Finite Group Databasebeta

← #28372272

Subgroups — #28372272

Order 13952.

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

normal subgroups only

This group has 758 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
1C22×#13182029C211yesmaximal, characteristic, complemented
2C2×#16812729C211yesmaximal, characteristic, complemented
3C2×#16812724C211yesmaximal, characteristic, complemented
4#20295186C214yesmaximal, complemented
5#20295169C214yesmaximal, complemented
6#20295034C211yesmaximal, characteristic, complemented
7#20295030C211yesmaximal, characteristic, complemented
8#20294990C211yesmaximal, characteristic, complemented
9#525×C109C211yesmaximal, characteristic, complemented
10C2×#13182029—24no—
11C2×#13182029—24no—
12C2×#13182029—22no—
13C2×#13182029C2214yes—
14C22×Dic218C2211yescharacteristic
15C22×D436C2211yescharacteristic
16C22×C22⋊D109—21no—
17C22×C4×D109—21no—
18#16812729—24no—
19#16812724—24no—
20C2×#13182019C2212yescomplemented
21C2×#13182017C2212yescomplemented
22C2×#13182016C2212yescomplemented
23C2×#13182014C2212yescomplemented
24C2×#13182013C2211yescharacteristic
25C2×#13182013C2211yescharacteristic
26#16812622C2214yescomplemented
27#16812619C2214yescomplemented
28C22⋊#11724784C2214yescomplemented
29C2×#41×C109C2211yescharacteristic
30C23×C4×C109C2211yescharacteristic
31#123×C109—21no—
32C22⋊Q8×C109C2212yescomplemented
33#121×C109C2212yescomplemented
34#120×C109—21no—
35C4×D4×C109—22no—
36#117×C109C2211yescharacteristic
37C23×D109—21no—
38#13182029—216no—
39#13182029—28no—
40#13182029—28no—
41C2×Dic218—22no—
42C2×Dic218—21no—
43C2×Dic218D414yescomplemented
44C2×Dic218C2312yes—
45C2×D436—22no—
46C2×D436—21no—
47C2×D436D414yescomplemented
48C2×D436C2312yes—
49C2×C22⋊D109—28no—
50C2×C22⋊D109—24no—
51C22×Dic109—21no—
52C2×C4×D109—28no—
53C2×C4×D109—24no—
54#13182019—24no—
55#13182017—24no—
56#13182016—24no—
57#13182014—24no—
58#13182013—22no—
59#13182013—22no—
60C2×#11724784C2312yes—
61C2×#11724784C2312yes—
62C24×C109D411yescharacteristic
63#41×C109—22no—
64#41×C109—22no—
65C2×Q8×C109C2311yescharacteristic
66C2×D4×C109—22no—
67C2×D4×C109—21no—
68C2×D4×C109C2311yescharacteristic
69C22×C4×C109—22no—
70C22×C4×C109—22no—
71C22×C4×C109—21no—
72C22×C4×C109D412yes—
73C22×C4×C109D411yescharacteristic
74C22×C4×C109C2312yes—
75C22×C4×C109C2311yescharacteristic
76C4⋊C4×C109—22no—
77C4⋊C4×C109C2312yes—
78C22⋊C4×C109—21no—
79C22⋊C4×C109—22no—
80C22⋊C4×C109—22no—
81C42×C109—21no—
82C23×C109—22no—
83C23×C109—21no—
84C23×C109—42no—
85C23×C109C2×D412yes—
86C23×C109C2×D411yescharacteristic
87C22×D109—24no—
88C22×D109—22no—
89C22×D109—44no—
90Q8×C109—21no—
91Q8×C109—21no—
92D4×C109—21no—
93D4×C109—21no—
94D4×C109—44no—
95D4×C109—42no—
96C2×C4×C109—22no—
97C2×C4×C109—24no—
98C2×C4×C109—22no—
99C2×C4×C109—22no—
100C2×C4×C109—22no—
101C2×C4×C109—21no—
102C2×C4×C109—42no—
103C2×C4×C109C2×D412yes—
104C2×C4×C109C2×D412yes—
105C2×C4×C109C2×D412yes—
106C2×C4×C109C2×D411yescharacteristic
107C2×C4×C109C2411yescharacteristic
108C22⋊D109—216no—
109C2×Dic109—24no—
110C2×Dic109—22no—
111D436—24no—
112D436—22no—
113D436C2×D414yes—
114C4×D109—216no—
115Dic218—24no—
116Dic218—22no—
117Dic218C2×D414yes—
118#11724784—22no—
119#11724784—22no—
120C22×C109—22no—
121C22×C109—22no—
122C22×C109—22no—
123C22×C109—22no—
124C22×C109—44no—
125C22×C109—41no—
126C22×C109—42no—
127C22×C109#12012yes—
128C22×C109C22×D411yescharacteristic
129C2×D109—24no—
130C2×D109—48no—
131C2×D109—44no—
132C4×C109—22no—
133C4×C109—41no—
134C4×C109—42no—
135C4×C109#12012yes—
136C4×C109C22×D411yescharacteristic
137C4×C109C22×D411yescharacteristic
138Dic109—24no—
139C2×C109—22no—
140C2×C109—42no—
141C2×C109—81no—
