Finite Group Databasebeta

← #38745518

Subgroups — #38745518

Order 25216.

428 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 217 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
1#27160978C212yesmaximal
2#27160965C212yesmaximal
3#27160965C212yesmaximal
4#27160965C212yesmaximal
5#27160965C212yesmaximal
6#27160947C211yesmaximal, characteristic
7C4⋊C4×Dic197C212yesmaximal
8C4×C4⋊Dic197C211yesmaximal, characteristic
9C2×#125×C197C211yesmaximal, characteristic
10C2×C4⋊Dic197C2212yes—
11C2×C4⋊Dic197C2212yes—
12C2×C4⋊Dic197C2212yes—
13C2×C4×Dic197C2212yes—
14C2×C4×Dic197C2211yescharacteristic
15C2×C4×Dic197C2211yescharacteristic
16C2×C4×Dic197C2212yes—
17#20263233C2212yes—
18#20263233C2212yes—
19#20263233C2212yes—
20#20263233C2212yes—
21#20263233C2211yescharacteristic
22#20263233C2211yescharacteristic
23#20263233C2212yes—
24#20263233C2212yes—
25#20263233C2212yes—
26#125×C197C418yescomplemented
27C2×C4⋊C4×C197C2212yes—
28C2×C4⋊C4×C197C2211yescharacteristic
29C2×C4⋊C4×C197C2211yescharacteristic
30C2×C4⋊C4×C197C2212yes—
31C2×C42×C197C2211yescharacteristic
32C22×Dic197C2312yes—
33C22×Dic197C2312yes—
34C22×Dic197C2312yes—
35C22×Dic197C2312yes—
36C4⋊Dic197—24no—
37C4⋊Dic197—24no—
38C4⋊Dic197—42no—
39C4×Dic197—24no—
40C4×Dic197—41no—
41C4×Dic197—41no—
42C4×Dic197—42no—
43C22×C4×C197C2311yescharacteristic
44C22×C4×C197C2311yescharacteristic
45C22×C4×C197C2311yescharacteristic
46C22×C4×C197C2312yes—
47C22×C4×C197C2312yes—
48C4⋊C4×C197C2×C418yes—
49C4⋊C4×C197C2×C414yes—
50C4⋊C4×C197C2×C414yes—
51C4⋊C4×C197C2×C418yes—
52C42×C197C2×C414yes—
53C2×Dic197—22no—
54C2×Dic197—22no—
55C2×Dic197—22no—
56C2×Dic197—22no—
57C2×Dic197—22no—
58C2×Dic197—22no—
59C2×Dic197—22no—
60C2×Dic197—22no—
61C2×Dic197—22no—
62C2×Dic197—22no—
63C2×Dic197—22no—
64C2×Dic197#4114yes—
65C23×C197C2411yescharacteristic
66C2×C4×C197—22no—
67C2×C4×C197—22no—
68C2×C4×C197—22no—
69C2×C4×C197—21no—
70C2×C4×C197—22no—
71C2×C4×C197—21no—
72C2×C4×C197—22no—
73C2×C4×C197—22no—
74C2×C4×C197C22×C412yes—
75C2×C4×C197C22×C412yes—
76C2×C4×C197C22×C412yes—
77C2×C4×C197C22×C414yes—
78C2×C4×C197C22×C414yes—
79C22×C197C23×C411yescharacteristic
80C22×C197C2×#4112yes—
81C22×C197H(5,𝔽2)11yescharacteristic
82C22×C1972−1+411yescharacteristic
83C22×C1972−1+412yes—
84C4×C197—24no—
85C4×C197—22no—
86C4×C197—22no—
87C4×C197—22no—
88C4×C197—24no—
89Dic197—24no—
90Dic197—42no—
91Dic197—42no—
92Dic197—42no—
93C2×C197C4×#4111yescharacteristic
94C2×C197#51911yescharacteristic
95C2×C197#52011yescharacteristic
96C2×C197#53912yes—
97C2×C197#54112yes—
98C197#233611yescharacteristic, complemented, Sylow
99#2336—1971nomaximal, Sylow
100C2×#125D19711yescharacteristic
101#389—1972no—
102#387—1972no—
103#387—1972no—
104#387—1972no—
105#387—1972no—
106#383—1971no—
107C4×C4⋊C4—1972no—
108C4×C4⋊C4—1971no—
109#125Dic19718yescomplemented
110C2×C4⋊C4—1972no—
111C2×C4⋊C4—1972no—
112C2×C4⋊C4—1972no—
113C2×C4⋊C4C2×D19712yes—
114C2×C4⋊C4C2×D19711yescharacteristic
115C2×C4⋊C4C2×D19711yescharacteristic
116C2×C4⋊C4C2×D19712yes—
117C2×C42—1971no—
118C2×C42—1971no—
119C2×C42—1972no—
120C2×C42—1972no—
121C2×C42C2×D19711yescharacteristic
122#95—1972no—
123#95—1972no—
124#95—1972no—
125#95—1971no—
126#95—1972no—
127#95—1971no—
128#95—1972no—
129#95—1972no—
130#95—1972no—
131C22×C4—1972no—
132C22×C4—1972no—
133C22×C4—1972no—
134C22×C4—1972no—
135C22×C4C22×D19711yescharacteristic
136C22×C4C22×D19711yescharacteristic
137C22×C4C22×D19712yes—
138C22×C4C22×D19712yes—
139C22×C4C22×D19711yescharacteristic
140C4⋊C4—7882no—
141C4⋊C4—3944no—
142C4⋊C4—3944no—
143C4⋊C4C2×Dic19718yes—
144C4⋊C4C2×Dic19714yes—
145C4⋊C4C2×Dic19714yes—
146C4⋊C4C2×Dic19718yes—
147C42—7881no—
148C42—7881no—
149C42—7882no—
150C42—3944no—
151C42C2×Dic19714yes—
152C23C23×D19711yescharacteristic
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—3942no—
162C2×C4—3942no—
163C2×C4—3942no—
164C2×C4—3942no—
165C2×C4—3942no—
166C2×C4—3942no—
167C2×C4—3942no—
168C2×C4—3942no—
169C2×C4—3942no—
170C2×C4—3942no—
171C2×C4—3942no—
172C2×C4—1974no—
173C2×C4C22×Dic19712yes—
174C2×C4C22×Dic19712yes—
175C2×C4C22×Dic19712yes—
176C2×C4C22×Dic19714yes—
177C2×C4C22×Dic19714yes—
178C22C23×Dic19711yescharacteristic
179C22#41×D19712yes—
180C22#2026337211yescharacteristic
181C22#2026337311yescharacteristic
182C22#2026337412yes—
183C4—24no—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—24no—
188C4—7882no—
189C4—7882no—
190C4—7882no—
191C4—3944no—
192C2#41×Dic19711yescharacteristic
193C2#2716155711yescharacteristic
194C2#2716156211yescharacteristic
195C2#2716167712yes—
196C2#2716168812yes—
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
FittingC2×#125×C197C2
socle, prime radical, cosocleC23×C197C24
hypercenterC2×#125D197
hypocenterC197#2336
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
26. #125×C197C4semidirect
98. C197#2336semidirect
109. #125Dic197semidirect

Maximal subgroups

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

subgroupindex
#271609782
#271609652
#271609472
C4⋊C4×Dic1972
C4×C4⋊Dic1972
C2×#125×C1972
#2336197