Finite Group Databasebeta

← #38745985

Subgroups — #38745985

Order 25216.

528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 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×#20263291C211yesmaximal, characteristic, complemented
2C2×#20263284C211yesmaximal, characteristic
3#27161014C211yesmaximal, characteristic
4#27161013C212yesmaximal, complemented
5#27161003C212yesmaximal
6#27160999C211yesmaximal, characteristic, complemented
7#27160992C211yesmaximal, characteristic
8#27160930C211yesmaximal, characteristic, complemented
9#27160930C212yesmaximal, complemented
10#27160926C212yesmaximal
11#400×C197C211yesmaximal, characteristic, complemented
12C22×Dic394C2211yescharacteristic
13C22×C4×D197C2211yescharacteristic
14#20263291—24no—
15#20263284—24no—
16C2×C4⋊Dic197C2211yescharacteristic
17C2×#15691337C2212yes—
18C2×#15691337C2212yes—
19C2×#15691337C2212yes—
20C2×#15691337C2212yes—
21C2×#15691335C2212yes—
22C2×#15691335C2212yes—
23C2×#15691335C2211yescharacteristic
24C2×#15691335C2211yescharacteristic
25C2×C4×Dic197C2212yes—
26C2×C4×Dic197C2212yes—
27C2×C4×Dic197C2211yescharacteristic
28C2×C4×Dic197C2211yescharacteristic
29#20263233C2212yes—
30#20263233C2212yes—
31#20263233C2211yescharacteristic
32C2×C4⋊C4×C197C2211yescharacteristic
33C2×C4⋊C4×C197C2211yescharacteristic
34#95×C197C2212yes—
35#95×C197C2211yescharacteristic
36#95×C197C2212yes—
37C23×D197C2311yescharacteristic
38C2×Dic394—24no—
39C2×Dic394—22no—
40C22×Dic197C2311yescharacteristic
41C22×Dic197C2311yescharacteristic
42C22×Dic197C2312yes—
43C22×Dic197C2312yes—
44C22×Dic197C2311yescharacteristic
45C2×C4×D197—22no—
46C2×C4×D197—22no—
47C2×C4×D197Q814yescomplemented
48C4⋊Dic197—22no—
49#15691337—24no—
50#15691337—24no—
51#15691337—24no—
52#15691337—24no—
53#15691335—22no—
54#15691335—24no—
55#15691335—24no—
56#15691335—22no—
57C4×Dic197—22no—
58C4×Dic197—24no—
59C4×Dic197—24no—
60C4×Dic197D414yes—
61C22×C4×C197C2312yes—
62C22×C4×C197C2311yescharacteristic
63C22×C4×C197C2312yes—
64C22×C4×C197C2311yescharacteristic
65C22×C4×C197C2311yescharacteristic
66C4⋊C4×C197—22no—
67C4⋊C4×C197—22no—
68C22×D197—21no—
69C22×D197—22no—
70C22×D197—22no—
71C22×D197—21no—
72C22×D197C2×Q812yes—
73Dic394—28no—
74C2×Dic197—22no—
75C2×Dic197—22no—
76C2×Dic197—22no—
77C2×Dic197—22no—
78C2×Dic197—22no—
79C2×Dic197—21no—
80C2×Dic197—22no—
81C2×Dic197C2×D412yes—
82C2×Dic197C2×D412yes—
83C2×Dic197C2×Q812yes—
84C2×Dic197#4114yes—
85C2×Dic197#4114yes—
86C2×Dic197#4112yes—
87C4×D197—28no—
88C23×C197C2411yescharacteristic
89C2×C4×C197—21no—
90C2×C4×C197—21no—
91C2×C4×C197—22no—
92C2×C4×C197—21no—
93C2×C4×C197—22no—
94C2×C4×C197—22no—
95C2×C4×C197—22no—
96C2×C4×C197—22no—
97C2×C4×C197—22no—
98C2×C4×C197—22no—
99C2×C4×C197—22no—
100C2×C4×C197C2×D412yes—
101C2×C4×C197C2×Q812yes—
102C22×C197C22×D411yescharacteristic
103C22×C197C22×Q811yescharacteristic
104C22×C197C2×#4111yescharacteristic
105C22×C197C2×#4112yes—
106C22×C197C2×#4112yes—
107C2×D197—22no—
108C2×D197—24no—
109C2×D197—41no—
110C2×D197—42no—
111C2×D197—41no—
112C4×C197—22no—
113C4×C197—22no—
114C4×C197—41no—
115C4×C197—42no—
116C4×C197—42no—
117Dic197—24no—
118Dic197—24no—
119Dic197—22no—
