Finite Group Databasebeta

← #487×D197 #38753020

Subgroups — #487×D197 #38753020

Order 25216.

424 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4⋊Q8×D197C211yesmaximal, characteristic
2C4⋊Q8×D197C211yesmaximal, characteristic
3#27161431C211yesmaximal, characteristic, complemented
4#27161430C211yesmaximal, characteristic, complemented
5#27161429C211yesmaximal, characteristic, complemented
6#27161428C214yesmaximal, complemented
7#103×D197C214yesmaximal
8C4×C8×D197C211yesmaximal, characteristic
9#487×C197C211yesmaximal, characteristic, complemented
10C2×Q8×D197—21no—
11C2×Q8×D197C2212yes—
12C2×C8×D197C2212yes—
13#20263326C2211yescharacteristic
14#20263326C2211yescharacteristic
15#20263325—21no—
16#20263325—21no—
17#20263325—21no—
18#20263325C2212yes—
19#20263324—21no—
20#20263324C2212yes—
21C4⋊C4×D197—21no—
22C4⋊C4×D197—21no—
23C4⋊C4×D197—21no—
24C4⋊C4×D197C2212yes—
25C42×D197C2211yescharacteristic
26#20263251C2214yes—
27#20263250C2214yes—
28#20263249C2214yes—
29C4×#13096802C2211yescharacteristic
30C8×Dic197C2212yes—
31C4⋊Q8×C197C2211yescharacteristic
32C4⋊Q8×C197C2211yescharacteristic
33#103×C197C2214yes—
34C4×C8×C197C2211yescharacteristic
35C2×Dic394—22no—
36C2×Dic394—22no—
37C2×Dic394—21no—
38C2×Dic394C2312yes—
39Q8×D197—28no—
40Q8×D197—44no—
41C2×C4×D197—22no—
42C2×C4×D197—22no—
43C2×C4×D197D411yescharacteristic
44C2×C4×D197D411yescharacteristic
45C2×C4×D197C2311yescharacteristic
46C2×#13096802C2312yes—
47C8×D197—24no—
48C4⋊Dic197—21no—
49C4⋊Dic197—21no—
50C4⋊Dic197—21no—
51C4⋊Dic197C2312yes—
52#15691337—22no—
53#15691337—22no—
54#15691337—22no—
55#15691337—22no—
56C4×Dic197D411yescharacteristic
57C4×Dic197D411yescharacteristic
58C4×Dic197C2311yescharacteristic
59C2×Q8×C197—21no—
60C2×Q8×C197C2312yes—
61C2×C8×C197C2312yes—
62C4⋊C4×C197—21no—
63C4⋊C4×C197—21no—
64C4⋊C4×C197—21no—
65C4⋊C4×C197C2312yes—
66C42×C197C2311yescharacteristic
67C22×D197C2×D411yescharacteristic
68Dic394—24no—
69Dic394—42no—
70Dic394—44no—
71Dic394—44no—
72C2×Dic197—22no—
73C2×Dic197—22no—
74C2×Dic197C2×D411yescharacteristic
75C2×Dic197C2×D411yescharacteristic
76C2×Dic197C2×D411yescharacteristic
77C4×D197—44no—
78C4×D197—44no—
79C4×D197SD1614yes—
80C4×D197Q1614yes—
81C4×D197#4114yes—
82#13096802—22no—
83Q8×C197—24no—
84Q8×C197—42no—
85C2×C4×C197—22no—
86C2×C4×C197—22no—
87C2×C4×C197C2×D411yescharacteristic
88C2×C4×C197C2×D411yescharacteristic
89C2×C4×C197C2411yescharacteristic
90C8×C197—22no—
91C22×C197C22×D411yescharacteristic
92C2×D197#12412yes—
93C2×D197C2×SD1612yes—
94C2×D197C2×Q1612yes—
95C4×C197—42no—
96C4×C197—42no—
97C4×C197C2×SD1612yes—
98C4×C197C2×Q1612yes—
99C4×C197C2×#4112yes—
100Dic197—42no—
101Dic197—42no—
102Dic197#12412yes—
103Dic197C2×SD1612yes—
104Dic197C2×Q1612yes—
105C2×C197C2×#12411yescharacteristic
