Finite Group Databasebeta

← #38742914

Subgroups — #38742914

Order 25216.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#20263284C211yesmaximal, characteristic
2C2×#20263260C211yesmaximal, characteristic, complemented
3C2×#20263257C211yesmaximal, characteristic, complemented
4C2×#20263238C211yesmaximal, characteristic
5#27160936C211yesmaximal, characteristic
6#27160841C211yesmaximal, characteristic, complemented
7#340×C197C211yesmaximal, characteristic, complemented
8C22×C4×D197C2211yescharacteristic
9C22×#13096802C2211yescharacteristic
10#20263284—22no—
11#20263284—22no—
12C2×C4⋊Dic197C2211yescharacteristic
13C2×C4⋊Dic197C2211yescharacteristic
14C2×#15691335—21no—
15C2×#15691335—21no—
16C2×C4×Dic197—21no—
17#20263260—22no—
18#20263257—22no—
19#20263238C414yescomplemented
20#20263233—21no—
21C22×C8×C197C2211yescharacteristic
22C2×C4⋊C4×C197C2211yescharacteristic
23C2×C4⋊C4×C197C2211yescharacteristic
24C23×D197D411yescharacteristic
25C22×Dic197—21no—
26C22×Dic197—21no—
27C22×Dic197D411yescharacteristic
28C2×C4×D197—21no—
29C2×C4×D197—21no—
30C2×C4×D197—21no—
31C2×C4×D197—21no—
32C2×C4×D197C2×C412yes—
33C2×C4×D197Q812yes—
34C2×#13096802—21no—
35C2×#13096802—21no—
36C2×#13096802C2×C412yes—
37C4⋊Dic197—21no—
38C4⋊Dic197—21no—
39C4⋊Dic197—21no—
40C4⋊Dic197D412yes—
41#15691335—24no—
42#15691335—42no—
43C4×Dic197—24no—
44C22×C4×C197—21no—
45C22×C4×C197—21no—
46C22×C4×C197C2311yescharacteristic
47C2×C8×C197—21no—
48C2×C8×C197—21no—
49C2×C8×C197C2×C412yes—
50C4⋊C4×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197—21no—
53C4⋊C4×C197D412yes—
54C22×D197—21no—
55C22×D197—21no—
56C22×D197—21no—
57C22×D197—21no—
58C22×D197—21no—
59C22×D197—21no—
60C22×D197C4⋊C412yes—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—22no—
64C2×Dic197—41no—
65C2×Dic197—41no—
66C2×Dic197—41no—
67C2×Dic197—41no—
68C2×Dic197—41no—
69C2×Dic197C4⋊C412yes—
70C4×D197—22no—
71C4×D197—22no—
72C4×D197—22no—
73C4×D197—22no—
74#13096802—22no—
75C23×C197C2×D411yescharacteristic
76C2×C4×C197—22no—
77C2×C4×C197—41no—
78C2×C4×C197—41no—
79C2×C4×C197—41no—
80C2×C4×C197—41no—
81C2×C4×C197—41no—
82C2×C4×C197C22×C411yescharacteristic
83C2×C4×C197C2×D411yescharacteristic
84C2×C4×C197C2×D411yescharacteristic
85C2×C4×C197C2×Q811yescharacteristic
86C2×C4×C197#4111yescharacteristic
87C2×C4×C197#4111yescharacteristic
88C8×C197—22no—
89C22×C197C2×C4⋊C411yescharacteristic
90C22×C197#12011yescharacteristic
91C22×C197#12311yescharacteristic
92C22×C197#13511yescharacteristic
93C22×C197#13511yescharacteristic
94C22×C197AGL(1,ℤ/8)11yescharacteristic
95C22×C197#13711yescharacteristic
96C2×D197—22no—
97C2×D197—22no—
98C2×D197—22no—
99C2×D197—41no—
100C2×D197—41no—
101C2×D197—41no—
102C2×D197—41no—
103C4×C197—42no—
104C4×C197—81no—
105C4×C197C4×D411yescharacteristic
106C4×C197C4×D411yescharacteristic
107C4×C197C22⋊Q811yescharacteristic
108C4×C197C22⋊Q811yescharacteristic
109Dic197—22no—
110Dic197—42no—
111Dic197—81no—
112C2×C197#38511yescharacteristic
113C2×C197#42711yescharacteristic
114C2×C197#42811yescharacteristic
115C2×C197#44911yescharacteristic
116C2×C197#45211yescharacteristic
117C2×C197#48211yescharacteristic
