Finite Group Databasebeta

← #497×D97 #27152172

Subgroups — #497×D97 #27152172

Order 12416.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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×Q16×D97C212yesmaximal, complemented
2C2×SD16×D97C212yesmaximal
3C4⋊Q8×D97C211yesmaximal, characteristic, complemented
4#124×D97C211yesmaximal, characteristic
5#20261130C211yesmaximal, characteristic, complemented
6#20261129C211yesmaximal, characteristic, complemented
7#20261128C212yesmaximal, complemented
8#20261127C211yesmaximal, characteristic, complemented
9#20261126C212yesmaximal, complemented
10#97×D97C211yesmaximal, characteristic, complemented
11#497×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97C2211yescharacteristic
13C2×Q8×D97C2212yes—
14C2×D4×D97C2211yescharacteristic
15C2×Dic388C2212yescomplemented
16C2×#13095501C2212yescomplemented
17C2×#13095501C2212yescomplemented
18Q16×D97—28no—
19C2×#13095500C2212yes—
20C2×#13095499C2212yes—
21C2×#13095498C2212yes—
22SD16×D97—28no—
23C2×C8×D97C2212yes—
24#15690810C2211yescharacteristic, complemented
25#15690809—22no—
26#15690808C2212yescomplemented
27#15690799C2211yescharacteristic
28#15690798C2211yescharacteristic
29#15690797C2211yescharacteristic
30#15690796—22no—
31C4⋊C4×D97—22no—
32C22⋊C4×D97—22no—
33C42×D97C2211yescharacteristic
34#15690721C2211yescharacteristic, complemented
35#15690720C2212yescomplemented
36C2×Q16×C97C2212yescomplemented
37C2×SD16×C97C2212yes—
38C4⋊Q8×C97C2211yescharacteristic, complemented
39#124×C97C2211yescharacteristic
40#97×C97C2211yescharacteristic, complemented
41C23×D97—21no—
42C2×Dic194—21no—
43C2×Dic194—22no—
44C2×Dic194C2311yescharacteristic
45C2×Dic194C2312yes—
46Q8×D97—24no—
47Q8×D97—24no—
48Q8×D97—24no—
49C2×D388C2311yescharacteristic
50C2×C22⋊D97—21no—
51D4×D97—24no—
52C2×C4×D97—21no—
53C2×C4×D97—22no—
54C2×C4×D97D412yes—
55C2×C4×D97C2311yescharacteristic
56Dic388—24no—
57#13095501—24no—
58#13095501—24no—
59#13095500—24no—
60#13095499—24no—
61#13095498—24no—
62C2×#11715895C2312yes—
63C8×D97D418yes—
64C4⋊Dic97—22no—
65#13095490—24no—
66C22⋊Dic97—22no—
67#13095488—22no—
68#13095488—22no—
69C4×Dic97D412yes—
70C4×Dic97C2311yescharacteristic
71C2×Q8×C97C2311yescharacteristic
72C2×Q8×C97C2312yes—
73C2×D4×C97C2311yescharacteristic
74Q16×C97—24no—
75SD16×C97—24no—
76C2×C8×C97C2312yes—
77C4⋊C4×C97—22no—
78C22⋊C4×C97—22no—
79C42×C97C2311yescharacteristic
80C23×C97—21no—
81C22×D97—21no—
82C22×D97—42no—
83C22×D97—44no—
84C22×D97C2×D411yescharacteristic
85Q8×C97—22no—
86Q8×C97—22no—
87Q8×C97—22no—
88D4×C97—22no—
89C2×C4×C97—21no—
90C2×C4×C97—22no—
91C2×C4×C97C2×D412yes—
92C2×C4×C97C2411yescharacteristic
93C22⋊D97—42no—
94C2×Dic97—21no—
95C2×Dic97—22no—
96C2×Dic97C2×D412yes—
97C2×Dic97C2×D411yescharacteristic
98D388—22no—
99C4×D97—24no—
100C4×D97—42no—
101C4×D97—44no—
102C4×D97C2×D414yes—
103Dic194—22no—
104Dic194—22no—
105Dic194—22no—
106Dic194—44no—
107Dic194—42no—
108C8×C97C2×D414yes—
109#11715895C2×D414yes—
110C22×C97—41no—
111C22×C97—42no—
112C22×C97C22×D411yescharacteristic
113C2×D97—41no—
114C2×D97—42no—
115C2×D97—84no—
116C2×D97#12712yes—
117C2×D97#13714yes—
118C4×C97—22no—
119C4×C97—41no—
120C4×C97—42no—
121C4×C97C22×D412yes—
122Dic97—22no—
123Dic97—42no—
124Dic97—41no—
125Dic97#12712yes—
126C2×C97—81no—
127C2×C97C2×#12711yescharacteristic
128C2×C97C2×#13712yes—
129D97—81no—
130D97#49714yesdirect product factor
131C2×#497—971nomaximal, Sylow
