Finite Group Databasebeta

← #38753096

Subgroups — #38753096

Order 25216.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#27161632C211yesmaximal, characteristic, complemented
2C2×#20263313C211yesmaximal, characteristic
3#27161437C214yesmaximal, complemented
4#27161432C214yesmaximal, complemented
5#27161089C211yesmaximal, characteristic, complemented
6#27161087C211yesmaximal, characteristic, complemented
7#27161084C211yesmaximal, characteristic
8M16×Dic197C211yesmaximal, characteristic, complemented
9#418×C197C211yesmaximal, characteristic
10C22×Dic394C2211yescharacteristic
11C2×#15691342C2211yescharacteristic
12#20263325C2212yescomplemented
13#20263317C2212yescomplemented
14#20263313—22no—
15#20263313C2212yes—
16#20263313C2212yes—
17#20263303—22no—
18C4×Dic394—22no—
19#20263285C2211yescharacteristic
20C2×C22⋊Dic197—22no—
21C2×C4×Dic197C2211yescharacteristic
22#20263251C2214yescomplemented
23#20263249C2214yescomplemented
24#20263247C2214yes—
25#20263236C2212yescomplemented
26C8×Dic197C2212yescomplemented
27C22×D4×C197C2211yescharacteristic
28C2×M16×C197C2211yescharacteristic
29#117×C197C2211yescharacteristic
30#102×C197C2214yes—
31C2×Dic394—22no—
32C2×Dic394—21no—
33C2×Dic394—22no—
34C2×Dic394—22no—
35C2×Dic394C2312yes—
36C22×Dic197—21no—
37C22×Dic197D412yes—
38#15691342—22no—
39C2×#13096802C2312yes—
40C4⋊Dic197—22no—
41C4⋊Dic197C2312yes—
42#15691337—22no—
43#15691337—22no—
44C22⋊Dic197—24no—
45C22⋊Dic197—24no—
46C22⋊Dic197—21no—
47C4×Dic197—21no—
48C4×Dic197D412yes—
49C4×Dic197C2311yescharacteristic
50C4×Dic197C2311yescharacteristic
51C24×C197—21no—
52C2×D4×C197—21no—
53C2×D4×C197—22no—
54C2×D4×C197—22no—
55C2×D4×C197C2312yes—
56C22×C4×C197C2311yescharacteristic
57M16×C197—22no—
58C2×C8×C197C2312yes—
59C4⋊C4×C197C2312yes—
60C22⋊C4×C197—21no—
61C42×C197—21no—
62Dic394—22no—
63Dic394—22no—
64Dic394—44no—
65Dic394—42no—
66C2×Dic197—22no—
67C2×Dic197—22no—
68C2×Dic197—42no—
69C2×Dic197—42no—
70C2×Dic197C2×D412yes—
71C2×Dic197C2×D412yes—
72#13096802—22no—
73C23×C197—22no—
74C23×C197—44no—
75C23×C197—42no—
76C23×C197C2×D411yescharacteristic
77D4×C197—22no—
78D4×C197—22no—
79D4×C197—42no—
80C2×C4×C197—22no—
81C2×C4×C197C2×D411yescharacteristic
82C2×C4×C197#4112yes—
83C2×C4×C197#4112yes—
84C2×C4×C197C2411yescharacteristic
85C8×C197—22no—
86C22×C197—22no—
87C22×C197—42no—
88C22×C197—44no—
89C22×C197—84no—
90C22×C197#12412yes—
91C22×C197C22×D411yescharacteristic
92C4×C197—42no—
93C4×C197#12412yes—
94C4×C197C2×#4111yescharacteristic
95C4×C197C2×#4111yescharacteristic
96Dic197—22no—
97Dic197—22no—
98Dic197—42no—
99Dic197—42no—
100C2×C197—22no—
101C2×C197—82no—
102C2×C197C2×#12411yescharacteristic
103C2×C197#57512yes—
104C197#314011yescharacteristic, complemented, Sylow
