Finite Group Databasebeta

← #38755847

Subgroups — #38755847

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×#20263349C211yesmaximal, characteristic, complemented
2#135×D197C211yesmaximal, characteristic, complemented
3#27161521C212yesmaximal, complemented
4#27161520C211yesmaximal, characteristic, complemented
5#27161517C211yesmaximal, characteristic, complemented
6#27161516C212yesmaximal, complemented
7#27161514C211yesmaximal, characteristic, complemented
8#27161512C212yesmaximal, complemented
9#27161510C212yesmaximal, complemented
10M32×D197C211yesmaximal, characteristic, complemented
11#510×C197C211yesmaximal, characteristic, complemented
12C2×#15691359—21no—
13#41×D197—21no—
14#20263349—21no—
15#20263349—21no—
16#20263349C2212yes—
17#20263349C2211yescharacteristic, complemented
18#20263349C2212yescomplemented
19#20263348C2211yescharacteristic
20#20263347—21no—
21#20263346—21no—
22#20263345C2211yescharacteristic, complemented
23Q16×D197C2211yescharacteristic, complemented
24C2×#15691345—21no—
25SD16×D197—21no—
26C2×D1576C2211yescharacteristic, complemented
27D8×D197C2211yescharacteristic, complemented
28C2×C8×D197C2211yescharacteristic
29#20263272C2212yescomplemented
30#20263271C2212yescomplemented
31#20263269C2212yescomplemented
32#20263269C2212yescomplemented
33D3152C2212yescomplemented
34#20263267C2212yescomplemented
35#20263266C2211yescharacteristic, complemented
36#20263265C2212yescomplemented
37C16×D197C2212yescomplemented
38#135×C197C2211yescharacteristic
39C2×Q16×C197C2211yescharacteristic, complemented
40Q32×C197C2212yescomplemented
41SD32×C197C2212yescomplemented
42M32×C197C2211yescharacteristic, complemented
43#15691359—21no—
44#15691359—22no—
45#15691359—22no—
46#15691359—41no—
47#15691359—41no—
48#15691358—21no—
49#15691358—41no—
50#15691357—21no—
51#15691357—41no—
52Q8×D197—21no—
53C2×D788—21no—
54C2×D788—41no—
55D4×D197—21no—
56D4×D197—41no—
57C2×C4×D197—41no—
58C2×C4×D197—41no—
59C2×C4×D197D411yescharacteristic
60Dic788C2311yescharacteristic
61#15691348—21no—
62#15691347—21no—
63#15691346—21no—
64#15691345—21no—
65#15691345—22no—
66#15691345—22no—
67D1576—21no—
68D1576C2312yes—
69D1576C2311yescharacteristic
70#15691343—21no—
71C2×#13096802D411yescharacteristic
72C8×D197D411yescharacteristic
73C8×D197D411yescharacteristic
74C8×D197C2311yescharacteristic
75C8×D197C2311yescharacteristic
76#15691333C2312yes—
77#41×C197—21no—
78C2×Q8×C197—21no—
79Q16×C197—21no—
80Q16×C197C2312yes—
81Q16×C197C2311yescharacteristic
82SD16×C197—21no—
83D8×C197C2311yescharacteristic
84C2×C8×C197C2311yescharacteristic
85C16×C197C2312yes—
86C22×D197—41no—
87C22×D197—41no—
88C22×D197D811yescharacteristic
89Dic394—21no—
90Dic394—41no—
91D788—21no—
92D788—22no—
93D788—41no—
94D788—42no—
95D788—42no—
96D788—41no—
97C22⋊D197—41no—
98C22⋊D197—41no—
99C22⋊D197—41no—
100C2×Dic197—41no—
101C2×Dic197D811yescharacteristic
102C4×D197—41no—
103C4×D197—41no—
104C4×D197—41no—
105C4×D197—42no—
106C4×D197—42no—
107C4×D197D811yescharacteristic
108C4×D197D811yescharacteristic
109C4×D197C2×D411yescharacteristic
110C4×D197C2×D411yescharacteristic
111#13096802C2×D411yescharacteristic
112#13096802C2×D411yescharacteristic
113Q8×C197—21no—
114Q8×C197—22no—
115Q8×C197—41no—
116D4×C197—21no—
117D4×C197—41no—
118C2×C4×C197—41no—
119C2×C4×C197—41no—
120C2×C4×C197C2×D411yescharacteristic
121C8×C197C2×D411yescharacteristic
122C8×C197C2411yescharacteristic
123C22×C197—41no—
124C22×C197C2×D811yescharacteristic
125C2×D197—41no—
126C2×D197—41no—
127C2×D197—42no—
128C2×D197—82no—
129C2×D197—82no—
130C2×D197C2×D811yescharacteristic
131C2×D197C2×D811yescharacteristic
132C4×C197—41no—
133C4×C197—42no—
134C4×C197C2×D811yescharacteristic
