Finite Group Databasebeta

← #38745983

Subgroups — #38745983

Order 25216.

528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 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×Q8×Dic197C211yesmaximal, characteristic, complemented
2C2×#20263284C211yesmaximal, characteristic
3#27161018C211yesmaximal, characteristic, complemented
4#27161016C211yesmaximal, characteristic
5#27161011C212yesmaximal, complemented
6#27161001C212yesmaximal
7#27160990C211yesmaximal, characteristic
8C4⋊C4×Dic197C212yesmaximal, complemented
9#27160930C211yesmaximal, characteristic, complemented
10#27160926C212yesmaximal
11#395×C197C211yesmaximal, characteristic, complemented
12C22×C4×D197C2211yescharacteristic
13Q8×Dic197—24no—
14#20263284—24no—
15C2×C4⋊Dic197C2212yes—
16C2×C4⋊Dic197C2212yes—
17C2×C4⋊Dic197C2211yescharacteristic
18C2×C4⋊Dic197C2211yescharacteristic
19C2×#15691337C2212yes—
20C2×#15691335C2212yes—
21C2×#15691335C2212yes—
22C2×#15691335C2211yescharacteristic
23C2×#15691335C2211yescharacteristic
24C2×C4×Dic197C2212yes—
25C2×C4×Dic197C2212yes—
26C2×C4×Dic197C2211yescharacteristic
27C2×C4×Dic197C2211yescharacteristic
28#20263233C2212yes—
29#20263233C2211yescharacteristic
30#20263233C2212yes—
31#20263233C2212yes—
32C22×Q8×C197C2211yescharacteristic
33C2×C4⋊C4×C197C2212yes—
34C2×C4⋊C4×C197C2211yescharacteristic
35#95×C197C2212yes—
36#95×C197C2211yescharacteristic
37C23×D197C2311yescharacteristic
38C22×Dic197C2312yes—
39C22×Dic197C2311yescharacteristic
40C22×Dic197C2312yes—
41C22×Dic197C2311yescharacteristic
42C22×Dic197C2311yescharacteristic
43C2×C4×D197—22no—
44C2×C4×D197—22no—
45C2×C4×D197Q814yescomplemented
46C4⋊Dic197—22no—
47C4⋊Dic197—24no—
48C4⋊Dic197—22no—
49C4⋊Dic197—24no—
50#15691337—24no—
51#15691335—22no—
52#15691335—22no—
53#15691335—24no—
54#15691335—24no—
55C4×Dic197—22no—
56C4×Dic197—24no—
57C4×Dic197—24no—
58C4×Dic197D414yes—
59C2×Q8×C197—22no—
60C2×Q8×C197—24no—
61C22×C4×C197C2311yescharacteristic
62C22×C4×C197C2311yescharacteristic
63C22×C4×C197C2311yescharacteristic
64C22×C4×C197C2312yes—
65C22×C4×C197C2312yes—
66C4⋊C4×C197—22no—
67C4⋊C4×C197—24no—
68C22×D197—21no—
69C22×D197—22no—
70C22×D197—22no—
71C22×D197—21no—
72C22×D197C2×Q812yes—
73C2×Dic197—22no—
74C2×Dic197—21no—
75C2×Dic197—22no—
76C2×Dic197—22no—
77C2×Dic197—22no—
78C2×Dic197—22no—
79C2×Dic197—22no—
80C2×Dic197—22no—
81C2×Dic197—22no—
82C2×Dic197C2×D412yes—
83C2×Dic197C2×D412yes—
84C2×Dic197C2×Q812yes—
85C2×Dic197#4112yes—
86C4×D197—28no—
87C23×C197C2411yescharacteristic
88Q8×C197—28no—
89C2×C4×C197—21no—
90C2×C4×C197—22no—
91C2×C4×C197—22no—
92C2×C4×C197—22no—
93C2×C4×C197—22no—
94C2×C4×C197—22no—
95C2×C4×C197—21no—
96C2×C4×C197—21no—
97C2×C4×C197—22no—
98C2×C4×C197C2×D412yes—
99C2×C4×C197C2×Q812yes—
100C2×C4×C197#4114yes—
101C2×C4×C197#4114yes—
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—24no—
115C4×C197—24no—
116C4×C197—41no—