132C97C2×#49711yescharacteristic, complemented, Sylow
133C22×Q16—972no—
134C22×SD16—972no—
135C2×C4⋊Q8—971no—
136C2×#124—971no—
137#497—972no—
138#497—972no—
139#497—971no—
140#497—971no—
141#497—971no—
142#497D9711yescharacteristic, direct product factor
143C2×#97—971no—
144C22×Q8—971no—
145C22×Q8—972no—
146C22×D4—971no—
147C2×Q16—972no—
148C2×Q16—972no—
149C2×Q16—972no—
150C2×Q16—1948no—
151C2×Q16C2×D9712yescomplemented
152C2×SD16—972no—
153C2×SD16—972no—
154C2×SD16—972no—
155C2×SD16—1948no—
156C2×SD16C2×D9712yes—
157C22×C8—972no—
158C4⋊Q8—972no—
159C4⋊Q8—971no—
160C4⋊Q8—1942no—
161C4⋊Q8C2×D9711yescharacteristic, complemented
162#124—971no—
163#124—971no—
164#124—971no—
165#124—1942no—
166#124C2×D9711yescharacteristic
167C2×C4⋊C4—1942no—
168C2×C22⋊C4—1942no—
169C2×C42—971no—
170#97—972no—
171#97—971no—
172#97C2×D9711yescharacteristic, complemented
173C24—1941no—
174C2×Q8—971no—
175C2×Q8—972no—
176C2×Q8—1941no—
177C2×Q8—1942no—
178C2×Q8—1944no—
179C2×Q8—1944no—
180C2×Q8—1944no—
181C2×Q8C22×D9711yescharacteristic
182C2×Q8C22×D9712yes—
183C2×D4—971no—
184C2×D4—1944no—
185C2×D4—1941no—
186C2×D4C22×D9711yescharacteristic
187C22×C4—971no—
188C22×C4—972no—
189C22×C4—1941no—
190C22×C4—1942no—
191Q16—24no—
192Q16—1944no—
193Q16—1944no—
194Q16—1944no—
195SD16—24no—
196SD16—1944no—
197SD16—1944no—
198SD16—1944no—
199C2×C8—972no—
200C2×C8—978no—
201C2×C8C22×D9712yes—
202C4⋊C4—22no—
203C4⋊C4—1944no—
204C4⋊C4—1942no—
205C22⋊C4—22no—
206C22⋊C4—1942no—
207C22⋊C4—1942no—
208C22⋊C4—1942no—
209C42—972no—
210C42—971no—
211C42C22×D9711yescharacteristic
212C23—21no—
213C23—971no—
214C23—3882no—
215C23—3884no—
216C23—1941no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—3882no—
221Q8—3884no—
222Q8—1942no—
223Q8—1942no—
224Q8—1942no—
225D4—22no—
226D4—3882no—
227D4—1942no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—971no—
231C2×C4—974no—
232C2×C4—972no—
233C2×C4—3882no—
234C2×C4—3884no—
235C2×C4—1944no—
236C2×C4—1941no—
237C2×C4—1942no—
238C2×C4D4×D9712yes—
239C2×C4C23×D9711yescharacteristic
240C8—974no—
241C8D4×D9714yes—
242C22—42no—
243C22—41no—
244C22—7764no—
245C22—974no—
246C22—972no—
247C22—3882no—
248C22—3881no—
249C22C2×D4×D9711yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—972no—
254C4—3881no—
255C4—3882no—
256C4—1942no—
257C4C2×D4×D9712yes—
258C2—81no—
259C2—7761no—
260C2—974no—
261C2#127×D9711yescharacteristic
262C2#137×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#497×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#497D97
hypocenterC97C2×#497
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D97C2semidirect
3. C4⋊Q8×D97C2semidirect
5. #20261130C2semidirect
6. #20261129C2semidirect
7. #20261128C2semidirect
8. #20261127C2semidirect
9. #20261126C2semidirect
10. #97×D97C2semidirect
11. #497×C97C2semidirect
15. C2×Dic388C22semidirect
16. C2×#13095501C22semidirect
17. C2×#13095501C22semidirect
24. #15690810C22semidirect
26. #15690808C22semidirect
34. #15690721C22semidirect
35. #15690720C22semidirect
36. C2×Q16×C97C22semidirect
38. C4⋊Q8×C97C22semidirect
40. #97×C97C22semidirect
130. D97#497direct (N × H)
132. C97C2×#497semidirect
142. #497D97direct (N × H)
151. C2×Q16C2×D97semidirect
161. C4⋊Q8C2×D97semidirect
172. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D972
C2×SD16×D972
C4⋊Q8×D972
#124×D972
#202611302
#202611292
#202611282
#202611272
#202611262
#97×D972
#497×C972
C2×#49797