120Dic197—41no—
121Dic197C22⋊Q814yes—
122C2×C197C2×C22⋊Q811yescharacteristic
123C2×C197#52511yescharacteristic
124C2×C197#52912yes—
125C2×C197#53211yescharacteristic
126C2×C197#53312yes—
127D197—42no—
128C197#239611yescharacteristic, complemented, Sylow
129#2396—1971nomaximal, Sylow
130C2×C4×Q8—1971no—
131C2×#117—1971no—
132#400D19711yescharacteristic, complemented
133#399—1971no—
134#398—1972no—
135#396—1972no—
136#395—1971no—
137#393—1971no—
138C4×C4⋊C4—1972no—
139C4×C4⋊C4—1971no—
140C4×C22⋊C4—1972no—
141C22×Q8—1971no—
142C23×C4—1971no—
143C4×Q8—3944no—
144#117—3944no—
145C2×C4⋊C4—1971no—
146C2×C4⋊C4—1972no—
147C2×C4⋊C4—1972no—
148C2×C4⋊C4—1972no—
149C2×C4⋊C4—1972no—
150C2×C4⋊C4C2×D19711yescharacteristic
151C2×C4⋊C4C2×D19711yescharacteristic
152C2×C22⋊C4—1971no—
153C2×C22⋊C4—1971no—
154C2×C22⋊C4—1972no—
155C2×C22⋊C4—1972no—
156C2×C42—1971no—
157C2×C42—1972no—
158C2×C42—1972no—
159C2×C42—1971no—
160#95—1972no—
161#95—1972no—
162#95—1971no—
163#95C2×D19712yes—
164#95C2×D19711yescharacteristic
165#95C2×D19712yes—
166C24—1971no—
167C2×Q8—3944no—
168C2×Q8—3942no—
169C22×C4—3942no—
170C22×C4—3942no—
171C22×C4—1971no—
172C22×C4—1972no—
173C22×C4—1972no—
174C22×C4—1971no—
175C22×C4—1974no—
176C22×C4—1971no—
177C22×C4C22×D19712yes—
178C22×C4C22×D19711yescharacteristic
179C22×C4C22×D19712yes—
180C22×C4C22×D19711yescharacteristic
181C22×C4C22×D19711yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—3944no—
185C4⋊C4—3944no—
186C4⋊C4—3944no—
187C4⋊C4—3944no—
188C4⋊C4—3942no—
189C22⋊C4—3944no—
190C22⋊C4—3944no—
191C22⋊C4—3942no—
192C22⋊C4—3942no—
193C42—3944no—
194C42—3942no—
195C42—3944no—
196C42—1974no—
197C23—3941no—
198C23—3942no—
199C23—3942no—
200C23—3941no—
201C23—1972no—
202C23C23×D19711yescharacteristic
203Q8—3948no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—3948no—
216C2×C4—3941no—
217C2×C4—3942no—
218C2×C4—3942no—
219C2×C4—3942no—
220C2×C4—3942no—
221C2×C4—3942no—
222C2×C4—3942no—
223C2×C4—1972no—
224C2×C4—1974no—
225C2×C4—1974no—
226C2×C4—1972no—
227C2×C4—1972no—
228C2×C4—1972no—
229C2×C4D4×D19712yes—
230C2×C4Q8×D19712yes—
231C22—7882no—
232C22—7881no—
233C22—7881no—
234C22—3944no—
235C22—3942no—
236C22C2×D4×D19711yescharacteristic
237C22C2×Q8×D19711yescharacteristic
238C22#41×D19711yescharacteristic
239C22#41×D19712yes—
240C22#41×D19712yes—
241C4—22no—
242C4—22no—
243C4—42no—
244C4—42no—
245C4—41no—
246C4—7881no—
247C4—3944no—
248C4—3942no—
249C4—3944no—
250C4—1974no—
251C2—7882no—
252C2C22⋊Q8×D19711yescharacteristic
253C2#2716158711yescharacteristic
254C2#2716161212yes—
255C2#2716162811yescharacteristic
256C2#2716163512yes—
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#400×C197C2
prime radicalC23×D197C23
hypercenter#400D197
hypocenterC197#2396
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#20263291C2semidirect
4. #27161013C2semidirect
6. #27160999C2semidirect
8. #27160930C2semidirect
9. #27160930C2semidirect
11. #400×C197C2semidirect
47. C2×C4×D197Q8semidirect
128. C197#2396semidirect
132. #400D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632912
C2×#202632842
#271610142
#271610132
#271610032
#271609992
#271609922
#271609302
#271609262
#400×C1972
#2396197