106C2×C197C22×SD1611yescharacteristic
107C2×C197C22×Q1611yescharacteristic
108D197#48714yesdirect product factor
109C197C2×#48711yescharacteristic, complemented, Sylow
110C2×#487—1971nomaximal, Sylow
111C2×C4⋊Q8—1971no—
112C2×C4⋊Q8—1971no—
113#487—1971no—
114#487—1971no—
115#487—1974no—
116#487—1971no—
117#487D19711yescharacteristic, direct product factor
118C2×#103—1974no—
119C2×C4×C8—1971no—
120C22×Q8—3941no—
121C22×Q8—1972no—
122C22×C8—1972no—
123C4⋊Q8—3941no—
124C4⋊Q8—3941no—
125C4⋊Q8—3941no—
126C4⋊Q8—3941no—
127C4⋊Q8—1972no—
128C4⋊Q8—1972no—
129C4⋊Q8—1971no—
130C4⋊Q8—1971no—
131C4⋊Q8C2×D19711yescharacteristic
132C4⋊Q8C2×D19711yescharacteristic
133C2×C4⋊C4—3941no—
134C2×C4⋊C4—3941no—
135C2×C4⋊C4—3941no—
136C2×C4⋊C4—1972no—
137C2×C42—1971no—
138#103—1974no—
139#103—1974no—
140#103—1974no—
141#103C2×D19714yes—
142C4×C8—1972no—
143C4×C8—1971no—
144C4×C8C2×D19711yescharacteristic
145C2×Q8—21no—
146C2×Q8—7884no—
147C2×Q8—3942no—
148C2×Q8—3942no—
149C2×Q8—3948no—
150C2×Q8—3941no—
151C2×Q8—1972no—
152C2×Q8C22×D19712yes—
153C22×C4—3942no—
154C22×C4—3942no—
155C22×C4—1971no—
156C22×C4—1971no—
157C22×C4—1971no—
158C2×C8—3944no—
159C2×C8—1972no—
160C2×C8C22×D19712yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—3942no—
165C4⋊C4—3942no—
166C4⋊C4—3942no—
167C4⋊C4—3942no—
168C4⋊C4—3941no—
169C4⋊C4—3941no—
170C4⋊C4—3941no—
171C4⋊C4—1972no—
172C4⋊C4C22×D19712yes—
173C42—1971no—
174C42—1971no—
175C42—1971no—
176C42C22×D19711yescharacteristic
177C23—1971no—
178Q8—24no—
179Q8—42no—
180Q8—7884no—
181Q8—7884no—
182Q8—7882no—
183Q8—3944no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—7884no—
187C2×C4—7884no—
188C2×C4—3942no—
189C2×C4—3942no—
190C2×C4—1971no—
191C2×C4—1971no—
192C2×C4—1971no—
193C2×C4—1974no—
194C2×C4—1974no—
195C2×C4—1974no—
196C2×C4D4×D19711yescharacteristic
197C2×C4D4×D19711yescharacteristic
198C2×C4C23×D19711yescharacteristic
199C8—22no—
200C8—3942no—
201C22—1972no—
202C22—1972no—
203C22—1972no—
204C22C2×D4×D19711yescharacteristic
205C4—42no—
206C4—42no—
207C4—7882no—
208C4—7882no—
209C4—1972no—
210C4—1972no—
211C4—1972no—
212C4SD16×D19712yes—
213C4Q16×D19712yes—
214C4#41×D19712yes—
215C2—1974no—
216C2#124×D19711yescharacteristic
217C2C2×SD16×D19711yescharacteristic
218C2C2×Q16×D19711yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#487×C197C2
socleC22×C197C22×D4
prime radicalC22×D197C2×D4
hypercenter#487D197
hypocenterC197C2×#487
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #27161431C2semidirect
4. #27161430C2semidirect
5. #27161429C2semidirect
6. #27161428C2semidirect
9. #487×C197C2semidirect
108. D197#487direct (N × H)
109. C197C2×#487semidirect
117. #487D197direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1972
#271614312
#271614302
#271614292
#271614282
#103×D1972
C4×C8×D1972
#487×C1972
C2×#487197