Finite Group Databasebeta

← #38741637

Subgroups — #38741637

Order 25216.

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

normal subgroups only

This group has 316 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, complemented
2#27161232C212yesmaximal, complemented
3#27160988C211yesmaximal, characteristic
4#27160817C212yesmaximal, complemented
5#323×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7C22×C22⋊D197—21no—
8C2×#15691347—22no—
9#20263305—21no—
10#20263305C2212yes—
11#20263303—22no—
12#20263303—22no—
13C2×C22⋊Dic197—21no—
14C2×#15691335—21no—
15#20263248—22no—
16#20263233C2211yescharacteristic
17C2×C22⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C23×D197—21no—
20C2×Dic394—21no—
21C2×Dic394D412yescomplemented
22C2×D788—22no—
23C2×D788—42no—
24C2×C22⋊D197—42no—
25C2×C22⋊D197—42no—
26C2×C22⋊D197—42no—
27C22×Dic197—21no—
28C22×Dic197—21no—
29C22×Dic197—21no—
30#15691347—44no—
31C4⋊Dic197—21no—
32C4⋊Dic197D412yescomplemented
33#15691337—22no—
34#15691337—22no—
35C22⋊Dic197—22no—
36C22⋊Dic197—22no—
37#15691335—42no—
38C24×C197D411yescharacteristic
39C22×C4×C197D411yescharacteristic
40C22×C4×C197C2311yescharacteristic
41C2×C8×C197—22no—
42C22⋊C4×C197—22no—
43C22×D197—22no—
44C22×D197—41no—
45C22×D197—41no—
46C22×D197—41no—
47C22×D197—41no—
48C22×D197—41no—
49C22×D197—41no—
50Dic394—42no—
51Dic394—81no—
52D788—24no—
53D788—42no—
54C22⋊D197—48no—
55C2×Dic197—22no—
56C2×Dic197—22no—
57C2×Dic197—22no—
58C2×Dic197—41no—
59C2×Dic197—41no—
60C2×Dic197—81no—
61C2×Dic197—81no—
62C23×C197—21no—
63C23×C197—21no—
64C23×C197—41no—
65C23×C197—41no—
66C23×C197SD1612yescomplemented
67C23×C197C2×D411yescharacteristic
68C2×C4×C197—21no—
69C2×C4×C197—41no—
70C2×C4×C197—41no—
71C2×C4×C197C2×D412yes—
72C8×C197—42no—
73C22×C197—21no—
74C22×C197—21no—
75C22×C197—22no—
76C22×C197—42no—
77C22×C197—41no—
78C22×C197—41no—
79C22×C197—81no—
80C22×C197#12011yescharacteristic
81C22×C197C2×SD1611yescharacteristic
82C22×C197#13711yescharacteristic
83C2×D197—42no—
84C2×D197—42no—
85C2×D197—42no—
86C2×D197—81no—
87C2×D197—81no—
88C2×D197—81no—
89C2×D197—81no—
90C4×C197—22no—
91C4×C197—41no—
92Dic197—42no—
93Dic197—42no—
94Dic197—42no—
95C2×C197—21no—
96C2×C197—21no—
97C2×C197—42no—
98C2×C197#44811yescharacteristic
99C2×C197#45311yescharacteristic
100C2×C197B(4,𝔽2)11yescharacteristic
101D197—82no—
102C197#159611yescharacteristic, complemented, Sylow
103#1596—1971nomaximal, Sylow
104#543—1971no—
105#450—1972no—
106#392—1971no—
107#328—1972no—
108#323D19711yescharacteristic, complemented
109C22×D4—3941no—
110C22×D4—1971no—
111C2×SD16—3942no—
112C22⋊Q8—3942no—
113C22⋊Q8—3942no—
114C22⋊Q8—3941no—
115C22⋊Q8—1972no—
116C2×C22⋊C4—3941no—
117C2×C22⋊C4—3941no—
118C2×C22⋊C4C2×D19711yescharacteristic
119#102—3942no—
120C22⋊C8C2×D19712yes—
121#95—1971no—
122C24—3941no—
123C24D78811yescharacteristic
124C2×Q8—3941no—
125C2×Q8—1972no—
126C2×D4—7882no—
127C2×D4—7882no—
128C2×D4—7882no—
129C2×D4—7882no—
130C2×D4—3942no—
131C22×C4—3941no—
132C22×C4—3941no—
133C22×C4—3941no—
134C22×C4D78811yescharacteristic
135C22×C4C22×D19711yescharacteristic
136SD16—7884no—
137C2×C8—22no—
138C4⋊C4—3942no—
139C4⋊C4—3941no—
140C4⋊C4—3942no—
141C4⋊C4—1972no—
142C22⋊C4—22no—
143C22⋊C4—7882no—
144C22⋊C4—3942no—
145C22⋊C4—3942no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—7881no—
151C23—7881no—
152C23—7881no—
153C23—7881no—
154C23—7881no—
155C23—7881no—
156C23—3942no—
157C23#1569134712yescomplemented
158C23C2×D78811yescharacteristic
159Q8—7882no—
160Q8—15761no—
161D4—7882no—
162D4—7888no—
163D4—3944no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—7881no—
168C2×C4—7881no—
169C2×C4—15761no—
170C2×C4—15761no—
171C2×C4—3942no—
172C2×C4—3942no—
173C2×C4—3942no—
174C2×C4D4×D19712yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—7882no—
184C22—7882no—
185C22—7882no—
186C22—15761no—
187C22—15761no—
188C22—15761no—
189C22—15761no—
190C22#2026329611yescharacteristic
191C22C2×#1569134711yescharacteristic
192C22#2026335911yescharacteristic
193C4—22no—
194C4—41no—
195C4—7882no—
196C4—7882no—
197C4—7882no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—15762no—
202C2#2716122011yescharacteristic
203C2#2716124911yescharacteristic
204C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#323×C197C2
socleC22×C197#120
prime radical#27160988C2
hypercenter#323D197
hypocenterC197#1596
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
1. #27161705C2semidirect
2. #27161232C2semidirect
4. #27160817C2semidirect
5. #323×C197C2semidirect
21. C2×Dic394D4semidirect
32. C4⋊Dic197D4semidirect
66. C23×C197SD16semidirect
102. C197#1596semidirect
108. #323D197semidirect
157. C23#15691347semidirect

Maximal subgroups

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

subgroupindex
#271617052
#271612322
#271609882
#271608172
#323×C1972
#1596197