Finite Group Databasebeta

← #38751192

Subgroups — #38751192

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×#20263374C211yesmaximal, characteristic
2#27161845C212yesmaximal, complemented
3#27161062C214yesmaximal, complemented
4#27161061C212yesmaximal, complemented
5C2×#20263245C211yesmaximal, characteristic, complemented
6#101×D197C214yesmaximal, complemented
7C2×#101×C197C211yesmaximal, characteristic, complemented
8#20263374—24no—
9#20263374—24no—
10C2×#15691359C412yes—
11C2×#15691359C2212yes—
12C2×#15691358C414yes—
13C2×#15691358C2211yescharacteristic
14C2×#15691358C2211yescharacteristic
15C2×Q8×D197C412yes—
16C2×Q8×D197C2212yes—
17#20263334—24no—
18#20263332—24no—
19C2×#15691342C2212yes—
20M16×D197C2218yes—
21#20263245C2214yes—
22#20263244C2214yes—
23#20263242C2214yes—
24C22×Q8×C197C2211yescharacteristic
25C2×M16×C197C2212yes—
26#101×C197C2214yes—
27#15691359—28no—
28#15691359—28no—
29#15691358—28no—
30#15691358—28no—
31#15691358—22no—
32#15691358—22no—
33#15691358D414yes—
34#15691358D414yes—
35C2×Dic394C2×C414yes—
36C2×Dic394C2×C411yescharacteristic
37C2×Dic394C2311yescharacteristic
38Q8×D197—28no—
39Q8×D197—28no—
40C2×D788C2×C414yes—
41C2×D788C2×C411yescharacteristic
42C2×D788C2311yescharacteristic
43C2×C22⋊D197C2×C412yes—
44C2×C22⋊D197C2×C412yes—
45C2×C4×D197C2×C414yes—
46C2×C4×D197C2×C412yes—
47C2×C4×D197C2×C414yes—
48C2×C4×D197C2312yes—
49#15691342—22no—
50#15691342C2314yes—
51#15691341—28no—
52C2×#13096802—22no—
53C8×D197—28no—
54C2×Q8×C197—22no—
55C2×Q8×C197—22no—
56C2×Q8×C197C2×C412yes—
57C2×Q8×C197C2312yes—
58C22×C4×C197C2×C412yes—
59C22×C4×C197C2311yescharacteristic
60M16×C197—22no—
61M16×C197C2314yes—
62C2×C8×C197—22no—
63C22×D197C22×C412yes—
64C22×D197C22×C412yes—
65Dic394—22no—
66Dic394—28no—
67Dic394C22⋊C414yes—
68D788—28no—
69D788—22no—
70D788C22⋊C414yes—
71C22⋊D197—22no—
72C22⋊D197—22no—
73C22⋊D197C22⋊C414yes—
74C22⋊D197C22⋊C414yes—
75C2×Dic197C22×C412yes—
76C2×Dic197C22×C412yes—
77C4×D197—28no—
78C4×D197—24no—
79C4×D197—28no—
80C4×D197C2×D418yes—
81#13096802—24no—
82C23×C197C22×C411yescharacteristic
83Q8×C197—24no—
84Q8×C197—24no—
85C2×C4×C197—22no—
86C2×C4×C197—22no—
87C2×C4×C197C22×C414yes—
88C2×C4×C197C22×C411yescharacteristic
89C2×C4×C197C2×D412yes—
90C2×C4×C197C2×D412yes—
91C2×C4×C197C2411yescharacteristic
92C8×C197—24no—
93C22×C197—41no—
94C22×C197C2×C22⋊C411yescharacteristic
95C22×C197C2×C22⋊C411yescharacteristic
96C22×C197C23×C411yescharacteristic
97C2×D197—22no—
98C2×D197—42no—
99C2×D197—42no—
100C2×D197C2×C22⋊C414yes—
101C4×C197—24no—
102C4×C197C2×C22⋊C412yes—
103C4×C197C22×D412yes—
104Dic197—22no—
105Dic197C2×C22⋊C414yes—
106C2×C197—21no—
107C2×C197—21no—
108C2×C197—21no—
109C2×C197C22×C22⋊C411yescharacteristic
110D197—24no—
111D197—42no—
