Finite Group Databasebeta

← #38741671

Subgroups — #38741671

Order 25216.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161705C211yesmaximal, characteristic
2#27161237C212yesmaximal, complemented
3#27160991C211yesmaximal, characteristic
4#27160817C212yesmaximal, complemented
5#323×C197C211yesmaximal, characteristic
6C22×Dic394C2211yescharacteristic
7C2×Dic788—22no—
8#20263305—21no—
9#20263305C2212yes—
10#20263303—22no—
11#20263303—22no—
12C2×C22⋊Dic197—21no—
13C2×C22⋊Dic197—21no—
14#20263251—22no—
15#20263233—21no—
16#20263233C2211yescharacteristic
17C2×C22⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C2×Dic394—21no—
20C2×Dic394—22no—
21C2×Dic394—42no—
22C2×Dic394D412yes—
23C22×Dic197—21no—
24C22×Dic197—21no—
25C22×Dic197—21no—
26C22×Dic197—21no—
27Dic788—44no—
28C4⋊Dic197—21no—
29C4⋊Dic197D412yes—
30#15691337—22no—
31#15691337—22no—
32C22⋊Dic197—22no—
33C22⋊Dic197—22no—
34C22⋊Dic197—42no—
35C24×C197D411yescharacteristic
36C22×C4×C197D411yescharacteristic
37C22×C4×C197C2311yescharacteristic
38C2×C8×C197—22no—
39C22⋊C4×C197—22no—
40Dic394—24no—
41Dic394—42no—
42Dic394—42no—
43Dic394—81no—
44C2×Dic197—22no—
45C2×Dic197—22no—
46C2×Dic197—22no—
47C2×Dic197—41no—
48C2×Dic197—41no—
49C2×Dic197—41no—
50C2×Dic197—41no—
51C2×Dic197—41no—
52C2×Dic197—81no—
53C2×Dic197—81no—
54C23×C197—21no—
55C23×C197—21no—
56C23×C197—41no—
57C23×C197—41no—
58C23×C197Q1612yescomplemented
59C23×C197C2×D411yescharacteristic
60C2×C4×C197—21no—
61C2×C4×C197—41no—
62C2×C4×C197—41no—
63C2×C4×C197C2×D412yes—
64C8×C197—42no—
65C22×C197—21no—
66C22×C197—21no—
67C22×C197—22no—
68C22×C197—42no—
69C22×C197—41no—
70C22×C197—41no—
71C22×C197—81no—
72C22×C197#12011yescharacteristic
73C22×C197C2×Q1611yescharacteristic
74C22×C197#13711yescharacteristic
75C4×C197—22no—
76C4×C197—41no—
77Dic197—42no—
78Dic197—42no—
79Dic197—42no—
80Dic197—81no—
81C2×C197—21no—
82C2×C197—21no—
83C2×C197—42no—
84C2×C197#45111yescharacteristic
85C2×C197#45511yescharacteristic
86C2×C197B(4,𝔽2)11yescharacteristic
87C197#160211yescharacteristic, complemented, Sylow
88#1602—1971nomaximal, Sylow
89#543—1971no—
90#451—1972no—
91#393—1971no—
92#328—1972no—
93#323D19711yescharacteristic
94C22×Q8—1971no—
95C2×Q16—3942no—
96C22⋊Q8—3942no—
97C22⋊Q8—3942no—
98C22⋊Q8—3941no—
99C22⋊Q8—1972no—
100C2×C22⋊C4—3941no—
101C2×C22⋊C4—3941no—
102C2×C22⋊C4C2×D19711yescharacteristic
103#103—3942no—
104C22⋊C8C2×D19712yes—
105#95—3941no—
106#95—1971no—
107C24D78811yescharacteristic
108C2×Q8—7882no—
109C2×Q8—3941no—
110C2×Q8—3942no—
111C2×Q8—1972no—
112C22×C4—3941no—
113C22×C4—3941no—
114C22×C4—3941no—
115C22×C4—3941no—
116C22×C4D78811yescharacteristic
117C22×C4C22×D19711yescharacteristic
118Q16—7884no—
119C2×C8—22no—
120C4⋊C4—3942no—
121C4⋊C4—3941no—
122C4⋊C4—3942no—
123C4⋊C4—1972no—
124C22⋊C4—22no—
125C22⋊C4—7882no—
126C22⋊C4—3942no—
127C22⋊C4—3942no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic78812yescomplemented
133C23C2×D78811yescharacteristic
134Q8—7882no—
135Q8—7882no—
136Q8—15761no—
137Q8—3944no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—7881no—
142C2×C4—7881no—
143C2×C4—7881no—
144C2×C4—7881no—
145C2×C4—7881no—
146C2×C4—15761no—
147C2×C4—15761no—
148C2×C4—3942no—
149C2×C4—3942no—
150C2×C4—3942no—
151C2×C4D4×D19712yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2026329611yescharacteristic
161C22C2×Dic78811yescharacteristic
162C22#2026335911yescharacteristic
163C4—22no—
164C4—41no—
165C4—7882no—
166C4—7882no—
167C4—7882no—
168C4—15761no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2716123711yescharacteristic
173C2#2716125911yescharacteristic
174C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#323×C197C2
socleC22×C197#120
prime radicalC24×C197D4
hypercenter#323D197
hypocenterC197#1602
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27161237C2semidirect
4. #27160817C2semidirect
58. C23×C197Q16semidirect
87. C197#1602semidirect
132. C23Dic788semidirect

Maximal subgroups

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

subgroupindex
#271617052
#271612372
#271609912
#271608172
#323×C1972
#1602197