Finite Group Databasebeta

← #38751065

Subgroups — #38751065

Order 25216.

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

normal subgroups only

This group has 644 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×#20263326C211yesmaximal, characteristic, complemented
2C2×#20263306C212yesmaximal
3C2×C42×D197C211yesmaximal, characteristic
4#27160984C212yesmaximal, complemented
5#27160978C214yesmaximal, complemented
6#27160936C214yesmaximal
7#384×C197C211yesmaximal, characteristic, complemented
8C22×Dic394C2212yes—
9C22×C4×D197C2211yescharacteristic
10C22×C4×D197C2212yes—
11#20263326—24no—
12#20263306—28no—
13C2×C4⋊Dic197C2214yes—
14C2×C4⋊Dic197C2214yes—
15C2×#15691335C2214yes—
16C2×C4×Dic197C2212yes—
17C2×C4×Dic197C2211yescharacteristic
18C42×D197—24no—
19#20263233C2218yes—
20C2×C4⋊C4×C197C2214yes—
21C2×C4⋊C4×C197C2212yes—
22C2×C42×C197C2211yescharacteristic
23C23×D197C2311yescharacteristic
24C2×Dic394—24no—
25C2×Dic394—24no—
26C2×Dic394—24no—
27C22×Dic197C2314yes—
28C22×Dic197C2311yescharacteristic
29C22×Dic197C2312yes—
30C2×C4×D197—22no—
31C2×C4×D197—24no—
32C2×C4×D197—24no—
33C2×C4×D197—22no—
34C2×C4×D197D414yes—
35C2×C4×D197Q818yescomplemented
36C4⋊Dic197—28no—
37C4⋊Dic197—28no—
38#15691335—28no—
39C4×Dic197—22no—
40C4×Dic197—24no—
41C22×C4×C197C2314yes—
42C22×C4×C197C2311yescharacteristic
43C22×C4×C197C2312yes—
44C4⋊C4×C197—24no—
45C4⋊C4×C197—28no—
46C42×C197D414yes—
47C22×D197—21no—
48C22×D197—22no—
49C22×D197—21no—
50C22×D197C2×D412yes—
51C22×D197C2×Q814yes—
52Dic394—48no—
53C2×Dic197—21no—
54C2×Dic197—22no—
55C2×Dic197—22no—
56C2×Dic197—21no—
57C2×Dic197—24no—
58C2×Dic197—24no—
59C2×Dic197—24no—
60C2×Dic197C2×D412yes—
61C2×Dic197C2×Q814yes—
62C4×D197—28no—
63C4×D197—216no—
64C23×C197C2411yescharacteristic
65C2×C4×C197—24no—
66C2×C4×C197—28no—
67C2×C4×C197C2×D412yes—
68C2×C4×C197C2×D412yes—
69C2×C4×C197C2×D414yes—
70C2×C4×C197C2×Q814yes—
71C2×C4×C197#4114yes—
72C2×C4×C197#4112yes—
73C22×C197C22×D411yescharacteristic
74C22×C197C22×D411yescharacteristic
75C22×C197C22×Q812yes—
76C22×C197C2×#4112yes—
77C22×C197C2×#4111yescharacteristic
78C2×D197—22no—
79C2×D197—24no—
80C2×D197—22no—
81C2×D197—24no—
82C2×D197C4⋊Q814yescomplemented
83C4×C197—44no—
84C4×C197#12114yes—
85C4×C197C22⋊Q818yescomplemented
86Dic197—22no—
87Dic197—24no—
88Dic197—44no—
89C2×C197C2×#12111yescharacteristic
90C2×C197C2×C22⋊Q812yes—
91C2×C197C2×C4⋊Q811yescharacteristic
92C2×C197#53211yescharacteristic
93C2×C197#53312yes—
94D197—24no—
95C197#285711yescharacteristic, complemented, Sylow
96#2857—1971nomaximal, Sylow
97C2×C4⋊Q8—1971no—
98C2×C22⋊Q8—1972no—
99C22×C42—1971no—
100#391—1972no—
101#389—1974no—
102#384D19711yescharacteristic, complemented
103#380—1974no—
104C22×Q8—1972no—
105C23×C4—1971no—
106C23×C4—1972no—
107C4⋊Q8—3944no—
108C22⋊Q8—3948no—
109C2×C4⋊C4—1974no—
110C2×C4⋊C4—1974no—
111C2×C4⋊C4C2×D19712yes—
112C2×C4⋊C4C2×D19714yes—
113C2×C22⋊C4—1974no—
114C2×C42—3944no—
115C2×C42—1972no—
116C2×C42—1971no—
117C2×C42C2×D19711yescharacteristic
118#95—1978no—
119C24—1971no—
120C2×Q8—3944no—
121C2×Q8—3944no—
122C2×Q8—3944no—
123C22×C4—3942no—
124C22×C4—3944no—
125C22×C4—3944no—
126C22×C4—3942no—
127C22×C4—1974no—
128C22×C4—1972no—
129C22×C4—1971no—
130C22×C4—1974no—
131C22×C4—1978no—
132C22×C4C22×D19714yes—
133C22×C4C22×D19711yescharacteristic
134C22×C4C22×D19712yes—
135C4⋊C4—24no—
136C4⋊C4—28no—
137C4⋊C4—3948no—
138C4⋊C4—3948no—
139C22⋊C4—3948no—
140C42—3942no—
141C42—3944no—
142C42D78814yes—
143C23—3941no—
144C23—3942no—
145C23—3941no—
146C23—1972no—
147C23—1974no—
148C23C23×D19711yescharacteristic
149Q8—7888no—
150C2×C4—24no—
151C2×C4—28no—
152C2×C4—3944no—
153C2×C4—3944no—
154C2×C4—3944no—
155C2×C4—3941no—
156C2×C4—3942no—
157C2×C4—3942no—
158C2×C4—3941no—
159C2×C4—3948no—
160C2×C4—39416no—
161C2×C4—1972no—
162C2×C4—1974no—
163C2×C4D4×D19712yes—
164C2×C4C2×D78812yes—
165C2×C4C2×D78814yes—
166C2×C4Q8×D19714yes—
167C2×C4#1569135714yes—
168C2×C4#1569135912yes—
169C22—3942no—
170C22—3944no—
171C22—3942no—
172C22—3944no—
173C22—1974no—
174C22C2×D4×D19711yescharacteristic
175C22C22×D78811yescharacteristic
176C22C2×Q8×D19712yes—
177C22C2×#1569135712yes—
178C22C2×#1569135911yescharacteristic
179C4—44no—
180C4—7884no—
181C4—3942no—
182C4—3944no—
183C4#2026330114yes—
184C4#2026330618yescomplemented
185C2—3944no—
186C2C2×#2026330111yescharacteristic
187C2C2×#2026330612yes—
188C2C4⋊Q8×D19711yescharacteristic
189C2#2716162711yescharacteristic
190C2#2716163412yes—
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#384×C197C2
prime radicalC23×D197C23
hypercenter#384D197
hypocenterC197#2857
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#20263326C2semidirect
4. #27160984C2semidirect
5. #27160978C2semidirect
7. #384×C197C2semidirect
35. C2×C4×D197Q8semidirect
82. C2×D197C4⋊Q8semidirect
85. C4×C197C22⋊Q8semidirect
95. C197#2857semidirect
102. #384D197semidirect
184. C4#20263306semidirect

Maximal subgroups

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

subgroupindex
C2×#202633262
C2×#202633062
C2×C42×D1972
#271609842
#271609782
#271609362
#384×C1972
#2857197