Finite Group Databasebeta

← #28373133

Subgroups — #28373133

Order 13952.

444 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#20295581C211yesmaximal, characteristic, complemented
2C4×D4×D109C211yesmaximal, characteristic
3#20295363C212yesmaximal, complemented
4#20295303C211yesmaximal, characteristic, complemented
5#20295232C211yesmaximal, characteristic, complemented
6#20295231C211yesmaximal, characteristic, complemented
7#20295178C212yesmaximal
8#20295125C212yesmaximal, complemented
9#20295123C212yesmaximal, complemented
10#20295082C211yesmaximal, characteristic, complemented
11#474×C109C211yesmaximal, characteristic, complemented
12C2×D4×D109C2211yescharacteristic
13C22×C4×D109—21no—
14C2×#13182018C2212yes—
15C2×#13182017C2212yes—
16#16812699C2211yescharacteristic, complemented
17#16812690—21no—
18#16812679C2212yes—
19#16812675—21no—
20Q8×Dic109—21no—
21#16812664C2212yescomplemented
22C4×D436C2211yescharacteristic
23D4×Dic109C2211yescharacteristic
24#16812661C2211yescharacteristic
25C4×C22⋊D109—21no—
26#16812659—21no—
27#16812657—21no—
28C4⋊C4×D109C2211yescharacteristic
29C22⋊C4×D109—21no—
30C42×D109C2211yescharacteristic
31#16812632C2212yescomplemented
32C4⋊#11724784C2211yescharacteristic, complemented
33#16812624C2212yescomplemented
34#16812622C2212yescomplemented
35#16812619C2212yes—
36#16812611C2212yes—
37#16812609C2212yescomplemented
38C4⋊Q8×C109C2211yescharacteristic, complemented
39C4×D4×C109C2211yescharacteristic
40#107×C109C2212yescomplemented
41C4⋊C8×C109C2211yescharacteristic, complemented
42#102×C109C2212yes—
43C23×D109—21no—
44C2×Dic218C2312yes—
45C2×D436C2311yescharacteristic
46C2×C22⋊D109—21no—
47D4×D109—22no—
48D4×D109—22no—
49C22×Dic109—21no—
50C2×C4×D109—21no—
51C2×C4×D109—21no—
52C2×C4×D109—21no—
53C2×C4×D109—44no—
54C2×C4×D109D411yescharacteristic
55C2×C4×D109D411yescharacteristic
56C2×C4×D109C2311yescharacteristic
57#13182018—24no—
58#13182017—24no—
59C2×#11724784C2312yes—
60C4⋊Dic109—21no—
61C4⋊Dic109—21no—
62C4⋊Dic109—22no—
63C4⋊Dic109C2311yescharacteristic
64#13182008—21no—
65#13182008—22no—
66C22⋊Dic109—21no—
67#13182006—22no—
68#13182006—21no—
69#13182006—21no—
70C4×Dic109—22no—
71C4×Dic109D411yescharacteristic
72C4×Dic109D411yescharacteristic
73C4×Dic109C2311yescharacteristic
74C2×Q8×C109—21no—
75C2×D4×C109C2311yescharacteristic
76C22×C4×C109—21no—
77C2×C8×C109C2312yes—
78C4⋊C4×C109—21no—
79C4⋊C4×C109C2312yes—
80C4⋊C4×C109C2311yescharacteristic
81C22⋊C4×C109—21no—
82C42×C109C2311yescharacteristic
83C23×C109—21no—
84C22×D109—24no—
85C22×D109—21no—
86C22×D109—42no—
87C22×D109—42no—
88C22×D109C2×D411yescharacteristic
89Q8×C109—42no—
90D4×C109—21no—
91D4×C109#4112yes—
92C2×C4×C109—21no—
93C2×C4×C109—22no—
94C2×C4×C109—42no—
95C2×C4×C109C2×D411yescharacteristic
96C2×C4×C109C2×D411yescharacteristic
97C2×C4×C109C2411yescharacteristic
98C22⋊D109—24no—
99C2×Dic109—22no—
100C2×Dic109—21no—
101C2×Dic109—42no—
102C2×Dic109C2×D411yescharacteristic
103C2×Dic109C2×D411yescharacteristic
104C2×Dic109C2×D411yescharacteristic
105D436—21no—
106D436#4112yes—
107C4×D109—22no—
108C4×D109—22no—
109C4×D109—21no—
110C4×D109—21no—
111C4×D109—42no—
112C4×D109—42no—
113C4×D109—42no—
114Dic218—22no—
115Dic218—22no—
116C8×C109—22no—
117#11724784—22no—
118C22×C109—22no—
119C22×C109—41no—
120C22×C109—41no—
121C22×C109C22×D411yescharacteristic
122C2×D109—21no—
123C2×D109—21no—
124C2×D109—22no—
125C2×D109—48no—
126C2×D109—41no—
127C2×D109—41no—
128C2×D109#13712yes—
129C4×C109—21no—
130C4×C109—41no—
131C4×C109—42no—
132C4×C109AGL(1,ℤ/8)12yes—
133C4×C109C2×#4111yescharacteristic
134C4×C109C2×#4111yescharacteristic
135Dic109—21no—
136Dic109—21no—
137Dic109—21no—
138Dic109—42no—
139Dic109—41no—
140C2×C109—42no—
141C2×C109#52511yescharacteristic
142C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C109C2×#13711yescharacteristic
