Finite Group Databasebeta

← #38741750

Subgroups — #38741750

Order 25216.

246 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 51 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#27161656C211yesmaximal, characteristic
2#27161398C211yesmaximal, characteristic, complemented
3#27161395C211yesmaximal, characteristic, complemented
4#27161019C211yesmaximal, characteristic, complemented
5#27160812C211yesmaximal, characteristic, complemented
6#27160804C211yesmaximal, characteristic, complemented
7#327×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9#20263304—21no—
10#20263303—21no—
11#20263300C2211yescharacteristic
12#20263299—21no—
13#20263298—21no—
14#20263298—21no—
15C2×C4⋊Dic197C2211yescharacteristic
16C2×#15691337C2211yescharacteristic
17#20263258—21no—
18#20263257—21no—
19#20263248—21no—
20#20263247—21no—
21C22⋊#13096802C2211yescharacteristic
22#20263233—21no—
23#20263233—21no—
24#121×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27#15691357—42no—
28#15691357—42no—
29C2×Dic394—21no—
30C2×D788D411yescharacteristic
31C2×C22⋊D197—21no—
32C2×C22⋊D197—21no—
33C2×C22⋊D197—21no—
34C22×Dic197—21no—
35C22×Dic197—21no—
36C22×Dic197D411yescharacteristic
37C22×Dic197D411yescharacteristic
38C2×C4×D197—21no—
39C2×#13096802—21no—
40C4⋊Dic197—21no—
41C4⋊Dic197—21no—
42C4⋊Dic197D411yescharacteristic
43#15691337—21no—
44#15691337—21no—
45#15691337—21no—
46C22⋊Dic197—21no—
47#15691335—21no—
48#15691335—21no—
49C2×D4×C197—21no—
50C2×D4×C197D411yescharacteristic
51C22×C4×C197—21no—
52C22×C4×C197C2311yescharacteristic
53C2×C8×C197—21no—
54C4⋊C4×C197D411yescharacteristic
55C22⋊C4×C197—21no—
56C22×D197—21no—
57Dic394—42no—
58D788—41no—
59C22⋊D197—42no—
60C22⋊D197—41no—
61C22⋊D197—41no—
62C22⋊D197—81no—
63C2×Dic197—21no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—21no—
67C2×Dic197—42no—
68C2×Dic197—41no—
69C2×Dic197—42no—
70C2×Dic197—41no—
71C2×Dic197SD1612yes—
72C4×D197—42no—
73#13096802—41no—
74C23×C197—21no—
75C23×C197C2×D411yescharacteristic
76D4×C197—41no—
77D4×C197—41no—
78D4×C197—41no—
79C2×C4×C197—21no—
80C2×C4×C197—41no—
81C2×C4×C197—41no—
82C2×C4×C197—41no—
83C2×C4×C197—41no—
84C2×C4×C197C2×D411yescharacteristic
85C2×C4×C197C2×D411yescharacteristic
86C8×C197—41no—
87C22×C197—21no—
88C22×C197—21no—
89C22×C197—41no—
90C22×C197—41no—
91C22×C197—41no—
92C22×C197#12011yescharacteristic
93C22×C197C2×SD1611yescharacteristic
94C22×C197AGL(1,ℤ/8)11yescharacteristic
95C2×D197—41no—
96C2×D197—41no—
97C2×D197—41no—
98C4×C197—21no—
99C4×C197—21no—
100C4×C197—41no—
101C4×C197—81no—
102Dic197—22no—
103Dic197—41no—
104Dic197—41no—
105Dic197—81no—
106C2×C197—21no—
107C2×C197—21no—
108C2×C197—81no—
109C2×C197#45011yescharacteristic
110C2×C197#45411yescharacteristic
111C2×C197#45811yescharacteristic
112D197—81no—
113C197#161611yescharacteristic, complemented, Sylow
114#1616—1971nomaximal, Sylow
115#537—1971no—
116#481—1971no—
117#481—1971no—
118#400—1971no—
119#327—1971no—
120#327D19711yescharacteristic, complemented
121#324—1971no—
122C2×#41—3941no—
123C22⋊Q8—3941no—
124C22⋊Q8—3941no—
125#121—3941no—
126#121—3941no—
127#121—3941no—
128#121—1971no—
129#121C2×D19711yescharacteristic
130C2×C4⋊C4—1971no—
131C2×C4⋊C4—1971no—
132#106—3941no—
133#106—3941no—
134#102—3941no—
135#102—3941no—
136C22⋊C8—1971no—
137C22⋊C8C2×D19711yescharacteristic
138#95—3941no—
139#95—3941no—
140#95C2×D19711yescharacteristic
141#41—7882no—
142#41—7882no—
143C2×Q8—3941no—
144C2×D4—21no—
145C2×D4—3941no—
146C2×D4—3941no—
147C2×D4—3941no—
148C2×D4—1971no—
149C2×D4D78811yescharacteristic
150C22×C4—21no—
151C22×C4—3941no—
152C22×C4—3941no—
153C22×C4—3941no—
154C22×C4—1971no—
155C22×C4—1971no—
156C22×C4C22×D19711yescharacteristic
157C2×C8—21no—
158C2×C8—3941no—
159C4⋊C4—3941no—
160C4⋊C4—3941no—
161C4⋊C4—3941no—
162C4⋊C4—3941no—
163C4⋊C4—3941no—
164C4⋊C4—1971no—
165C4⋊C4D78811yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—3941no—
168C22⋊C4—3941no—
169C22⋊C4—3941no—
170C23—21no—
171C23—3941no—
172C23D4×D19711yescharacteristic
173Q8—7882no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—7881no—
178D4—7881no—
179D4—7881no—
180D4—7882no—
181D4—15761no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—7882no—
188C2×C4—7882no—
189C2×C4—7881no—
190C2×C4—7882no—
191C2×C4—7881no—
192C2×C4—3941no—
193C2×C4—3941no—
194C2×C4—3941no—
195C2×C4—3941no—
196C2×C4—1972no—
197C2×C4D4×D19711yescharacteristic
198C2×C4C2×D78811yescharacteristic
199C8—41no—
200C8—7881no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—7881no—
207C22—7881no—
208C22—7881no—
209C22#2026329611yescharacteristic
210C22SD16×D19711yescharacteristic
211C22#2026335011yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—7881no—
217C4—7881no—
218C4—15761no—
219C4—3942no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—15761no—
224C2#2716123111yescharacteristic
225C2#2716125311yescharacteristic
226C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197#120
prime radical#27161656C2
hypercenter#327D197
hypocenterC197#1616
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. #27161398C2semidirect
3. #27161395C2semidirect
4. #27161019C2semidirect
5. #27160812C2semidirect
6. #27160804C2semidirect
7. #327×C197C2semidirect
113. C197#1616semidirect
120. #327D197semidirect

Maximal subgroups

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

subgroupindex
#271616562
#271613982
#271613952
#271610192
#271608122
#271608042
#327×C1972
#1616197