Finite Group Databasebeta

← #38751193

Subgroups — #38751193

Order 25216.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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×#20263373C211yesmaximal, characteristic
2#27161845C212yesmaximal, complemented
3#27161063C211yesmaximal, characteristic, complemented
4#27161060C214yesmaximal, complemented
5C2×#20263244C212yesmaximal, complemented
6#101×D197C214yesmaximal, complemented
7#413×C197C211yesmaximal, characteristic, complemented
8#20263373—24no—
9#20263373—24no—
10C2×#15691358C412yes—
11C2×#15691358C2211yescharacteristic
12C2×#15691357C414yes—
13C2×#15691357C2212yes—
14C22×Dic394C412yes—
15C22×Dic394C2211yescharacteristic
16C2×Q8×D197C2212yes—
17#20263334—24no—
18#20263332—24no—
19C2×#15691342C2212yes—
20M16×D197C2218yes—
21#20263245C2212yes—
22#20263244C2218yes—
23#20263243C2212yes—
24C2×#41×C197C2211yescharacteristic
25C2×M16×C197C2212yes—
26#101×C197C2212yes—
27#100×C197C2212yes—
28#15691358—24no—
29#15691358—24no—
30#15691358D414yes—
31#15691358D414yes—
32#15691357—216no—
33#15691357—28no—
34C2×Dic394—24no—
35C2×Dic394—24no—
36C2×Dic394—22no—
37C2×Dic394—22no—
38C2×Dic394C2×C414yes—
39C2×Dic394C2×C412yes—
40C2×Dic394C2×C411yescharacteristic
41C2×Dic394C2312yes—
42Q8×D197—28no—
43C2×D788C2×C411yescharacteristic
44C2×C22⋊D197C2×C414yes—
45C2×C22⋊D197C2×C412yes—
46C22×Dic197C2×C414yes—
47C22×Dic197C2×C412yes—
48C2×C4×D197C2×C414yes—
49C2×C4×D197C2312yes—
50#15691342—22no—
51#15691342C2314yes—
52#15691341—28no—
53C2×#13096802—22no—
54C8×D197—28no—
55#41×C197—22no—
56#41×C197—22no—
57C2×Q8×C197C2311yescharacteristic
58C2×D4×C197C2×C412yes—
59C2×D4×C197C2311yescharacteristic
60C22×C4×C197C2×C412yes—
61C22×C4×C197C2311yescharacteristic
62M16×C197—22no—
63M16×C197C2314yes—
64C2×C8×C197—22no—
65C22×D197C22×C412yes—
66Dic394—22no—
67Dic394—22no—
68Dic394—24no—
69Dic394—28no—
70Dic394C22⋊C412yes—
71Dic394C22⋊C412yes—
72D788C22⋊C412yes—
73D788C22⋊C412yes—
74C22⋊D197—28no—
75C22⋊D197C22⋊C418yes—
76C2×Dic197—24no—
77C2×Dic197—28no—
78C2×Dic197C22×C414yes—
79C2×Dic197C22×C412yes—
80C4×D197—28no—
81C4×D197C2×D418yes—
82#13096802—24no—
83C23×C197C22×C412yes—
84C23×C197C22×C411yescharacteristic
85Q8×C197—22no—
86D4×C197—24no—
87D4×C197—22no—
88C2×C4×C197—22no—
89C2×C4×C197—22no—
90C2×C4×C197C22×C412yes—
91C2×C4×C197C22×C411yescharacteristic
92C2×C4×C197C2×D412yes—
93C2×C4×C197C2×D412yes—
94C2×C4×C197C2411yescharacteristic
95C8×C197—24no—
96C22×C197—22no—
97C22×C197—41no—
98C22×C197—42no—
99C22×C197C2×C22⋊C412yes—
100C22×C197C23×C411yescharacteristic
101C2×D197—42no—
102C2×D197C2×C22⋊C414yes—
103C4×C197—22no—
104C4×C197C2×C22⋊C411yescharacteristic
105C4×C197C2×C22⋊C411yescharacteristic
106C4×C197C22×D412yes—
107Dic197—24no—
108Dic197C2×C22⋊C414yes—
109C2×C197—21no—
110C2×C197—22no—
111C2×C197—42no—
112C2×C197C22×C22⋊C411yescharacteristic
