Finite Group Databasebeta

← #38741712

Subgroups — #38741712

Order 25216.

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

normal subgroups only

This group has 216 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#27161407C211yesmaximal, characteristic, complemented
3#27161393C211yesmaximal, characteristic
4#27161011C211yesmaximal, 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⋊Dic197—21no—
16C2×C4⋊Dic197C2211yescharacteristic
17C2×#15691337C2211yescharacteristic
18#20263260—21no—
19#20263257—21no—
20#20263250—21no—
21#20263248—21no—
22C22⋊#13096802C2211yescharacteristic
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⋊Dic197—42no—
39C4⋊Dic197D411yescharacteristic
40#15691337—21no—
41#15691337—21no—
42#15691337—21no—
43#15691335—21no—
44#15691335—21no—
45#15691335—21no—
46#15691335—21no—
47C4×Dic197—21no—
48C2×Q8×C197D411yescharacteristic
49C22×C4×C197—21no—
50C22×C4×C197C2311yescharacteristic
51C2×C8×C197—21no—
52C4⋊C4×C197—21no—
53C4⋊C4×C197D411yescharacteristic
54C22⋊C4×C197—21no—
55C22×D197—21no—
56D788—41no—
57C22⋊D197—41no—
58C22⋊D197—41no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—21no—
64C2×Dic197—41no—
65C2×Dic197—41no—
66C2×Dic197—41no—
67C2×Dic197—41no—
68C4×D197—81no—
69#13096802—41no—
70C23×C197C2×D411yescharacteristic
71Q8×C197—41no—
72C2×C4×C197—22no—
73C2×C4×C197—21no—
74C2×C4×C197—21no—
75C2×C4×C197—41no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197C2×D411yescharacteristic
79C2×C4×C197C2×D411yescharacteristic
80C8×C197—41no—
81C22×C197—21no—
82C22×C197—21no—
83C22×C197#12011yescharacteristic
84C22×C197#13511yescharacteristic
85C22×C197AGL(1,ℤ/8)11yescharacteristic
86C2×D197—41no—
87C2×D197—41no—
88C2×D197—41no—
89C4×C197—21no—
90C4×C197—21no—
91C4×C197—42no—
92C4×C197—41no—
93C4×C197—41no—
94Dic197—41no—
95Dic197—41no—
96Dic197—41no—
97Dic197—81no—
98C2×C197—21no—
99C2×C197—21no—
100C2×C197#44911yescharacteristic
101C2×C197#45611yescharacteristic
102C2×C197#45811yescharacteristic
103D197—81no—
104C197#160911yescharacteristic, complemented, Sylow
105#1609—1971nomaximal, Sylow
106#539—1971no—
107#483—1971no—
108#480—1971no—
109#398—1971no—
110#328D19711yescharacteristic, complemented
111#327—1971no—
112#324—1971no—
113#126—3941no—
114#125—3941no—
115#123—3941no—
116#123—3941no—
117C22⋊Q8—3941no—
118C22⋊Q8C2×D19711yescharacteristic
119#121—1971no—
120C4×D4—3941no—
121C2×C4⋊C4—3941no—
122C2×C4⋊C4—1971no—
123C2×C4⋊C4—1971no—
124#107—3941no—
125#106—3941no—
126#103—3941no—
127#102—3941no—
128C22⋊C8—1971no—
129C22⋊C8C2×D19711yescharacteristic
130#95—3941no—
131#95C2×D19711yescharacteristic
132C2×Q8D78811yescharacteristic
133C2×D4—3941no—
134C2×D4—1971no—
135C22×C4—21no—
136C22×C4—3941no—
137C22×C4—3941no—
138C22×C4—1971no—
139C22×C4—1971no—
140C22×C4C22×D19711yescharacteristic
141C2×C8—21no—
142C2×C8—3941no—
143C4⋊C4—21no—
144C4⋊C4—7882no—
145C4⋊C4—3941no—
146C4⋊C4—3941no—
147C4⋊C4—3941no—
148C4⋊C4—3941no—
149C4⋊C4—3941no—
150C4⋊C4—3941no—
151C4⋊C4—3941no—
152C4⋊C4—1971no—
153C4⋊C4D78811yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—3941no—
156C22⋊C4—3941no—
157C22⋊C4—3941no—
158C22⋊C4—3941no—
159C42—3941no—
160C23—3941no—
161C23D4×D19711yescharacteristic
162Q8—41no—
163D4—7881no—
164D4—7881no—
165D4—7881no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—7881no—
173C2×C4—7881no—
174C2×C4—7881no—
175C2×C4—7881no—
176C2×C4—15761no—
177C2×C4—3941no—
178C2×C4—3941no—
179C2×C4—3941no—
180C2×C4—3941no—
181C2×C4—3941no—
182C2×C4D4×D19711yescharacteristic
183C2×C4C2×D78811yescharacteristic
184C8—41no—
185C8—7881no—
186C22—21no—
187C22—21no—
188C22—7881no—
189C22—7881no—
190C22—7881no—
191C22#2026329611yescharacteristic
192C22#2026334911yescharacteristic
193C22#2026335311yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—7881no—
200C4—7881no—
201C4—7881no—
202C4—15761no—
203C2—21no—
204C2—21no—
205C2—15761no—
206C2#2716122711yescharacteristic
207C2#2716126311yescharacteristic
208C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197#120
prime radical#20263300C22
hypercenter#328D197
hypocenterC197#1609
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. #27161407C2semidirect
4. #27161011C2semidirect
6. #27160804C2semidirect
7. #328×C197C2semidirect
104. C197#1609semidirect
110. #328D197semidirect

Maximal subgroups

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

subgroupindex
#271616782
#271614072
#271613932
#271610112
#271608122
#271608042
#328×C1972
#1609197