Finite Group Databasebeta

← #38752242

Subgroups — #38752242

Order 25216.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20263357C211yesmaximal, characteristic
2C22×#15691348C211yesmaximal, characteristic, complemented
3#27161655C211yesmaximal, characteristic
4#27161301C214yesmaximal, complemented
5#27161289C214yesmaximal, complemented
6#27161106C211yesmaximal, characteristic, complemented
7#27161087C211yesmaximal, characteristic, complemented
8#27161073C211yesmaximal, characteristic
9#519×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11C22×Dic394C2211yescharacteristic
12#20263357—24no—
13C2×#15691348—24no—
14C2×#15691348C2212yes—
15C2×#15691348C2214yes—
16C2×#15691345C2212yes—
17C2×#15691342C2211yescharacteristic
18C22×#13096802C2211yescharacteristic
19#20263326C2212yes—
20#20263314C2212yes—
21#20263310—22no—
22#20263306—22no—
23C4⋊#13096802C2214yes—
24#20263249C2212yes—
25#20263249C2214yes—
26#20263246C2212yes—
27C22×Q8×C197C2211yescharacteristic
28C4×Q8×C197—22no—
29C4×Q8×C197C2214yes—
30#117×C197—22no—
31#117×C197C2211yescharacteristic
32#117×C197C2211yescharacteristic
33#15691358—22no—
34#15691358—22no—
35C2×Dic394—21no—
36C2×Dic394—22no—
37C2×Dic394—22no—
38C2×Dic394C2312yes—
39C2×Dic394C2311yescharacteristic
40C2×D788C2311yescharacteristic
41C2×C22⋊D197—21no—
42C22×Dic197—21no—
43C2×C4×D197—21no—
44#15691348—28no—
45#15691348—24no—
46#15691345—24no—
47#15691342—22no—
48C2×#13096802—22no—
49C2×#13096802C2312yes—
50C2×#13096802C2312yes—
51C4⋊Dic197—24no—
52#15691335—24no—
53C2×Q8×C197—21no—
54C2×Q8×C197—22no—
55C2×Q8×C197D414yes—
56C2×Q8×C197C2312yes—
57C22×C4×C197—21no—
58C22×C4×C197C2311yescharacteristic
59C4⋊C4×C197—24no—
60C4⋊C4×C197D412yes—
61C4⋊C4×C197C2312yes—
62C22⋊C4×C197—21no—
63C22⋊C4×C197D412yes—
64C42×C197—24no—
65C42×C197—21no—
66C42×C197C2312yes—
67C22×D197—21no—
68Dic394—22no—
69Dic394—21no—
70Dic394—22no—
71Dic394—21no—
72Dic394—42no—
73D788—21no—
74D788—21no—
75C22⋊D197—42no—
76C2×Dic197—22no—
77C2×Dic197—21no—
78C2×Dic197—42no—
79C4×D197—42no—
80#13096802—22no—
81#13096802—22no—
82C23×C197C2×D411yescharacteristic
83Q8×C197—24no—
84Q8×C197—22no—
85Q8×C197C2×D414yes—
86C2×C4×C197—24no—
87C2×C4×C197—22no—
88C2×C4×C197—22no—
89C2×C4×C197C2×D414yes—
90C2×C4×C197C2×D411yescharacteristic
91C2×C4×C197C2×D412yes—
92C2×C4×C197#4112yes—
93C2×C4×C197C2411yescharacteristic
94C22×C197—22no—
95C22×C197#12112yes—
96C22×C197C22×D411yescharacteristic
97C2×D197—41no—
98C2×D197—42no—
99C4×C197—24no—
100C4×C197—24no—
101C4×C197—42no—
102C4×C197#12112yes—
103C4×C197C22×D411yescharacteristic
104C4×C197C2×#4111yescharacteristic
105Dic197—41no—
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#519D19711yescharacteristic, complemented
117#464—1974no—
118#462—1974no—
119#424—1971no—
120#419—1971no—
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×M16—1971no—
131C22×C8—1971no—
132C4⋊Q8—1972no—
133#124—1972no—
134#123—3942no—
135C22⋊Q8—3942no—
136C4×Q8—22no—
137C4×Q8C2×D19714yes—
138#117—22no—
139#117C2×D19711yescharacteristic
140#117C2×D19711yescharacteristic
141C4⋊C8—1974no—
142#103—1972no—
143#103—1974no—
144#102—1972no—
145#41—3942no—
146#41—3942no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3942no—
150C2×Q8—3942no—
151C2×Q8—3941no—
152C2×Q8—1972no—
153C2×Q8—1971no—
154C2×Q8D78814yes—
155C2×Q8C22×D19712yes—
156C2×D4—3941no—
157C2×D4—1971no—
158C22×C4—21no—
159C22×C4—3941no—
160C22×C4—3941no—
161C22×C4C22×D19711yescharacteristic
162Q16—3948no—
163Q16—3944no—
164SD16—3944no—
165M16—3942no—
166C2×C8—3942no—
167C2×C8—1972no—
168C2×C8—1972no—
169C4⋊C4—24no—
170C4⋊C4—3944no—
171C4⋊C4C22⋊D19712yes—
172C4⋊C4C22×D19712yes—
173C22⋊C4—21no—
174C22⋊C4—3944no—
175C22⋊C4C22⋊D19712yes—
176C42—24no—
177C42—21no—
178C42C22×D19712yes—
179C23—3941no—
180C23C2×C22⋊D19711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—7882no—
184Q8—3942no—
185Q8—3941no—
186Q8—3942no—
187Q8—3941no—
188Q8C2×D78814yes—
189D4—7882no—
190D4—3941no—
191D4—3941no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—7882no—
196C2×C4—7882no—
197C2×C4—3942no—
198C2×C4—3941no—
199C2×C4C2×C22⋊D19714yes—
200C2×C4C2×C22⋊D19711yescharacteristic
201C2×C4C2×D78812yes—
202C2×C4#1569135812yes—
203C2×C4C23×D19711yescharacteristic
204C8—3942no—
205C8—3942no—
206C22—22no—
207C22—7882no—
208C22—7881no—
209C22#2026330012yes—
210C22C22×C22⋊D19711yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—7881no—
215C4—7882no—
216C4#2026330012yes—
217C4C22×D78811yescharacteristic
218C4C2×#1569135811yescharacteristic
219C2—22no—
220C2—15761no—
221C2C2×#2026330011yescharacteristic
222C2#2716189412yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#519×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691358C22
hypercenter#519D197
hypocenterC197#3044
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#15691348C2semidirect
4. #27161301C2semidirect
5. #27161289C2semidirect
6. #27161106C2semidirect
7. #27161087C2semidirect
9. #519×C197C2semidirect
111. C197#3044semidirect
116. #519D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633572
C22×#156913482
#271616552
#271613012
#271612892
#271611062
#271610872
#271610732
#519×C1972
#3044197