144D109—22no—
145D109—42no—
146#3110—1091nomaximal, Sylow
147C109#311011yescharacteristic, complemented, Sylow
148#533—1091no—
149C2×C4×D4—1091no—
150#484—1092no—
151#474—1091no—
152#474D10911yescharacteristic, complemented
153#461—1091no—
154#461—1091no—
155#450—1092no—
156#440—1092no—
157#440—1092no—
158#432—1091no—
159C22×D4—1091no—
160C23×C4—2181no—
161C2×SD16—1092no—
162C2×SD16—1092no—
163C4⋊Q8—1091no—
164C4⋊Q8C2×D10911yescharacteristic, complemented
165#125—2181no—
166C22⋊Q8—1092no—
167C22⋊Q8—2181no—
168C4×Q8—1092no—
169C4×Q8—2181no—
170C4×D4—1091no—
171C4×D4—1091no—
172C4×D4—1091no—
173C4×D4—2181no—
174C4×D4—2181no—
175C4×D4C2×D10911yescharacteristic
176#117—2181no—
177C2×C4⋊C4—1091no—
178C2×C22⋊C4—2181no—
179C2×C42—1091no—
180#107—1092no—
181#107C2×D10912yescomplemented
182C4⋊C8—1091no—
183C4⋊C8C2×D10911yescharacteristic, complemented
184#103—1092no—
185#103—1092no—
186#102—1092no—
187#102C2×D10912yes—
188C22⋊C8—1092no—
189#97—1092no—
190C24—2181no—
191C2×Q8—21no—
192C2×Q8—1092no—
193C2×D4—1091no—
194C2×D4—2181no—
195C2×D4—2182no—
196C2×D4—2182no—
197C2×D4C22×D10911yescharacteristic
198C22×C4—21no—
199C22×C4—1091no—
200C22×C4—1091no—
201C22×C4—1091no—
202C22×C4—4364no—
203C22×C4—2181no—
204C22×C4—2181no—
205C22×C4—2181no—
206C22×C4—2181no—
207SD16—2184no—
208SD16—2184no—
209C2×C8—1092no—
210C2×C8C22×D10912yes—
211C4⋊C4—21no—
212C4⋊C4—1091no—
213C4⋊C4—2181no—
214C4⋊C4—2182no—
215C4⋊C4—2182no—
216C4⋊C4—2181no—
217C4⋊C4—2181no—
218C4⋊C4C22×D10912yes—
219C4⋊C4C22×D10911yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—2181no—
222C22⋊C4—2182no—
223C22⋊C4—2181no—
224C22⋊C4—2181no—
225C42—1091no—
226C42—1091no—
227C42—1091no—
228C42—2182no—
229C42C22×D10911yescharacteristic
230C23—21no—
231C23—1091no—
232C23—4362no—
233C23—4362no—
234C23—2184no—
235C23—2181no—
236Q8—42no—
237Q8—2182no—
238Q8—2182no—
239D4—21no—
240D4—1092no—
241D4—2184no—
242D4—2181no—
243D4#1318203012yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—1091no—
248C2×C4—1091no—
249C2×C4—1091no—
250C2×C4—4362no—
251C2×C4—4362no—
252C2×C4—4362no—
253C2×C4—4362no—
254C2×C4—2182no—
255C2×C4—2182no—
256C2×C4—2181no—
257C2×C4—2182no—
258C2×C4—2181no—
259C2×C4—2181no—
260C2×C4D4×D10911yescharacteristic
261C2×C4D4×D10911yescharacteristic
262C2×C4C23×D10911yescharacteristic
263C8—22no—
264C8—2182no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—1092no—
269C22—4368no—
270C22—4361no—
271C22—4361no—
272C22—2182no—
273C22—2181no—
274C22—2181no—
275C22C2×D4×D10911yescharacteristic
276C4—21no—
277C4—41no—
278C4—42no—
279C4—4362no—
280C4—4361no—
281C4—2181no—
282C4—2181no—
283C4—2181no—
284C4#1681272312yes—
285C4#41×D10911yescharacteristic
286C4C2×#1318203011yescharacteristic
287C2—42no—
288C2—4362no—
289C2—2182no—
290C2#2029553911yescharacteristic
291C2C2×#1681272311yescharacteristic
292C2#137×D10911yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#474×C109C2
socleC22×C109C22×D4
prime radicalC2×D4×D109C22
hypercenter#474D109
hypocenterC109#3110
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20295581C2semidirect
3. #20295363C2semidirect
4. #20295303C2semidirect
5. #20295232C2semidirect
6. #20295231C2semidirect
8. #20295125C2semidirect
9. #20295123C2semidirect
10. #20295082C2semidirect
11. #474×C109C2semidirect
16. #16812699C22semidirect
21. #16812664C22semidirect
31. #16812632C22semidirect
32. C4⋊#11724784C22semidirect
33. #16812624C22semidirect
34. #16812622C22semidirect
37. #16812609C22semidirect
38. C4⋊Q8×C109C22semidirect
40. #107×C109C22semidirect
41. C4⋊C8×C109C22semidirect
147. C109#3110semidirect
152. #474D109semidirect
164. C4⋊Q8C2×D109semidirect
181. #107C2×D109semidirect
183. C4⋊C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955812
C4×D4×D1092
#202953632
#202953032
#202952322
#202952312
#202951782
#202951252
#202951232
#202950822
#474×C1092
#3110109