Finite Group Databasebeta

← #38743853

Subgroups — #38743853

Order 25216.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20263326C211yesmaximal, characteristic
2C2×#20263250C212yesmaximal
3#27160841C212yesmaximal
4#27160834C211yesmaximal, characteristic
5#391×C197C211yesmaximal, characteristic
6C22×Dic394—21no—
7C22×#13096802C2212yes—
8#20263326—24no—
9C2×C4⋊Dic197—21no—
10C2×C4⋊Dic197C2212yes—
11#20263250—24no—
12C22×Q8×C197C2211yescharacteristic
13C2×C4⋊C4×C197C2211yescharacteristic
14C2×C42×C197C2211yescharacteristic
15#95×C197—21no—
16#95×C197—21no—
17C2×Dic394—24no—
18C2×Dic394—42no—
19C2×Dic394—42no—
20C22×Dic197—22no—
21C2×#13096802—22no—
22C2×#13096802—22no—
23C2×#13096802—22no—
24C4⋊Dic197—24no—
25C4⋊Dic197—22no—
26C4⋊Dic197D414yes—
27C2×Q8×C197—21no—
28C2×Q8×C197—22no—
29C2×Q8×C197—22no—
30C2×Q8×C197Q812yescomplemented
31C22×C4×C197—21no—
32C22×C4×C197—21no—
33C22×C4×C197D411yescharacteristic
34C22×C4×C197D411yescharacteristic
35C22×C4×C197C2311yescharacteristic
36C4⋊C4×C197—22no—
37C42×C197—22no—
38Dic394—48no—
39C2×Dic197—24no—
40C2×Dic197—42no—
41C2×Dic197—42no—
42#13096802—42no—
43C23×C197C2×D411yescharacteristic
44Q8×C197—24no—
45Q8×C197—42no—
46C2×C4×C197—21no—
47C2×C4×C197—21no—
48C2×C4×C197—21no—
49C2×C4×C197—21no—
50C2×C4×C197—22no—
51C2×C4×C197—41no—
52C2×C4×C197—41no—
53C2×C4×C197—41no—
54C2×C4×C197—41no—
55C2×C4×C197—41no—
56C2×C4×C197SD1612yes—
57C2×C4×C197Q1612yes—
58C2×C4×C197C2×D412yes—
59C2×C4×C197C2×Q811yescharacteristic
60C2×C4×C197#4112yes—
61C2×C4×C197#4111yescharacteristic
62C22×C197#12011yescharacteristic
63C22×C197C22⋊Q811yescharacteristic
64C22×C197#12411yescharacteristic
65C22×C197C2×SD1611yescharacteristic
66C22×C197C2×Q1611yescharacteristic
67C22×C197AGL(1,ℤ/8)11yescharacteristic
68C22×C197#13711yescharacteristic
69C4×C197—22no—
70C4×C197—22no—
71C4×C197—42no—
72C4×C197—81no—
73C4×C197C22⋊Q812yes—
74C4×C197#12412yes—
75Dic197—44no—
76C2×C197#39311yescharacteristic
77C2×C197#45011yescharacteristic
78C2×C197#45111yescharacteristic
79C2×C197#47511yescharacteristic
80C2×C197#47711yescharacteristic
81C2×C197#48711yescharacteristic
82C2×C197#48911yescharacteristic
83C197#202411yescharacteristic, complemented, Sylow
84#2024—1971nomaximal, Sylow
85C2×C4⋊Q8—1971no—
86C2×#103—1972no—
87#391D19711yescharacteristic
88#340—1972no—
89#336—1971no—
90C22×Q8—3941no—
91C22×Q8C2×D19711yescharacteristic
92C22×C8—1972no—
93C4⋊Q8—3944no—
94C2×C4⋊C4—3941no—
95C2×C4⋊C4—1972no—
96C2×C4⋊C4C2×D19711yescharacteristic
97C2×C42C2×D19711yescharacteristic
98#103—3944no—
99#95—21no—
100#95—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—7882no—
105C2×Q8—7882no—
106C2×Q8—3944no—
107C2×Q8Dic39412yescomplemented
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—3942no—
111C22×C4C22⋊D19711yescharacteristic
112C22×C4C22⋊D19711yescharacteristic
113C22×C4C22×D19711yescharacteristic
114C2×C8—3942no—
115C2×C8—3942no—
116C2×C8—3942no—
117C4⋊C4—22no—
118C4⋊C4—3944no—
119C4⋊C4—3942no—
120C4⋊C4—1974no—
121C42—22no—
122C23C2×C22⋊D19711yescharacteristic
123Q8—24no—
124Q8—42no—
125Q8—7888no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—7882no—
137C2×C4—7882no—
138C2×C4—3944no—
139C2×C4#1569134512yes—
140C2×C4#1569134812yes—
141C2×C4D4×D19712yes—
142C2×C4C2×Dic39411yescharacteristic
143C2×C4#1569135712yes—
144C2×C4#1569135811yescharacteristic
145C8—7882no—
146C22#2026329411yescharacteristic
147C22#2026330511yescharacteristic
148C22#2026331311yescharacteristic
149C22C2×#1569134511yescharacteristic
150C22C2×#1569134811yescharacteristic
151C22#2026335111yescharacteristic
152C22#2026335611yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—81no—
157C4—7884no—
158C4#2026330312yes—
159C4#2026331212yes—
160C2#2716099111yescharacteristic
161C2#2716122911yescharacteristic
162C2#2716123411yescharacteristic
163C2#2716136411yescharacteristic
164C2#2716137411yescharacteristic
165C2#2716142911yescharacteristic
166C2#2716143811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D197
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#391×C197C2
socle, prime radicalC23×C197C2×D4
hypercenter#391D197
hypocenterC197#2024
LCSG ▷ C22×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C197Q8semidirect
83. C197#2024semidirect
107. C2×Q8Dic394semidirect

Maximal subgroups

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

subgroupindex
C2×#202633262
C2×#202632502
#271608412
#271608342
#391×C1972
#2024197