Finite Group Databasebeta

← #38743759

Subgroups — #38743759

Order 25216.

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

normal subgroups only

This group has 370 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×#20263356C212yesmaximal, complemented
2C2×#20263253C212yesmaximal, complemented
3#27160984C211yesmaximal, characteristic, complemented
4#27160848C211yesmaximal, characteristic, complemented
5#537×C197C211yesmaximal, characteristic
6C22×Dic394C2211yescharacteristic
7#20263356—24no—
8#20263356—24no—
9C2×#15691348—22no—
10C2×#15691346—22no—
11C2×#15691342C2212yes—
12C2×C4×Dic197C2211yescharacteristic
13#20263253—24no—
14#20263233—22no—
15C2×#41×C197C2212yes—
16C22⋊Q8×C197—22no—
17#121×C197—22no—
18#121×C197—22no—
19C2×C4⋊C4×C197C2211yescharacteristic
20C2×Dic394—22no—
21C2×Dic394—22no—
22C2×Dic394—21no—
23C2×Dic394D412yescomplemented
24C22×Dic197—21no—
25C22×Dic197—21no—
26#15691348—28no—
27#15691346—28no—
28#15691342—22no—
29#15691342D414yescomplemented
30C2×#13096802—22no—
31C4×Dic197—22no—
32#41×C197—22no—
33#41×C197—22no—
34#41×C197—22no—
35#41×C197—22no—
36C2×Q8×C197D412yes—
37C2×D4×C197—22no—
38C2×D4×C197—22no—
39C2×D4×C197D412yes—
40C22×C4×C197—22no—
41C22×C4×C197D412yes—
42C22×C4×C197C2311yescharacteristic
43C4⋊C4×C197—22no—
44C4⋊C4×C197—22no—
45C22⋊C4×C197—24no—
46Dic394—24no—
47Dic394—42no—
48C2×Dic197—22no—
49C2×Dic197—22no—
50C2×Dic197—41no—
51C2×Dic197—41no—
52C2×Dic197—41no—
53C2×Dic197—41no—
54#13096802—24no—
55C23×C197—22no—
56C23×C197C2×D411yescharacteristic
57Q8×C197—22no—
58Q8×C197—22no—
59D4×C197—22no—
60D4×C197—22no—
61D4×C197—42no—
62D4×C197—42no—
63D4×C197—82no—
64C2×C4×C197—22no—
65C2×C4×C197—22no—
66C2×C4×C197—42no—
67C2×C4×C197—42no—
68C2×C4×C197—42no—
69C2×C4×C197C2×D411yescharacteristic
70C2×C4×C197C2×D412yes—
71C2×C4×C197C2×D412yes—
72C2×C4×C197#4111yescharacteristic
73C22×C197—21no—
74C22×C197—21no—
75C22×C197—42no—
76C22×C197—42no—
77C22×C197—42no—
78C22×C197#12011yescharacteristic
79C22×C197#12111yescharacteristic
80C22×C197#12711yescharacteristic
81C4×C197—42no—
82C4×C197—42no—
83C4×C197#12012yes—
84C4×C197#12112yes—
85Dic197—42no—
86Dic197—42no—
87C2×C197—21no—
88C2×C197—21no—
89C2×C197—82no—
90C2×C197#39211yescharacteristic
91C2×C197#45511yescharacteristic
92C2×C197#45611yescharacteristic
93C197#200711yescharacteristic, complemented, Sylow
94#2007—1971nomaximal, Sylow
95C2×#137—1972no—
96#537D19711yescharacteristic
97C2×#104—1972no—
98#391—1971no—
99#343—1971no—
100C2×#41C2×D19712yes—
101C22×Q8—1971no—
102#137—3944no—
103#137—3944no—
104C2×Q16—3942no—
105C2×SD16—3942no—
106C2×M16—1972no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D19711yescharacteristic
111C2×C42—1971no—
112#104—3944no—
113#95—3942no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—3942no—
119C2×Q8—3942no—
120C2×Q8—3941no—
121C2×Q8—1972no—
122C2×Q8C22⋊D19712yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D19712yes—
126C22×C4—22no—
127C22×C4—3941no—
128C22×C4—3941no—
129C22×C4C22⋊D19712yes—
130C22×C4C22×D19711yescharacteristic
131Q16—3948no—
132SD16—3948no—
133M16—3942no—
134M16—1974no—
135C2×C8—3942no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—3942no—
140C23—22no—
141C23C2×C22⋊D19711yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—7882no—
145Q8—3944no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—7881no—
157C2×C4—7881no—
158C2×C4—7881no—
159C2×C4—7881no—
160C2×C4—3942no—
161C2×C4—3942no—
162C2×C4D4×D19711yescharacteristic
163C2×C4D4×D19712yes—
164C2×C4C2×C22⋊D19712yes—
165C2×C4#1569135711yescharacteristic
166C8—3944no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#2026329511yescharacteristic
173C22#2026329711yescharacteristic
174C22#2026332211yescharacteristic
175C4—42no—
176C4—42no—
177C4—7882no—
178C4—7882no—
179C4#2026329412yes—
180C4#2026329812yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2716098711yescharacteristic
185C2#2716125711yescharacteristic
186C2#2716126211yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting, prime radical#537×C197C2
socleC22×C197#120
hypercenter#537D197
hypocenterC197#2007
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×#20263356C2semidirect
2. C2×#20263253C2semidirect
3. #27160984C2semidirect
4. #27160848C2semidirect
23. C2×Dic394D4semidirect
29. #15691342D4semidirect
93. C197#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#202633562
C2×#202632532
#271609842
#271608482
#537×C1972
#2007197