105#3140—1971nomaximal, Sylow
106#533—1971no—
107C2×#124—1971no—
108#489—1974no—
109#488—1974no—
110#419—1971no—
111#419—1971no—
112#418—1971no—
113#418D19711yescharacteristic
114C4×M16—1971no—
115C22×Q8—1971no—
116C22×D4C2×D19711yescharacteristic
117C2×M16—1971no—
118C2×M16C2×D19711yescharacteristic
119C4⋊Q8—1972no—
120#125—1972no—
121#124—3942no—
122#124—1972no—
123#124—1972no—
124C22⋊Q8—3942no—
125C4×Q8—3942no—
126#117—1971no—
127#117C2×D19711yescharacteristic
128C2×C22⋊C4—3942no—
129C2×C42—1971no—
130#103—1974no—
131#103—1974no—
132#102—1974no—
133#102C2×D19714yes—
134#97—1972no—
135C4×C8—1972no—
136C24—21no—
137C2×Q8—3942no—
138C2×Q8—3942no—
139C2×Q8—3942no—
140C2×Q8—3941no—
141C2×Q8—1972no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C22×D19712yes—
146C22×C4—3941no—
147C22×C4—1972no—
148C22×C4C22×D19711yescharacteristic
149M16—22no—
150M16—3942no—
151C2×C8—1972no—
152C2×C8C22×D19712yes—
153C4⋊C4—3942no—
154C4⋊C4—3942no—
155C4⋊C4—3942no—
156C4⋊C4—1972no—
157C4⋊C4C22×D19712yes—
158C22⋊C4—21no—
159C22⋊C4—3941no—
160C22⋊C4—3944no—
161C22⋊C4—3944no—
162C42—21no—
163C42—3941no—
164C42—1972no—
165C42—1971no—
166C42—1971no—
167C23—22no—
168C23—44no—
169C23—42no—
170C23D4×D19711yescharacteristic
171Q8—7884no—
172Q8—7882no—
173Q8—3942no—
174Q8—3942no—
175D4—22no—
176D4—22no—
177D4—42no—
178C2×C4—22no—
179C2×C4—7882no—
180C2×C4—7882no—
181C2×C4—3942no—
182C2×C4—3942no—
183C2×C4—1972no—
184C2×C4—1972no—
185C2×C4D4×D19711yescharacteristic
186C2×C4#1569135712yes—
187C2×C4#1569135812yes—
188C2×C4C23×D19711yescharacteristic
189C8—22no—
190C8—3942no—
191C22—22no—
192C22—44no—
193C22—42no—
194C22—84no—
195C22#2026331212yes—
196C22C2×D4×D19711yescharacteristic
197C4—42no—
198C4—7882no—
199C4—7882no—
200C4—3942no—
201C4—3942no—
202C4#2026331212yes—
203C4C2×#1569135711yescharacteristic
204C4C2×#1569135811yescharacteristic
205C2—22no—
206C2—82no—
207C2C2×#2026331211yescharacteristic
208C2#2716187512yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#418×C197C2
socleC22×C197C22×D4
prime radicalC22×D4×C197C22
hypercenter#418D197
hypocenterC197#3140
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #27161632C2semidirect
3. #27161437C2semidirect
4. #27161432C2semidirect
5. #27161089C2semidirect
6. #27161087C2semidirect
8. M16×Dic197C2semidirect
12. #20263325C22semidirect
13. #20263317C22semidirect
22. #20263251C22semidirect
23. #20263249C22semidirect
25. #20263236C22semidirect
26. C8×Dic197C22semidirect
104. C197#3140semidirect

Maximal subgroups

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

subgroupindex
#271616322
C2×#202633132
#271614372
#271614322
#271610892
#271610872
#271610842
M16×Dic1972
#418×C1972
#3140197