Finite Group Databasebeta

← #38743035

Subgroups — #38743035

Order 25216.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20263285C211yesmaximal, characteristic, complemented
2C2×#20263247C212yesmaximal, complemented
3C2×C22⋊#13096802C211yesmaximal, characteristic, complemented
4#27160841C212yesmaximal, complemented
5C2×#121×C197C211yesmaximal, characteristic
6C22×#13096802C2212yes—
7#20263285—24no—
8C2×C4⋊Dic197C2212yes—
9C2×C22⋊Dic197—21no—
10C2×C4×Dic197—21no—
11#20263247—24no—
12C22⋊#13096802—22no—
13C22×D4×C197—21no—
14C22×D4×C197C2211yescharacteristic
15C23×C4×C197C2211yescharacteristic
16#121×C197—22no—
17#121×C197C414yescomplemented
18C2×C4⋊C4×C197C2211yescharacteristic
19C2×C22⋊C4×C197—21no—
20C22×Dic197—22no—
21C2×#13096802—22no—
22C2×#13096802—22no—
23C2×#13096802—22no—
24C4⋊Dic197—22no—
25C4⋊Dic197D414yescomplemented
26C22⋊Dic197—24no—
27C4×Dic197—24no—
28C24×C197—21no—
29C24×C197D411yescharacteristic
30C2×D4×C197—24no—
31C2×D4×C197—22no—
32C2×D4×C197—21no—
33C2×D4×C197—22no—
34C2×D4×C197—44no—
35C2×D4×C197C2×C412yes—
36C22×C4×C197—22no—
37C22×C4×C197—22no—
38C22×C4×C197—21no—
39C22×C4×C197C2×C412yes—
40C22×C4×C197D411yescharacteristic
41C22×C4×C197D412yes—
42C22×C4×C197C2311yescharacteristic
43C4⋊C4×C197—21no—
44C4⋊C4×C197C2×C412yes—
45C22⋊C4×C197—24no—
46C2×Dic197—24no—
47C2×Dic197—42no—
48C2×Dic197—42no—
49#13096802—42no—
50C23×C197—21no—
51C23×C197—21no—
52C23×C197—22no—
53C23×C197—22no—
54C23×C197—22no—
55C23×C197—41no—
56C23×C197—41no—
57C23×C197—42no—
58C23×C197—42no—
59C23×C197C22⋊C412yes—
60C23×C197C2×D411yescharacteristic
61D4×C197—24no—
62D4×C197—48no—
63D4×C197—42no—
64C2×C4×C197—21no—
65C2×C4×C197—21no—
66C2×C4×C197—24no—
67C2×C4×C197—24no—
68C2×C4×C197—22no—
69C2×C4×C197—42no—
70C2×C4×C197C22⋊C412yes—
71C2×C4×C197C22×C411yescharacteristic
72C2×C4×C197C2×D412yes—
73C2×C4×C197C2×D411yescharacteristic
74C2×C4×C197#4112yes—
75C22×C197—22no—
76C22×C197—22no—
77C22×C197—22no—
78C22×C197—42no—
79C22×C197—42no—
80C22×C197—42no—
81C22×C197—42no—
82C22×C197—42no—
83C22×C197—82no—
84C22×C197—82no—
85C22×C197C2×C22⋊C411yescharacteristic
86C22×C197#12011yescharacteristic
87C22×C197#12311yescharacteristic
88C22×C197#13512yes—
89C22×C197AGL(1,ℤ/8)12yes—
90C4×C197—22no—
91C4×C197—42no—
92C4×C197C4×D412yes—
93C4×C197#12112yes—
94Dic197—44no—
95C2×C197—42no—
96C2×C197—82no—
97C2×C197#38611yescharacteristic
98C2×C197#41611yescharacteristic
99C2×C197#41811yescharacteristic
100C2×C197#44912yes—
101C2×C197#48212yes—
102C197#187511yescharacteristic, complemented, Sylow
103#1875—1971nomaximal, Sylow
104C2×#121D19711yescharacteristic
105C2×#117—1971no—
106C2×#102—1972no—
107C2×C22⋊C8—1971no—
108#340—1972no—
109C22×D4—21no—
110C22×D4C2×D19711yescharacteristic
111C23×C4C2×D19711yescharacteristic
112C22×C8—1972no—
113#121—22no—
114#121Dic19714yescomplemented
115#117—3944no—
116C2×C4⋊C4—1972no—
117C2×C4⋊C4C2×D19711yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—3941no—
120C2×C42—3941no—
121#102—3944no—
122C22⋊C8—3942no—
123C24—21no—
124C24C22⋊D19711yescharacteristic
125C2×D4—24no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—44no—
130C2×D4C2×Dic19712yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—3942no—
135C22×C4C2×Dic19712yes—
136C22×C4C22⋊D19711yescharacteristic
137C22×C4C22⋊D19712yes—
138C22×C4C22×D19711yescharacteristic
139C2×C8—3942no—
140C2×C8—3942no—
141C2×C8—3942no—
142C4⋊C4—21no—
143C4⋊C4—3942no—
144C4⋊C4—1974no—
145C4⋊C4C2×Dic19712yes—
146C22⋊C4—24no—
147C22⋊C4—3944no—
148C42—3944no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—42no—
157C23—42no—
158C23C22⋊Dic19712yes—
159C23C2×C22⋊D19711yescharacteristic
160D4—24no—
161D4—48no—
162D4—42no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—42no—
169C2×C4—7882no—
170C2×C4—7882no—
171C2×C4—3944no—
172C2×C4C22⋊Dic19712yes—
173C2×C4C22×Dic19711yescharacteristic
174C2×C4D4×D19712yes—
175C2×C4C2×C22⋊D19711yescharacteristic
176C2×C4#1569135712yes—
177C8—7882no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—82no—
187C22—82no—
188C22C2×C22⋊Dic19711yescharacteristic
189C22#2026329411yescharacteristic
190C22#2026330711yescharacteristic
191C22#2026334612yes—
192C22#2026335212yes—
193C4—22no—
194C4—42no—
195C4—7884no—
196C4D4×Dic19712yes—
197C4#2026329812yes—
198C2—42no—
199C2—82no—
200C2#2716095911yescharacteristic
201C2#2716107511yescharacteristic
202C2#2716108411yescharacteristic
203C2#2716122212yes—
204C2#2716140012yes—
subgroupG/H
centerC23C2×C22⋊D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting, prime radicalC2×#121×C197C2
socleC23×C197C2×D4
cosocleC22×C4×C197C23
hypercenterC2×#121D197
hypocenterC197#1875
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#20263285C2semidirect
2. C2×#20263247C2semidirect
3. C2×C22⋊#13096802C2semidirect
4. #27160841C2semidirect
17. #121×C197C4semidirect
25. C4⋊Dic197D4semidirect
102. C197#1875semidirect
114. #121Dic197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632852
C2×#202632472
C2×C22⋊#130968022
#271608412
C2×#121×C1972
#1875197