Finite Group Databasebeta

← #28372824

Subgroups — #28372824

Order 13952.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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#20295329C212yesmaximal, complemented
4#20295328C212yesmaximal, complemented
5#20295317C211yesmaximal, characteristic, complemented
6#20295316C211yesmaximal, characteristic, complemented
7#20295303C211yesmaximal, characteristic, complemented
8#20295067C212yesmaximal, complemented
9#20295048C211yesmaximal, characteristic, complemented
10#20295029C212yesmaximal
11#474×C109C211yesmaximal, characteristic, complemented
12C2×D4×D109C2211yescharacteristic
13C22×C4×D109—21no—
14C2×#13182012C2212yes—
15#16812699C2211yescharacteristic, complemented
16#16812690C2212yescomplemented
17#16812679—21no—
18#16812675—21no—
19Q8×Dic109—21no—
20#16812664—21no—
21C4×D436C2211yescharacteristic
22D4×Dic109C2211yescharacteristic
23#16812661C2211yescharacteristic
24C4×C22⋊D109—21no—
25#16812659—21no—
26#16812657C2212yes—
27C4⋊C4×D109C2211yescharacteristic
28C22⋊C4×D109—21no—
29C42×D109C2211yescharacteristic
30#16812632C2212yescomplemented
31#16812631C2212yescomplemented
32#16812630C2212yescomplemented
33C4⋊#11724784C2211yescharacteristic, complemented
34#16812628C2212yescomplemented
35#16812621C2212yes—
36#16812620C2212yes—
37#16812619C2212yes—
38C4⋊Q8×C109C2211yescharacteristic, complemented
39C4×D4×C109C2211yescharacteristic
40#107×C109C2212yescomplemented
41C4⋊C8×C109C2211yescharacteristic, complemented
42#102×C109C2212yes—
43C23×D109—21no—
44C2×Dic218—21no—
45C2×D436C2311yescharacteristic
46C2×C22⋊D109—21no—
47D4×D109—22no—
48D4×D109Q814yescomplemented
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#13182012—24no—
58C2×#11724784C2312yes—
59C4⋊Dic109—21no—
60C4⋊Dic109—22no—
61C4⋊Dic109C2312yes—
62C4⋊Dic109C2311yescharacteristic
63#13182008—21no—
64#13182008—22no—
65C22⋊Dic109—21no—
66#13182006—22no—
67#13182006—21no—
68#13182006—21no—
69C4×Dic109—22no—
70C4×Dic109D411yescharacteristic
71C4×Dic109D411yescharacteristic
72C4×Dic109C2311yescharacteristic
73C2×Q8×C109—21no—
74C2×D4×C109C2311yescharacteristic
75C22×C4×C109—21no—
76C2×C8×C109C2312yes—
77C4⋊C4×C109—21no—
78C4⋊C4×C109C2312yes—
79C4⋊C4×C109C2311yescharacteristic
80C22⋊C4×C109—21no—
81C42×C109C2311yescharacteristic
82C23×C109—21no—
83C22×D109—21no—
84C22×D109—24no—
85C22×D109—42no—
86C22×D109—42no—
87C22×D109C2×D411yescharacteristic
88Q8×C109—42no—
89D4×C109—21no—
90D4×C109C2×Q812yes—
91C2×C4×C109—21no—
92C2×C4×C109—22no—
93C2×C4×C109—42no—
94C2×C4×C109C2×D411yescharacteristic
95C2×C4×C109C2×D411yescharacteristic
96C2×C4×C109C2411yescharacteristic
97C22⋊D109—24no—
98C2×Dic109—22no—
99C2×Dic109—21no—
100C2×Dic109—42no—
101C2×Dic109C2×D411yescharacteristic
102C2×Dic109C2×D411yescharacteristic
103C2×Dic109C2×D411yescharacteristic
104D436—21no—
105D436C2×Q812yes—
106C4×D109—22no—
107C4×D109—41no—
108C4×D109—42no—
109C4×D109—42no—
110C4×D109—42no—
111C4×D109C2×Q812yes—
112C4×D109#4112yes—
113Dic218—42no—
114C8×C109—22no—
115#11724784—22no—
116C22×C109—22no—
117C22×C109—41no—
118C22×C109—41no—
119C22×C109C22×D411yescharacteristic
120C2×D109—21no—
121C2×D109—21no—
122C2×D109—22no—
123C2×D109—48no—
124C2×D109—41no—
125C2×D109—41no—
126C2×D109C22⋊Q812yes—
127C4×C109—21no—
128C4×C109—41no—
129C4×C109—42no—
130C4×C109AGL(1,ℤ/8)12yes—
131C4×C109C22×Q811yescharacteristic
132C4×C109C2×#4111yescharacteristic
133Dic109—21no—
134Dic109—21no—
135Dic109—41no—
136Dic109—42no—
137Dic109C22⋊Q812yes—
138C2×C109—42no—
139C2×C109C2×C22⋊Q811yescharacteristic
140C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C109#57511yescharacteristic
142D109—22no—
143D109—42no—
