Finite Group Databasebeta

← #38744376

Subgroups — #38744376

Order 25216.

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

normal subgroups only

This group has 258 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#27161628C211yesmaximal, characteristic
2#27161051C212yesmaximal, complemented
3#27160874C211yesmaximal, characteristic, complemented
4#27160871C212yesmaximal, complemented
5#354×C197C211yesmaximal, characteristic, complemented
6C2×#15691359C411yescharacteristic, complemented
7C2×#15691357C2211yescharacteristic
8#20263324C411yescharacteristic, complemented
9#20263322C2211yescharacteristic
10#20263314C411yescharacteristic, complemented
11#20263299—22no—
12#20263286—22no—
13#20263282—22no—
14C42×D197C411yescharacteristic, complemented
15#20263241C2212yes—
16#20263240—22no—
17#124×C197C2211yescharacteristic
18#99×C197C2212yes—
19#15691359—41no—
20#15691359—41no—
21#15691357—41no—
22#15691357—41no—
23C2×Dic394C2×C411yescharacteristic
24C2×D788—21no—
25C2×D788C2×C411yescharacteristic
26C2×C22⋊D197—22no—
27C2×C22⋊D197D412yes—
28C22×Dic197D412yes—
29C2×C4×D197—21no—
30C2×C4×D197—21no—
31C2×C4×D197C2×C411yescharacteristic
32#15691337—22no—
33#15691337—42no—
34C22⋊Dic197—22no—
35#15691335—22no—
36C4×Dic197—21no—
37C4×Dic197—42no—
38C4×Dic197C2×C411yescharacteristic
39C2×Q8×C197C2×C411yescharacteristic
40C2×D4×C197C2311yescharacteristic
41C22⋊C4×C197—22no—
42C22⋊C4×C197—22no—
43C42×C197C2×C411yescharacteristic
44C22×D197—21no—
45C22×D197C22⋊C411yescharacteristic
46Dic394—41no—
47D788—41no—
48D788—42no—
49C22⋊D197—42no—
50C22⋊D197—44no—
51C2×Dic197—21no—
52C2×Dic197—22no—
53C2×Dic197—22no—
54C2×Dic197—42no—
55C2×Dic197C22⋊C412yes—
56C2×Dic197C22⋊C411yescharacteristic
57C4×D197—21no—
58C4×D197—21no—
59C4×D197—42no—
60C4×D197—41no—
61C4×D197—41no—
62C23×C197C2×D412yes—
63Q8×C197—41no—
64D4×C197—41no—
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—42no—
68C2×C4×C197C22×C411yescharacteristic
69C22×C197—22no—
70C22×C197—42no—
71C22×C197C2×C22⋊C411yescharacteristic
72C2×D197—21no—
73C2×D197—41no—
74C2×D197—42no—
75C2×D197—41no—
76C4×C197—21no—
77C4×C197—41no—
78C4×C197—41no—
79C4×C197—82no—
80Dic197—22no—
81Dic197—22no—
82Dic197—82no—
83Dic197#9911yescharacteristic
84Dic197#9911yescharacteristic
85C2×C197—21no—
86C2×C197—42no—
87C2×C197C2×#9911yescharacteristic
88D197—41no—
89D197—81no—
90C197#212611yescharacteristic, complemented, Sylow
91#2126—1971nomaximal, Sylow
92#532—1971no—
93#410—1972no—
94#354—1971no—
95#354D19711yescharacteristic, complemented
96#353—1972no—
97C2×#41—1971no—
98C2×#41—1971no—
99C4⋊Q8—1971no—
100#127—1971no—
101#124—1971no—
102#124C2×D19711yescharacteristic
103#121—3942no—
104C4×D4—3942no—
105#117—3942no—
106C2×C42—1971no—
107#99—3942no—
108#99—1972no—
109#99C2×D19712yes—
110#41—7881no—
111#41—7881no—
112#41—7881no—
113#41—7881no—
114C2×Q8—1971no—
115C2×Q8C4×D19711yescharacteristic
116C2×D4—3941no—
117C2×D4—3942no—
118C2×D4—1972no—
119C2×D4—1971no—
120C2×D4C22×D19711yescharacteristic
121C22×C4—3941no—
122C22×C4—3941no—
123C22×C4—1972no—
124C22×C4—1971no—
125C4⋊C4—7882no—
126C4⋊C4—3942no—
127C22⋊C4—22no—
128C22⋊C4—22no—
129C22⋊C4—3942no—
130C22⋊C4—3942no—
131C42—7882no—
132C42—3941no—
133C42—1971no—
134C42C4×D19711yescharacteristic
135C23—3941no—
136C23—1971no—
137C23D4×D19712yes—
138Q8—41no—
139Q8—7881no—
140D4—41no—
141D4—7881no—
142D4—7882no—
143D4—7882no—
144D4—7884no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—7882no—
149C2×C4—7881no—
150C2×C4—7881no—
151C2×C4—7882no—
152C2×C4—3941no—
153C2×C4—3941no—
154C2×C4—3941no—
155C2×C4—3942no—
156C2×C4—3942no—
157C2×C4—1972no—
158C2×C4—1971no—
159C2×C4C2×C4×D19711yescharacteristic
160C22—22no—
161C22—42no—
162C22—7881no—
163C22—7882no—
164C22—7881no—
165C22—3941no—
166C22C22⋊C4×D19711yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—82no—
171C4—15762no—
172C4—3942no—
173C4—3942no—
174C4—1971no—
175C4—1971no—
176C2—21no—
177C2—42no—
178C2—7881no—
179C2—15761no—
180C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#354×C197C2
socleC2×C197C2×#99
prime radical#27161628C2
cosocleC2×D4×C197C23
hypercenter#354D197
hypocenterC197#2126
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. #27161051C2semidirect
3. #27160874C2semidirect
4. #27160871C2semidirect
5. #354×C197C2semidirect
6. C2×#15691359C4semidirect
8. #20263324C4semidirect
10. #20263314C4semidirect
14. C42×D197C4semidirect
90. C197#2126semidirect
95. #354D197semidirect

Maximal subgroups

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

subgroupindex
#271616282
#271610512
#271608742
#271608712
#354×C1972
#2126197