Finite Group Databasebeta

← #38744380

Subgroups — #38744380

Order 25216.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161647C211yesmaximal, characteristic
2#99×D197C212yesmaximal, complemented
3#27160874C211yesmaximal, characteristic, complemented
4#27160871C212yesmaximal, complemented
5#354×C197C211yesmaximal, characteristic, complemented
6C2×#15691359C411yescharacteristic, complemented
7C2×D4×D197C2211yescharacteristic
8#20263323C411yescharacteristic, complemented
9#20263322C2211yescharacteristic
10#20263315C411yescharacteristic, complemented
11#20263299—22no—
12#20263296—22no—
13#20263283C411yescharacteristic, complemented
14C22⋊C4×D197—22no—
15#20263241C2212yes—
16#20263240—22no—
17#124×C197C2211yescharacteristic
18#99×C197C2212yes—
19C23×D197D412yes—
20#15691359—42no—
21C2×D788—21no—
22C2×D788—22no—
23C2×D788C2×C411yescharacteristic
24C2×D788C2×C411yescharacteristic, complemented
25C2×C22⋊D197—22no—
26C2×C22⋊D197D412yes—
27D4×D197—42no—
28C2×C4×D197—21no—
29C2×C4×D197—42no—
30C2×C4×D197C2×C411yescharacteristic
31#15691337—21no—
32C22⋊Dic197—22no—
33#15691335—21no—
34#15691335—22no—
35#15691335—42no—
36C4×Dic197C2×C411yescharacteristic, complemented
37C2×Q8×C197C2×C411yescharacteristic, complemented
38C2×D4×C197C2311yescharacteristic
39C22⋊C4×C197—22no—
40C22⋊C4×C197—22no—
41C42×C197C2×C411yescharacteristic, complemented
42C22×D197—22no—
43C22×D197—22no—
44C22×D197—21no—
45C22×D197—82no—
46C22×D197C22⋊C412yes—
47C22×D197C22⋊C411yescharacteristic
48D788—41no—
49D788—41no—
50D788—42no—
51D788—82no—
52C22⋊D197—42no—
53C22⋊D197—82no—
54C2×Dic197—21no—
55C2×Dic197—42no—
56C2×Dic197C22⋊C411yescharacteristic
57C4×D197—22no—
58C4×D197—42no—
59C4×D197—84no—
60C23×C197C2×D412yes—
61Q8×C197—41no—
62D4×C197—41no—
63C2×C4×C197—21no—
64C2×C4×C197—21no—
65C2×C4×C197—42no—
66C2×C4×C197C22×C411yescharacteristic
67C22×C197—22no—
68C22×C197—42no—
69C22×C197C2×C22⋊C411yescharacteristic
70C2×D197—22no—
71C2×D197—41no—
72C2×D197—44no—
73C2×D197—42no—
74C2×D197—42no—
75C2×D197—41no—
76C2×D197—82no—
77C2×D197#9912yes—
78C4×C197—21no—
79C4×C197—41no—
80C4×C197—41no—
81C4×C197—82no—
82Dic197—21no—
83Dic197—41no—
84Dic197—82no—
85C2×C197—21no—
86C2×C197—42no—
87C2×C197C2×#9911yescharacteristic
88D197—22no—
89D197—42no—
90D197—81no—
91C197#212711yescharacteristic, complemented, Sylow
92#2127—1971nomaximal, Sylow
93#535—1971no—
94C2×#99—1972no—
95#354—1971no—
96#354D19711yescharacteristic, complemented
97#353—1972no—
98C2×#41—1971no—
99C22×D4—1971no—
100#127—1971no—
101#127—1971no—
102#124—1971no—
103#124C2×D19711yescharacteristic
104#121—3942no—
105#120—3942no—
106#117—1971no—
107C2×C22⋊C4—3942no—
108#99—3942no—
109#99—1972no—
110#99C2×D19712yes—
111C24—1972no—
112#41—7882no—
113C2×Q8C4×D19711yescharacteristic, complemented
114C2×D4—7882no—
115C2×D4—3941no—
116C2×D4—3942no—
117C2×D4—3942no—
118C2×D4—1972no—
119C2×D4—1971no—
120C2×D4—1971no—
121C2×D4C22×D19711yescharacteristic
122C22×C4—7882no—
123C22×C4—3941no—
124C22×C4—1971no—
125C4⋊C4—3941no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—7882no—
129C22⋊C4—3941no—
130C22⋊C4—3942no—
131C22⋊C4—3942no—
132C42—1971no—
133C42C4×D19711yescharacteristic, complemented
134C23—15762no—
135C23—3942no—
136C23—3942no—
137C23—3941no—
138C23—1972no—
139C23—1971no—
140C23D4×D19712yes—
141Q8—41no—
142D4—41no—
143D4—7881no—
144D4—7881no—
145D4—7882no—
146D4—7882no—
147D4—15762no—
148D4—15762no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—7882no—
153C2×C4—7882no—
154C2×C4—15764no—
155C2×C4—3942no—
156C2×C4—3941no—
157C2×C4—1971no—
158C2×C4C2×C4×D19711yescharacteristic
159C22—22no—
160C22—42no—
161C22—7882no—
162C22—7882no—
163C22—7881no—
164C22—7884no—
165C22—7881no—
166C22—15762no—
167C22—3942no—
168C22—1972no—
169C22C22⋊C4×D19711yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—7881no—
175C4—15762no—
176C4—3941no—
177C2—21no—
178C2—42no—
179C2—7882no—
180C2—15761no—
181C2—3942no—
182C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#354×C197C2
socleC2×C197C2×#99
prime radical#27161647C2
cosocleC2×D4×C197C23
hypercenter#354D197
hypocenterC197#2127
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #99×D197C2semidirect
3. #27160874C2semidirect
4. #27160871C2semidirect
5. #354×C197C2semidirect
6. C2×#15691359C4semidirect
8. #20263323C4semidirect
10. #20263315C4semidirect
13. #20263283C4semidirect
24. C2×D788C2×C4semidirect
36. C4×Dic197C2×C4semidirect
37. C2×Q8×C197C2×C4semidirect
41. C42×C197C2×C4semidirect
91. C197#2127semidirect
96. #354D197semidirect
113. C2×Q8C4×D197semidirect
133. C42C4×D197semidirect

Maximal subgroups

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

subgroupindex
#271616472
#99×D1972
#271608742
#271608712
#354×C1972
#2127197