Finite Group Databasebeta

← #38741746

Subgroups — #38741746

Order 25216.

208 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161678C211yesmaximal, characteristic
2#27161412C211yesmaximal, characteristic, complemented
3#27161398C211yesmaximal, characteristic
4#27161019C211yesmaximal, characteristic, complemented
5#27160812C211yesmaximal, characteristic
6#27160804C211yesmaximal, characteristic, complemented
7#328×C197C211yesmaximal, characteristic, complemented
8#20263321—21no—
9#20263317—21no—
10#20263311—21no—
11#20263310—21no—
12#20263302—21no—
13#20263300C2211yescharacteristic
14#20263286—21no—
15C2×C4⋊Dic197C2211yescharacteristic
16C2×#15691337C2211yescharacteristic
17#20263259—21no—
18#20263258—21no—
19#20263250—21no—
20#20263248—21no—
21C22⋊#13096802C2211yescharacteristic
22#20263233—21no—
23#20263233—21no—
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C2×D788D411yescharacteristic
28C2×C22⋊D197—21no—
29C22×Dic197—21no—
30C22×Dic197D411yescharacteristic
31C22×Dic197D411yescharacteristic
32C2×C4×D197—21no—
33C2×#13096802—21no—
34C4⋊Dic197—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197—21no—
38C4⋊Dic197D411yescharacteristic
39#15691337—21no—
40#15691337—21no—
41#15691337—21no—
42#15691335—21no—
43#15691335—21no—
44#15691335—21no—
45#15691335—21no—
46C4×Dic197—21no—
47C2×Q8×C197D411yescharacteristic
48C22×C4×C197—21no—
49C22×C4×C197C2311yescharacteristic
50C2×C8×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197D411yescharacteristic
53C22⋊C4×C197—21no—
54C22×D197—21no—
55D788—41no—
56C22⋊D197—41no—
57C22⋊D197—41no—
58C2×Dic197—21no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—41no—
64C2×Dic197—42no—
65C2×Dic197—41no—
66C4×D197—81no—
67#13096802—41no—
68C23×C197C2×D411yescharacteristic
69Q8×C197—41no—
70C2×C4×C197—21no—
71C2×C4×C197—21no—
72C2×C4×C197—41no—
73C2×C4×C197—41no—
74C2×C4×C197—41no—
75C2×C4×C197—41no—
76C2×C4×C197C2×D411yescharacteristic
77C2×C4×C197C2×D411yescharacteristic
78C8×C197—41no—
79C22×C197—21no—
80C22×C197—21no—
81C22×C197#12011yescharacteristic
82C22×C197#13511yescharacteristic
83C22×C197#13711yescharacteristic
84C2×D197—41no—
85C2×D197—41no—
86C2×D197—41no—
87C4×C197—21no—
88C4×C197—21no—
89C4×C197—41no—
90C4×C197—41no—
91C4×C197—81no—
92Dic197—41no—
93Dic197—41no—
94Dic197—41no—
95Dic197—81no—
96C2×C197—21no—
97C2×C197—21no—
98C2×C197#45211yescharacteristic
99C2×C197#45411yescharacteristic
100C2×C197#45811yescharacteristic
101D197—81no—
102C197#161511yescharacteristic, complemented, Sylow
103#1615—1971nomaximal, Sylow
104#539—1971no—
105#484—1971no—
106#481—1971no—
107#400—1971no—
108#328D19711yescharacteristic, complemented
109#327—1971no—
110#324—1971no—
111#126—3941no—
112#125—3941no—
113#123—3941no—
114#123—3941no—
115C22⋊Q8—3941no—
116C22⋊Q8C2×D19711yescharacteristic
117#121—1971no—
118C4×D4—3941no—
119C2×C4⋊C4—1971no—
120C2×C4⋊C4—1971no—
121#107—3941no—
122#106—3941no—
123#103—3941no—
124#102—3941no—
125C22⋊C8—1971no—
126C22⋊C8C2×D19711yescharacteristic
127#95—3941no—
128#95—3941no—
129#95C2×D19711yescharacteristic
130C2×Q8D78811yescharacteristic
131C2×D4—3941no—
132C2×D4—1971no—
133C22×C4—21no—
134C22×C4—3941no—
135C22×C4—3941no—
136C22×C4—1971no—
137C22×C4—1971no—
138C22×C4C22×D19711yescharacteristic
139C2×C8—21no—
140C2×C8—3941no—
141C4⋊C4—21no—
142C4⋊C4—3941no—
143C4⋊C4—3941no—
144C4⋊C4—3941no—
145C4⋊C4—3941no—
146C4⋊C4—3941no—
147C4⋊C4—3941no—
148C4⋊C4—3941no—
149C4⋊C4—1971no—
150C4⋊C4D78811yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—3941no—
153C22⋊C4—3941no—
154C22⋊C4—3941no—
155C22⋊C4—3941no—
156C42—3941no—
157C23—3941no—
158C23D4×D19711yescharacteristic
159Q8—41no—
160D4—7881no—
161D4—7881no—
162D4—7881no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—7881no—
170C2×C4—7882no—
171C2×C4—7881no—
172C2×C4—15761no—
173C2×C4—3941no—
174C2×C4—3941no—
175C2×C4—3941no—
176C2×C4—3941no—
177C2×C4—3941no—
178C2×C4D4×D19711yescharacteristic
179C2×C4C2×D78811yescharacteristic
180C8—41no—
181C8—7881no—
182C22—21no—
183C22—21no—
184C22—7881no—
185C22—7881no—
186C22—7881no—
187C22#2026329611yescharacteristic
188C22#2026334711yescharacteristic
189C22#2026335811yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—81no—
195C4—7881no—
196C4—7881no—
197C4—7881no—
198C4—15761no—
199C2—21no—
200C2—21no—
201C2—15761no—
202C2#2716124411yescharacteristic
203C2#2716125311yescharacteristic
204C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197#120
prime radical#20263300C22
hypercenter#328D197
hypocenterC197#1615
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27161412C2semidirect
4. #27161019C2semidirect
6. #27160804C2semidirect
7. #328×C197C2semidirect
102. C197#1615semidirect
108. #328D197semidirect

Maximal subgroups

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

subgroupindex
#271616782
#271614122
#271613982
#271610192
#271608122
#271608042
#328×C1972
#1615197