295169C2semidirect
5. #20295164C2semidirect
6. #20295034C2semidirect
7. #20295020C2semidirect
8. #20294993C2semidirect
9. #535×C109C2semidirect
13. C2×#13182029C22semidirect
17. C22×Dic218C22semidirect
26. #16812622C22semidirect
27. #16812622C22semidirect
28. #16812619C22semidirect
29. C22⋊#11724784C22semidirect
40. #13182029D4semidirect
48. C2×Dic218D4semidirect
146. C109#3010semidirect
150. #535D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127462
C2×#168127252
C22×#131820172
#202951692
#202951642
#202950342
#202950202
#202949932
#535×C1092
#3010109