Finite Group Databasebeta

← #38751061

Subgroups — #38751061

Order 25216.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#20263324C211yesmaximal, characteristic, complemented
2C2×#20263306C211yesmaximal, characteristic
3C2×#20263302C211yesmaximal, characteristic
4C2×C42×D197C211yesmaximal, characteristic
5#27160985C211yesmaximal, characteristic, complemented
6#27160976C214yesmaximal, complemented
7#27160948C211yesmaximal, characteristic, complemented
8#27160938C214yesmaximal
9#391×C197C211yesmaximal, characteristic, complemented
10C22×Dic394C2211yescharacteristic
11C22×C4×D197C2211yescharacteristic
12C22×C4×D197C2212yes—
13#20263324—24no—
14#20263306—24no—
15#20263302—24no—
16C2×C4⋊Dic197C2211yescharacteristic
17C2×C4⋊Dic197C2212yes—
18C2×#15691337C2214yes—
19C2×#15691337C2214yes—
20C2×#15691337C2212yes—
21C2×#15691335C2212yes—
22C2×#15691335C2212yes—
23C2×C4×Dic197C2212yes—
24C2×C4×Dic197C2211yescharacteristic
25C42×D197—24no—
26#20263233C2214yes—
27C22×Q8×C197C2211yescharacteristic
28C2×C4⋊C4×C197C2211yescharacteristic
29C2×C42×C197C2211yescharacteristic
30#95×C197C2214yes—
31C23×D197C2311yescharacteristic
32C2×Dic394—22no—
33C2×Dic394—22no—
34C2×Dic394—22no—
35C22×Dic197C2312yes—
36C22×Dic197C2312yes—
37C22×Dic197C2311yescharacteristic
38C22×Dic197C2312yes—
39C2×C4×D197—24no—
40C2×C4×D197—24no—
41C2×C4×D197—22no—
42C2×C4×D197—22no—
43C2×C4×D197D414yes—
44C2×C4×D197Q818yescomplemented
45C4⋊Dic197—24no—
46C4⋊Dic197—22no—
47#15691337—24no—
48#15691337—28no—
49#15691337—28no—
50#15691335—24no—
51#15691335—24no—
52C4×Dic197—24no—
53C4×Dic197D414yes—
54C2×Q8×C197—22no—
55C2×Q8×C197—22no—
56C2×Q8×C197—22no—
57C22×C4×C197C2312yes—
58C22×C4×C197C2312yes—
59C22×C4×C197C2312yes—
60C22×C4×C197C2311yescharacteristic
61C4⋊C4×C197—22no—
62C42×C197—22no—
63C22×D197—21no—
64C22×D197—22no—
65C22×D197—21no—
66C22×D197C2×D412yes—
67C22×D197C2×Q814yes—
68Dic394—44no—
69C2×Dic197—21no—
70C2×Dic197—21no—
71C2×Dic197—22no—
72C2×Dic197—22no—
73C2×Dic197—22no—
74C2×Dic197—24no—
75C2×Dic197—22no—
76C2×Dic197C2×D412yes—
77C2×Dic197C2×D414yes—
78C2×Dic197C2×Q814yes—
79C2×Dic197#4114yes—
80C4×D197—28no—
81C4×D197—216no—
82C23×C197C2411yescharacteristic
83Q8×C197—44no—
84C2×C4×C197—22no—
85C2×C4×C197—22no—
86C2×C4×C197—22no—
87C2×C4×C197—22no—
88C2×C4×C197—22no—
89C2×C4×C197—22no—
90C2×C4×C197—24no—
91C2×C4×C197C2×D412yes—
92C2×C4×C197C2×D412yes—
93C2×C4×C197C2×Q814yes—
94C2×C4×C197#4112yes—
95C22×C197C22×D411yescharacteristic
96C22×C197C22×D411yescharacteristic
97C22×C197C22×Q812yes—
98C22×C197C2×#4112yes—
99C22×C197C2×#4111yescharacteristic
100C2×D197—22no—
101C2×D197—24no—
102C2×D197—22no—
103C2×D197—24no—
104C2×D197C4⋊Q814yescomplemented
105C4×C197—24no—
106C4×C197—42no—
107C4×C197—42no—
108C4×C197#12114yes—
109Dic197—22no—
110Dic197—42no—
111Dic197—42no—
112Dic197C22⋊Q818yescomplemented
113C2×C197C2×#12111yescharacteristic
114C2×C197C2×C22⋊Q812yes—
115C2×C197C2×C4⋊Q811yescharacteristic
116C2×C197#53211yescharacteristic