135C4×C197C22×D411yescharacteristic
136Dic197—41no—
137Dic197C2×D811yescharacteristic
138Dic197C2×D811yescharacteristic
139C2×C197—21no—
140C2×C197—81no—
141C2×C197C22×D811yescharacteristic
142D197—21no—
143D197—21no—
144D197—81no—
145D197—82no—
146C197#341411yescharacteristic, complemented, Sylow
147#3414—1971nomaximal, Sylow
148C2×#135—1971no—
149C2×#135—1971no—
150#510—1971no—
151#510—1972no—
152#510D19711yescharacteristic, complemented
153#509—1971no—
154#509—1972no—
155#509—1971no—
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—1972no—
166#135—1971no—
167#135—1972no—
168#135—1971no—
169#135—1971no—
170#135C2×D19711yescharacteristic
171C2×Q16—1971no—
172C2×Q16C2×D19711yescharacteristic, complemented
173C2×SD16—3941no—
174C2×SD16—3941no—
175C2×D8—1971no—
176C2×D8—1971no—
177C22×C8—1971no—
178Q32—1972no—
179Q32C2×D19712yescomplemented
180SD32—1972no—
181SD32—1972no—
182SD32—1972no—
183SD32C2×D19712yescomplemented
184D16—1972no—
185D16—1972no—
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—3941no—
196#41—3942no—
197#41—3941no—
198#41—3942no—
199#41—3941no—
200C2×Q8—21no—
201C2×Q8—3941no—
202C2×D4—7881no—
203C2×D4—7881no—
204C2×D4—3941no—
205C2×D4—3941no—
206C22×C4—7881no—
207C22×C4—7881no—
208C22×C4—1971no—
209Q16—21no—
210Q16—3941no—
211Q16—1971no—
212Q16C22×D19711yescharacteristic
213Q16C22×D19712yes—
214SD16—21no—
215SD16—3941no—
216SD16—3942no—
217SD16—3941no—
218SD16—3942no—
219SD16—3941no—
220D8—3941no—
221D8—3941no—
222D8—1972no—
223D8—1971no—
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—7881no—
234C23—7881no—
235C23—1971no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—7881no—
240Q8—3941no—
241D4—21no—
242D4—41no—
243D4—7881no—
244D4—7882no—
245D4—7882no—
246D4—7881no—
247D4—7881no—
248D4—7881no—
249D4—7881no—
250D4—3941no—
251D4—3942no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—7881no—
255C2×C4—7881no—
256C2×C4—7882no—
257C2×C4—7882no—
258C2×C4—7881no—
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—7881no—
272C22—7882no—
273C22—7881no—
274C22—15762no—
275C22—15762no—
276C22—1971no—
277C22—1971no—
278C22D8×D19711yescharacteristic
279C4—41no—
280C4—42no—
281C4—7881no—
282C4—1971no—
283C4—1971no—
284C4D8×D19711yescharacteristic
285C4C2×D4×D19711yescharacteristic
286C2—21no—
287C2—81no—
288C2—15762no—
289C2—15761no—
290C2—3941no—
291C2—3941no—
292C2C2×D8×D19711yescharacteristic
subgroupG/H
centerC2C2×D8×D197
commutator, cosocleC8×C197C24
FrattiniC8C23×D197
Fitting#510×C197C2
socleC2×C197C22×D8
hypercenter#510D197
hypocenterC197#3414
LCSG ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#20263349C2semidirect
2. #135×D197C2semidirect
3. #27161521C2semidirect
4. #27161520C2semidirect
5. #27161517C2semidirect
6. #27161516C2semidirect
7. #27161514C2semidirect
8. #27161512C2semidirect
9. #27161510C2semidirect
10. M32×D197C2semidirect
11. #510×C197C2semidirect
17. #20263349C22semidirect
18. #20263349C22semidirect
22. #20263345C22semidirect
23. Q16×D197C22semidirect
26. C2×D1576C22semidirect
27. D8×D197C22semidirect
29. #20263272C22semidirect
30. #20263271C22semidirect
31. #20263269C22semidirect
32. #20263269C22semidirect
33. D3152C22semidirect
34. #20263267C22semidirect
35. #20263266C22semidirect
36. #20263265C22semidirect
37. C16×D197C22semidirect
39. C2×Q16×C197C22semidirect
40. Q32×C197C22semidirect
41. SD32×C197C22semidirect
42. M32×C197C22semidirect
146. C197#3414semidirect
152. #510D197semidirect
172. C2×Q16C2×D197semidirect
179. Q32C2×D197semidirect
183. SD32C2×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×#202633492
#135×D1972
#271615212
#271615202
#271615172
#271615162
#271615142
#271615122
#271615102
M32×D1972
#510×C1972
#3414197