Finite Group Databasebeta

← #38741715

Subgroups — #38741715

Order 25216.

254 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 57 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#27161390C212yesmaximal, complemented
2#27161009C211yesmaximal, characteristic, complemented
3#27160813C212yesmaximal, complemented
4#27160805C211yesmaximal, characteristic, complemented
5#537×C197C211yesmaximal, characteristic
6C2×C4⋊Dic197C2211yescharacteristic
7C2×#15691337—21no—
8#20263259—22no—
9#20263247—22no—
10C22⋊#13096802C2212yes—
11#20263233—21no—
12#20263233C2211yescharacteristic
13C2×#41×C197—21no—
14C22⋊Q8×C197—22no—
15#121×C197—22no—
16#121×C197—21no—
17#121×C197C2212yes—
18C2×C4⋊C4×C197C2211yescharacteristic
19C22×Dic197—21no—
20C22×Dic197—21no—
21C2×#13096802—22no—
22C4⋊Dic197—22no—
23#15691337—42no—
24#41×C197—42no—
25#41×C197—42no—
26C2×Q8×C197—21no—
27C2×D4×C197—21no—
28C2×D4×C197—22no—
29C2×D4×C197—21no—
30C2×D4×C197D412yes—
31C22×C4×C197—22no—
32C22×C4×C197D411yescharacteristic
33C22×C4×C197D411yescharacteristic
34C22×C4×C197C2311yescharacteristic
35C4⋊C4×C197—21no—
36C4⋊C4×C197—22no—
37C4⋊C4×C197D412yes—
38C22⋊C4×C197—22no—
39C22⋊C4×C197—22no—
40C2×Dic197—22no—
41C2×Dic197—41no—
42C2×Dic197—41no—
43C2×Dic197—41no—
44C2×Dic197—41no—
45C2×Dic197—41no—
46#13096802—42no—
47C23×C197—22no—
48C23×C197C2×D411yescharacteristic
49Q8×C197—42no—
50D4×C197—42no—
51D4×C197—42no—
52D4×C197—42no—
53D4×C197—42no—
54D4×C197—81no—
55C2×C4×C197—21no—
56C2×C4×C197—21no—
57C2×C4×C197—21no—
58C2×C4×C197—22no—
59C2×C4×C197—41no—
60C2×C4×C197—44no—
61C2×C4×C197—41no—
62C2×C4×C197D812yes—
63C2×C4×C197C2×D412yes—
64C22×C197—21no—
65C22×C197—21no—
66C22×C197—42no—
67C22×C197—42no—
68C22×C197—42no—
69C22×C197#12011yescharacteristic
70C22×C197C2×D811yescharacteristic
71C22×C197AGL(1,ℤ/8)11yescharacteristic
72C4×C197—22no—
73C4×C197—22no—
74C4×C197—42no—
75C4×C197—41no—
76Dic197—42no—
77Dic197—81no—
78C2×C197—21no—
79C2×C197—21no—
80C2×C197—82no—
81C2×C197#44711yescharacteristic
82C2×C197#45611yescharacteristic
83C2×C197#45811yescharacteristic
84C197#161011yescharacteristic, complemented, Sylow
85#1610—1971nomaximal, Sylow
86#537D19711yescharacteristic
87#480—1972no—
88#398—1971no—
89#327—1972no—
90#324—1971no—
91C2×#41—21no—
92C22⋊Q8—22no—
93#121—22no—
94#121—21no—
95#121C2×D19712yes—
96C2×C4⋊C4—3941no—
97C2×C4⋊C4—1971no—
98C2×C4⋊C4C2×D19711yescharacteristic
99#107—3942no—
100#102—3942no—
101C22⋊C8—1972no—
102#95—3941no—
103#95—1971no—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C22⋊D19712yes—
111C22×C4—22no—
112C22×C4—3941no—
113C22×C4—3941no—
114C22×C4C22⋊D19711yescharacteristic
115C22×C4C22⋊D19711yescharacteristic
116C22×C4C22×D19711yescharacteristic
117C2×C8—3942no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—7882no—
121C4⋊C4—3942no—
122C4⋊C4C22⋊D19712yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×C22⋊D19711yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—44no—
139C2×C4—41no—
140C2×C4—7881no—
141C2×C4—7881no—
142C2×C4—7881no—
143C2×C4—7881no—
144C2×C4—7881no—
145C2×C4—3942no—
146C2×C4#1569134312yes—
147C2×C4C2×C22⋊D19712yes—
148C8—7882no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#2026329511yescharacteristic
155C22C2×#1569134311yescharacteristic
156C22#2026335211yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—7882no—
162C4—15761no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2716121311yescharacteristic
167C2#2716126211yescharacteristic
168C2#2716126911yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting, prime radical#537×C197C2
socleC22×C197#120
hypercenter#537D197
hypocenterC197#1610
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #27161390C2semidirect
2. #27161009C2semidirect
3. #27160813C2semidirect
4. #27160805C2semidirect
84. C197#1610semidirect

Maximal subgroups

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

subgroupindex
#271613902
#271610092
#271608132
#271608052
#537×C1972
#1610197