Finite Group Databasebeta

← #38741649

Subgroups — #38741649

Order 25216.

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

normal subgroups only

This group has 310 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#27161660C211yesmaximal, characteristic, complemented
2#27161215C212yesmaximal, complemented
3#27160996C211yesmaximal, characteristic, complemented
4#27160814C212yesmaximal, complemented
5#324×C197C211yesmaximal, characteristic, complemented
6C2×#15691358—21no—
7C22×D788C2211yescharacteristic
8C2×D1576—22no—
9#20263306—22no—
10#20263301—22no—
11#20263300—21no—
12#20263300C2212yes—
13C2×#15691335—21no—
14C2×#15691335—21no—
15#20263248—22no—
16#20263233C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C23×D197—21no—
20#15691358—42no—
21#15691358—42no—
22C2×Dic394—21no—
23C2×D788—21no—
24C2×D788—21no—
25C2×D788—22no—
26C2×D788—42no—
27C2×D788D412yes—
28C2×C22⋊D197—22no—
29C22×Dic197—21no—
30C2×C4×D197—22no—
31D1576—44no—
32C4⋊Dic197—21no—
33C4⋊Dic197D412yes—
34#15691335—22no—
35#15691335—22no—
36#15691335—42no—
37#15691335—42no—
38C22×C4×C197D411yescharacteristic
39C22×C4×C197D411yescharacteristic
40C22×C4×C197C2311yescharacteristic
41C2×C8×C197—22no—
42C4⋊C4×C197—22no—
43C22×D197—22no—
44C22×D197—22no—
45C22×D197—41no—
46C22×D197—41no—
47C22×D197—41no—
48C22×D197—41no—
49C22×D197—41no—
50C22×D197—41no—
51Dic394—42no—
52D788—24no—
53D788—42no—
54D788—42no—
55D788—42no—
56D788—81no—
57C22⋊D197—42no—
58C22⋊D197—42no—
59C2×Dic197—22no—
60C2×Dic197—41no—
61C2×Dic197—41no—
62C2×Dic197—41no—
63C4×D197—44no—
64C23×C197C2×D411yescharacteristic
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—21no—
68C2×C4×C197—41no—
69C2×C4×C197—41no—
70C2×C4×C197D812yescomplemented
71C2×C4×C197C2×D412yes—
72C8×C197—42no—
73C22×C197—21no—
74C22×C197—21no—
75C22×C197#12011yescharacteristic
76C22×C197C2×D811yescharacteristic
77C22×C197AGL(1,ℤ/8)11yescharacteristic
78C2×D197—42no—
79C2×D197—42no—
80C2×D197—42no—
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—41no—
91Dic197—42no—
92Dic197—81no—
93C2×C197—21no—
94C2×C197—21no—
95C2×C197#44711yescharacteristic
96C2×C197#45411yescharacteristic
97C2×C197B(4,𝔽2)11yescharacteristic
98D197—82no—
99D197—82no—
100C197#159811yescharacteristic, complemented, Sylow
101#1598—1971nomaximal, Sylow
102#537—1971no—
103#447—1972no—
104#394—1971no—
105#327—1972no—
106#324D19711yescharacteristic, complemented
107C2×#41—3941no—
108C22×D4—1971no—
109C2×D8—3942no—
110C22⋊Q8—3942no—
111#121—3942no—
112#121—3941no—
113#121—1972no—
114C2×C4⋊C4C2×D19711yescharacteristic
115C2×C22⋊C4—3941no—
116C2×C22⋊C4—3941no—
117#102—3942no—
118C22⋊C8C2×D19712yes—
119#95—1971no—
120C24—3941no—
121#41—7882no—
122#41—7882no—
123C2×Q8—3941no—
124C2×D4—7882no—
125C2×D4—3941no—
126C2×D4—3941no—
127C2×D4—3942no—
128C2×D4—3942no—
129C2×D4—1972no—
130C22×C4—3942no—
131C22×C4—3941no—
132C22×C4D78811yescharacteristic
133C22×C4D78811yescharacteristic
134C22×C4C22×D19711yescharacteristic
135D8—7884no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—3941no—
139C4⋊C4—1972no—
140C22⋊C4—7882no—
141C22⋊C4—7882no—
142C22⋊C4—3942no—
143C22⋊C4—3942no—
144C23—7881no—
145C23—7881no—
146C23—7881no—
147C23—7881no—
148C23—7881no—
149C23—7881no—
150C23—3942no—
151C23—3942no—
152C23C2×D78811yescharacteristic
153Q8—7882no—
154D4—7882no—
155D4—7882no—
156D4—7882no—
157D4—7882no—
158D4—7882no—
159D4—15761no—
160D4—3944no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—7884no—
167C2×C4—7881no—
168C2×C4—7881no—
169C2×C4—7881no—
170C2×C4—3942no—
171C2×C4D157612yescomplemented
172C2×C4D4×D19712yes—
173C8—42no—
174C22—21no—
175C22—21no—
176C22—7882no—
177C22—7882no—
178C22—7882no—
179C22—7882no—
180C22—7882no—
181C22—7882no—
182C22—15761no—
183C22—15761no—
184C22—15761no—
185C22—15761no—
186C22#2026329611yescharacteristic
187C22C2×D157611yescharacteristic
188C22#2026335411yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—7882no—
193C4—15761no—
194C2—21no—
195C2—21no—
196C2—15762no—
197C2—15762no—
198C2#2716121511yescharacteristic
199C2#2716125411yescharacteristic
200C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#324×C197C2
socleC22×C197#120
hypercenter#324D197
hypocenterC197#1598
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #27161660C2semidirect
2. #27161215C2semidirect
3. #27160996C2semidirect
4. #27160814C2semidirect
5. #324×C197C2semidirect
70. C2×C4×C197D8semidirect
100. C197#1598semidirect
106. #324D197semidirect
171. C2×C4D1576semidirect

Maximal subgroups

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

subgroupindex
#271616602
#271612152
#271609962
#271608142
#324×C1972
#1598197