Finite Group Databasebeta

← #38744367

Subgroups — #38744367

Order 25216.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#27161650C211yesmaximal, characteristic
2#27161053C211yesmaximal, characteristic
3#27160870C214yesmaximal, complemented
4#413×C197C211yesmaximal, characteristic
5C22×Dic394C411yescharacteristic, complemented
6#20263324—21no—
7#20263313—21no—
8#20263307C412yescomplemented
9#20263307C2212yes—
10#20263305—21no—
11#20263305—21no—
12#20263285—21no—
13#20263285C411yescharacteristic, complemented
14#20263241—21no—
15#20263241C2212yes—
16C2×#41×C197C2211yescharacteristic
17C2×M16×C197—21no—
18#101×C197—21no—
19#100×C197C2212yes—
20C2×Dic394—21no—
21C2×Dic394—21no—
22C2×Dic394—41no—
23C2×Dic394—41no—
24C22×Dic197C2×C412yes—
25C4⋊Dic197—21no—
26C4⋊Dic197—41no—
27#15691337—22no—
28#15691337—22no—
29C22⋊Dic197—22no—
30C22⋊Dic197—22no—
31C22⋊Dic197—22no—
32C22⋊Dic197C2×C412yes—
33C4×Dic197—21no—
34C4×Dic197—41no—
35#41×C197—41no—
36#41×C197—41no—
37C2×Q8×C197D411yescharacteristic
38C2×D4×C197C2×C411yescharacteristic
39C2×D4×C197D411yescharacteristic
40C2×D4×C197C2311yescharacteristic
41C22×C4×C197C2×C411yescharacteristic
42C22×C4×C197D411yescharacteristic
43C22×C4×C197D411yescharacteristic
44M16×C197—22no—
45M16×C197—41no—
46C2×C8×C197—41no—
47Dic394—42no—
48Dic394—41no—
49Dic394—41no—
50C2×Dic197—22no—
51C2×Dic197—22no—
52C2×Dic197—42no—
53C2×Dic197—42no—
54C23×C197C22⋊C411yescharacteristic
55C23×C197C22×C411yescharacteristic
56C23×C197C2×D411yescharacteristic
57Q8×C197—41no—
58D4×C197—41no—
59D4×C197—41no—
60D4×C197—41no—
61C2×C4×C197—21no—
62C2×C4×C197—21no—
63C2×C4×C197—21no—
64C2×C4×C197—21no—
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197C22⋊C411yescharacteristic
68C2×C4×C197C22⋊C411yescharacteristic
69C2×C4×C197C22⋊C411yescharacteristic
70C2×C4×C197C2×D411yescharacteristic
71C8×C197—42no—
72C22×C197—21no—
73C22×C197—21no—
74C22×C197—21no—
75C22×C197—41no—
76C22×C197—41no—
77C22×C197—41no—
78C22×C197C2×C22⋊C411yescharacteristic
79C4×C197—21no—
80C4×C197—21no—
81C4×C197—21no—
82C4×C197#9911yescharacteristic
83C4×C197#9911yescharacteristic
84Dic197—42no—
85Dic197—42no—
86Dic197—82no—
87C2×C197—21no—
88C2×C197—41no—
89C2×C197—41no—
90C2×C197—41no—
91C2×C197C2×#9911yescharacteristic
92C197#212311yescharacteristic, complemented, Sylow
93#2123—1971nomaximal, Sylow
94#536—1971no—
95#413D19711yescharacteristic
96#410—1971no—
97#352—1974no—
98C2×#41C2×D19711yescharacteristic
99C22×Q8—1971no—
100C2×M16—21no—
101C4⋊Q8—3941no—
102#124—3941no—
103#123—1972no—
104#123—1972no—
105C22⋊Q8—3941no—
106C22⋊Q8—3941no—
107#117—3941no—
108#117—1971no—
109#101—21no—
110#100C2×D19712yes—
111#99—3941no—
112#99—1972no—
113#41—41no—
114#41—41no—
115C2×Q8—7881no—
116C2×Q8—7881no—
117C2×Q8—3941no—
118C2×Q8—3941no—
119C2×Q8C22⋊D19711yescharacteristic
120C2×D4C4×D19711yescharacteristic
121C2×D4D78811yescharacteristic
122C2×D4C22×D19711yescharacteristic
123C22×C4—1972no—
124C22×C4C4×D19711yescharacteristic
125C22×C4C22⋊D19711yescharacteristic
126C22×C4D78811yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C4⋊C4—7881no—
131C4⋊C4—3942no—
132C4⋊C4—3941no—
133C4⋊C4—3942no—
134C22⋊C4—3942no—
135C22⋊C4—3942no—
136C22⋊C4—3942no—
137C22⋊C4—1972no—
138C42—7881no—
139C42—3941no—
140C23#1569133511yescharacteristic
141C23C2×C4×D19711yescharacteristic
142C23C2×D78811yescharacteristic
143Q8—41no—
144Q8—7882no—
145Q8—7881no—
146Q8—7881no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—7882no—
157C2×C4—7882no—
158C2×C4—3942no—
159C2×C4—3942no—
160C2×C4#1569133511yescharacteristic
161C2×C4#1569133511yescharacteristic
162C2×C4#1569133511yescharacteristic
163C2×C4C2×C22⋊D19711yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×#1569133511yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—7882no—
176C4—7882no—
177C4—15762no—
178C4#2026324011yescharacteristic
179C4#2026324011yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#2026324011yescharacteristic
subgroupG/H
centerC2C2×#20263240
commutatorC23×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#413×C197C2
socleC2×C197C2×#99
prime radicalC2×#41×C197C22
cosocleC2×D4×C197C23
hypercenter#413D197
hypocenterC197#2123
LCSG ▷ C23×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #27160870C2semidirect
5. C22×Dic394C4semidirect
8. #20263307C4semidirect
13. #20263285C4semidirect
92. C197#2123semidirect

Maximal subgroups

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

subgroupindex
#271616502
#271610532
#271608702
#413×C1972
#2123197