Finite Group Databasebeta

← #38741727

Subgroups — #38741727

Order 25216.

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

normal subgroups only

This group has 232 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#27161666C211yesmaximal, characteristic
2#27161410C211yesmaximal, characteristic, complemented
3#27161390C211yesmaximal, characteristic, complemented
4#27161006C211yesmaximal, characteristic, complemented
5#27160815C211yesmaximal, characteristic, complemented
6#27160802C211yesmaximal, characteristic, complemented
7#327×C197C211yesmaximal, characteristic, complemented
8#20263321—21no—
9#20263312—21no—
10#20263310—21no—
11#20263305C2211yescharacteristic
12#20263298—21no—
13#20263294—21no—
14D4×Dic197—21no—
15C2×C4⋊Dic197C2211yescharacteristic
16C2×#15691335—21no—
17C2×#15691335C2211yescharacteristic
18#20263259—21no—
19#20263258—21no—
20#20263251—21no—
21#20263247—21no—
22C22⋊#13096802C2211yescharacteristic
23#20263233—21no—
24#121×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C23×D197D411yescharacteristic
28C2×Dic394D411yescharacteristic
29C2×C22⋊D197—21no—
30C2×C22⋊D197—21no—
31C22×Dic197—21no—
32C22×Dic197—21no—
33C22×Dic197D411yescharacteristic
34C2×#13096802—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197—21no—
38C4⋊Dic197D411yescharacteristic
39#15691337—21no—
40C22⋊Dic197—21no—
41C22⋊Dic197—21no—
42#15691335—21no—
43#15691335—21no—
44#15691335—21no—
45#15691335—21no—
46#15691335—42no—
47C4×Dic197—21no—
48C2×D4×C197—21no—
49C2×D4×C197D411yescharacteristic
50C22×C4×C197—21no—
51C22×C4×C197C2311yescharacteristic
52C2×C8×C197—21no—
53C4⋊C4×C197D411yescharacteristic
54C22⋊C4×C197—21no—
55C22×D197—21no—
56C22×D197—21no—
57C22×D197—21no—
58C22×D197—41no—
59C22×D197—41no—
60Dic394—41no—
61C22⋊D197—81no—
62C22⋊D197—81no—
63C2×Dic197—21no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—41no—
67C2×Dic197—41no—
68C2×Dic197—41no—
69C2×Dic197—41no—
70C2×Dic197—81no—
71#13096802—41no—
72C23×C197—21no—
73C23×C197C2×D411yescharacteristic
74D4×C197—41no—
75D4×C197—41no—
76D4×C197—41no—
77C2×C4×C197—21no—
78C2×C4×C197—41no—
79C2×C4×C197—41no—
80C2×C4×C197—41no—
81C2×C4×C197—41no—
82C2×C4×C197C2×D411yescharacteristic
83C2×C4×C197C2×D411yescharacteristic
84C8×C197—41no—
85C22×C197—21no—
86C22×C197—21no—
87C22×C197—41no—
88C22×C197—41no—
89C22×C197—41no—
90C22×C197#12011yescharacteristic
91C22×C197#13511yescharacteristic
92C22×C197AGL(1,ℤ/8)11yescharacteristic
93C2×D197—42no—
94C2×D197—41no—
95C2×D197—41no—
96C2×D197—41no—
97C2×D197—81no—
98C4×C197—21no—
99C4×C197—21no—
100C4×C197—41no—
101C4×C197—81no—
102Dic197—41no—
103Dic197—41no—
104Dic197—41no—
105Dic197—81no—
106C2×C197—21no—
107C2×C197—21no—
108C2×C197—81no—
109C2×C197#44911yescharacteristic
110C2×C197#45511yescharacteristic
111C2×C197#45811yescharacteristic
112D197—81no—
113C197#161211yescharacteristic, complemented, Sylow
114#1612—1971nomaximal, Sylow
115#538—1971no—
116#483—1971no—
117#480—1971no—
118#397—1971no—
119#328—1971no—
120#327D19711yescharacteristic, complemented
121#323—1971no—
122#126—3941no—
123#124—3941no—
124#123—3941no—
125C22⋊Q8—1971no—
126#121—3941no—
127#121C2×D19711yescharacteristic
128#120—3941no—
129C4×D4—3941no—
130C2×C4⋊C4—1971no—
131C2×C22⋊C4—3941no—
132C2×C22⋊C4—1971no—
133#107—3941no—
134#106—3941no—
135#103—3941no—
136#102—3941no—
137C22⋊C8—1971no—
138C22⋊C8C2×D19711yescharacteristic
139#95—3941no—
140#95C2×D19711yescharacteristic
141C24—1971no—
142C2×Q8—1971no—
143C2×D4—21no—
144C2×D4—3941no—
145C2×D4—3941no—
146C2×D4D78811yescharacteristic
147C22×C4—21no—
148C22×C4—3941no—
149C22×C4—3941no—
150C22×C4—1971no—
151C22×C4C22×D19711yescharacteristic
152C2×C8—21no—
153C2×C8—3941no—
154C4⋊C4—3941no—
155C4⋊C4—3941no—
156C4⋊C4—3941no—
157C4⋊C4—3941no—
158C4⋊C4—1971no—
159C4⋊C4D78811yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—7882no—
162C22⋊C4—3941no—
163C22⋊C4—3941no—
164C22⋊C4—3941no—
165C22⋊C4—3941no—
166C22⋊C4—3941no—
167C22⋊C4—3941no—
168C42—3941no—
169C23—21no—
170C23—7881no—
171C23—7881no—
172C23—3941no—
173C23—3941no—
174C23—3941no—
175C23D4×D19711yescharacteristic
176Q8—7881no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—15761no—
181D4—15761no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—7881no—
188C2×C4—7881no—
189C2×C4—7881no—
190C2×C4—7881no—
191C2×C4—15761no—
192C2×C4—3941no—
193C2×C4—3941no—
194C2×C4—3941no—
195C2×C4D4×D19711yescharacteristic
196C2×C4C2×D78811yescharacteristic
197C8—41no—
198C8—7881no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—7882no—
205C22—7881no—
206C22—7881no—
207C22—7881no—
208C22—15761no—
209C22#2026329611yescharacteristic
210C22#2026334711yescharacteristic
211C22#2026335011yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—7881no—
217C4—7881no—
218C4—7881no—
219C4—15761no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—15761no—
224C2#2716122511yescharacteristic
225C2#2716125811yescharacteristic
226C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197#120
prime radical#27161666C2
hypercenter#327D197
hypocenterC197#1612
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27161410C2semidirect
3. #27161390C2semidirect
4. #27161006C2semidirect
5. #27160815C2semidirect
6. #27160802C2semidirect
7. #327×C197C2semidirect
113. C197#1612semidirect
120. #327D197semidirect

Maximal subgroups

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

subgroupindex
#271616662
#271614102
#271613902
#271610062
#271608152
#271608022
#327×C1972
#1612197