117Dic197—22no—
118Dic197—42no—
119Dic197—41no—
120Dic197—42no—
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#400—1971no—
133#399—1971no—
134#398—1972no—
135#396—1972no—
136#395D19711yescharacteristic, complemented
137#393—1971no—
138C4×C4⋊C4—1972no—
139C4×C4⋊C4—1971no—
140C4×C22⋊C4—1972no—
141C22×Q8C2×D19711yescharacteristic
142C23×C4—1971no—
143C4×Q8—3944no—
144#117—3944no—
145C2×C4⋊C4—1971no—
146C2×C4⋊C4—1972no—
147C2×C4⋊C4—1971no—
148C2×C4⋊C4—1972no—
149C2×C4⋊C4—1972no—
150C2×C4⋊C4C2×D19711yescharacteristic
151C2×C4⋊C4C2×D19712yes—
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#95—1972no—
164#95C2×D19712yes—
165#95C2×D19711yescharacteristic
166C24—1971no—
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—3942no—
170C22×C4—3942no—
171C22×C4—1972no—
172C22×C4—1971no—
173C22×C4—1972no—
174C22×C4—1971no—
175C22×C4—1974no—
176C22×C4—1971no—
177C22×C4C22×D19711yescharacteristic
178C22×C4C22×D19711yescharacteristic
179C22×C4C22×D19711yescharacteristic
180C22×C4C22×D19712yes—
181C22×C4C22×D19712yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—3942no—
185C4⋊C4—3944no—
186C4⋊C4—3942no—
187C4⋊C4—3944no—
188C4⋊C4—3944no—
189C22⋊C4—3942no—
190C22⋊C4—3942no—
191C22⋊C4—3944no—
192C22⋊C4—3944no—
193C42—3944no—
194C42—3942no—
195C42—3944no—
196C42—1974no—
197C23—3941no—
198C23—3941no—
199C23—3942no—
200C23—3942no—
201C23—1972no—
202C23C23×D19711yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—3948no—
214C2×C4—3941no—
215C2×C4—3942no—
216C2×C4—3942no—
217C2×C4—3942no—
218C2×C4—3942no—
219C2×C4—3942no—
220C2×C4—3942no—
221C2×C4—3942no—
222C2×C4—3942no—
223C2×C4—1972no—
224C2×C4—1972no—
225C2×C4—1972no—
226C2×C4—1972no—
227C2×C4D4×D19712yes—
228C2×C4Q8×D19712yes—
229C2×C4#1569135714yes—
230C2×C4#1569135914yes—
231C22—7882no—
232C22—7881no—
233C22—7881no—
234C22—3944no—
235C22—3942no—
236C22C2×D4×D19711yescharacteristic
237C22C2×Q8×D19711yescharacteristic
238C22#41×D19711yescharacteristic
239C22C2×#1569135712yes—
240C22C2×#1569135912yes—
241C4—24no—
242C4—22no—
243C4—22no—
244C4—24no—
245C4—41no—
246C4—7882no—
247C4—7881no—
248C4—7882no—
249C4—3942no—
250C4—1974no—
251C2—7882no—
252C2C22⋊Q8×D19711yescharacteristic
253C2#2716158511yescharacteristic
254C2#2716161512yes—
255C2#2716163011yescharacteristic
256C2#2716163412yes—
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#395×C197C2
prime radicalC23×D197C23
hypercenter#395D197
hypocenterC197#2396
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic197C2semidirect
3. #27161018C2semidirect
5. #27161011C2semidirect
8. C4⋊C4×Dic197C2semidirect
9. #27160930C2semidirect
11. #395×C197C2semidirect
45. C2×C4×D197Q8semidirect
128. C197#2396semidirect
136. #395D197semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1972
C2×#202632842
#271610182
#271610162
#271610112
#271610012
#271609902
C4⋊C4×Dic1972
#271609302
#271609262
#395×C1972
#2396197