Finite Group Databasebeta

← #38752508

Subgroups — #38752508

Order 25216.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#27161632C211yesmaximal, characteristic, complemented
2C2×#20263288C211yesmaximal, characteristic
3#27161382C212yesmaximal, complemented
4#27161378C212yesmaximal, complemented
5#27161368C212yesmaximal, complemented
6#27161358C212yesmaximal, complemented
7#27161119C211yesmaximal, characteristic, complemented
8#27161102C211yesmaximal, characteristic, complemented
9#27161085C211yesmaximal, characteristic
10#27161083C211yesmaximal, characteristic
11#428×C197C211yesmaximal, characteristic, complemented
12C22×D788C2211yescharacteristic
13C22×C4×D197—21no—
14C2×#15691342C2211yescharacteristic
15#20263325C2211yescharacteristic, complemented
16#20263325C2211yescharacteristic, complemented
17#20263317C2211yescharacteristic, complemented
18#20263317C2211yescharacteristic, complemented
19#20263303—21no—
20#20263303—21no—
21C4×Dic394—21no—
22C4×Dic394—21no—
23#20263288—21no—
24#20263288—21no—
25#20263288C2212yes—
26#20263288C2212yes—
27#20263285C2211yescharacteristic
28C2×#15691335—21no—
29C2×C4×Dic197C2211yescharacteristic
30#20263259C2212yescomplemented
31#20263257C2212yescomplemented
32#20263255C2212yescomplemented
33#20263255C2212yescomplemented
34#20263248C2212yes—
35#20263248C2212yes—
36#20263246C2212yes—
37#20263246C2212yes—
38C2×M16×C197C2211yescharacteristic
39#117×C197C2211yescharacteristic
40C2×C4⋊C4×C197C2211yescharacteristic
41#107×C197C2212yescomplemented
42#106×C197C2212yescomplemented
43C23×D197—21no—
44C2×Dic394—21no—
45C2×Dic394—21no—
46C2×D788—21no—
47C2×D788—22no—
48C2×D788Q814yescomplemented
49C2×D788C2312yes—
50C22×Dic197D411yescharacteristic
51C22×Dic197D411yescharacteristic
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197—44no—
55#15691342—22no—
56C2×#13096802C2312yes—
57C4⋊Dic197—21no—
58C4⋊Dic197—21no—
59C4⋊Dic197C2311yescharacteristic
60C4⋊Dic197C2311yescharacteristic
61#15691337—21no—
62#15691337—21no—
63#15691337—21no—
64#15691337—21no—
65C22⋊Dic197—21no—
66#15691335—22no—
67#15691335—22no—
68C4×Dic197—21no—
69C4×Dic197D411yescharacteristic
70C4×Dic197D411yescharacteristic
71C4×Dic197C2311yescharacteristic
72C4×Dic197C2311yescharacteristic
73C22×C4×C197—21no—
74C22×C4×C197C2311yescharacteristic
75M16×C197—22no—
76C2×C8×C197C2312yes—
77C4⋊C4×C197—21no—
78C4⋊C4×C197C2311yescharacteristic
79C4⋊C4×C197C2312yes—
80C4⋊C4×C197C2311yescharacteristic
81C22⋊C4×C197—21no—
82C42×C197—21no—
83C22×D197—24no—
84C22×D197—22no—
85C22×D197—42no—
86C22×D197—42no—
87Dic394—42no—
88Dic394—42no—
89D788—24no—
90D788—22no—
91D788C2×Q814yes—
92C2×Dic197—22no—
93C2×Dic197—21no—
94C2×Dic197—21no—
95C2×Dic197—41no—
96C2×Dic197C2×D411yescharacteristic
97C2×Dic197C2×D411yescharacteristic
98C2×Dic197C2×D411yescharacteristic
99C2×Dic197C2×D411yescharacteristic
100C4×D197—44no—
101C4×D197—44no—
102#13096802—22no—
103C23×C197C2×D411yescharacteristic
104C2×C4×C197—21no—
105C2×C4×C197—21no—
106C2×C4×C197—22no—
107C2×C4×C197—42no—
108C2×C4×C197C2×D411yescharacteristic
109C2×C4×C197C2×Q812yes—
110C2×C4×C197#4112yes—
111C2×C4×C197C2411yescharacteristic
112C8×C197—22no—
113C22×C197—21no—
114C22×C197—21no—
115C22×C197C22⋊Q812yes—
116C22×C197C22×D411yescharacteristic
117C2×D197—24no—
118C2×D197—48no—
119C2×D197—42no—
120C2×D197—42no—
121C4×C197—42no—
122C4×C197—41no—
123C4×C197—41no—
124C4×C197C22⋊Q812yes—
125C4×C197C22×Q811yescharacteristic
126C4×C197C2×#4111yescharacteristic
127Dic197—21no—
128Dic197—21no—
129Dic197—21no—
130Dic197—41no—
131Dic197—41no—
132Dic197AGL(1,ℤ/8)12yes—
133C2×C197—22no—
134C2×C197C2×C22⋊Q811yescharacteristic
