Finite Group Databasebeta

← #38741762

Subgroups — #38741762

Order 25216.

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

normal subgroups only

This group has 240 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#27161667C211yesmaximal, characteristic
2#27161415C211yesmaximal, characteristic, complemented
3#27161398C211yesmaximal, characteristic, complemented
4#27161015C211yesmaximal, characteristic, complemented
5#27160817C211yesmaximal, characteristic, complemented
6#27160814C211yesmaximal, characteristic, complemented
7#323×C197C211yesmaximal, characteristic, complemented
8#20263319—21no—
9#20263312—21no—
10#20263308—21no—
11#20263305C2211yescharacteristic
12#20263300C2211yescharacteristic
13#20263299—21no—
14#20263295—21no—
15C4×C22⋊D197—21no—
16C2×C4⋊Dic197C2211yescharacteristic
17C2×#15691337—21no—
18C2×C22⋊Dic197—21no—
19#20263260—21no—
20#20263258—21no—
21#20263251—21no—
22#20263248—21no—
23#20263233C2211yescharacteristic
24C2×C22⋊C4×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26C22⋊C8×C197C2211yescharacteristic
27C2×Dic394D411yescharacteristic
28C2×D788D411yescharacteristic
29C2×C22⋊D197—21no—
30C2×C22⋊D197—21no—
31C2×C22⋊D197—21no—
32C22×Dic197—21no—
33C22×Dic197—21no—
34C2×C4×D197—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197D411yescharacteristic
38C4⋊Dic197D411yescharacteristic
39#15691337—21no—
40#15691337—21no—
41#15691337—42no—
42C22⋊Dic197—21no—
43C22⋊Dic197—21no—
44C22⋊Dic197—21no—
45C22⋊Dic197—42no—
46#15691335—21no—
47#15691335—21no—
48C4×Dic197—21no—
49C24×C197D411yescharacteristic
50C22×C4×C197D411yescharacteristic
51C22×C4×C197C2311yescharacteristic
52C2×C8×C197—21no—
53C2×C8×C197—21no—
54C22⋊C4×C197—21no—
55C22⋊C4×C197—21no—
56C22×D197—21no—
57Dic394—41no—
58D788—41no—
59C22⋊D197—41no—
60C22⋊D197—41no—
61C22⋊D197—81no—
62C22⋊D197—81no—
63C2×Dic197—22no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—21no—
67C2×Dic197—41no—
68C2×Dic197—41no—
69C2×Dic197—41no—
70C2×Dic197—41no—
71C2×Dic197—41no—
72C4×D197—81no—
73C23×C197—21no—
74C23×C197—21no—
75C23×C197—21no—
76C23×C197—41no—
77C23×C197—41no—
78C23×C197C2×D411yescharacteristic
79C2×C4×C197—21no—
80C2×C4×C197—41no—
81C2×C4×C197—41no—
82C2×C4×C197C2×D411yescharacteristic
83C2×C4×C197C2×D411yescharacteristic
84C8×C197—41no—
85C8×C197—41no—
86C22×C197—21no—
87C22×C197—21no—
88C22×C197—42no—
89C22×C197—41no—
90C22×C197—41no—
91C22×C197—41no—
92C22×C197—81no—
93C22×C197#12011yescharacteristic
94C22×C197#13511yescharacteristic
95C22×C197#13711yescharacteristic
96C2×D197—41no—
97C2×D197—41no—
98C2×D197—41no—
99C4×C197—21no—
100C4×C197—21no—
101C4×C197—41no—
102Dic197—42no—
103Dic197—41no—
104Dic197—41no—
105Dic197—41no—
106Dic197—81no—
107C2×C197—21no—
108C2×C197—21no—
109C2×C197—81no—
110C2×C197#45211yescharacteristic
111C2×C197#45311yescharacteristic
112C2×C197#45811yescharacteristic
113D197—81no—
114C197#161811yescharacteristic, complemented, Sylow
115#1618—1971nomaximal, Sylow
116#538—1971no—
117#484—1971no—
118#481—1971no—
119#399—1971no—
120#328—1971no—
121#327—1971no—
122#323D19711yescharacteristic, complemented
123#126—3941no—
124#124—3941no—
125#123—3941no—
126C22⋊Q8—1971no—
127#121—3941no—
128#121—1971no—
129#120—3941no—
130C4×D4—3941no—
131C2×C4⋊C4—3941no—
132C2×C4⋊C4—1971no—
133C2×C22⋊C4—3941no—
134C2×C22⋊C4C2×D19711yescharacteristic
135#107—3941no—
136#106—3941no—
137#103—3941no—
138#102—3941no—
139C22⋊C8C2×D19711yescharacteristic
140C22⋊C8C2×D19711yescharacteristic
141#95—1971no—
142C24D78811yescharacteristic
143C2×Q8—1971no—
144C2×D4—3941no—
145C2×D4—3941no—
146C2×D4—3941no—
147C2×D4—1971no—
148C22×C4—3941no—
149C22×C4—3941no—
150C22×C4—3941no—
151C22×C4D78811yescharacteristic
152C22×C4C22×D19711yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—7882no—
156C4⋊C4—3941no—
157C4⋊C4—3941no—
158C4⋊C4—3941no—
159C4⋊C4—3941no—
160C4⋊C4—1971no—
161C4⋊C4—1971no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—7882no—
165C22⋊C4—3941no—
166C22⋊C4—3941no—
167C22⋊C4—3941no—
168C22⋊C4—3941no—
169C22⋊C4—3941no—
170C42—3941no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—3941no—
177C23C2×D78811yescharacteristic
178Q8—7881no—
179D4—7881no—
180D4—7881no—
181D4—7881no—
182D4—15761no—
183D4—15761no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—7881no—
188C2×C4—7881no—
189C2×C4—7881no—
190C2×C4—7881no—
191C2×C4—7881no—
192C2×C4—15761no—
193C2×C4—3942no—
194C2×C4—3941no—
195C2×C4—3941no—
196C2×C4—3941no—
197C2×C4D4×D19711yescharacteristic
198C2×C4D4×D19711yescharacteristic
199C8—41no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—7881no—
209C22—7881no—
210C22—7881no—
211C22#2026329611yescharacteristic
212C22#2026334811yescharacteristic
213C22#2026335911yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—7882no—
218C4—7881no—
219C4—7881no—
220C4—7881no—
221C4—15761no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—15761no—
226C2#2716124311yescharacteristic
227C2#2716124911yescharacteristic
228C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#323×C197C2
socleC22×C197#120
prime radical#27161667C2
hypercenter#323D197
hypocenterC197#1618
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
2. #27161415C2semidirect
3. #27161398C2semidirect
4. #27161015C2semidirect
5. #27160817C2semidirect
6. #27160814C2semidirect
7. #323×C197C2semidirect
114. C197#1618semidirect
122. #323D197semidirect

Maximal subgroups

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

subgroupindex
#271616672
#271614152
#271613982
#271610152
#271608172
#271608142
#323×C1972
#1618197