Finite Group Databasebeta

← #38743102

Subgroups — #38743102

Order 25216.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20263324C211yesmaximal, characteristic
2#27161556C211yesmaximal, characteristic
3#27161088C211yesmaximal, characteristic
4C2×#20263244C211yesmaximal, characteristic
5#27160840C211yesmaximal, characteristic
6#27160838C211yesmaximal, characteristic
7#419×C197C211yesmaximal, characteristic
8C22×Dic394C2211yescharacteristic
9C2×#15691342C2211yescharacteristic
10#20263324—22no—
11#20263324C414yescomplemented
12C4×Dic394—22no—
13#20263291—22no—
14#20263285C2211yescharacteristic
15#20263282—22no—
16C2×#15691337—22no—
17C2×C4×Dic197C2211yescharacteristic
18#20263250—22no—
19#20263244—22no—
20C22×Q8×C197C2211yescharacteristic
21C2×M16×C197C2211yescharacteristic
22#117×C197C2211yescharacteristic
23#103×C197—22no—
24C2×Dic394—21no—
25C2×Dic394—21no—
26C2×Dic394—21no—
27C2×Dic394—21no—
28C2×Dic394C2×C412yes—
29C2×Dic394D412yes—
30C22×Dic197—21no—
31C22×Dic197D412yes—
32#15691342—21no—
33#15691342—21no—
34C2×#13096802—21no—
35C4⋊Dic197D412yes—
36#15691337—24no—
37#15691337—28no—
38C22⋊Dic197—21no—
39C4×Dic197—24no—
40C4×Dic197—21no—
41C4×Dic197—21no—
42C4×Dic197C2×C412yes—
43C2×Q8×C197—21no—
44C2×Q8×C197—21no—
45C2×Q8×C197—21no—
46C2×Q8×C197—21no—
47C2×Q8×C197—21no—
48C2×Q8×C197C2×C412yes—
49C22×C4×C197—21no—
50C22×C4×C197C2311yescharacteristic
51M16×C197—21no—
52M16×C197—21no—
53C2×C8×C197—21no—
54C4⋊C4×C197D412yes—
55C22⋊C4×C197—21no—
56C42×C197—21no—
57Dic394—22no—
58Dic394—22no—
59Dic394—22no—
60Dic394—22no—
61C2×Dic197—24no—
62C2×Dic197—22no—
63C2×Dic197—22no—
64C2×Dic197—22no—
65C2×Dic197—22no—
66C2×Dic197C22⋊C414yes—
67#13096802—41no—
68C23×C197C2×D411yescharacteristic
69Q8×C197—22no—
70Q8×C197—22no—
71Q8×C197—41no—
72Q8×C197—41no—
73C2×C4×C197—22no—
74C2×C4×C197—22no—
75C2×C4×C197—41no—
76C2×C4×C197—41no—
77C2×C4×C197C22×C411yescharacteristic
78C2×C4×C197C2×D411yescharacteristic
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197#4111yescharacteristic
82C2×C4×C197#4111yescharacteristic
83C8×C197—41no—
84C22×C197—22no—
85C22×C197C2×C22⋊C411yescharacteristic
86C22×C197#12011yescharacteristic
87C22×C197#12311yescharacteristic
88C4×C197—42no—
89C4×C197—42no—
90C4×C197C4×D411yescharacteristic
91C4×C197C4×D411yescharacteristic
92C4×C197#12111yescharacteristic
93C4×C197#12111yescharacteristic
94Dic197—24no—
95Dic197—24no—
96Dic197—42no—
97C2×C197—21no—
98C2×C197—21no—
99C2×C197#38611yescharacteristic
100C2×C197#45612yes—
101C197#188611yescharacteristic, complemented, Sylow
102#1886—1971nomaximal, Sylow
103C2×C4⋊Q8—1971no—
104#519—1971no—
105#419—1971no—
106#419D19711yescharacteristic
107C2×#101—1971no—
108#339—1971no—
109#339—1971no—
110C22×Q8—1971no—
111C22×Q8C2×D19711yescharacteristic
112C2×M16—1971no—
113C2×M16C2×D19711yescharacteristic
114C4⋊Q8—3942no—
115C4⋊Q8—1974no—
116C4×Q8—3942no—
117C4×Q8—3942no—
118#117—3942no—
119#117—1971no—
120#117C2×D19711yescharacteristic
121C2×C4⋊C4—3942no—
122C2×C42—1971no—
123#103—22no—
124#103—3942no—
125#101—3942no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—3941no—
132C2×Q8—3941no—
133C2×Q8—3941no—
134C2×Q8—3941no—
135C2×Q8—1972no—
136C2×Q8—1972no—
137C2×Q8C4×D19712yes—
138C22×C4—21no—
139C22×C4—3941no—
140C22×C4—1972no—
141C22×C4C22×D19711yescharacteristic
142M16—21no—
143M16—21no—
144M16—3941no—
145M16—3941no—
146C2×C8—21no—
147C2×C8—3941no—
148C4⋊C4—3944no—
149C4⋊C4—3948no—
150C4⋊C4—1972no—
151C4⋊C4C22⋊D19712yes—
152C22⋊C4—21no—
153C22⋊C4—3941no—
154C42—21no—
155C42—3944no—
156C42—3941no—
157C42—3941no—
158C42—1972no—
159C23D4×D19711yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—3942no—
165Q8—3942no—
166Q8—3942no—
167Q8—3942no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—3942no—
173C2×C4—3944no—
174C2×C4—3942no—
175C2×C4—3942no—
176C2×C4—3942no—
177C2×C4—1974no—
178C2×C4C2×C4×D19711yescharacteristic
179C2×C4D4×D19711yescharacteristic
180C2×C4D4×D19711yescharacteristic
181C2×C4C2×C22⋊D19711yescharacteristic
182C2×C4#1569135711yescharacteristic
183C2×C4#1569135811yescharacteristic
184C8—41no—
185C8—7881no—
186C22—22no—
187C22C22⋊C4×D19711yescharacteristic
188C22#2026329411yescharacteristic
189C22#2026330811yescharacteristic
190C4—42no—
191C4—42no—
192C4—7882no—
193C4—3944no—
194C4—3944no—
195C4#2026328611yescharacteristic
196C4C4×C22⋊D19711yescharacteristic
197C4#2026329711yescharacteristic
198C4#2026329911yescharacteristic
199C2—21no—
200C2—21no—
201C2#2716095511yescharacteristic
202C2#2716126012yes—
subgroupG/H
centerC22#20263294
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#419×C197C2
socleC22×C197#120
prime radicalC23×C197C2×D4
cosocleC22×C4×C197C23
hypercenter#419D197
hypocenterC197#1886
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
11. #20263324C4semidirect
101. C197#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#202633242
#271615562
#271610882
C2×#202632442
#271608402
#271608382
#419×C1972
#1886197