142C2×C109C2×#12011yescharacteristic
143C2×C109#57512yes—
144D109—44no—
145#3005—1091nomaximal, Sylow
146C109#300511yescharacteristic, complemented, Sylow
147C22×#41—1091no—
148C2×#137—1091no—
149C2×AGL(1,ℤ/8)—1091no—
150#525D10911yescharacteristic, complemented
151#452—1094no—
152#449—1094no—
153#419—1091no—
154#418—1091no—
155#407—1091no—
156C2×#41—1094no—
157C2×#41—2184no—
158C2×#41—2184no—
159C2×#41—2182no—
160C2×#41C2×D10911yescharacteristic
161C22×Q8—1091no—
162C22×D4—1091no—
163C22×D4—2181no—
164C23×C4—2181no—
165C23×C4C2×D10911yescharacteristic
166#137—2184no—
167AGL(1,ℤ/8)—2184no—
168C2×Q16—1092no—
169C2×SD16—1092no—
170C2×SD16—1092no—
171C2×D8—1092no—
172C2×M16—1091no—
173C2×M16—1091no—
174#123—21no—
175C22⋊Q8C2×D10912yescomplemented
176#121C2×D10912yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D10911yescharacteristic
180#103—1094no—
181#102—1094no—
182C22⋊C8—1094no—
183C24—2181no—
184C24C22⋊D10911yescharacteristic
185#41—22no—
186#41—22no—
187#41—2188no—
188#41—2188no—
189#41—21816no—
190C2×Q8—1094no—
191C2×Q8—1092no—
192C2×Q8—2182no—
193C2×Q8—2181no—
194C2×Q8C22×D10911yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—1094no—
198C2×D4—1092no—
199C2×D4—2188no—
200C2×D4—2184no—
201C2×D4—2182no—
202C2×D4—2181no—
203C2×D4C22×D10911yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—2188no—
208C22×C4—2181no—
209C22×C4—2184no—
210C22×C4C22⋊D10912yes—
211C22×C4C22⋊D10911yescharacteristic
212C22×C4C22×D10912yes—
213C22×C4C22×D10911yescharacteristic
214Q16—2184no—
215SD16—2184no—
216SD16—2184no—
217D8—2184no—
218M16—2182no—
219M16—2182no—
220C2×C8—1092no—
221C2×C8—1092no—
222C4⋊C4—22no—
223C4⋊C4C22×D10912yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—4364no—
232C23—2184no—
233C23—2182no—
234C23C2×C22⋊D10912yes—
235C23C2×C22⋊D10911yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—1094no—
239Q8—2184no—
240Q8—2182no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—1094no—
246D4—21816no—
247D4—2184no—
248D4—2182no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—21816no—
257C2×C4—2184no—
258C2×C4—2182no—
259C2×C4D4×D10912yes—
260C2×C4D4×D10912yes—
261C2×C4C2×C22⋊D10912yes—
262C2×C4C2×C22⋊D10911yescharacteristic
263C2×C4C23×D10911yescharacteristic
264C8—2182no—
265C8—2182no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—4368no—
274C22—4364no—
275C22—2184no—
276C22#1681266712yes—
277C22C22×C22⋊D10911yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—2184no—
282C4#1681266712yes—
283C4C2×D4×D10911yescharacteristic
284C4C2×D4×D10911yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—4364no—
289C2C2×#1681266711yescharacteristic
290C2#2029582412yes—
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#525×C109C2
socleC22×C109C22×D4
hypercenter#525D109
hypocenterC109#3005
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#13182029C2semidirect
2. C2×#16812729C2semidirect
3. C2×#16812724C2semidirect
4. #20295186C2semidirect
5. #20295169C2semidirect
6. #20295034C2semidirect
7. #20295030C2semidirect
8. #20294990C2semidirect
9. #525×C109C2semidirect
20. C2×#13182019C22semidirect
21. C2×#13182017C22semidirect
22. C2×#13182016C22semidirect
23. C2×#13182014C22semidirect
26. #16812622C22semidirect
27. #16812619C22semidirect
28. C22⋊#11724784C22semidirect
32. C22⋊Q8×C109C22semidirect
33. #121×C109C22semidirect
43. C2×Dic218D4semidirect
47. C2×D436D4semidirect
146. C109#3005semidirect
150. #525D109semidirect
175. C22⋊Q8C2×D109semidirect
176. #121C2×D109semidirect

Maximal subgroups

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

subgroupindex
C22×#131820292
C2×#168127292
C2×#168127242
#202951862
#202951692
#202950342
#202950302
#202949902
#525×C1092
#3005109