Finite Group Databasebeta

← #38743758

Subgroups — #38743758

Order 25216.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20263357C212yesmaximal, complemented
2#27161660C211yesmaximal, characteristic
3C2×#20263254C212yesmaximal, complemented
4#27160848C211yesmaximal, characteristic, complemented
5#391×C197C211yesmaximal, characteristic, complemented
6C2×#15691358C2212yes—
7#20263357—24no—
8#20263357—24no—
9C2×#15691348—22no—
10C2×#15691345—22no—
11C2×#15691342C2212yes—
12#20263306—22no—
13#20263301—22no—
14#20263300—22no—
15#20263254—24no—
16C22×Q8×C197C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C2×C42×C197C2211yescharacteristic
19#95×C197—22no—
20#15691358—22no—
21#15691358—22no—
22#15691358—22no—
23#15691358—22no—
24C2×Dic394D412yes—
25C2×D788—22no—
26C2×D788D412yes—
27C2×C22⋊D197—22no—
28C2×C4×D197—22no—
29#15691348—28no—
30#15691345—28no—
31#15691342—22no—
32#15691342D414yescomplemented
33C2×#13096802—22no—
34C4⋊Dic197—22no—
35#15691335—24no—
36C2×Q8×C197—21no—
37C2×Q8×C197—22no—
38C2×Q8×C197—22no—
39C2×Q8×C197D412yescomplemented
40C22×C4×C197—21no—
41C22×C4×C197—21no—
42C22×C4×C197D412yes—
43C22×C4×C197C2311yescharacteristic
44C4⋊C4×C197—22no—
45C42×C197—22no—
46C22×D197—22no—
47Dic394—22no—
48Dic394—22no—
49D788—22no—
50D788—22no—
51D788—82no—
52C22⋊D197—42no—
53C22⋊D197—42no—
54C2×Dic197—22no—
55C4×D197—42no—
56C4×D197—42no—
57#13096802—24no—
58C23×C197C2×D411yescharacteristic
59Q8×C197—24no—
60Q8×C197—42no—
61C2×C4×C197—22no—
62C2×C4×C197—22no—
63C2×C4×C197—22no—
64C2×C4×C197—41no—
65C2×C4×C197—41no—
66C2×C4×C197—41no—
67C2×C4×C197—41no—
68C2×C4×C197—42no—
69C2×C4×C197C2×D412yes—
70C2×C4×C197C2×D412yes—
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197#4111yescharacteristic
73C22×C197—21no—
74C22×C197—21no—
75C22×C197#12011yescharacteristic
76C22×C197#12111yescharacteristic
77C22×C197#12711yescharacteristic
78C2×D197—42no—
79C2×D197—42no—
80C2×D197—42no—
81C4×C197—42no—
82C4×C197—42no—
83C4×C197—42no—
84C4×C197#12012yes—
85C4×C197#12112yes—
86Dic197—42no—
87C2×C197—21no—
88C2×C197—21no—
89C2×C197#39211yescharacteristic
90C2×C197#45511yescharacteristic
91C2×C197#45611yescharacteristic
92D197—82no—
93C197#200711yescharacteristic, complemented, Sylow
94#2007—1971nomaximal, Sylow
95C2×#137—1972no—
96#537—1971no—
97C2×#104—1972no—
98#391D19711yescharacteristic, complemented
99#343—1971no—
100C2×#41—1972no—
101C22×Q8C2×D19711yescharacteristic
102#137—3944no—
103#137—3944no—
104C2×Q16—3942no—
105C2×SD16—3942no—
106C2×M16—1972no—
107C22⋊Q8—3942no—
108#121—3942no—
109#121—3942no—
110C2×C4⋊C4C2×D19711yescharacteristic
111C2×C42C2×D19711yescharacteristic
112#104—3944no—
113#95—22no—
114#41—3942no—
115#41—3942no—
116#41—3942no—
117#41—3942no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—1972no—
122C2×Q8D78812yescomplemented
123C2×D4—3942no—
124C2×D4—3942no—
125C2×D4—1972no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—3942no—
129C22×C4C22⋊D19712yes—
130C22×C4C22×D19711yescharacteristic
131Q16—3948no—
132SD16—3948no—
133M16—3942no—
134M16—1974no—
135C2×C8—3942no—
136C4⋊C4—22no—
137C4⋊C4—3942no—
138C22⋊C4—3944no—
139C42—22no—
140C23—3942no—
141C23C2×C22⋊D19711yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—3942no—
145Q8—3942no—
146D4—7882no—
147D4—7882no—
148D4—15762no—
149D4—3942no—
150D4—3942no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—7882no—
160C2×C4—7882no—
161C2×C4—3942no—
162C2×C4D4×D19712yes—
163C2×C4D4×D19712yes—
164C2×C4C2×D78811yescharacteristic
165C2×C4#1569135811yescharacteristic
166C8—3944no—
167C22—21no—
168C22—21no—
169C22—7882no—
170C22—7882no—
171C22—7882no—
172C22#2026329411yescharacteristic
173C22#2026330011yescharacteristic
174C22#2026332211yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—7882no—
179C4#2026329612yes—
180C4#2026329912yes—
181C2—21no—
182C2—21no—
183C2—15762no—
184C2#2716098811yescharacteristic
185C2#2716125611yescharacteristic
186C2#2716126111yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#391×C197C2
socleC22×C197#120
prime radical#27161660C2
hypercenter#391D197
hypocenterC197#2007
LCSG ▷ C22×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#20263357C2semidirect
3. C2×#20263254C2semidirect
4. #27160848C2semidirect
5. #391×C197C2semidirect
32. #15691342D4semidirect
39. C2×Q8×C197D4semidirect
93. C197#2007semidirect
98. #391D197semidirect
122. C2×Q8D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633572
#271616602
C2×#202632542
#271608482
#391×C1972
#2007197