Finite Group Databasebeta

← #38742992

Subgroups — #38742992

Order 25216.

374 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#27161554C211yesmaximal, characteristic
2C2×#20263241C211yesmaximal, characteristic
3#27160952C212yesmaximal, complemented
4#27160845C212yesmaximal, complemented
5#386×C197C211yesmaximal, characteristic
6D4×Dic197—22no—
7D4×Dic197—22no—
8#20263285—21no—
9C2×#15691337—22no—
10C2×C22⋊Dic197—21no—
11C2×C22⋊Dic197C2212yes—
12C2×C22⋊Dic197C2212yes—
13#20263241C414yescomplemented
14#20263233—22no—
15C22×D4×C197C2211yescharacteristic
16C23×C4×C197C2211yescharacteristic
17C2×C22⋊C4×C197—21no—
18C2×C22⋊C4×C197C2211yescharacteristic
19#95×C197—21no—
20C22×Dic197—21no—
21C22×Dic197—21no—
22C22×Dic197—22no—
23C22×Dic197D412yes—
24C22×Dic197D412yes—
25C4⋊Dic197—21no—
26C4⋊Dic197—21no—
27#15691337—44no—
28C22⋊Dic197—22no—
29C22⋊Dic197—22no—
30C22⋊Dic197—22no—
31C22⋊Dic197—22no—
32C22⋊Dic197C2×C414yescomplemented
33C4×Dic197—22no—
34C24×C197D411yescharacteristic
35C24×C197C2311yescharacteristic
36C2×D4×C197—22no—
37C2×D4×C197—22no—
38C2×D4×C197C2×C412yes—
39C2×D4×C197Q812yes—
40C22×C4×C197—21no—
41C22×C4×C197—22no—
42C22×C4×C197—21no—
43C22×C4×C197—42no—
44C22×C4×C197D411yescharacteristic
45C22⋊C4×C197—22no—
46C22⋊C4×C197—42no—
47C2×Dic197—24no—
48C2×Dic197—22no—
49C2×Dic197—22no—
50C2×Dic197—42no—
51C2×Dic197—42no—
52C2×Dic197—42no—
53C2×Dic197—42no—
54C2×Dic197—81no—
55C2×Dic197—81no—
56C23×C197—21no—
57C23×C197—21no—
58C23×C197—41no—
59C23×C197—41no—
60C23×C197—41no—
61C23×C197—41no—
62C23×C197C4⋊C412yescomplemented
63C23×C197C22×C411yescharacteristic
64C23×C197C2×D412yes—
65C23×C197C2×D411yescharacteristic
66C23×C197C2×Q811yescharacteristic
67C23×C197#4112yes—
68D4×C197—24no—
69D4×C197—24no—
70C2×C4×C197—21no—
71C2×C4×C197—21no—
72C2×C4×C197—22no—
73C2×C4×C197—24no—
74C2×C4×C197—41no—
75C2×C4×C197—42no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197—41no—
79C2×C4×C197—41no—
80C2×C4×C197C4⋊C412yescomplemented
81C22×C197—21no—
82C22×C197—22no—
83C22×C197—21no—
84C22×C197—22no—
85C22×C197—22no—
86C22×C197—41no—
87C22×C197—41no—
88C22×C197—42no—
89C22×C197—42no—
90C22×C197—41no—
91C22×C197—41no—
92C22×C197—81no—
93C22×C197C2×C4⋊C411yescharacteristic
94C22×C197C4×D412yes—
95C22×C197#12011yescharacteristic
96C22×C197C22⋊Q812yes—
97C22×C197#12311yescharacteristic
98C4×C197—22no—
99C4×C197—42no—
100C4×C197—81no—
101Dic197—44no—
102Dic197—42no—
103Dic197—42no—
104C2×C197—22no—
105C2×C197—21no—
106C2×C197—21no—
107C2×C197—22no—
108C2×C197—42no—
109C2×C197#38511yescharacteristic
110C2×C197B(4,𝔽2)11yescharacteristic
111C2×C197#45811yescharacteristic
112C197#186711yescharacteristic, complemented, Sylow
113#1867—1971nomaximal, Sylow
114#518—1971no—
115C2×#99—1971no—
116#386D19711yescharacteristic
117#385—1972no—
118#342—1972no—
119C22×D4C2×D19711yescharacteristic
120C23×C4C2×D19711yescharacteristic
121C4×D4—3942no—
122C4×D4—3942no—
123#117—3941no—
124C2×C4⋊C4—3942no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—3941no—
127C2×C22⋊C4—1972no—
128C2×C22⋊C4—1972no—
129C2×C22⋊C4C2×D19711yescharacteristic
130#99—1974no—
131#95—21no—
132#95—3942no—
133C24C22⋊D19711yescharacteristic
134C24C22×D19711yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4C4×D19712yes—
138C2×D4Dic39412yes—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—42no—
143C22×C4—3942no—
144C22×C4—3941no—
145C22×C4—3941no—
146C22×C4—1972no—
147C22×C4—1972no—
148C22×C4C22⋊D19711yescharacteristic
149C4⋊C4—7884no—
150C4⋊C4—3941no—
151C4⋊C4—3941no—
152C22⋊C4—22no—
153C22⋊C4—42no—
154C22⋊C4—3942no—
155C22⋊C4—3942no—
156C22⋊C4—3942no—
157C22⋊C4—3942no—
158C22⋊C4—1974no—
159C42—3942no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23#1569133712yescomplemented
167C23C2×C4×D19711yescharacteristic
168C23D4×D19712yes—
169C23C2×C22⋊D19711yescharacteristic
170C23C2×Dic39411yescharacteristic
171C23#1569135712yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—7882no—
185C2×C4—7882no—
186C2×C4—7882no—
187C2×C4—7882no—
188C2×C4—15761no—
189C2×C4—15761no—
190C2×C4—3944no—
191C2×C4—3942no—
192C2×C4—3942no—
193C2×C4#1569133712yescomplemented
194C22—21no—
195C22—22no—
196C22—21no—
197C22—22no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×#1569133711yescharacteristic
207C22#2026328612yes—
208C22#2026329411yescharacteristic
209C22#2026330312yes—
210C22#2026330711yescharacteristic
211C4—22no—
212C4—42no—
213C4—81no—
214C4—7884no—
215C4—7882no—
216C4—7882no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#2716095111yescharacteristic
223C2#2716126511yescharacteristic
224C2#2716126811yescharacteristic
subgroupG/H
centerC22#20263294
commutatorC23×C197C22×C4
FrattiniC24C22×D197
Fitting#386×C197C2
socleC22×C197#120
prime radicalC22×D4×C197C22
cosocleC24×C197C23
hypercenter#386D197
hypocenterC197#1867
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #27160952C2semidirect
4. #27160845C2semidirect
13. #20263241C4semidirect
32. C22⋊Dic197C2×C4semidirect
62. C23×C197C4⋊C4semidirect
80. C2×C4×C197C4⋊C4semidirect
112. C197#1867semidirect
166. C23#15691337semidirect
193. C2×C4#15691337semidirect

Maximal subgroups

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

subgroupindex
#271615542
C2×#202632412
#271609522
#271608452
#386×C1972
#1867197