Finite Group Databasebeta

← #38742046

Subgroups — #38742046

Order 25216.

184 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161489C211yesmaximal, characteristic, complemented
2#27161449C211yesmaximal, characteristic, complemented
3#27161438C211yesmaximal, characteristic
4#27160821C211yesmaximal, characteristic, complemented
5#27160821C211yesmaximal, characteristic, complemented
6#27160819C211yesmaximal, characteristic
7#563×C197C211yesmaximal, characteristic
8#20263326C2211yescharacteristic
9#20263259—21no—
10#20263257—21no—
11#20263250—21no—
12#20263249—21no—
13#20263249—21no—
14#20263247—21no—
15#20263237C2211yescharacteristic
16#20263237C2211yescharacteristic
17#20263237C2211yescharacteristic
18C4⋊Q8×C197—21no—
19#126×C197—21no—
20#126×C197—21no—
21#125×C197C2211yescharacteristic
22#125×C197C2211yescharacteristic
23#124×C197C2211yescharacteristic
24#123×C197—21no—
25C22⋊Q8×C197—21no—
26C22⋊Q8×C197—21no—
27C2×Dic394—21no—
28C2×#13096802—21no—
29C2×#13096802—21no—
30C2×#13096802—21no—
31C4⋊Dic197—21no—
32C4⋊Dic197—21no—
33C2×Q8×C197—21no—
34C2×Q8×C197D411yescharacteristic
35C2×D4×C197D411yescharacteristic
36C22×C4×C197—21no—
37C4⋊C4×C197—21no—
38C4⋊C4×C197—21no—
39C4⋊C4×C197—21no—
40C4⋊C4×C197—21no—
41C4⋊C4×C197—21no—
42C4⋊C4×C197—21no—
43C4⋊C4×C197D411yescharacteristic
44C4⋊C4×C197D411yescharacteristic
45C4⋊C4×C197D411yescharacteristic
46C4⋊C4×C197D411yescharacteristic
47C22⋊C4×C197—21no—
48C22⋊C4×C197—21no—
49C22⋊C4×C197—21no—
50C22⋊C4×C197—21no—
51C22⋊C4×C197—21no—
52C42×C197—21no—
53C42×C197C2311yescharacteristic
54Dic394—42no—
55C2×Dic197—41no—
56#13096802—22no—
57#13096802—41no—
58#13096802—41no—
59C23×C197—21no—
60Q8×C197—41no—
61Q8×C197—81no—
62D4×C197—41no—
63C2×C4×C197—21no—
64C2×C4×C197—21no—
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—21no—
68C2×C4×C197—81no—
69C2×C4×C197C2×D411yescharacteristic
70C2×C4×C197C2×D411yescharacteristic
71C2×C4×C197C2×D411yescharacteristic
72C22×C197—41no—
73C22×C197—41no—
74C22×C197—41no—
75C22×C197#12011yescharacteristic
76C4×C197—21no—
77C4×C197—21no—
78C4×C197—21no—
79C4×C197—41no—
80C4×C197—41no—
81C4×C197—41no—
82C4×C197—41no—
83C4×C197—41no—
84Dic197—81no—
85C2×C197—81no—
86C2×C197#45411yescharacteristic
87C2×C197#45511yescharacteristic
88C2×C197#45611yescharacteristic
89C197#166411yescharacteristic, complemented, Sylow
90#1664—1971nomaximal, Sylow
91#563D19711yescharacteristic
92#501—1971no—
93#491—1971no—
94#489—1971no—
95#330—1971no—
96#330—1971no—
97#329—1971no—
98C4⋊Q8—21no—
99C4⋊Q8—1971no—
100#126—21no—
101#126—21no—
102#125C2×D19711yescharacteristic
103#125C2×D19711yescharacteristic
104#124C2×D19711yescharacteristic
105#123—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#107—3941no—
109#106—3941no—
110#103—3941no—
111#103—3941no—
112#103—3941no—
113#102—3941no—
114#97—1971no—
115#97—1971no—
116#97—1971no—
117C2×Q8—21no—
118C2×Q8—3941no—
119C2×Q8C22⋊D19711yescharacteristic
120C2×D4C22⋊D19711yescharacteristic
121C22×C4—21no—
122C2×C8—3941no—
123C2×C8—3941no—
124C2×C8—3941no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3941no—
132C4⋊C4—3941no—
133C4⋊C4C22⋊D19711yescharacteristic
134C4⋊C4C22⋊D19711yescharacteristic
135C4⋊C4C22⋊D19711yescharacteristic
136C4⋊C4C22⋊D19711yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C22×D19711yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—81no—
147Q8—7882no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—81no—
155C2×C4—7881no—
156C2×C4C2×C22⋊D19711yescharacteristic
157C2×C4C2×C22⋊D19711yescharacteristic
158C2×C4C2×C22⋊D19711yescharacteristic
159C8—7881no—
160C8—7881no—
161C8—3942no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#2026329511yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—15761no—
175C2—81no—
176C2#2716125211yescharacteristic
177C2#2716125711yescharacteristic
178C2#2716126211yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#563×C197C2
socleC22×C197#120
prime radicalC2×D4×C197D4
hypercenter#563D197
hypocenterC197#1664
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #27161489C2semidirect
2. #27161449C2semidirect
4. #27160821C2semidirect
5. #27160821C2semidirect
89. C197#1664semidirect

Maximal subgroups

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

subgroupindex
#271614892
#271614492
#271614382
#271608212
#271608192
#563×C1972
#1664197