112C197#288911yescharacteristic, complemented, Sylow
113#2889—1971nomaximal, Sylow
114C2×2−1+4—1971no—
115#568—1972no—
116#413—1972no—
117#413—1974no—
118C2×#101—1971no—
119C2×#101—1974no—
120C2×#101D19711yescharacteristic, complemented
1212−1+4—3944no—
1222−1+4—3944no—
123C2×#41—1972no—
124C2×#41—1972no—
125C2×#41—1974no—
126C2×#41—1971no—
127C2×#41—1971no—
128C22×Q8—1972no—
129C22×Q8—1972no—
130C22×Q8C2×D19711yescharacteristic
131#131—3944no—
132#131—3944no—
133C2×M16—1972no—
134C2×M16—1978no—
135C2×M16C2×D19712yes—
136#101—1974no—
137#101—1974no—
138#101C2×D19714yes—
139#100—1974no—
140#41—3948no—
141#41—3948no—
142#41—3948no—
143#41—3948no—
144#41—3942no—
145#41—3942no—
146#41—1974no—
147#41—1974no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—3948no—
151C2×Q8—3948no—
152C2×Q8—1974no—
153C2×Q8—1971no—
154C2×Q8—1971no—
155C2×Q8C4×D19712yes—
156C2×Q8C22×D19712yes—
157C2×D4—1972no—
158C2×D4—1972no—
159C2×D4—1974no—
160C2×D4—1971no—
161C2×D4—1971no—
162C22×C4—1974no—
163C22×C4—1974no—
164C22×C4—1972no—
165C22×C4—1972no—
166C22×C4C4×D19712yes—
167C22×C4C22×D19711yescharacteristic
168M16—22no—
169M16—3948no—
170M16—3942no—
171M16—1974no—
172M16C22×D19714yes—
173C2×C8—22no—
174C2×C8—3948no—
175C2×C8—3942no—
176C23—1972no—
177C23—1972no—
178C23C2×C4×D19711yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—3948no—
182Q8—3942no—
183Q8—1974no—
184D4—3948no—
185D4—3942no—
186D4—3942no—
187D4—3942no—
188D4—1974no—
189D4—1974no—
190D4—1974no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—3948no—
194C2×C4—3948no—
195C2×C4—3944no—
196C2×C4—1972no—
197C2×C4—1972no—
198C2×C4—1978no—
199C2×C4C2×C4×D19714yes—
200C2×C4C2×C4×D19711yescharacteristic
201C2×C4D4×D19712yes—
202C2×C4D4×D19712yes—
203C2×C4C23×D19711yescharacteristic
204C8—24no—
205C8—3944no—
206C22—41no—
207C22—7882no—
208C22—7882no—
209C22—3942no—
210C22—1974no—
211C22C22⋊C4×D19711yescharacteristic
212C22C22⋊C4×D19711yescharacteristic
213C22C22×C4×D19711yescharacteristic
214C4—24no—
215C4—3942no—
216C4—1974no—
217C4C22⋊C4×D19712yes—
218C4C2×D4×D19712yes—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—7882no—
223C2—3944no—
224C2C2×C22⋊C4×D19711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D197
commutatorC22×C197C23×C4
FrattiniC2×C4C23×D197
FittingC2×#101×C197C2
socleC2×C197C22×C22⋊C4
prime radicalC2×#20263374C2
cosocleC2×C4×C197C24
hypercenterC2×#101D197
hypocenterC197#2889
LCSG ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #27161845C2semidirect
3. #27161062C2semidirect
4. #27161061C2semidirect
5. C2×#20263245C2semidirect
6. #101×D197C2semidirect
7. C2×#101×C197C2semidirect
112. C197#2889semidirect
120. C2×#101D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633742
#271618452
#271610622
#271610612
C2×#202632452
#101×D1972
C2×#101×C1972
#2889197