Finite Group Databasebeta

← #38755844

Subgroups — #38755844

Order 25216.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20263345C211yesmaximal, characteristic, complemented
2#135×D197C211yesmaximal, characteristic, complemented
3#27161522C211yesmaximal, characteristic, complemented
4#27161519C211yesmaximal, characteristic, complemented
5#27161518C212yesmaximal, complemented
6#27161515C211yesmaximal, characteristic, complemented
7#27161513C212yesmaximal, complemented
8#27161510C212yesmaximal, complemented
9#27161508C212yesmaximal, complemented
10M32×D197C211yesmaximal, characteristic, complemented
11#509×C197C211yesmaximal, characteristic, complemented
12C2×#15691357—21no—
13#41×D197—21no—
14#20263349C2211yescharacteristic, complemented
15#20263348C2211yescharacteristic
16#20263347—21no—
17#20263346—21no—
18#20263345—21no—
19#20263345—21no—
20#20263345C2212yes—
21#20263345C2211yescharacteristic, complemented
22#20263345C2212yescomplemented
23C2×Dic788C2211yescharacteristic, complemented
24Q16×D197C2211yescharacteristic, complemented
25C2×#15691346—21no—
26SD16×D197—21no—
27D8×D197C2211yescharacteristic, complemented
28C2×C8×D197C2211yescharacteristic
29Dic1576C2212yescomplemented
30#20263272C2212yescomplemented
31#20263271C2212yescomplemented
32#20263270C2212yescomplemented
33#20263270C2212yescomplemented
34#20263267C2212yescomplemented
35#20263266C2211yescharacteristic, complemented
36#20263265C2212yescomplemented
37C16×D197C2212yescomplemented
38#135×C197C2211yescharacteristic
39C2×D8×C197C2211yescharacteristic, complemented
40SD32×C197C2212yescomplemented
41D16×C197C2212yescomplemented
42M32×C197C2211yescharacteristic, complemented
43#15691359—21no—
44#15691358—21no—
45#15691358—41no—
46#15691357—22no—
47#15691357—22no—
48#15691357—21no—
49#15691357—41no—
50#15691357—41no—
51#15691357—41no—
52C2×Dic394—21no—
53Q8×D197—21no—
54C2×C22⋊D197—41no—
55D4×D197—21no—
56D4×D197—41no—
57C22×Dic197—41no—
58C2×C4×D197—41no—
59C2×C4×D197D411yescharacteristic
60Dic788—21no—
61Dic788C2311yescharacteristic
62Dic788C2312yes—
63#15691348—21no—
64#15691347—21no—
65#15691346—21no—
66#15691346—22no—
67#15691346—22no—
68#15691345—21no—
69D1576C2311yescharacteristic
70#15691343—21no—
71C2×#13096802D411yescharacteristic
72C8×D197D411yescharacteristic
73C8×D197D411yescharacteristic
74C8×D197C2311yescharacteristic
75C8×D197C2311yescharacteristic
76#15691333C2312yes—
77#41×C197—21no—
78C2×D4×C197—21no—
79Q16×C197C2311yescharacteristic
80SD16×C197—21no—
81D8×C197—21no—
82D8×C197C2311yescharacteristic
83D8×C197C2312yes—
84C2×C8×C197C2311yescharacteristic
85C16×C197C2312yes—
86C22×D197—41no—
87C22×D197D811yescharacteristic
88Dic394—22no—
89Dic394—21no—
90Dic394—41no—
91Dic394—41no—
92D788—21no—
93D788—41no—
94C22⋊D197—41no—
95C22⋊D197—41no—
96C22⋊D197—42no—
97C22⋊D197—42no—
98C22⋊D197—41no—
99C2×Dic197—41no—
100C2×Dic197—41no—
101C2×Dic197—42no—
102C2×Dic197—42no—
103C2×Dic197D811yescharacteristic
104C4×D197—41no—
105C4×D197—41no—
106C4×D197—41no—
107C4×D197D811yescharacteristic
108C4×D197D811yescharacteristic
109C4×D197C2×D411yescharacteristic
110C4×D197C2×D411yescharacteristic
111#13096802C2×D411yescharacteristic
112#13096802C2×D411yescharacteristic
113C23×C197—41no—
114Q8×C197—21no—
115D4×C197—22no—
116D4×C197—21no—
117D4×C197—41no—
118D4×C197—41no—
119C2×C4×C197—41no—
120C2×C4×C197C2×D411yescharacteristic
121C8×C197C2×D411yescharacteristic
122C8×C197C2411yescharacteristic
123C22×C197—41no—
124C22×C197—42no—
125C22×C197—82no—
126C22×C197C2×D811yescharacteristic
127C2×D197—41no—
128C2×D197—41no—
129C2×D197—82no—
130C2×D197C2×D811yescharacteristic
131C2×D197C2×D811yescharacteristic
132C4×C197—41no—
133C4×C197C2×D811yescharacteristic
134C4×C197C22×D411yescharacteristic
135Dic197—41no—
136Dic197—42no—
137Dic197C2×D811yescharacteristic
138Dic197C2×D811yescharacteristic
139C2×C197—21no—
140C2×C197—81no—
141C2×C197—82no—
142C2×C197C22×D811yescharacteristic
143D197—21no—
144D197—21no—
145D197—81no—
