Finite Group Databasebeta

← #38741401

Subgroups — #38741401

Order 25216.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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#27161634C211yesmaximal, characteristic
2#27161101C211yesmaximal, characteristic
3C4⋊C8×D197C211yesmaximal, characteristic
4#27160827C211yesmaximal, characteristic, complemented
5#27160827C211yesmaximal, characteristic, complemented
6#27160825C211yesmaximal, characteristic, complemented
7#332×C197C211yesmaximal, characteristic, complemented
8C2×#15691341—21no—
9C2×C8×D197—21no—
10#20263326C2211yescharacteristic
11#20263317—21no—
12#20263306C412yes—
13#20263302—21no—
14Q8×Dic197—21no—
15#20263291C412yes—
16#20263284—21no—
17C42×D197C2211yescharacteristic
18C4⋊#13096802C2211yescharacteristic
19C4⋊#13096802C2211yescharacteristic
20#20263255—21no—
21#20263255—21no—
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25C2×Dic394C2×C412yes—
26C2×C4×D197C2×C411yescharacteristic
27C2×C4×D197D411yescharacteristic
28C2×C4×D197D411yescharacteristic
29#15691341—42no—
30C2×#13096802—21no—
31C2×#13096802—21no—
32C8×D197—42no—
33C4⋊Dic197—22no—
34C4⋊Dic197—21no—
35C4⋊Dic197—21no—
36#15691337—22no—
37#15691335—22no—
38C4×Dic197—22no—
39C4×Dic197C2×C411yescharacteristic
40C4×Dic197D411yescharacteristic
41C4×Dic197D411yescharacteristic
42C2×Q8×C197—21no—
43C2×C8×C197—21no—
44C2×C8×C197—21no—
45C4⋊C4×C197—21no—
46C4⋊C4×C197C2×C412yes—
47C42×C197C2311yescharacteristic
48C22×D197C22⋊C411yescharacteristic
49Dic394—42no—
50C2×Dic197—22no—
51C2×Dic197C22⋊C411yescharacteristic
52C2×Dic197C22⋊C411yescharacteristic
53C2×Dic197C22⋊C411yescharacteristic
54C4×D197—21no—
55C4×D197—21no—
56C4×D197—41no—
57C4×D197SD1612yes—
58C4×D197Q1612yes—
59#13096802—41no—
60#13096802—41no—
61Q8×C197—41no—
62Q8×C197—41no—
63C2×C4×C197—22no—
64C2×C4×C197C22×C411yescharacteristic
65C2×C4×C197C2×D411yescharacteristic
66C2×C4×C197C2×D411yescharacteristic
67C8×C197—41no—
68C8×C197—41no—
69C22×C197C2×C22⋊C411yescharacteristic
70C2×D197—21no—
71C2×D197—21no—
72C2×D197#10312yes—
73C4×C197—21no—
74C4×C197—42no—
75C4×C197C2×SD1611yescharacteristic
76C4×C197C2×Q1611yescharacteristic
77C4×C197AGL(1,ℤ/8)11yescharacteristic
78C4×C197#13711yescharacteristic
79Dic197—21no—
80Dic197—21no—
81Dic197—42no—
82Dic197#10312yes—
83C2×C197#41311yescharacteristic
84C2×C197C2×#10311yescharacteristic
85C2×C197#41711yescharacteristic
86D197—22no—
87C197#155011yescharacteristic, complemented, Sylow
88#1550—1971nomaximal, Sylow
89#533—1971no—
90#423—1971no—
91C2×C4⋊C8—1971no—
92#333—1971no—
93#333—1971no—
94#332—1971no—
95#332D19711yescharacteristic, complemented
96C2×M16—3941no—
97C22×C8—3941no—
98C4⋊Q8—1971no—
99C4⋊Q8C2×D19711yescharacteristic
100#125—3941no—
101C22⋊Q8—3941no—
102C22⋊Q8—1972no—
103C4×Q8—3941no—
104C4×Q8—1972no—
105#117—3941no—
106C2×C42—1971no—
107C4⋊C8—3941no—
108C4⋊C8—3941no—
109C4⋊C8—1971no—
110C4⋊C8—1971no—
111C4⋊C8C2×D19711yescharacteristic
112C4⋊C8C2×D19711yescharacteristic
113C2×Q8—21no—
114C2×Q8—1972no—
115C22×C4—1971no—
116C22×C4—1971no—
117C22×C4—1971no—
118M16—7882no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—7882no—
122C2×C8—3941no—
123C2×C8—3941no—
124C4⋊C4—21no—
125C4⋊C4—3942no—
126C4⋊C4—3942no—
127C4⋊C4—3941no—
128C4⋊C4—3941no—
129C4⋊C4C4×D19712yes—
130C22⋊C4—3942no—
131C42—3942no—
132C42—1971no—
133C42—1971no—
134C42—1971no—
135C42C22×D19711yescharacteristic
136C23—1971no—
137Q8—41no—
138Q8—41no—
139Q8—7882no—
140C2×C4—22no—
141C2×C4—7881no—
142C2×C4—3942no—
143C2×C4—3941no—
144C2×C4—3941no—
145C2×C4—1971no—
146C2×C4—1971no—
147C2×C4—1971no—
148C2×C4—1972no—
149C2×C4—1972no—
150C2×C4C2×C4×D19711yescharacteristic
151C2×C4D4×D19711yescharacteristic
152C2×C4D4×D19711yescharacteristic
153C8—41no—
154C8—41no—
155C8—7881no—
156C8—7881no—
157C22—3941no—
158C22—3941no—
159C22—1972no—
160C22C22⋊C4×D19711yescharacteristic
161C4—21no—
162C4—42no—
163C4—7882no—
164C4—3941no—
165C4—3941no—
166C4—1972no—
167C4SD16×D19711yescharacteristic
168C4Q16×D19711yescharacteristic
169C4#2026335011yescharacteristic
170C4#2026335511yescharacteristic
171C2—3942no—
172C2#2716106211yescharacteristic
173C2#103×D19711yescharacteristic
174C2#2716107711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#332×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC22×D197C22⋊C4
cosocleC42×C197C23
hypercenter#332D197
hypocenterC197#1550
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #27160827C2semidirect
5. #27160827C2semidirect
6. #27160825C2semidirect
7. #332×C197C2semidirect
87. C197#1550semidirect
95. #332D197semidirect

Maximal subgroups

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

subgroupindex
#271616342
#271611012
C4⋊C8×D1972
#271608272
#271608252
#332×C1972
#1550197