Finite Group Databasebeta

← #38753003

Subgroups — #38753003

Order 25216.

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

normal subgroups only

This group has 382 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#27161684C211yesmaximal, characteristic
2#27161403C212yesmaximal, complemented
3#27161386C212yesmaximal, complemented
4#27161279C212yesmaximal, complemented
5#27161218C212yesmaximal, complemented
6#27161177C212yesmaximal, complemented
7C4×#15691345C212yesmaximal, complemented
8#27161140C211yesmaximal, characteristic, complemented
9#519×C197C211yesmaximal, characteristic, complemented
10C2×#15691345C2212yes—
11C2×#15691345C2212yes—
12#20263323C2211yescharacteristic
13#20263318C2211yescharacteristic
14#20263311—22no—
15#20263300C2212yes—
16#20263299—22no—
17C4×D788C2212yes—
18#20263259C2212yes—
19#20263257C2212yes—
20C4⋊#13096802C2212yes—
21#20263250C2212yes—
22#20263250C2212yes—
23#20263246C2212yes—
24#20263246C2212yes—
25C22⋊#13096802C2212yes—
26#20263237C2211yescharacteristic
27C4×#13096802C2211yescharacteristic
28C22×Q8×C197C2211yescharacteristic
29C4×Q8×C197—22no—
30C4×Q8×C197C2212yes—
31C4×Q8×C197C2212yes—
32#117×C197—22no—
33#117×C197C2211yescharacteristic
34#117×C197C2211yescharacteristic
35C2×D788—22no—
36C2×D788C2312yes—
37C2×C22⋊D197—22no—
38C2×C4×D197—22no—
39#15691345—24no—
40#15691345—24no—
41C2×#13096802C2312yes—
42C2×#13096802C2312yes—
43C4⋊Dic197C2312yes—
44#15691337—22no—
45#15691335—22no—
46#15691335—22no—
47C2×Q8×C197—24no—
48C2×Q8×C197—21no—
49C2×Q8×C197C2312yes—
50C22×C4×C197—21no—
51C22×C4×C197C2311yescharacteristic
52C4⋊C4×C197—22no—
53C4⋊C4×C197—22no—
54C4⋊C4×C197D412yes—
55C4⋊C4×C197C2312yes—
56C22⋊C4×C197—21no—
57C22⋊C4×C197D412yes—
58C42×C197—22no—
59C42×C197—22no—
60C42×C197—21no—
61C42×C197C2311yescharacteristic
62C42×C197C2311yescharacteristic
63C22×D197—22no—
64D788—22no—
65D788—22no—
66D788—82no—
67C22⋊D197—82no—
68C2×Dic197—22no—
69C4×D197—82no—
70#13096802—22no—
71#13096802—22no—
72C23×C197C2×D411yescharacteristic
73Q8×C197—24no—
74Q8×C197—22no—
75Q8×C197#4114yes—
76C2×C4×C197—24no—
77C2×C4×C197—22no—
78C2×C4×C197—22no—
79C2×C4×C197—22no—
80C2×C4×C197C2×D412yes—
81C2×C4×C197C2×D412yes—
82C2×C4×C197C2×D411yescharacteristic
83C2×C4×C197C2411yescharacteristic
84C22×C197—21no—
85C22×C197—22no—
86C22×C197C22×D411yescharacteristic
87C2×D197—42no—
88C2×D197—42no—
89C2×D197—42no—
90C4×C197—24no—
91C4×C197—21no—
92C4×C197—22no—
93C4×C197—22no—
94C4×C197—42no—
95C4×C197C2×#4112yes—
96Dic197—42no—
97C2×C197—41no—
98C2×C197#52511yescharacteristic
99C2×C197#57712yes—
100D197—82no—
101C197#312611yescharacteristic, complemented, Sylow
102#3126—1971nomaximal, Sylow
103#540—1971no—
104#519D19711yescharacteristic, complemented
105#482—1972no—
106#479—1972no—
107#460—1972no—
108#448—1972no—
109#440—1972no—
110C4×SD16—1972no—
111#433—1971no—
112C22×Q8C2×D19711yescharacteristic
113C2×SD16—1972no—
114C2×SD16—1972no—
115#127—1971no—
116#125—1971no—
117#123—3942no—
118#121—3942no—
119#121—1972no—
120C4×Q8—22no—
121C4×Q8C2×D19712yes—
122C4×Q8C2×D19712yes—
123C4×D4—1972no—
124#117—22no—
125#117C2×D19711yescharacteristic
126#117C2×D19711yescharacteristic
127#107—1972no—
128#106—1972no—
129C4⋊C8—1972no—
130#103—1972no—
131#103—1972no—
132#102—1972no—
133#102—1972no—
134C22⋊C8—1972no—
135#97—1971no—
136C4×C8—1971no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D19712yes—
140C2×D4—3942no—
141C2×D4—3942no—
142C2×D4—1972no—
143C22×C4—21no—
144C22×C4—3942no—
145C22×C4C22×D19711yescharacteristic
146SD16—3944no—
147SD16—3944no—
148C2×C8—1972no—
149C2×C8—1972no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3942no—
153C4⋊C4—1972no—
154C4⋊C4C22⋊D19712yes—
155C4⋊C4C22×D19712yes—
156C22⋊C4—21no—
157C22⋊C4—3942no—
158C22⋊C4—3942no—
159C22⋊C4C22⋊D19712yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D19711yescharacteristic
164C42C22×D19711yescharacteristic
165C23—3942no—
166C23C2×C22⋊D19711yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1569135814yes—
170D4—15762no—
171D4—15762no—
172D4—3942no—
173D4—3942no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—15762no—
179C2×C4—3942no—
180C2×C4C2×C22⋊D19712yes—
181C2×C4C2×C22⋊D19712yes—
182C2×C4C2×C22⋊D19711yescharacteristic
183C2×C4C23×D19711yescharacteristic
184C8—3942no—
185C8—3942no—
186C22—22no—
187C22—21no—
188C22—7882no—
189C22—7882no—
190C22—7882no—
191C22C22×C22⋊D19711yescharacteristic
192C4—22no—
193C4—22no—
194C4—24no—
195C4—21no—
196C4—42no—
197C4—7882no—
198C4C2×#1569135812yes—
199C2—41no—
200C2—15762no—
201C2#2716159011yescharacteristic
202C2#2716188712yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#519×C197C2
socleC22×C197C22×D4
prime radical#27161684C2
hypercenter#519D197
hypocenterC197#3126
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #27161403C2semidirect
3. #27161386C2semidirect
4. #27161279C2semidirect
5. #27161218C2semidirect
6. #27161177C2semidirect
7. C4×#15691345C2semidirect
8. #27161140C2semidirect
9. #519×C197C2semidirect
101. C197#3126semidirect
104. #519D197semidirect

Maximal subgroups

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

subgroupindex
#271616842
#271614032
#271613862
#271612792
#271612182
#271611772
C4×#156913452
#271611402
#519×C1972
#3126197