Finite Group Databasebeta

← #38741717

Subgroups — #38741717

Order 25216.

254 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161660C211yesmaximal, characteristic
2#27161393C212yesmaximal, complemented
3#27161009C211yesmaximal, characteristic, complemented
4#27160814C212yesmaximal, complemented
5#324×C197C211yesmaximal, characteristic, complemented
6C2×#15691358—21no—
7#20263306—22no—
8#20263301—22no—
9#20263300—21no—
10#20263300C2212yes—
11C2×C4⋊Dic197C2211yescharacteristic
12C2×#15691337—21no—
13#20263260—22no—
14#20263248—22no—
15#20263233—21no—
16#20263233C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19#15691358—42no—
20#15691358—42no—
21C2×Dic394—21no—
22C2×D788—21no—
23C2×D788—21no—
24C2×D788D412yes—
25C2×C22⋊D197—22no—
26C22×Dic197—21no—
27C22×Dic197—21no—
28C2×C4×D197—22no—
29C4⋊Dic197—21no—
30C4⋊Dic197—22no—
31C4⋊Dic197D412yes—
32#15691337—42no—
33#15691335—22no—
34#15691335—22no—
35C22×C4×C197D411yescharacteristic
36C22×C4×C197D411yescharacteristic
37C22×C4×C197C2311yescharacteristic
38C2×C8×C197—22no—
39C4⋊C4×C197—22no—
40C22×D197—22no—
41Dic394—42no—
42D788—42no—
43D788—42no—
44D788—81no—
45C22⋊D197—42no—
46C22⋊D197—42no—
47C2×Dic197—22no—
48C2×Dic197—22no—
49C2×Dic197—41no—
50C2×Dic197—41no—
51C2×Dic197—41no—
52C2×Dic197—41no—
53C2×Dic197—41no—
54C4×D197—44no—
55C23×C197C2×D411yescharacteristic
56C2×C4×C197—21no—
57C2×C4×C197—21no—
58C2×C4×C197—21no—
59C2×C4×C197—41no—
60C2×C4×C197—41no—
61C2×C4×C197D812yes—
62C2×C4×C197C2×D412yes—
63C8×C197—42no—
64C22×C197—21no—
65C22×C197—21no—
66C22×C197#12011yescharacteristic
67C22×C197C2×D811yescharacteristic
68C22×C197AGL(1,ℤ/8)11yescharacteristic
69C2×D197—42no—
70C2×D197—42no—
71C2×D197—42no—
72C4×C197—22no—
73C4×C197—22no—
74C4×C197—41no—
75Dic197—42no—
76Dic197—42no—
77Dic197—81no—
78C2×C197—21no—
79C2×C197—21no—
80C2×C197#44711yescharacteristic
81C2×C197#45611yescharacteristic
82C2×C197#45811yescharacteristic
83D197—82no—
84C197#161011yescharacteristic, complemented, Sylow
85#1610—1971nomaximal, Sylow
86#537—1971no—
87#480—1972no—
88#398—1971no—
89#327—1972no—
90#324D19711yescharacteristic, complemented
91C2×#41—3941no—
92C22⋊Q8—3942no—
93#121—3942no—
94#121—3941no—
95#121—1972no—
96C2×C4⋊C4—3941no—
97C2×C4⋊C4—1971no—
98C2×C4⋊C4C2×D19711yescharacteristic
99#107—3942no—
100#102—3942no—
101C22⋊C8C2×D19712yes—
102#95—3941no—
103#95—1971no—
104#41—7882no—
105#41—7882no—
106C2×Q8—3941no—
107C2×D4—3941no—
108C2×D4—3941no—
109C2×D4—3942no—
110C2×D4—1972no—
111C22×C4—3942no—
112C22×C4—3941no—
113C22×C4—3941no—
114C22×C4D78811yescharacteristic
115C22×C4D78811yescharacteristic
116C22×C4C22×D19711yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—7882no—
120C4⋊C4—3941no—
121C4⋊C4—3942no—
122C4⋊C4—1972no—
123C22⋊C4—3942no—
124C22⋊C4—3942no—
125C23—3942no—
126C23C2×D78811yescharacteristic
127Q8—7882no—
128D4—7882no—
129D4—7882no—
130D4—7882no—
131D4—7882no—
132D4—15761no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—7884no—
139C2×C4—7881no—
140C2×C4—7881no—
141C2×C4—7881no—
142C2×C4—7881no—
143C2×C4—7881no—
144C2×C4—3942no—
145C2×C4—3942no—
146C2×C4D157612yes—
147C2×C4D4×D19712yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—7882no—
152C22—7882no—
153C22—7882no—
154C22#2026329611yescharacteristic
155C22C2×D157611yescharacteristic
156C22#2026335411yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—7882no—
161C4—7882no—
162C4—15761no—
163C2—21no—
164C2—21no—
165C2—15762no—
166C2#2716121511yescharacteristic
167C2#2716126411yescharacteristic
168C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#324×C197C2
socleC22×C197#120
prime radical#27161660C2
hypercenter#324D197
hypocenterC197#1610
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27161393C2semidirect
3. #27161009C2semidirect
4. #27160814C2semidirect
5. #324×C197C2semidirect
84. C197#1610semidirect
90. #324D197semidirect

Maximal subgroups

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

subgroupindex
#271616602
#271613932
#271610092
#271608142
#324×C1972
#1610197