Finite Group Databasebeta

← #38741654

Subgroups — #38741654

Order 25216.

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

normal subgroups only

This group has 266 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#27161710C211yesmaximal, characteristic, complemented
2#27161232C212yesmaximal
3#27160996C211yesmaximal, characteristic
4#27160817C212yesmaximal, complemented
5#324×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7C2×#15691347—22no—
8#20263325—22no—
9#20263317—22no—
10#20263305—21no—
11#20263305C2212yes—
12#20263285—21no—
13C2×#15691335—21no—
14C2×#15691335—21no—
15#20263248—22no—
16#20263233C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C23×D197—21no—
20C2×Dic394—21no—
21C2×Dic394D412yes—
22C2×D788—22no—
23C2×D788—42no—
24C22×Dic197—21no—
25#15691347—44no—
26C4⋊Dic197—21no—
27C4⋊Dic197—21no—
28C4⋊Dic197—21no—
29C4⋊Dic197D412yes—
30#15691337—22no—
31#15691337—22no—
32C22⋊Dic197—22no—
33#15691335—42no—
34#15691335—42no—
35C4×Dic197—22no—
36C22×C4×C197D411yescharacteristic
37C22×C4×C197D411yescharacteristic
38C22×C4×C197C2311yescharacteristic
39C2×C8×C197—22no—
40C4⋊C4×C197—22no—
41C22×D197—22no—
42C22×D197—41no—
43C22×D197—41no—
44C22×D197—41no—
45C22×D197—41no—
46C22×D197—41no—
47C22×D197—41no—
48Dic394—42no—
49Dic394—81no—
50D788—24no—
51D788—42no—
52C2×Dic197—22no—
53C2×Dic197—22no—
54C2×Dic197—41no—
55C2×Dic197—41no—
56C2×Dic197—41no—
57C23×C197C2×D411yescharacteristic
58C2×C4×C197—21no—
59C2×C4×C197—21no—
60C2×C4×C197—21no—
61C2×C4×C197—41no—
62C2×C4×C197—41no—
63C2×C4×C197SD1612yescomplemented
64C2×C4×C197C2×D412yes—
65C8×C197—42no—
66C22×C197—21no—
67C22×C197—21no—
68C22×C197#12011yescharacteristic
69C22×C197C2×SD1611yescharacteristic
70C22×C197#13711yescharacteristic
71C2×D197—42no—
72C2×D197—42no—
73C2×D197—42no—
74C2×D197—81no—
75C2×D197—81no—
76C2×D197—81no—
77C2×D197—81no—
78C4×C197—22no—
79C4×C197—22no—
80C4×C197—41no—
81Dic197—42no—
82Dic197—42no—
83Dic197—81no—
84C2×C197—21no—
85C2×C197—21no—
86C2×C197#44811yescharacteristic
87C2×C197#45411yescharacteristic
88C2×C197B(4,𝔽2)11yescharacteristic
89D197—82no—
90C197#159911yescharacteristic, complemented, Sylow
91#1599—1971nomaximal, Sylow
92#544—1971no—
93#450—1972no—
94#394—1971no—
95#328—1972no—
96#324D19711yescharacteristic, complemented
97C22×D4—1971no—
98C2×SD16—3942no—
99C4⋊Q8—3942no—
100#125—3942no—
101C22⋊Q8—3941no—
102C22⋊Q8—1972no—
103#117—3941no—
104C2×C4⋊C4C2×D19711yescharacteristic
105C2×C22⋊C4—3941no—
106C2×C22⋊C4—3941no—
107#102—3942no—
108C22⋊C8C2×D19712yes—
109#95—1971no—
110C24—3941no—
111C2×Q8—3941no—
112C2×Q8—1972no—
113C2×D4—7882no—
114C2×D4—3942no—
115C22×C4—3941no—
116C22×C4D78811yescharacteristic
117C22×C4D78811yescharacteristic
118C22×C4C22×D19711yescharacteristic
119SD16—7884no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—3942no—
123C4⋊C4—3941no—
124C4⋊C4—3941no—
125C4⋊C4—3941no—
126C4⋊C4—3942no—
127C4⋊C4—1972no—
128C22⋊C4—7882no—
129C22⋊C4—7882no—
130C22⋊C4—3942no—
131C42—3942no—
132C23—7881no—
133C23—7881no—
134C23—7881no—
135C23—7881no—
136C23—7881no—
137C23—7881no—
138C23—3942no—
139C23C2×D78811yescharacteristic
140Q8—7882no—
141Q8—15761no—
142D4—7882no—
143D4—3944no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—7881no—
150C2×C4—7881no—
151C2×C4—7881no—
152C2×C4—3942no—
153C2×C4—3942no—
154C2×C4#1569134712yescomplemented
155C2×C4D4×D19712yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—7882no—
160C22—7882no—
161C22—7882no—
162C22—15761no—
163C22—15761no—
164C22—15761no—
165C22—15761no—
166C22#2026329611yescharacteristic
167C22C2×#1569134711yescharacteristic
168C22#2026335911yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—7882no—
173C4—7882no—
174C4—15761no—
175C2—21no—
176C2—21no—
177C2—15762no—
178C2#2716122011yescharacteristic
179C2#2716125411yescharacteristic
180C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#324×C197C2
socleC22×C197#120
prime radicalC22×D788C22
hypercenter#324D197
hypocenterC197#1599
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #27161710C2semidirect
4. #27160817C2semidirect
5. #324×C197C2semidirect
63. C2×C4×C197SD16semidirect
90. C197#1599semidirect
96. #324D197semidirect
154. C2×C4#15691347semidirect

Maximal subgroups

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

subgroupindex
#271617102
#271612322
#271609962
#271608172
#324×C1972
#1599197