146C197#341411yescharacteristic, complemented, Sylow
147#3414—1971nomaximal, Sylow
148C2×#135—1971no—
149C2×#135—1971no—
150#510—1971no—
151#510—1972no—
152#510—1971no—
153#509—1971no—
154#509—1972no—
155#509D19711yescharacteristic, complemented
156#508—1972no—
157#508—1972no—
158C2×M32—1971no—
159C2×#41—3941no—
160C2×#41—3941no—
161#135—3941no—
162#135—3941no—
163#135—3941no—
164#135—3941no—
165#135—1971no—
166#135—1972no—
167#135—1972no—
168#135—1971no—
169#135—1971no—
170#135C2×D19711yescharacteristic
171C2×Q16—1971no—
172C2×Q16—1971no—
173C2×SD16—3941no—
174C2×SD16—3941no—
175C2×D8—1971no—
176C2×D8C2×D19711yescharacteristic, complemented
177C22×C8—1971no—
178Q32—1972no—
179Q32—1972no—
180SD32—1972no—
181SD32—1972no—
182SD32—1972no—
183SD32C2×D19712yescomplemented
184D16—1972no—
185D16C2×D19712yescomplemented
186M32—1971no—
187M32—1972no—
188M32C2×D19711yescharacteristic, complemented
189C2×C16—1972no—
190#41—21no—
191#41—7881no—
192#41—7881no—
193#41—7881no—
194#41—7881no—
195#41—3942no—
196#41—3942no—
197#41—3941no—
198#41—3941no—
199#41—3941no—
200C2×Q8—3941no—
201C2×Q8—3941no—
202C2×D4—21no—
203C2×D4—7881no—
204C2×D4—7881no—
205C2×D4—3941no—
206C22×C4—7881no—
207C22×C4—7881no—
208C22×C4—1971no—
209Q16—3941no—
210Q16—3941no—
211Q16—1971no—
212Q16—1972no—
213Q16C22×D19711yescharacteristic
214SD16—21no—
215SD16—3942no—
216SD16—3942no—
217SD16—3941no—
218SD16—3941no—
219SD16—3941no—
220D8—21no—
221D8—3941no—
222D8—1971no—
223D8C22×D19712yes—
224D8C22×D19711yescharacteristic
225C2×C8—1971no—
226C2×C8—1971no—
227C2×C8—1971no—
228C2×C8—1971no—
229C2×C8—1971no—
230C2×C8C22×D19711yescharacteristic
231C16—1972no—
232C16C22×D19712yes—
233C23—41no—
234C23—7881no—
235C23—1971no—
236Q8—21no—
237Q8—7881no—
238Q8—7881no—
239Q8—3942no—
240Q8—3941no—
241D4—22no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—7881no—
246D4—7881no—
247D4—7882no—
248D4—7882no—
249D4—7881no—
250D4—7881no—
251D4—3941no—
252C2×C4—41no—
253C2×C4—7881no—
254C2×C4—7881no—
255C2×C4—7881no—
256C2×C4—7881no—
257C2×C4—7882no—
258C2×C4—7882no—
259C2×C4—7881no—
260C2×C4—1971no—
261C2×C4—1971no—
262C2×C4—1971no—
263C2×C4—1971no—
264C2×C4—1971no—
265C2×C4D4×D19711yescharacteristic
266C8—1971no—
267C8—1971no—
268C8D4×D19711yescharacteristic
269C8C23×D19711yescharacteristic
270C22—41no—
271C22—42no—
272C22—82no—
273C22—7881no—
274C22—7881no—
275C22—15762no—
276C22—1971no—
277C22—1971no—
278C22D8×D19711yescharacteristic
279C4—41no—
280C4—7881no—
281C4—7882no—
282C4—1971no—
283C4—1971no—
284C4D8×D19711yescharacteristic
285C4C2×D4×D19711yescharacteristic
286C2—21no—
287C2—82no—
288C2—81no—
289C2—15761no—
290C2—3941no—
291C2—3941no—
292C2C2×D8×D19711yescharacteristic
subgroupG/H
centerC2C2×D8×D197
commutator, cosocleC8×C197C24
FrattiniC8C23×D197
Fitting#509×C197C2
socleC2×C197C22×D8
hypercenter#509D197
hypocenterC197#3414
LCSG ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. C2×#20263345C2semidirect
2. #135×D197C2semidirect
3. #27161522C2semidirect
4. #27161519C2semidirect
5. #27161518C2semidirect
6. #27161515C2semidirect
7. #27161513C2semidirect
8. #27161510C2semidirect
9. #27161508C2semidirect
10. M32×D197C2semidirect
11. #509×C197C2semidirect
14. #20263349C22semidirect
21. #20263345C22semidirect
22. #20263345C22semidirect
23. C2×Dic788C22semidirect
24. Q16×D197C22semidirect
27. D8×D197C22semidirect
29. Dic1576C22semidirect
30. #20263272C22semidirect
31. #20263271C22semidirect
32. #20263270C22semidirect
33. #20263270C22semidirect
34. #20263267C22semidirect
35. #20263266C22semidirect
36. #20263265C22semidirect
37. C16×D197C22semidirect
39. C2×D8×C197C22semidirect
40. SD32×C197C22semidirect
41. D16×C197C22semidirect
42. M32×C197C22semidirect
146. C197#3414semidirect
155. #509D197semidirect
176. C2×D8C2×D197semidirect
183. SD32C2×D197semidirect
185. D16C2×D197semidirect
188. M32C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633452
#135×D1972
#271615222
#271615192
#271615182
#271615152
#271615132
#271615102
#271615082
M32×D1972
#509×C1972
#3414197