Finite Group Databasebeta

← #38741705

Subgroups — #38741705

Order 25216.

228 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161705C211yesmaximal, characteristic
2#27161410C212yesmaximal, complemented
3#27161004C211yesmaximal, characteristic
4#27160817C212yesmaximal, complemented
5#323×C197C211yesmaximal, characteristic
6#20263305—21no—
7#20263305C2212yes—
8#20263303—22no—
9#20263303—22no—
10C2×C4⋊Dic197C2211yescharacteristic
11C2×C22⋊Dic197—21no—
12C2×C22⋊Dic197—21no—
13#20263258—22no—
14#20263251—22no—
15#20263233—21no—
16#20263233C2211yescharacteristic
17C2×C22⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C2×Dic394—21no—
20C2×Dic394D412yes—
21C22×Dic197—21no—
22C22×Dic197—21no—
23C22×Dic197—21no—
24C22×Dic197—21no—
25C4⋊Dic197—21no—
26C4⋊Dic197—22no—
27C4⋊Dic197D412yes—
28#15691337—22no—
29#15691337—22no—
30C22⋊Dic197—22no—
31C22⋊Dic197—22no—
32C22⋊Dic197—42no—
33C24×C197D411yescharacteristic
34C22×C4×C197D411yescharacteristic
35C22×C4×C197C2311yescharacteristic
36C2×C8×C197—22no—
37C22⋊C4×C197—22no—
38Dic394—42no—
39Dic394—81no—
40C2×Dic197—22no—
41C2×Dic197—22no—
42C2×Dic197—41no—
43C2×Dic197—41no—
44C2×Dic197—41no—
45C2×Dic197—41no—
46C2×Dic197—41no—
47C2×Dic197—41no—
48C2×Dic197—81no—
49C2×Dic197—81no—
50C23×C197—21no—
51C23×C197—21no—
52C23×C197—41no—
53C23×C197—41no—
54C23×C197SD1612yes—
55C23×C197C2×D411yescharacteristic
56C2×C4×C197—21no—
57C2×C4×C197—41no—
58C2×C4×C197—41no—
59C2×C4×C197C2×D412yes—
60C8×C197—42no—
61C22×C197—21no—
62C22×C197—21no—
63C22×C197—22no—
64C22×C197—42no—
65C22×C197—41no—
66C22×C197—41no—
67C22×C197—81no—
68C22×C197#12011yescharacteristic
69C22×C197C2×SD1611yescharacteristic
70C22×C197#13711yescharacteristic
71C4×C197—22no—
72C4×C197—41no—
73Dic197—42no—
74Dic197—42no—
75Dic197—81no—
76Dic197—81no—
77C2×C197—21no—
78C2×C197—21no—
79C2×C197—42no—
80C2×C197#44811yescharacteristic
81C2×C197#45511yescharacteristic
82C2×C197#45811yescharacteristic
83C197#160811yescharacteristic, complemented, Sylow
84#1608—1971nomaximal, Sylow
85#543—1971no—
86#483—1972no—
87#397—1971no—
88#328—1972no—
89#323D19711yescharacteristic
90C22⋊Q8—3942no—
91C22⋊Q8—3942no—
92C22⋊Q8—3941no—
93C22⋊Q8—1972no—
94C2×C4⋊C4—1971no—
95C2×C22⋊C4—3941no—
96C2×C22⋊C4—3941no—
97C2×C22⋊C4C2×D19711yescharacteristic
98#106—3942no—
99#103—3942no—
100C22⋊C8C2×D19712yes—
101#95—3941no—
102#95—1971no—
103C24D78811yescharacteristic
104C2×Q8—3941no—
105C2×Q8—1972no—
106C22×C4—3941no—
107C22×C4—3941no—
108C22×C4—3941no—
109C22×C4—3941no—
110C22×C4D78811yescharacteristic
111C22×C4C22×D19711yescharacteristic
112C2×C8—22no—
113C4⋊C4—3942no—
114C4⋊C4—3941no—
115C4⋊C4—3942no—
116C4⋊C4—3942no—
117C4⋊C4—1972no—
118C22⋊C4—22no—
119C22⋊C4—7882no—
120C22⋊C4—3942no—
121C22⋊C4—3942no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1569134712yes—
127C23C2×D78811yescharacteristic
128Q8—7882no—
129Q8—15761no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—7881no—
134C2×C4—7881no—
135C2×C4—7881no—
136C2×C4—7881no—
137C2×C4—7881no—
138C2×C4—7881no—
139C2×C4—15761no—
140C2×C4—15761no—
141C2×C4—3942no—
142C2×C4—3942no—
143C2×C4D4×D19712yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#2026329611yescharacteristic
153C22C2×#1569134711yescharacteristic
154C22#2026335911yescharacteristic
155C4—22no—
156C4—41no—
157C4—7882no—
158C4—7882no—
159C4—15761no—
160C4—15761no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#2716122011yescharacteristic
165C2#2716125911yescharacteristic
166C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#323×C197C2
socleC22×C197#120
prime radicalC24×C197D4
hypercenter#323D197
hypocenterC197#1608
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27161410C2semidirect
4. #27160817C2semidirect
83. C197#1608semidirect

Maximal subgroups

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

subgroupindex
#271617052
#271614102
#271610042
#271608172
#323×C1972
#1608197