Finite Group Databasebeta

← #38752241

Subgroups — #38752241

Order 25216.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20263359C211yesmaximal, characteristic
2C22×#15691348C211yesmaximal, characteristic, complemented
3#27161653C211yesmaximal, characteristic
4#27161556C211yesmaximal, characteristic, complemented
5#27161304C214yesmaximal, complemented
6#27161288C214yesmaximal, complemented
7#27161107C211yesmaximal, characteristic, complemented
8#27161073C211yesmaximal, characteristic
9#419×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11C22×Dic394C2211yescharacteristic
12#20263359—24no—
13C2×Dic788C2212yes—
14C2×#15691348—24no—
15C2×#15691348C2214yes—
16C2×#15691347C2212yes—
17C22×#13096802C2211yescharacteristic
18#20263324C2212yes—
19#20263315C2212yes—
20#20263309—22no—
21#20263302—22no—
22C4×Dic394—22no—
23#20263291C2214yes—
24#20263285C2211yescharacteristic
25#20263282—22no—
26#20263255C2214yes—
27#20263249C2212yes—
28#20263246C2212yes—
29C22×Q8×C197C2211yescharacteristic
30C2×M16×C197C2211yescharacteristic
31#117×C197C2211yescharacteristic
32#103×C197C2214yes—
33#15691358—22no—
34#15691358—22no—
35C2×Dic394—21no—
36C2×Dic394—22no—
37C2×Dic394D414yes—
38C2×Dic394C2311yescharacteristic
39C2×Dic394C2312yes—
40C2×D788C2311yescharacteristic
41C2×C22⋊D197—21no—
42C22×Dic197—21no—
43C2×C4×D197—21no—
44Dic788—24no—
45#15691348—28no—
46#15691347—24no—
47C2×#13096802—22no—
48C2×#13096802C2312yes—
49C4⋊Dic197D412yes—
50#15691337—24no—
51#15691337—24no—
52C22⋊Dic197D412yes—
53#15691335—24no—
54C4×Dic197—24no—
55C4×Dic197C2312yes—
56C2×Q8×C197—21no—
57C2×Q8×C197—22no—
58C2×Q8×C197—22no—
59C2×Q8×C197C2312yes—
60C22×C4×C197—21no—
61C22×C4×C197C2311yescharacteristic
62M16×C197—22no—
63C2×C8×C197C2312yes—
64C4⋊C4×C197C2312yes—
65C22⋊C4×C197—21no—
66C42×C197—21no—
67C22×D197—21no—
68Dic394—24no—
69Dic394—22no—
70Dic394—21no—
71Dic394—21no—
72Dic394C2×D414yes—
73D788—21no—
74D788—21no—
75C22⋊D197—42no—
76C2×Dic197—24no—
77C2×Dic197—22no—
78C2×Dic197—21no—
79C2×Dic197C2×D414yes—
80C4×D197—42no—
81#13096802—22no—
82C23×C197C2×D411yescharacteristic
83Q8×C197—22no—
84Q8×C197—22no—
85Q8×C197—42no—
86C2×C4×C197—22no—
87C2×C4×C197—22no—
88C2×C4×C197—42no—
89C2×C4×C197C2×D411yescharacteristic
90C2×C4×C197C2×D412yes—
91C2×C4×C197#4112yes—
92C2×C4×C197C2411yescharacteristic
93C8×C197—22no—
94C22×C197—22no—
95C22×C197#12112yes—
96C22×C197C22×D411yescharacteristic
97C2×D197—41no—
98C2×D197—42no—
99C4×C197—42no—
100C4×C197—42no—
101C4×C197#12112yes—
102C4×C197C22×D411yescharacteristic
103C4×C197C2×#4111yescharacteristic
104Dic197—24no—
105Dic197—24no—
106Dic197—41no—
107C2×C197—22no—
108C2×C197C2×#12111yescharacteristic
109C2×C197#57812yes—
110D197—81no—
111C197#304411yescharacteristic, complemented, Sylow
112#3044—1971nomaximal, Sylow
113C2×#137—1971no—
114C22×Q16—1971no—
115#536—1971no—
116#519—1971no—
117#464—1974no—
118#462—1974no—
119#424—1971no—
120#419D19711yescharacteristic, complemented
121#416—1971no—
122C2×#41—1971no—
123C22×Q8—1971no—
124C22×Q8C2×D19711yescharacteristic
125#137—3944no—
126C2×Q16—3944no—
127C2×Q16—1974no—
128C2×Q16—1972no—
129C2×SD16—1972no—
130C2×M16C2×D19711yescharacteristic
131C22×C8—1971no—
132C4⋊Q8—1972no—
133#124—1972no—
134#123—3942no—
135C22⋊Q8—3942no—
136C4×Q8—3942no—
137C4×Q8—1974no—
138#117—3942no—
139#117—1971no—
140#117C2×D19711yescharacteristic
141C4⋊C8—1974no—
142#103—1972no—
143#103C2×D19714yes—
144#102—1972no—
145#41—3942no—
146#41—3942no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—3941no—
151C2×Q8—3942no—
152C2×Q8—1974no—
153C2×Q8—1971no—
154C2×Q8—1972no—
155C2×Q8C22×D19712yes—
156C2×D4—3941no—
157C2×D4—1971no—
158C22×C4—21no—
159C22×C4—3941no—
160C22×C4—3941no—
161C22×C4C22×D19711yescharacteristic
162Q16—3944no—
163Q16—3948no—
164SD16—3944no—
165M16—22no—
166C2×C8—3942no—
167C2×C8—1972no—
168C2×C8C22×D19712yes—
169C4⋊C4—3944no—
170C4⋊C4—3944no—
171C4⋊C4—1972no—
172C4⋊C4C22×D19712yes—
173C22⋊C4—21no—
174C22⋊C4—3944no—
175C22⋊C4—1972no—
176C42—21no—
177C42—3944no—
178C42—1972no—
179C23—3941no—
180C23D4×D19711yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—42no—
184Q8—3941no—
185Q8—3941no—
186Q8—3944no—
187Q8—3942no—
188Q8—1974no—
189D4—7882no—
190D4—3941no—
191D4—3941no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—7882no—
196C2×C4—3941no—
197C2×C4—3944no—
198C2×C4—3942no—
199C2×C4—1974no—
200C2×C4D4×D19711yescharacteristic
201C2×C4D4×D19712yes—
202C2×C4#1569135812yes—
203C2×C4C23×D19711yescharacteristic
204C8—22no—
205C8—3942no—
206C22—22no—
207C22—7882no—
208C22—7881no—
209C22#2026329912yes—
210C22C2×D4×D19711yescharacteristic
211C4—42no—
212C4—42no—
213C4—7881no—
214C4—3944no—
215C4—3944no—
216C4#2026329912yes—
217C4C2×D4×D19711yescharacteristic
218C4C2×#1569135811yescharacteristic
219C2—22no—
220C2—15761no—
221C2C2×#2026329911yescharacteristic
222C2#2716189512yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#419×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691358C22
hypercenter#419D197
hypocenterC197#3044
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#15691348C2semidirect
4. #27161556C2semidirect
5. #27161304C2semidirect
6. #27161288C2semidirect
7. #27161107C2semidirect
9. #419×C197C2semidirect
111. C197#3044semidirect
120. #419D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633592
C22×#156913482
#271616532
#271615562
#271613042
#271612882
#271611072
#271610732
#419×C1972
#3044197