118C2×C197#48511yescharacteristic
119D197—42no—
120C197#185411yescharacteristic, complemented, Sylow
121#1854—1971nomaximal, Sylow
122C2×#117—1971no—
123C2×#107—1971no—
124C2×#106—1971no—
125C2×C22⋊C8—1971no—
126#380—1971no—
127#340—1971no—
128#340D19711yescharacteristic, complemented
129C23×C4—1971no—
130C22×C8—1971no—
131C22×C8C2×D19711yescharacteristic
132#117—3942no—
133#117—3942no—
134C2×C4⋊C4—1971no—
135C2×C4⋊C4—1971no—
136C2×C4⋊C4C2×D19711yescharacteristic
137C2×C4⋊C4C2×D19711yescharacteristic
138C2×C22⋊C4—3941no—
139C2×C22⋊C4—3941no—
140C2×C42—3941no—
141#107—3942no—
142#106—3942no—
143C22⋊C8—1974no—
144#95—3941no—
145C24—1971no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3941no—
149C22×C4—3941no—
150C22×C4—3941no—
151C22×C4—3941no—
152C22×C4—3941no—
153C22×C4—3941no—
154C22×C4—1971no—
155C22×C4—1972no—
156C22×C4—1972no—
157C22×C4C22×D19711yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—3941no—
161C2×C8—3941no—
162C2×C8—1972no—
163C2×C8C4×D19712yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—3941no—
168C4⋊C4—3941no—
169C4⋊C4—3941no—
170C4⋊C4—1972no—
171C4⋊C4D78812yes—
172C22⋊C4—7882no—
173C22⋊C4—3944no—
174C42—3944no—
175C23—3941no—
176C23—3941no—
177C23—3941no—
178C23—3941no—
179C23—3941no—
180C23—3941no—
181C23—1972no—
182C23D4×D19711yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—7881no—
190C2×C4—7881no—
191C2×C4—7881no—
192C2×C4—7881no—
193C2×C4—7881no—
194C2×C4—3942no—
195C2×C4—3941no—
196C2×C4—3941no—
197C2×C4—3942no—
198C2×C4—3942no—
199C2×C4—3942no—
200C2×C4—3942no—
201C2×C4—1972no—
202C2×C4C2×C4×D19711yescharacteristic
203C2×C4D4×D19711yescharacteristic
204C2×C4C2×D78811yescharacteristic
205C2×C4Q8×D19711yescharacteristic
206C2×C4#1569135711yescharacteristic
207C2×C4#1569135811yescharacteristic
208C8—22no—
209C8—3942no—
210C22—7881no—
211C22—7881no—
212C22—7881no—
213C22—7881no—
214C22—3942no—
215C22—3942no—
216C22—3942no—
217C22C4⋊C4×D19711yescharacteristic
218C22#2026329611yescharacteristic
219C22#2026330811yescharacteristic
220C22#2026334511yescharacteristic
221C22#2026334911yescharacteristic
222C22#2026335311yescharacteristic
223C22#2026335511yescharacteristic
224C4—42no—
225C4—81no—
226C4—7882no—
227C4—15761no—
228C4—3942no—
229C4#2026328611yescharacteristic
230C4C4×D78811yescharacteristic
231C4#2026330411yescharacteristic
232C4#2026330611yescharacteristic
233C2—7882no—
234C2#2716095311yescharacteristic
235C2#2716111611yescharacteristic
236C2#2716111811yescharacteristic
237C2#2716122711yescharacteristic
238C2#2716124211yescharacteristic
239C2#2716140111yescharacteristic
240C2#2716141911yescharacteristic
subgroupG/H
centerC23D4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#340×C197C2
socleC23×C197C2×D4
prime radicalC23×D197D4
cosocleC22×C4×C197C23
hypercenter#340D197
hypocenterC197#1854
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#20263260C2semidirect
3. C2×#20263257C2semidirect
6. #27160841C2semidirect
7. #340×C197C2semidirect
19. #20263238C4semidirect
120. C197#1854semidirect
128. #340D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632842
C2×#202632602
C2×#202632572
C2×#202632382
#271609362
#271608412
#340×C1972
#1854197