144#3077—1091nomaximal, Sylow
145C109#307711yescharacteristic, complemented, Sylow
146#533—1091no—
147C2×C4×D4—1091no—
148#478—1092no—
149#478—1092no—
150#476—1091no—
151#476—1091no—
152#474—1091no—
153#474D10911yescharacteristic, complemented
154#428—1092no—
155#423—1091no—
156#418—1092no—
157C22×D4—1091no—
158C23×C4—2181no—
159C2×M16—1092no—
160C4⋊Q8—1091no—
161C4⋊Q8C2×D10911yescharacteristic, complemented
162#125—1092no—
163C22⋊Q8—2181no—
164C22⋊Q8—2181no—
165C4×Q8—2181no—
166C4×Q8—2181no—
167C4×D4—1091no—
168C4×D4—1091no—
169C4×D4—1091no—
170C4×D4—2181no—
171C4×D4—2181no—
172C4×D4C2×D10911yescharacteristic
173#117—1092no—
174C2×C4⋊C4—1091no—
175C2×C22⋊C4—2181no—
176C2×C42—1091no—
177#107—1092no—
178#107C2×D10912yescomplemented
179#106—1092no—
180#106—1092no—
181C4⋊C8—1092no—
182C4⋊C8—1091no—
183C4⋊C8C2×D10911yescharacteristic, complemented
184#102—1092no—
185#102—1092no—
186#102—1092no—
187#102C2×D10912yes—
188C24—2181no—
189C2×Q8—21no—
190C2×Q8—2181no—
191C2×D4—1094no—
192C2×D4—1091no—
193C2×D4—2182no—
194C2×D4—2181no—
195C2×D4C22×D10911yescharacteristic
196C22×C4—21no—
197C22×C4—1091no—
198C22×C4—1091no—
199C22×C4—1091no—
200C22×C4—4364no—
201C22×C4—2181no—
202C22×C4—2181no—
203C22×C4—2181no—
204C22×C4—2181no—
205M16—2184no—
206C2×C8—1092no—
207C2×C8C22×D10912yes—
208C4⋊C4—21no—
209C4⋊C4—1092no—
210C4⋊C4—1091no—
211C4⋊C4—2181no—
212C4⋊C4—2182no—
213C4⋊C4—2182no—
214C4⋊C4—2181no—
215C4⋊C4C22×D10912yes—
216C4⋊C4C22×D10911yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—2181no—
219C22⋊C4—2182no—
220C22⋊C4—2181no—
221C22⋊C4—2181no—
222C42—1091no—
223C42—1091no—
224C42—1091no—
225C42—2182no—
226C42C22×D10911yescharacteristic
227C23—21no—
228C23—1091no—
229C23—4362no—
230C23—4362no—
231C23—2184no—
232C23—2181no—
233Q8—42no—
234Q8—4362no—
235D4—21no—
236D4—1092no—
237D4—2184no—
238D4—2181no—
239D4Q8×D10912yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—1091no—
244C2×C4—1092no—
245C2×C4—1092no—
246C2×C4—1091no—
247C2×C4—1091no—
248C2×C4—4361no—
249C2×C4—4362no—
250C2×C4—4362no—
251C2×C4—4362no—
252C2×C4—4362no—
253C2×C4—2182no—
254C2×C4—2181no—
255C2×C4—2182no—
256C2×C4D4×D10911yescharacteristic
257C2×C4D4×D10911yescharacteristic
258C2×C4C23×D10911yescharacteristic
259C8—22no—
260C8—2182no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—1092no—
265C22—4368no—
266C22—4361no—
267C22—4361no—
268C22—2182no—
269C22—2181no—
270C22—2181no—
271C22C2×D4×D10911yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—1092no—
276C4—4361no—
277C4—4362no—
278C4—2181no—
279C4—2181no—
280C4#1681272312yes—
281C4C2×Q8×D10911yescharacteristic
282C4#41×D10911yescharacteristic
283C2—42no—
284C2—4362no—
285C2—2182no—
286C2C22⋊Q8×D10911yescharacteristic
287C2C2×#1681272311yescharacteristic
288C2#2029582511yescharacteristic
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#3077
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. #20295329C2semidirect
4. #20295328C2semidirect
5. #20295317C2semidirect
6. #20295316C2semidirect
7. #20295303C2semidirect
8. #20295067C2semidirect
9. #20295048C2semidirect
11. #474×C109C2semidirect
15. #16812699C22semidirect
16. #16812690C22semidirect
30. #16812632C22semidirect
31. #16812631C22semidirect
32. #16812630C22semidirect
33. C4⋊#11724784C22semidirect
34. #16812628C22semidirect
38. C4⋊Q8×C109C22semidirect
40. #107×C109C22semidirect
41. C4⋊C8×C109C22semidirect
48. D4×D109Q8semidirect
145. C109#3077semidirect
153. #474D109semidirect
161. C4⋊Q8C2×D109semidirect
178. #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
#202953292
#202953282
#202953172
#202953162
#202953032
#202950672
#202950482
#202950292
#474×C1092
#3077109