Finite Group Databasebeta

← #38744385

Subgroups — #38744385

Order 25216.

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

normal subgroups only

This group has 238 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#27161649C211yesmaximal, characteristic
2#27161051C212yesmaximal
3#27160874C211yesmaximal, characteristic, complemented
4#27160873C212yesmaximal, complemented
5#354×C197C211yesmaximal, characteristic, complemented
6C2×#15691357C2211yescharacteristic
7C2×Q8×D197C411yescharacteristic, complemented
8#20263326C411yescharacteristic, complemented
9#20263324C411yescharacteristic, complemented
10#20263313C2211yescharacteristic
11#20263308—22no—
12#20263303—22no—
13#20263283C411yescharacteristic, complemented
14#20263282—22no—
15#20263241C2212yes—
16#20263240—22no—
17#124×C197C2211yescharacteristic
18#99×C197C2212yes—
19#15691357—42no—
20C2×Dic394—21no—
21C2×Dic394C2×C411yescharacteristic
22C2×Dic394C2×C411yescharacteristic
23Q8×D197—42no—
24C2×C22⋊D197D412yes—
25C22×Dic197D412yes—
26C2×C4×D197—21no—
27C2×C4×D197C2×C411yescharacteristic
28C4⋊Dic197—22no—
29#15691337—21no—
30#15691337—22no—
31#15691337—42no—
32C22⋊Dic197—22no—
33C22⋊Dic197—22no—
34#15691335—21no—
35C4×Dic197—42no—
36C4×Dic197C2×C411yescharacteristic
37C2×Q8×C197C2×C411yescharacteristic
38C2×D4×C197C2311yescharacteristic
39C22⋊C4×C197—22no—
40C22⋊C4×C197—22no—
41C42×C197C2×C411yescharacteristic
42C22×D197C22⋊C411yescharacteristic
43Dic394—41no—
44Dic394—42no—
45Dic394—41no—
46C22⋊D197—42no—
47C2×Dic197—22no—
48C2×Dic197—22no—
49C2×Dic197—21no—
50C2×Dic197—21no—
51C2×Dic197—42no—
52C2×Dic197C22⋊C412yes—
53C2×Dic197C22⋊C411yescharacteristic
54C4×D197—22no—
55C4×D197—42no—
56C23×C197C2×D412yes—
57Q8×C197—41no—
58D4×C197—41no—
59C2×C4×C197—21no—
60C2×C4×C197—21no—
61C2×C4×C197—42no—
62C2×C4×C197C22×C411yescharacteristic
63C22×C197—22no—
64C22×C197—42no—
65C22×C197C2×C22⋊C411yescharacteristic
66C2×D197—21no—
67C2×D197—41no—
68C4×C197—21no—
69C4×C197—41no—
70C4×C197—41no—
71C4×C197—82no—
72Dic197—22no—
73Dic197—41no—
74Dic197—41no—
75Dic197—82no—
76Dic197#9912yes—
77C2×C197—21no—
78C2×C197—42no—
79C2×C197C2×#9911yescharacteristic
80D197—41no—
81C197#212911yescharacteristic, complemented, Sylow
82#2129—1971nomaximal, Sylow
83#536—1971no—
84#410—1972no—
85#354—1971no—
86#354—1972no—
87#354D19711yescharacteristic, complemented
88C2×#41—1971no—
89C22×Q8—1971no—
90C4⋊Q8—1971no—
91C4⋊Q8—1971no—
92#124—1971no—
93#124C2×D19711yescharacteristic
94#123—3942no—
95C22⋊Q8—3942no—
96#117—3942no—
97#117—1971no—
98#99—3942no—
99#99—1972no—
100#99C2×D19712yes—
101#41—7882no—
102C2×Q8—7882no—
103C2×Q8—3941no—
104C2×Q8—1971no—
105C2×Q8—1971no—
106C2×Q8C4×D19711yescharacteristic
107C2×D4—1972no—
108C2×D4C22×D19711yescharacteristic
109C22×C4—3941no—
110C22×C4—1972no—
111C22×C4—1971no—
112C4⋊C4—7882no—
113C4⋊C4—3941no—
114C4⋊C4—3942no—
115C4⋊C4—3942no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—3941no—
119C22⋊C4—3942no—
120C22⋊C4—3942no—
121C42—7882no—
122C42—1971no—
123C42C4×D19711yescharacteristic
124C23—1971no—
125C23D4×D19712yes—
126Q8—41no—
127Q8—7881no—
128Q8—7881no—
129Q8—7882no—
130D4—41no—
131D4—7882no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—7882no—
136C2×C4—7882no—
137C2×C4—3942no—
138C2×C4—3941no—
139C2×C4—3941no—
140C2×C4—3942no—
141C2×C4—3942no—
142C2×C4—1972no—
143C2×C4—1971no—
144C2×C4C2×C4×D19711yescharacteristic
145C22—22no—
146C22—42no—
147C22—7881no—
148C22—3941no—
149C22C22⋊C4×D19711yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—7881no—
155C4—7881no—
156C4—15762no—
157C4—3942no—
158C4—1972no—
159C2—21no—
160C2—42no—
161C2—7881no—
162C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#354×C197C2
socleC2×C197C2×#99
prime radicalC2×#15691357C22
cosocleC2×D4×C197C23
hypercenter#354D197
hypocenterC197#2129
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
3. #27160874C2semidirect
4. #27160873C2semidirect
5. #354×C197C2semidirect
7. C2×Q8×D197C4semidirect
8. #20263326C4semidirect
9. #20263324C4semidirect
13. #20263283C4semidirect
81. C197#2129semidirect
87. #354D197semidirect

Maximal subgroups

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

subgroupindex
#271616492
#271610512
#271608742
#271608732
#354×C1972
#2129197