Finite Group Databasebeta

← #38752243

Subgroups — #38752243

Order 25216.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×Dic788C211yesmaximal, characteristic, complemented
3#27161655C211yesmaximal, characteristic
4#27161556C211yesmaximal, characteristic, complemented
5#27161302C214yesmaximal, complemented
6#27161290C214yesmaximal, complemented
7#27161089C211yesmaximal, characteristic, complemented
8#27161076C211yesmaximal, characteristic
9#424×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11C22×Dic394C2211yescharacteristic
12C22×Dic394C2211yescharacteristic
13#20263359—24no—
14C2×Dic788—24no—
15C2×Dic788C2214yes—
16C2×Dic788C2212yes—
17C2×#15691347C2212yes—
18#20263326C2212yes—
19#20263314C2212yes—
20#20263310—22no—
21#20263306—22no—
22C4×Dic394C2214yes—
23#20263291—22no—
24#20263285C2211yescharacteristic
25#20263282—22no—
26#20263251C2214yes—
27#20263251C2212yes—
28#20263248C2212yes—
29C2×M16×C197C2211yescharacteristic
30C22×C8×C197C2211yescharacteristic
31#117×C197C2211yescharacteristic
32C4⋊C8×C197C2214yes—
33#15691358—22no—
34#15691358—22no—
35C2×Dic394—21no—
36C2×Dic394—21no—
37C2×Dic394—22no—
38C2×Dic394—22no—
39C2×Dic394—22no—
40C2×Dic394D414yes—
41C2×Dic394C2312yes—
42C2×Dic394C2311yescharacteristic
43C2×Dic394C2312yes—
44C2×D788C2311yescharacteristic
45C2×C22⋊D197—21no—
46C22×Dic197—21no—
47C22×Dic197—21no—
48C2×C4×D197—21no—
49Dic788—28no—
50Dic788—24no—
51#15691347—24no—
52C4⋊Dic197—24no—
53C4⋊Dic197C2312yes—
54#15691337—24no—
55C22⋊Dic197—21no—
56#15691335—24no—
57C4×Dic197—24no—
58C4×Dic197—21no—
59C22×C4×C197C2311yescharacteristic
60M16×C197—22no—
61C2×C8×C197—22no—
62C2×C8×C197C2312yes—
63C2×C8×C197C2312yes—
64C4⋊C4×C197D412yes—
65C22⋊C4×C197D412yes—
66C42×C197C2312yes—
67C22×D197—21no—
68Dic394—24no—
69Dic394—22no—
70Dic394—22no—
71Dic394—21no—
72Dic394—22no—
73Dic394—21no—
74Dic394—42no—
75Dic394C2×D414yes—
76D788—21no—
77D788—21no—
78C22⋊D197—42no—
79C2×Dic197—24no—
80C2×Dic197—22no—
81C2×Dic197—22no—
82C2×Dic197—22no—
83C2×Dic197—21no—
84C2×Dic197—42no—
85C4×D197—42no—
86C23×C197C2×D411yescharacteristic
87C2×C4×C197C2×D412yes—
88C2×C4×C197C2×D414yes—
89C2×C4×C197C2×D411yescharacteristic
90C2×C4×C197#4112yes—
91C2×C4×C197C2411yescharacteristic
92C8×C197—22no—
93C8×C197—22no—
94C22×C197—22no—
95C22×C197#12112yes—
96C22×C197C22×D411yescharacteristic
97C2×D197—41no—
98C2×D197—42no—
99C4×C197—24no—
100C4×C197#12112yes—
101C4×C197C22×D411yescharacteristic
102C4×C197C2×#4111yescharacteristic
103Dic197—24no—
104Dic197—41no—
105Dic197—42no—
106Dic197—42no—
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#424D19711yescharacteristic, complemented
120#419—1971no—
121#416—1971no—
122C2×#41—1971no—
123C22×Q8—1971no—
124C22×Q8—1971no—
125#137—3944no—
126C2×Q16—3944no—
127C2×Q16—1974no—
128C2×Q16—1972no—
129C2×SD16—1972no—
130C2×M16C2×D19711yescharacteristic
131C22×C8C2×D19711yescharacteristic
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⋊C8C2×D19714yes—
142#103—1972no—
143#103—1974no—
144#102—1972no—
145#41—3942no—
146#41—3942no—
147C2×Q8—3942no—
148C2×Q8—3942no—
149C2×Q8—3941no—
150C2×Q8—3941no—
151C2×Q8—3942no—
152C2×Q8—1974no—
153C2×Q8—1972no—
154C2×Q8—1971no—
155C2×Q8—1972no—
156C2×D4—3941no—
157C2×D4—1971no—
158C22×C4—3941no—
159C22×C4—3941no—
160C22×C4—3941no—
161C22×C4C22×D19711yescharacteristic
162Q16—3948no—
163Q16—3944no—
164SD16—3944no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D19712yes—
168C2×C8C22×D19712yes—
169C4⋊C4—3944no—
170C4⋊C4—3944no—
171C4⋊C4—1972no—
172C4⋊C4D78812yes—
173C22⋊C4—3941no—
174C22⋊C4—3944no—
175C22⋊C4D78812yes—
176C42—3944no—
177C42—3941no—
178C42C22×D19712yes—
179C23—3941no—
180C23C2×D78811yescharacteristic
181Q8—7882no—
182Q8—3942no—
183Q8—3941no—
184Q8—3942no—
185Q8—3941no—
186Q8—3944no—
187Q8—3942no—
188Q8—1974no—
189D4—7882no—
190D4—3941no—
191D4—3941no—
192C2×C4—7882no—
193C2×C4—7882no—
194C2×C4—3942no—
195C2×C4—3942no—
196C2×C4—3941no—
197C2×C4—3944no—
198C2×C4—3942no—
199C2×C4D4×D19712yes—
200C2×C4C2×D78814yes—
201C2×C4C2×D78811yescharacteristic
202C2×C4#1569135912yes—
203C2×C4C23×D19711yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—7882no—
208C22—7881no—
209C22#2026330112yes—
210C22C22×D78811yescharacteristic
211C4—24no—
212C4—7882no—
213C4—7881no—
214C4—7882no—
215C4—3944no—
216C4#2026330112yes—
217C4C2×D4×D19711yescharacteristic
218C4C2×#1569135911yescharacteristic
219C2—22no—
220C2—15761no—
221C2C2×#2026330111yescharacteristic
222C2#2716189612yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#424×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691358C22
hypercenter#424D197
hypocenterC197#3044
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic788C2semidirect
4. #27161556C2semidirect
5. #27161302C2semidirect
6. #27161290C2semidirect
7. #27161089C2semidirect
9. #424×C197C2semidirect
111. C197#3044semidirect
119. #424D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633592
C22×Dic7882
#271616552
#271615562
#271613022
#271612902
#271610892
#271610762
#424×C1972
#3044197