Finite Group Databasebeta

← #38752998

Subgroups — #38752998

Order 25216.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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#27161556C211yesmaximal, characteristic, complemented
2#27161400C212yesmaximal, complemented
3#27161383C212yesmaximal, complemented
4#27161276C212yesmaximal, complemented
5#27161217C212yesmaximal, complemented
6#27161179C212yesmaximal, complemented
7C4×#15691346C212yesmaximal, complemented
8#27161140C211yesmaximal, characteristic, complemented
9#540×C197C211yesmaximal, characteristic
10C22×Dic394C2211yescharacteristic
11C2×#15691346C2212yes—
12C2×#15691346C2212yes—
13C4×Dic394C2212yes—
14C4×Dic394C2212yes—
15#20263291—22no—
16#20263285C2211yescharacteristic
17#20263282—22no—
18#20263259C2212yes—
19#20263257C2212yes—
20C4⋊#13096802C2212yes—
21#20263249C2212yes—
22#20263249C2212yes—
23#20263247C2212yes—
24#20263247C2212yes—
25C22⋊#13096802C2212yes—
26#20263237C2211yescharacteristic
27C4×#13096802C2211yescharacteristic
28#127×C197C2211yescharacteristic
29#125×C197C2211yescharacteristic
30#123×C197—22no—
31#121×C197—22no—
32#121×C197C2212yes—
33C4×D4×C197C2212yes—
34#117×C197C2211yescharacteristic
35C2×Dic394—24no—
36C2×Dic394—21no—
37C2×Dic394C2312yes—
38C22×Dic197—21no—
39#15691346—24no—
40#15691346—24no—
41C2×#13096802C2312yes—
42C2×#13096802C2312yes—
43C4⋊Dic197C2312yes—
44#15691337—22no—
45#15691337—22no—
46C22⋊Dic197—21no—
47C4×Dic197—22no—
48C4×Dic197—22no—
49C4×Dic197—21no—
50C2×D4×C197—22no—
51C2×D4×C197—22no—
52C2×D4×C197C2312yes—
53C22×C4×C197—22no—
54C22×C4×C197C2311yescharacteristic
55C4⋊C4×C197—22no—
56C4⋊C4×C197D412yes—
57C4⋊C4×C197C2312yes—
58C22⋊C4×C197—22no—
59C22⋊C4×C197—22no—
60C22⋊C4×C197D412yes—
61C42×C197C2311yescharacteristic
62C42×C197C2311yescharacteristic
63Dic394—24no—
64Dic394—22no—
65Dic394#4114yes—
66C2×Dic197—24no—
67C2×Dic197—22no—
68C2×Dic197—22no—
69#13096802—22no—
70#13096802—22no—
71C23×C197—22no—
72C23×C197C2×D411yescharacteristic
73D4×C197—22no—
74D4×C197—22no—
75D4×C197—82no—
76D4×C197—82no—
77C2×C4×C197—22no—
78C2×C4×C197—22no—
79C2×C4×C197—82no—
80C2×C4×C197C2×D412yes—
81C2×C4×C197C2×D412yes—
82C2×C4×C197C2×D411yescharacteristic
83C2×C4×C197C2411yescharacteristic
84C22×C197—21no—
85C22×C197—22no—
86C22×C197—42no—
87C22×C197—42no—
88C22×C197—42no—
89C22×C197C22×D411yescharacteristic
90C4×C197—21no—
91C4×C197—22no—
92C4×C197—22no—
93C4×C197—42no—
94C4×C197C2×#4112yes—
95Dic197—24no—
96Dic197—42no—
97C2×C197—41no—
98C2×C197—82no—
99C2×C197#52511yescharacteristic
100C2×C197#57712yes—
101C197#312611yescharacteristic, complemented, Sylow
102#3126—1971nomaximal, Sylow
103#540D19711yescharacteristic
104#519—1971no—
105#482—1972no—
106#479—1972no—
107#460—1972no—
108#448—1972no—
109#440—1972no—
110C4×SD16—1972no—
111#433—1971no—
112C22×Q8—1971no—
113C2×SD16—1972no—
114C2×SD16—1972no—
115#127C2×D19711yescharacteristic
116#125C2×D19711yescharacteristic
117#123—22no—
118#121—22no—
119#121C2×D19712yes—
120C4×Q8—3942no—
121C4×Q8—1972no—
122C4×Q8—1972no—
123C4×D4C2×D19712yes—
124#117—3942no—
125#117—1971no—
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—3944no—
138C2×Q8—3941no—
139C2×Q8—1972no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4C22×D19712yes—
143C22×C4—22no—
144C22×C4—3941no—
145C22×C4C22×D19711yescharacteristic
146SD16—3944no—
147SD16—3944no—
148C2×C8—1972no—
149C2×C8—1972no—
150C4⋊C4—22no—
151C4⋊C4—3942no—
152C4⋊C4—3942no—
153C4⋊C4—1972no—
154C4⋊C4C22⋊D19712yes—
155C4⋊C4C22×D19712yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—3941no—
159C22⋊C4C22⋊D19712yes—
160C42—3942no—
161C42—3942no—
162C42—3941no—
163C42C22×D19711yescharacteristic
164C42C22×D19711yescharacteristic
165C23—22no—
166C23C2×C22⋊D19711yescharacteristic
167Q8—3944no—
168Q8—3942no—
169Q8—1974no—
170D4—22no—
171D4—22no—
172D4—82no—
173D4—82no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—82no—
177C2×C4—3942no—
178C2×C4—3944no—
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—42no—
189C22—42no—
190C22—42no—
191C22C22×C22⋊D19711yescharacteristic
192C4—22no—
193C4—22no—
194C4—21no—
195C4—42no—
196C4—7882no—
197C4—3944no—
198C4#41×D19712yes—
199C2—41no—
200C2—82no—
201C2#2716158611yescharacteristic
202C2#2716189012yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radical#540×C197C2
socleC22×C197C22×D4
hypercenter#540D197
hypocenterC197#3126
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #27161556C2semidirect
2. #27161400C2semidirect
3. #27161383C2semidirect
4. #27161276C2semidirect
5. #27161217C2semidirect
6. #27161179C2semidirect
7. C4×#15691346C2semidirect
8. #27161140C2semidirect
101. C197#3126semidirect

Maximal subgroups

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

subgroupindex
#271615562
#271614002
#271613832
#271612762
#271612172
#271611792
C4×#156913462
#271611402
#540×C1972
#3126197