Finite Group Databasebeta

← #38743122

Subgroups — #38743122

Order 25216.

386 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20263309C211yesmaximal, characteristic
2#27161552C211yesmaximal, characteristic
3C2×#20263240C211yesmaximal, characteristic
4#27160942C211yesmaximal, characteristic, complemented
5#27160846C211yesmaximal, characteristic, complemented
6#27160845C211yesmaximal, characteristic, complemented
7#381×C197C211yesmaximal, characteristic, complemented
8C22×C22⋊D197C2211yescharacteristic
9#20263309—22no—
10#20263309C414yescomplemented
11C4×C22⋊D197—22no—
12#20263286—22no—
13#20263282—21no—
14C2×#15691337—21no—
15C2×C22⋊Dic197C2211yescharacteristic
16C2×C22⋊Dic197C2211yescharacteristic
17C2×C22⋊Dic197C2211yescharacteristic
18C2×#15691335—21no—
19C22⋊C4×D197—21no—
20#20263240—22no—
21#20263233—22no—
22C23×C4×C197C2211yescharacteristic
23C2×C22⋊C4×C197C2211yescharacteristic
24C2×C22⋊C4×C197C2211yescharacteristic
25#95×C197—22no—
26C23×D197D411yescharacteristic
27C2×C22⋊D197—21no—
28C2×C22⋊D197—21no—
29C2×C22⋊D197—21no—
30C2×C22⋊D197—21no—
31C2×C22⋊D197C2×C412yes—
32C2×C22⋊D197D412yes—
33C22×Dic197—21no—
34C22×Dic197—21no—
35C22×Dic197D411yescharacteristic
36C22×Dic197D412yes—
37C2×C4×D197—22no—
38#15691337—22no—
39#15691337—24no—
40C22⋊Dic197—21no—
41C22⋊Dic197—21no—
42C22⋊Dic197—21no—
43C22⋊Dic197—21no—
44C22⋊Dic197—21no—
45C22⋊Dic197C2×C412yescomplemented
46#15691335—22no—
47#15691335—24no—
48C4×Dic197—22no—
49C24×C197C2311yescharacteristic
50C22×C4×C197—21no—
51C22×C4×C197—21no—
52C22×C4×C197—21no—
53C22×C4×C197—21no—
54C22×C4×C197—21no—
55C22×C4×C197—21no—
56C22×C4×C197—21no—
57C22×C4×C197C2×C412yescomplemented
58C22×C4×C197D412yes—
59C22⋊C4×C197—21no—
60C22⋊C4×C197—21no—
61C22⋊C4×C197—21no—
62C22⋊C4×C197—21no—
63C22×D197—21no—
64C22×D197—21no—
65C22×D197—42no—
66C22×D197C22⋊C412yes—
67C22⋊D197—22no—
68C22⋊D197—22no—
69C22⋊D197—22no—
70C22⋊D197—22no—
71C2×Dic197—21no—
72C2×Dic197—21no—
73C2×Dic197—22no—
74C2×Dic197—22no—
75C2×Dic197—42no—
76C2×Dic197—42no—
77C2×Dic197—42no—
78C2×Dic197—41no—
79C2×Dic197C22⋊C412yes—
80C4×D197—82no—
81C23×C197—42no—
82C23×C197C22×C411yescharacteristic
83C23×C197C2×D411yescharacteristic
84C23×C197C2×D411yescharacteristic
85C23×C197C2×D411yescharacteristic
86C23×C197C2×D411yescharacteristic
87C23×C197#4111yescharacteristic
88C23×C197#4111yescharacteristic
89C2×C4×C197—22no—
90C2×C4×C197—22no—
91C2×C4×C197—22no—
92C2×C4×C197—22no—
93C2×C4×C197—41no—
94C2×C4×C197—41no—
95C2×C4×C197—41no—
96C2×C4×C197—41no—
97C2×C4×C197—42no—
98C2×C4×C197—42no—
99C2×C4×C197—41no—
100C22×C197—22no—
101C22×C197—22no—
102C22×C197—22no—
103C22×C197—41no—
104C22×C197—41no—
105C22×C197—41no—
106C22×C197—41no—
107C22×C197C2×C22⋊C411yescharacteristic
108C22×C197C4×D411yescharacteristic
109C22×C197C4×D411yescharacteristic
110C22×C197#12011yescharacteristic
111C22×C197#12111yescharacteristic
112C22×C197#12111yescharacteristic
113C22×C197#12311yescharacteristic
114C2×D197—22no—
115C2×D197—42no—
116C2×D197—42no—
117C2×D197—81no—
118C4×C197—42no—
119C4×C197—42no—
120C4×C197—81no—
121Dic197—22no—
122Dic197—42no—
123Dic197—42no—
124Dic197—81no—
125C2×C197—21no—