135C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C197#57511yescharacteristic
137D197—44no—
138C197#307711yescharacteristic, complemented, Sylow
139#3077—1971nomaximal, Sylow
140#533—1971no—
141C2×C4×D4—1971no—
142#478—1972no—
143#478—1972no—
144#476—1972no—
145#474—1972no—
146#428—1971no—
147#428D19711yescharacteristic, complemented
148#423—1971no—
149#418—1971no—
150#418—1971no—
151C22×D4—1971no—
152C23×C4—3941no—
153C2×M16—1971no—
154C2×M16C2×D19711yescharacteristic
155C4⋊Q8—1971no—
156C4⋊Q8—1971no—
157#125—1971no—
158#125—1971no—
159C22⋊Q8—3941no—
160C22⋊Q8—3941no—
161C4×Q8—3941no—
162C4×Q8—3941no—
163C4×D4—3941no—
164C4×D4—3941no—
165C4×D4—1972no—
166C4×D4—1972no—
167#117—1971no—
168#117C2×D19711yescharacteristic
169C2×C4⋊C4C2×D19711yescharacteristic
170C2×C22⋊C4—3941no—
171C2×C42—1971no—
172#107—1972no—
173#107C2×D19712yescomplemented
174#106—1972no—
175#106C2×D19712yescomplemented
176C4⋊C8—1972no—
177C4⋊C8—1972no—
178#102—1972no—
179#102—1972no—
180#102—1972no—
181#102—1972no—
182C24—3941no—
183C2×Q8—3941no—
184C2×Q8—3941no—
185C2×D4—3941no—
186C2×D4—3942no—
187C2×D4—1974no—
188C2×D4—1972no—
189C22×C4—21no—
190C22×C4—7884no—
191C22×C4—3942no—
192C22×C4—3942no—
193C22×C4—1971no—
194C22×C4—1971no—
195C22×C4C22×D19711yescharacteristic
196M16—22no—
197M16—3942no—
198C2×C8—1972no—
199C2×C8C22×D19712yes—
200C4⋊C4—21no—
201C4⋊C4—3941no—
202C4⋊C4—3941no—
203C4⋊C4—3941no—
204C4⋊C4—3941no—
205C4⋊C4—3941no—
206C4⋊C4—3941no—
207C4⋊C4—1971no—
208C4⋊C4—1971no—
209C4⋊C4C22×D19711yescharacteristic
210C4⋊C4C22×D19711yescharacteristic
211C4⋊C4C22×D19712yes—
212C22⋊C4—21no—
213C22⋊C4—3942no—
214C22⋊C4—3942no—
215C22⋊C4—3941no—
216C42—21no—
217C42—3941no—
218C42—1971no—
219C42—1971no—
220C42—1971no—
221C42—1971no—
222C23—7882no—
223C23—7882no—
224C23—3944no—
225C23—3942no—
226C23D4×D19711yescharacteristic
227Q8—7882no—
228Q8—7882no—
229D4—3944no—
230D4—3942no—
231D4—1974no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—7884no—
237C2×C4—7884no—
238C2×C4—7881no—
239C2×C4—3941no—
240C2×C4—3941no—
241C2×C4—3942no—
242C2×C4—1971no—
243C2×C4—1971no—
244C2×C4—1971no—
245C2×C4—1971no—
246C2×C4D4×D19711yescharacteristic
247C2×C4Q8×D19712yes—
248C2×C4#1569135812yes—
249C2×C4C23×D19711yescharacteristic
250C8—22no—
251C8—3942no—
252C22—21no—
253C22—21no—
254C22—7888no—
255C22—7882no—
256C22—7882no—
257C22—3944no—
258C22#2026330412yes—
259C22C2×D4×D19711yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—7881no—
264C4—7881no—
265C4—3941no—
266C4—3941no—
267C4—3941no—
268C4—1972no—
269C4#2026330412yes—
270C4C2×Q8×D19711yescharacteristic
271C4C2×#1569135811yescharacteristic
272C2—22no—
273C2—7884no—
274C2C2×#2026330411yescharacteristic
275C2AGL(1,ℤ/8)×D19711yescharacteristic
276C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#428×C197C2
socleC22×C197C22×D4
prime radicalC22×D788C22
hypercenter#428D197
hypocenterC197#3077
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #27161632C2semidirect
3. #27161382C2semidirect
4. #27161378C2semidirect
5. #27161368C2semidirect
6. #27161358C2semidirect
7. #27161119C2semidirect
8. #27161102C2semidirect
11. #428×C197C2semidirect
15. #20263325C22semidirect
16. #20263325C22semidirect
17. #20263317C22semidirect
18. #20263317C22semidirect
30. #20263259C22semidirect
31. #20263257C22semidirect
32. #20263255C22semidirect
33. #20263255C22semidirect
41. #107×C197C22semidirect
42. #106×C197C22semidirect
48. C2×D788Q8semidirect
138. C197#3077semidirect
147. #428D197semidirect
173. #107C2×D197semidirect
175. #106C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616322
C2×#202632882
#271613822
#271613782
#271613682
#271613582
#271611192
#271611022
#271610852
#271610832
#428×C1972
#3077197