Finite Group Databasebeta

← #38741049

Subgroups — #38741049

Order 25216.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×#20263288C211yesmaximal, characteristic
2#27161041C212yesmaximal, complemented
3#27160805C211yesmaximal, characteristic, complemented
4#27160802C212yesmaximal, complemented
5#324×C197C211yesmaximal, characteristic, complemented
6C22×D788C411yescharacteristic
7C22×C4×D197C411yescharacteristic
8C22×C4×D197C2211yescharacteristic
9C2×#15691341—22no—
10#20263288—22no—
11#20263288—22no—
12C2×#15691335C411yescharacteristic
13C2×#15691335C2211yescharacteristic
14C2×C4×Dic197C411yescharacteristic
15C22⋊#13096802C2212yes—
16#20263238—22no—
17C2×C4⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C23×D197C2×C411yescharacteristic
20C23×D197C2×C411yescharacteristic
21C2×D788—22no—
22C2×D788—22no—
23C2×D788—22no—
24C22×Dic197C2×C411yescharacteristic
25C22×Dic197C2×C411yescharacteristic
26C2×C4×D197—21no—
27C2×C4×D197—21no—
28C2×C4×D197—42no—
29C2×C4×D197—42no—
30C2×C4×D197D412yes—
31C2×C4×D197D412yes—
32#15691341—44no—
33C2×#13096802—22no—
34#15691335—22no—
35#15691335—22no—
36C4×Dic197—21no—
37C4×Dic197—21no—
38C22×C4×C197C2×C411yescharacteristic
39C22×C4×C197C2×C411yescharacteristic
40C22×C4×C197C2311yescharacteristic
41C2×C8×C197—22no—
42C4⋊C4×C197—22no—
43C22×D197—21no—
44C22×D197—21no—
45C22×D197—21no—
46C22×D197—21no—
47C22×D197—41no—
48C22×D197—41no—
49C22×D197—41no—
50C22×D197—41no—
51C22×D197C22⋊C412yes—
52C22×D197M1612yescomplemented
53D788—28no—
54C2×Dic197—22no—
55C2×Dic197—21no—
56C2×Dic197—41no—
57C2×Dic197C22⋊C411yescharacteristic
58C2×Dic197C22⋊C411yescharacteristic
59C4×D197—24no—
60C4×D197—42no—
61C4×D197—42no—
62C4×D197—42no—
63#13096802—42no—
64C23×C197C22×C411yescharacteristic
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—21no—
68C2×C4×C197—41no—
69C2×C4×C197—41no—
70C2×C4×C197M1612yescomplemented
71C2×C4×C197C2×D412yes—
72C8×C197—42no—
73C22×C197—21no—
74C22×C197—21no—
75C22×C197C2×C22⋊C411yescharacteristic
76C22×C197C2×M1611yescharacteristic
77C22×C197C2×M1611yescharacteristic
78C2×D197—22no—
79C2×D197—22no—
80C2×D197—22no—
81C2×D197—22no—
82C2×D197—42no—
83C2×D197—42no—
84C2×D197—42no—
85C2×D197—41no—
86C2×D197—41no—
87C2×D197—42no—
88C4×C197—22no—
89C4×C197—22no—
90C4×C197—41no—
91Dic197—21no—
92Dic197—41no—
93Dic197#9912yes—
94C2×C197—21no—
95C2×C197—21no—
96C2×C197#40711yescharacteristic
97C2×C197C2×#9911yescharacteristic
98C2×C197#41311yescharacteristic
99D197—42no—
100D197—42no—
101C197#146511yescharacteristic, complemented, Sylow
102#1465—1971nomaximal, Sylow
103C2×C4×D4—1971no—
104#407—1972no—
105#324—1971no—
106#324D19711yescharacteristic, complemented
107#323—1972no—
108C22×D4—1971no—
109C23×C4—1971no—
110C23×C4—1971no—
111C2×M16—3942no—
112C4×D4—3942no—
113C4×D4—3942no—
114C2×C4⋊C4C2×D19711yescharacteristic
115C2×C22⋊C4—1971no—
116C2×C22⋊C4—1971no—
117C2×C42—1971no—
118C22⋊C8—3942no—
119C22⋊C8—1972no—
120C22⋊C8C2×D19712yes—
121C24—1971no—
122C24—1971no—
123C2×D4—3942no—
124C2×D4—3942no—
125C2×D4—3942no—
126C22×C4—7882no—
127C22×C4—7882no—
128C22×C4—3941no—
129C22×C4—3941no—
130C22×C4—1971no—
131C22×C4—1972no—
132C22×C4—1971no—
133C22×C4—1972no—
134C22×C4C4×D19711yescharacteristic
135C22×C4C4×D19711yescharacteristic
136C22×C4C22×D19711yescharacteristic
137M16—7884no—
138C2×C8—22no—
139C2×C8—3942no—
140C4⋊C4—22no—
141C22⋊C4—3942no—
142C22⋊C4—3942no—
143C42—3941no—
144C42—3941no—
145C23—7881no—
146C23—7881no—
147C23—7881no—
148C23—7881no—
149C23—3941no—
150C23—3941no—
151C23—3941no—
152C23—3941no—
153C23—1972no—
154C23—1972no—
155C23C2×C4×D19711yescharacteristic
156D4—3948no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—7882no—
163C2×C4—7882no—
164C2×C4—7882no—
165C2×C4—7881no—
166C2×C4—3944no—
167C2×C4—3941no—
168C2×C4—3942no—
169C2×C4—1971no—
170C2×C4—1971no—
171C2×C4#1569134112yescomplemented
172C2×C4D4×D19712yes—
173C8—42no—
174C8—7882no—
175C22—21no—
176C22—21no—
177C22—7882no—
178C22—7882no—
179C22—7882no—
180C22—7882no—
181C22—7881no—
182C22—7881no—
183C22—3942no—
184C22—3942no—
185C22—3942no—
186C22—3942no—
187C22C22⋊C4×D19711yescharacteristic
188C22M16×D19711yescharacteristic
189C22C2×#1569134111yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—7881no—
194C4—3941no—
195C4—1972no—
196C2—21no—
197C2—21no—
198C2—7882no—
199C2—7882no—
200C2#2716104111yescharacteristic
201C2#99×D19711yescharacteristic
202C2#2716106211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC23×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#324×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC22×D788C4
cosocleC22×C4×C197C23
hypercenter#324D197
hypocenterC197#1465
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27161041C2semidirect
3. #27160805C2semidirect
4. #27160802C2semidirect
5. #324×C197C2semidirect
52. C22×D197M16semidirect
70. C2×C4×C197M16semidirect
101. C197#1465semidirect
106. #324D197semidirect
171. C2×C4#15691341semidirect

Maximal subgroups

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

subgroupindex
C2×#202632882
#271610412
#271608052
#271608022
#324×C1972
#1465197