Finite Group Databasebeta

← #38741647

Subgroups — #38741647

Order 25216.

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

normal subgroups only

This group has 310 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#27161212C212yesmaximal, complemented
2#27160996C211yesmaximal, characteristic, complemented
3#27160813C212yesmaximal, complemented
4#27160805C211yesmaximal, characteristic, complemented
5#537×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7C2×#15691343—22no—
8C2×#15691335—21no—
9C2×#15691335—21no—
10#20263246—22no—
11C22⋊#13096802C2212yes—
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
19C23×D197—21no—
20C2×D788—22no—
21C2×D788—42no—
22C22×Dic197—21no—
23#15691343—44no—
24C2×#13096802—22no—
25#15691335—42no—
26#15691335—42no—
27#41×C197—42no—
28#41×C197—42no—
29C2×Q8×C197—21no—
30C2×D4×C197—21no—
31C2×D4×C197—22no—
32C2×D4×C197—21no—
33C2×D4×C197D412yes—
34C22×C4×C197—22no—
35C22×C4×C197D411yescharacteristic
36C22×C4×C197D411yescharacteristic
37C22×C4×C197C2311yescharacteristic
38C4⋊C4×C197—21no—
39C4⋊C4×C197—22no—
40C4⋊C4×C197D412yes—
41C22⋊C4×C197—22no—
42C22⋊C4×C197—22no—
43C22×D197—22no—
44C22×D197—41no—
45C22×D197—41no—
46C22×D197—41no—
47C22×D197—41no—
48C22×D197—41no—
49C22×D197—41no—
50D788—24no—
51D788—42no—
52C2×Dic197—41no—
53C2×Dic197—41no—
54C2×Dic197—41no—
55#13096802—42no—
56C23×C197—22no—
57C23×C197C2×D411yescharacteristic
58Q8×C197—42no—
59D4×C197—42no—
60D4×C197—42no—
61D4×C197—42no—
62D4×C197—42no—
63D4×C197—81no—
64C2×C4×C197—21no—
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—22no—
68C2×C4×C197—41no—
69C2×C4×C197—44no—
70C2×C4×C197—41no—
71C2×C4×C197D812yescomplemented
72C2×C4×C197C2×D412yes—
73C22×C197—21no—
74C22×C197—21no—
75C22×C197—42no—
76C22×C197—42no—
77C22×C197—42no—
78C22×C197#12011yescharacteristic
79C22×C197C2×D811yescharacteristic
80C22×C197AGL(1,ℤ/8)11yescharacteristic
81C2×D197—42no—
82C2×D197—42no—
83C2×D197—42no—
84C2×D197—81no—
85C2×D197—81no—
86C2×D197—81no—
87C2×D197—81no—
88C4×C197—22no—
89C4×C197—22no—
90C4×C197—42no—
91C4×C197—41no—
92Dic197—81no—
93C2×C197—21no—
94C2×C197—21no—
95C2×C197—82no—
96C2×C197#44711yescharacteristic
97C2×C197#45411yescharacteristic
98C2×C197B(4,𝔽2)11yescharacteristic
99D197—82no—
100C197#159811yescharacteristic, complemented, Sylow
101#1598—1971nomaximal, Sylow
102#537D19711yescharacteristic, complemented
103#447—1972no—
104#394—1971no—
105#327—1972no—
106#324—1971no—
107C2×#41—21no—
108C22×D4—1971no—
109C2×D8—3942no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D19712yes—
114C2×C4⋊C4C2×D19711yescharacteristic
115C2×C22⋊C4—3941no—
116C2×C22⋊C4—3941no—
117#102—3942no—
118C22⋊C8—1972no—
119#95—1971no—
120C24—3941no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—7882no—
128C2×D4—3942no—
129C2×D4C22⋊D19712yes—
130C22×C4—22no—
131C22×C4—3941no—
132C22×C4C22⋊D19711yescharacteristic
133C22×C4C22⋊D19711yescharacteristic
134C22×C4C22×D19711yescharacteristic
135D8—7884no—
136C2×C8—3942no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D19712yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—7882no—
143C22⋊C4—7882no—
144C23—22no—
145C23—7881no—
146C23—7881no—
147C23—7881no—
148C23—7881no—
149C23—7881no—
150C23—7881no—
151C23—3942no—
152C23C2×C22⋊D19711yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—7882no—
160D4—3944no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—7881no—
169C2×C4—7881no—
170C2×C4—7881no—
171C2×C4#1569134312yescomplemented
172C2×C4C2×C22⋊D19712yes—
173C8—7882no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—7882no—
180C22—7882no—
181C22—7882no—
182C22—15761no—
183C22—15761no—
184C22—15761no—
185C22—15761no—
186C22#2026329511yescharacteristic
187C22C2×#1569134311yescharacteristic
188C22#2026335211yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—41no—
193C4—15761no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—15762no—
198C2#2716121311yescharacteristic
199C2#2716125211yescharacteristic
200C2#2716126611yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#537×C197C2
socleC22×C197#120
hypercenter#537D197
hypocenterC197#1598
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. #27161212C2semidirect
2. #27160996C2semidirect
3. #27160813C2semidirect
4. #27160805C2semidirect
5. #537×C197C2semidirect
71. C2×C4×C197D8semidirect
100. C197#1598semidirect
102. #537D197semidirect
171. C2×C4#15691343semidirect

Maximal subgroups

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

subgroupindex
#271612122
#271609962
#271608132
#271608052
#537×C1972
#1598197