126C2×C197—21no—
127C2×C197—21no—
128C2×C197—21no—
129C2×C197—21no—
130C2×C197—21no—
131C2×C197#38611yescharacteristic
132C2×C197#45812yes—
133D197—42no—
134C197#189011yescharacteristic, complemented, Sylow
135#1890—1971nomaximal, Sylow
136C2×#123—1971no—
137#518—1971no—
138C2×#99—1971no—
139#381—1971no—
140#381D19711yescharacteristic, complemented
141#342—1971no—
142#342—1971no—
143C22×D4—1971no—
144C23×C4C2×D19711yescharacteristic
145#123—3942no—
146#123—1974no—
147C4×D4—3942no—
148C4×D4—3942no—
149#117—3941no—
150C2×C4⋊C4—3941no—
151C2×C22⋊C4—3941no—
152C2×C22⋊C4—3941no—
153C2×C22⋊C4—1971no—
154C2×C22⋊C4—1971no—
155C2×C22⋊C4—1971no—
156C2×C22⋊C4C2×D19711yescharacteristic
157C2×C22⋊C4C2×D19711yescharacteristic
158#99—3942no—
159#95—22no—
160#95—3942no—
161C24—1971no—
162C24C22×D19711yescharacteristic
163C2×D4—3941no—
164C2×D4—3941no—
165C2×D4—3941no—
166C2×D4—3941no—
167C2×D4—1972no—
168C2×D4—1972no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—3942no—
177C22×C4—3941no—
178C22×C4—3941no—
179C22×C4—1972no—
180C22×C4—1971no—
181C22×C4C4×D19712yescomplemented
182C22×C4C22⋊D19712yes—
183C4⋊C4—3942no—
184C4⋊C4—3944no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—3942no—
190C22⋊C4—3944no—
191C22⋊C4—3941no—
192C22⋊C4—3941no—
193C22⋊C4—3941no—
194C22⋊C4—3941no—
195C22⋊C4—3941no—
196C22⋊C4—1972no—
197C42—3942no—
198C23—42no—
199C23—7882no—
200C23—3941no—
201C23—3941no—
202C23—1972no—
203C23C2×C4×D19711yescharacteristic
204C23D4×D19711yescharacteristic
205C23D4×D19711yescharacteristic
206C23D4×D19711yescharacteristic
207C23C2×C22⋊D19711yescharacteristic
208C23#1569135711yescharacteristic
209C23#1569135811yescharacteristic
210D4—3942no—
211D4—3942no—
212D4—3942no—
213D4—3942no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—7882no—
226C2×C4—7882no—
227C2×C4—7882no—
228C2×C4—7881no—
229C2×C4—15762no—
230C2×C4—3942no—
231C2×C4—3942no—
232C2×C4—3941no—
233C2×C4—3941no—
234C2×C4—1972no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—7882no—
243C22—7882no—
244C22—15761no—
245C22—3942no—
246C22C22⋊C4×D19711yescharacteristic
247C22#2026328611yescharacteristic
248C22C4×C22⋊D19711yescharacteristic
249C22#2026329411yescharacteristic
250C22#2026329711yescharacteristic
251C22#2026329911yescharacteristic
252C22#2026330811yescharacteristic
253C4—42no—
254C4—42no—
255C4—81no—
256C4—7882no—
257C4—7882no—
258C4—15761no—
259C4—3942no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—7882no—
267C2#2716095511yescharacteristic
268C2#2716126812yes—
subgroupG/H
centerC22#20263294
commutatorC23×C197C22×C4
FrattiniC24C22×D197
Fitting#381×C197C2
socleC22×C197#120
prime radicalC22×C22⋊D197C22
cosocleC24×C197C23
hypercenter#381D197
hypocenterC197#1890
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
4. #27160942C2semidirect
5. #27160846C2semidirect
6. #27160845C2semidirect
7. #381×C197C2semidirect
10. #20263309C4semidirect
45. C22⋊Dic197C2×C4semidirect
57. C22×C4×C197C2×C4semidirect
134. C197#1890semidirect
140. #381D197semidirect
181. C22×C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633092
#271615522
C2×#202632402
#271609422
#271608462
#271608452
#381×C1972
#1890197