113D197—24no—
114C197#288911yescharacteristic, complemented, Sylow
115#2889—1971nomaximal, Sylow
116C2×2−1+4—1971no—
117#568—1972no—
118#413—1971no—
119#413—1974no—
120#413D19711yescharacteristic, complemented
121C2×#101—1972no—
122C2×#101—1974no—
1232−1+4—3944no—
1242−1+4—3944no—
125C2×#41—1974no—
126C2×#41—1972no—
127C2×#41—1972no—
128C2×#41—1971no—
129C2×#41C2×D19711yescharacteristic
130C22×Q8—1972no—
131C22×Q8—1972no—
132C22×Q8—1971no—
133#131—3944no—
134#131—3944no—
135C2×M16—1972no—
136C2×M16—1978no—
137C2×M16C2×D19712yes—
138#101—1972no—
139#101—1978no—
140#101C2×D19712yes—
141#100—1972no—
142#100C2×D19712yes—
143#41—22no—
144#41—22no—
145#41—3944no—
146#41—3944no—
147#41—39416no—
148#41—3948no—
149#41—1974no—
150#41—1974no—
151C2×Q8—3944no—
152C2×Q8—3944no—
153C2×Q8—3948no—
154C2×Q8—3942no—
155C2×Q8—3942no—
156C2×Q8—1972no—
157C2×Q8—1974no—
158C2×Q8—1971no—
159C2×Q8—1972no—
160C2×Q8C22×D19711yescharacteristic
161C2×D4—1974no—
162C2×D4—1972no—
163C2×D4—1971no—
164C2×D4C4×D19712yes—
165C2×D4C22×D19711yescharacteristic
166C22×C4—1972no—
167C22×C4—1974no—
168C22×C4—1974no—
169C22×C4—1972no—
170C22×C4C4×D19712yes—
171C22×C4C22×D19711yescharacteristic
172M16—22no—
173M16—3948no—
174M16—3942no—
175M16—1974no—
176M16C22×D19714yes—
177C2×C8—22no—
178C2×C8—3948no—
179C2×C8—3942no—
180C23—1972no—
181C23C2×C4×D19712yes—
182C23C2×C4×D19711yescharacteristic
183Q8—22no—
184Q8—3948no—
185Q8—3942no—
186Q8—3942no—
187Q8—3944no—
188Q8—1972no—
189Q8—1972no—
190D4—24no—
191D4—22no—
192D4—3948no—
193D4—1978no—
194D4—1972no—
195D4—1972no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—3948no—
199C2×C4—3944no—
200C2×C4—3948no—
201C2×C4—1974no—
202C2×C4—1972no—
203C2×C4—1978no—
204C2×C4C2×C4×D19712yes—
205C2×C4C2×C4×D19711yescharacteristic
206C2×C4D4×D19712yes—
207C2×C4D4×D19712yes—
208C2×C4C23×D19711yescharacteristic
209C8—24no—
210C8—3944no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—7882no—
215C22—1974no—
216C22C22⋊C4×D19712yes—
217C22C22×C4×D19711yescharacteristic
218C4—22no—
219C4—3944no—
220C4—1974no—
221C4C22⋊C4×D19711yescharacteristic
222C4C22⋊C4×D19711yescharacteristic
223C4C2×D4×D19712yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—3944no—
228C2C2×C22⋊C4×D19711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D197
commutatorC22×C197C23×C4
FrattiniC2×C4C23×D197
Fitting#413×C197C2
socleC2×C197C22×C22⋊C4
prime radicalC2×#20263373C2
cosocleC2×C4×C197C24
hypercenter#413D197
hypocenterC197#2889
LCSG ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #27161845C2semidirect
3. #27161063C2semidirect
4. #27161060C2semidirect
5. C2×#20263244C2semidirect
6. #101×D197C2semidirect
7. #413×C197C2semidirect
114. C197#2889semidirect
120. #413D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633732
#271618452
#271610632
#271610602
C2×#202632442
#101×D1972
#413×C1972
#2889197