Finite Group Databasebeta

← #38743756

Subgroups — #38743756

Order 25216.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20263354C212yesmaximal, complemented
2#27161660C211yesmaximal, characteristic, complemented
3C2×#20263252C212yesmaximal, complemented
4#27160980C211yesmaximal, characteristic, complemented
5#343×C197C211yesmaximal, characteristic, complemented
6C2×#15691358C2212yes—
7C22×D788C2211yescharacteristic
8#20263354—24no—
9#20263354—24no—
10C2×#15691347—22no—
11C2×D1576—22no—
12#20263306—22no—
13#20263301—22no—
14#20263300—22no—
15C2×#15691335—22no—
16C2×C4×Dic197C2211yescharacteristic
17#20263252—24no—
18C2×M16×C197C2212yes—
19C2×C4⋊C4×C197C2211yescharacteristic
20C23×D197—21no—
21#15691358—22no—
22#15691358—22no—
23#15691358—22no—
24#15691358—22no—
25C2×Dic394D412yes—
26C2×D788—22no—
27C2×D788—22no—
28C2×D788—22no—
29C2×D788—21no—
30C2×D788D412yes—
31C2×D788D412yescomplemented
32C2×C22⋊D197—22no—
33C22×Dic197—21no—
34C2×C4×D197—22no—
35#15691347—28no—
36D1576—28no—
37C4⋊Dic197—22no—
38#15691335—24no—
39#15691335—44no—
40C4×Dic197—22no—
41C22×C4×C197D412yes—
42C22×C4×C197C2311yescharacteristic
43M16×C197—22no—
44M16×C197D414yescomplemented
45C2×C8×C197—22no—
46C4⋊C4×C197—22no—
47C22×D197—22no—
48C22×D197—22no—
49C22×D197—42no—
50C22×D197—42no—
51C22×D197—41no—
52C22×D197—41no—
53Dic394—22no—
54Dic394—22no—
55D788—22no—
56D788—24no—
57D788—22no—
58D788—42no—
59D788—82no—
60C22⋊D197—42no—
61C22⋊D197—42no—
62C2×Dic197—22no—
63C2×Dic197—22no—
64C2×Dic197—41no—
65C2×Dic197—41no—
66C4×D197—42no—
67C4×D197—42no—
68C23×C197C2×D411yescharacteristic
69C2×C4×C197—22no—
70C2×C4×C197—42no—
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197C2×D412yes—
73C2×C4×C197C2×D412yes—
74C2×C4×C197#4111yescharacteristic
75C8×C197—24no—
76C22×C197—21no—
77C22×C197—21no—
78C22×C197#12011yescharacteristic
79C22×C197#12111yescharacteristic
80C22×C197#12711yescharacteristic
81C2×D197—44no—
82C2×D197—42no—
83C2×D197—42no—
84C2×D197—42no—
85C2×D197—42no—
86C2×D197—82no—
87C2×D197—81no—
88C2×D197—81no—
89C4×C197—42no—
90C4×C197#12012yes—
91C4×C197#12112yes—
92Dic197—42no—
93Dic197—42no—
94C2×C197—21no—
95C2×C197—21no—
96C2×C197#39211yescharacteristic
97C2×C197#45311yescharacteristic
98C2×C197#45411yescharacteristic
99D197—82no—
100D197—82no—
101C197#200611yescharacteristic, complemented, Sylow
102#2006—1971nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1972no—
104#537—1971no—
105C2×#104—1972no—
106#390—1971no—
107#343D19711yescharacteristic, complemented
108C2×#41—1972no—
109C22×D4—1971no—
110AGL(1,ℤ/8)—3944no—
111AGL(1,ℤ/8)—3944no—
112C2×SD16—3942no—
113C2×D8—3942no—
114C2×M16C2×D19712yes—
115C22⋊Q8—3942no—
116#121—3942no—
117#121—3942no—
118C2×C4⋊C4C2×D19711yescharacteristic
119C2×C22⋊C4—3942no—
120C2×C42—1971no—
121#104—3944no—
122C24—3941no—
123#41—3942no—
124#41—3942no—
125#41—3942no—
126#41—3942no—
127C2×Q8—1972no—
128C2×D4—3942no—
129C2×D4—3942no—
130C2×D4—3942no—
131C2×D4—3942no—
132C2×D4—3941no—
133C2×D4—1972no—
134C2×D4—1972no—
135C22×C4—3941no—
136C22×C4—3942no—
137C22×C4D78812yes—
138C22×C4C22×D19711yescharacteristic
139SD16—3948no—
140D8—3948no—
141M16—22no—
142M16D78814yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—3942no—
146C22⋊C4—7884no—
147C22⋊C4—3944no—
148C42—3942no—
149C23—7882no—
150C23—7882no—
151C23—7881no—
152C23—7881no—
153C23—3942no—
154C23—3942no—
155C23C2×D78811yescharacteristic
156Q8—3942no—
157Q8—3942no—
158D4—7882no—
159D4—7882no—
160D4—7882no—
161D4—15762no—
162D4—3942no—
163D4—3944no—
164D4—3942no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—7881no—
168C2×C4—7881no—
169C2×C4—7882no—
170C2×C4—7882no—
171C2×C4—3942no—
172C2×C4—3942no—
173C2×C4D4×D19711yescharacteristic
174C2×C4D4×D19712yes—
175C2×C4C2×D78812yes—
176C2×C4#1569135911yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—7884no—
181C22—7882no—
182C22—7882no—
183C22—7882no—
184C22—7882no—
185C22—15762no—
186C22—15761no—
187C22—15761no—
188C22#2026329611yescharacteristic
189C22#2026330111yescharacteristic
190C22#2026332311yescharacteristic
191C4—42no—
192C4—7882no—
193C4—7882no—
194C4#2026329612yes—
195C4#2026330112yes—
196C2—21no—
197C2—21no—
198C2—15762no—
199C2—15762no—
200C2#2716098911yescharacteristic
201C2#2716124911yescharacteristic
202C2#2716125411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#343×C197C2
socleC22×C197#120
hypercenter#343D197
hypocenterC197#2006
LCSG ▷ C22×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#20263354C2semidirect
2. #27161660C2semidirect
3. C2×#20263252C2semidirect
4. #27160980C2semidirect
5. #343×C197C2semidirect
31. C2×D788D4semidirect
44. M16×C197D4semidirect
101. C197#2006semidirect
107. #343D197semidirect
142. M16D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
#271616602
C2×#202632522
#271609802
#343×C1972
#2006197