Finite Group Databasebeta

← #38743754

Subgroups — #38743754

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×#20263351C212yesmaximal, complemented
2C2×#20263253C212yesmaximal, complemented
3#27160980C211yesmaximal, characteristic, complemented
4#27160848C211yesmaximal, characteristic, complemented
5#537×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7#20263351—24no—
8#20263351—24no—
9C2×#15691345—22no—
10C2×#15691343—22no—
11C2×#15691342C2212yes—
12C2×#15691335—22no—
13C2×C4×Dic197C2211yescharacteristic
14#20263253—24no—
15C2×#41×C197C2212yes—
16C22⋊Q8×C197—22no—
17#121×C197—22no—
18#121×C197—22no—
19C2×C4⋊C4×C197C2211yescharacteristic
20C23×D197—21no—
21C2×D788—22no—
22C2×D788—22no—
23C2×D788—21no—
24C2×D788D412yescomplemented
25C22×Dic197—21no—
26#15691345—28no—
27#15691343—28no—
28#15691342—22no—
29#15691342D414yescomplemented
30C2×#13096802—22no—
31#15691335—44no—
32C4×Dic197—22no—
33#41×C197—22no—
34#41×C197—22no—
35#41×C197—22no—
36#41×C197—22no—
37C2×Q8×C197D412yes—
38C2×D4×C197—22no—
39C2×D4×C197—22no—
40C2×D4×C197D412yes—
41C22×C4×C197—22no—
42C22×C4×C197D412yes—
43C22×C4×C197C2311yescharacteristic
44C4⋊C4×C197—22no—
45C4⋊C4×C197—22no—
46C22⋊C4×C197—24no—
47C22×D197—22no—
48C22×D197—42no—
49C22×D197—42no—
50C22×D197—41no—
51C22×D197—41no—
52D788—24no—
53D788—42no—
54C2×Dic197—22no—
55C2×Dic197—41no—
56C2×Dic197—41no—
57#13096802—24no—
58C23×C197—22no—
59C23×C197C2×D411yescharacteristic
60Q8×C197—22no—
61Q8×C197—22no—
62D4×C197—22no—
63D4×C197—22no—
64D4×C197—42no—
65D4×C197—42no—
66D4×C197—82no—
67C2×C4×C197—22no—
68C2×C4×C197—22no—
69C2×C4×C197—42no—
70C2×C4×C197—42no—
71C2×C4×C197—42no—
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197C2×D412yes—
74C2×C4×C197C2×D412yes—
75C2×C4×C197#4111yescharacteristic
76C22×C197—21no—
77C22×C197—21no—
78C22×C197—42no—
79C22×C197—42no—
80C22×C197—42no—
81C22×C197#12011yescharacteristic
82C22×C197#12111yescharacteristic
83C22×C197#12711yescharacteristic
84C2×D197—44no—
85C2×D197—42no—
86C2×D197—82no—
87C2×D197—81no—
88C2×D197—81no—
89C4×C197—42no—
90C4×C197—42no—
91C4×C197#12012yes—
92C4×C197#12112yes—
93Dic197—42no—
94C2×C197—21no—
95C2×C197—21no—
96C2×C197—82no—
97C2×C197#39211yescharacteristic
98C2×C197#45311yescharacteristic
99C2×C197#45411yescharacteristic
100D197—82no—
101C197#200611yescharacteristic, complemented, Sylow
102#2006—1971nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1972no—
104#537D19711yescharacteristic, complemented
105C2×#104—1972no—
106#390—1971no—
107#343—1971no—
108C2×#41C2×D19712yes—
109C22×D4—1971no—
110AGL(1,ℤ/8)—3944no—
111AGL(1,ℤ/8)—3944no—
112C2×SD16—3942no—
113C2×D8—3942no—
114C2×M16—1972no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D19711yescharacteristic
119C2×C22⋊C4—3942no—
120C2×C42—1971no—
121#104—3944no—
122C24—3941no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D19712yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—3942no—
131C2×D4—3942no—
132C2×D4—3941no—
133C2×D4—1972no—
134C2×D4C22⋊D19712yes—
135C22×C4—22no—
136C22×C4—3941no—
137C22×C4C22⋊D19712yes—
138C22×C4C22×D19711yescharacteristic
139SD16—3948no—
140D8—3948no—
141M16—3942no—
142M16—1974no—
143C2×C8—3942no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—7884no—
148C42—3942no—
149C23—22no—
150C23—7882no—
151C23—7882no—
152C23—7881no—
153C23—7881no—
154C23—3942no—
155C23C2×C22⋊D19711yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—42no—
162D4—82no—
163D4—7882no—
164D4—3944no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—7881no—
171C2×C4—7881no—
172C2×C4—3942no—
173C2×C4D4×D19711yescharacteristic
174C2×C4D4×D19712yes—
175C2×C4C2×C22⋊D19712yes—
176C2×C4#1569135711yescharacteristic
177C8—3944no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—7884no—
184C22—7882no—
185C22—15762no—
186C22—15761no—
187C22—15761no—
188C22#2026329511yescharacteristic
189C22#2026329711yescharacteristic
190C22#2026332211yescharacteristic
191C4—42no—
192C4—42no—
193C4—7882no—
194C4#2026329412yes—
195C4#2026329812yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—15762no—
200C2#2716098711yescharacteristic
201C2#2716124711yescharacteristic
202C2#2716125211yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#537×C197C2
socleC22×C197#120
hypercenter#537D197
hypocenterC197#2006
LCSG ▷ C22×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#20263351C2semidirect
2. C2×#20263253C2semidirect
3. #27160980C2semidirect
4. #27160848C2semidirect
5. #537×C197C2semidirect
24. C2×D788D4semidirect
29. #15691342D4semidirect
101. C197#2006semidirect
104. #537D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633512
C2×#202632532
#271609802
#271608482
#537×C1972
#2006197