117C2×C197#53312yes—
118D197—24no—
119C197#285711yescharacteristic, complemented, Sylow
120#2857—1971nomaximal, Sylow
121C2×C4⋊Q8—1971no—
122C2×C22⋊Q8—1971no—
123C2×C22⋊Q8—1971no—
124C22×C42—1971no—
125#391—1971no—
126#391D19711yescharacteristic, complemented
127#389—1974no—
128#384—1971no—
129#380—1974no—
130C22×Q8—1971no—
131C22×Q8C2×D19711yescharacteristic
132C23×C4—1971no—
133C23×C4—1972no—
134C4⋊Q8—3944no—
135C22⋊Q8—3944no—
136C22⋊Q8—3944no—
137C2×C4⋊C4—1972no—
138C2×C4⋊C4—1971no—
139C2×C4⋊C4—1972no—
140C2×C4⋊C4—1974no—
141C2×C4⋊C4—1974no—
142C2×C4⋊C4C2×D19711yescharacteristic
143C2×C22⋊C4—1972no—
144C2×C22⋊C4—1972no—
145C2×C42—3944no—
146C2×C42—1972no—
147C2×C42—1971no—
148C2×C42C2×D19711yescharacteristic
149#95—1974no—
150#95C2×D19714yes—
151C24—1971no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—3942no—
156C2×Q8—3942no—
157C2×Q8—3942no—
158C22×C4—3944no—
159C22×C4—3944no—
160C22×C4—3942no—
161C22×C4—3942no—
162C22×C4—1972no—
163C22×C4—1972no—
164C22×C4—1971no—
165C22×C4—1974no—
166C22×C4—1978no—
167C22×C4—1972no—
168C22×C4C22×D19712yes—
169C22×C4C22×D19712yes—
170C22×C4C22×D19712yes—
171C22×C4C22×D19711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—3948no—
174C4⋊C4—3944no—
175C4⋊C4—3948no—
176C4⋊C4—3944no—
177C4⋊C4—3942no—
178C22⋊C4—3944no—
179C22⋊C4—3944no—
180C42—22no—
181C42—3944no—
182C42—1974no—
183C23—3941no—
184C23—3942no—
185C23—3941no—
186C23—1972no—
187C23—1974no—
188C23C23×D19711yescharacteristic
189Q8—44no—
190Q8—7884no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—39416no—
199C2×C4—3942no—
200C2×C4—3942no—
201C2×C4—3942no—
202C2×C4—3944no—
203C2×C4—3942no—
204C2×C4—3941no—
205C2×C4—3941no—
206C2×C4—3948no—
207C2×C4—1972no—
208C2×C4—1974no—
209C2×C4—1974no—
210C2×C4—1974no—
211C2×C4D4×D19712yes—
212C2×C4D4×D19712yes—
213C2×C4Q8×D19714yes—
214C2×C4#1569135812yes—
215C22—3942no—
216C22—3944no—
217C22—3942no—
218C22—3944no—
219C22—1974no—
220C22C2×D4×D19711yescharacteristic
221C22C2×D4×D19711yescharacteristic
222C22C2×Q8×D19712yes—
223C22#41×D19712yes—
224C22C2×#1569135811yescharacteristic
225C4—24no—
226C4—42no—
227C4—42no—
228C4—7882no—
229C4—7882no—
230C4—3942no—
231C4—1978no—
232C4#2026329914yes—
233C2—3944no—
234C2C2×#2026329911yescharacteristic
235C2C22⋊Q8×D19712yes—
236C2C4⋊Q8×D19711yescharacteristic
237C2#2716162811yescharacteristic
238C2#2716163312yes—
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#391×C197C2
prime radicalC23×D197C23
hypercenter#391D197
hypocenterC197#2857
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#20263324C2semidirect
5. #27160985C2semidirect
6. #27160976C2semidirect
7. #27160948C2semidirect
9. #391×C197C2semidirect
44. C2×C4×D197Q8semidirect
104. C2×D197C4⋊Q8semidirect
112. Dic197C22⋊Q8semidirect
119. C197#2857semidirect
126. #391D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633242
C2×#202633062
C2×#202633022
C2×C42×D1972
#271609852
#271609762
#271609482
#271609382
#391×C1972
#2857197