Finite Group Databasebeta

← #38755360

Subgroups — #38755360

Order 25216.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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#27161799C211yesmaximal, characteristic, complemented
2#27161790C211yesmaximal, characteristic, complemented
3#27161432C212yesmaximal, complemented
4#27161431C211yesmaximal, characteristic, complemented
5#27161389C212yesmaximal, complemented
6#27161293C212yesmaximal
7#27161285C211yesmaximal, characteristic, complemented
8#27161203C211yesmaximal, characteristic, complemented
9#27161166C212yesmaximal, complemented
10C4×#15691343C211yesmaximal, characteristic
11#476×C197C211yesmaximal, characteristic, complemented
12C2×#15691347C2212yes—
13C2×#15691343C2211yescharacteristic
14#20263326C2211yescharacteristic
15#20263319—22no—
16#20263317C2212yes—
17#20263313C2212yes—
18#20263306—21no—
19#20263305—21no—
20#20263302—22no—
21C4×Dic394C2211yescharacteristic
22#20263291C2212yes—
23C4×D788C2211yescharacteristic
24#20263288C2212yes—
25#20263285—21no—
26C4⋊C4×D197—21no—
27#20263259C2211yescharacteristic
28#20263255C2212yes—
29#20263251C2212yes—
30#20263249C2212yes—
31#20263249C2212yes—
32#20263247C2211yescharacteristic
33#20263246C2211yescharacteristic
34C4×#13096802C2211yescharacteristic
35C8×Dic197C2212yes—
36C4⋊Q8×C197C2211yescharacteristic
37C4×D4×C197C2211yescharacteristic
38#106×C197C2212yes—
39C4⋊C8×C197C2211yescharacteristic
40#102×C197C2212yes—
41C2×Dic394D411yescharacteristic
42C2×Dic394C2312yes—
43C2×D788C2311yescharacteristic
44C2×C4×D197—22no—
45C2×C4×D197—21no—
46#15691347—24no—
47#15691343—22no—
48C2×#13096802C2311yescharacteristic
49C2×#13096802C2311yescharacteristic
50C4⋊Dic197—22no—
51C4⋊Dic197—21no—
52C4⋊Dic197—21no—
53C4⋊Dic197D411yescharacteristic
54C4⋊Dic197C2311yescharacteristic
55#15691337—22no—
56#15691337—22no—
57#15691337D412yes—
58C22⋊Dic197—22no—
59C22⋊Dic197—22no—
60#15691335—22no—
61#15691335—21no—
62C4×Dic197—22no—
63C4×Dic197C2312yes—
64C2×Q8×C197—21no—
65C2×D4×C197C2311yescharacteristic
66C22×C4×C197—21no—
67C2×C8×C197C2312yes—
68C4⋊C4×C197—21no—
69C4⋊C4×C197C2312yes—
70C4⋊C4×C197C2311yescharacteristic
71C22⋊C4×C197—21no—
72C42×C197C2311yescharacteristic
73C22×D197—21no—
74Dic394—22no—
75Dic394—22no—
76Dic394—22no—
77D788—21no—
78D788#4112yes—
79C2×Dic197—22no—
80C2×Dic197—21no—
81C2×Dic197C2×D412yes—
82C2×Dic197C2×D412yes—
83C4×D197—42no—
84C4×D197—82no—
85#13096802—21no—
86#13096802—21no—
87C23×C197—21no—
88Q8×C197—81no—
89D4×C197—21no—
90D4×C197—21no—
91C2×C4×C197—21no—
92C2×C4×C197—22no—
93C2×C4×C197—42no—
94C2×C4×C197C2×D411yescharacteristic
95C2×C4×C197C2×D411yescharacteristic
96C2×C4×C197C2411yescharacteristic
97C8×C197—22no—
98C22×C197—41no—
99C22×C197—41no—
100C22×C197—41no—
101C22×C197C22×D411yescharacteristic
102C2×D197—22no—
103C2×D197—41no—
104C2×D197—41no—
105C4×C197—21no—
106C4×C197—41no—
107C4×C197—42no—
108C4×C197#13512yes—
109C4×C197C2×#4111yescharacteristic
110C4×C1972−1+411yescharacteristic
111Dic197—22no—
112Dic197—22no—
113Dic197—42no—
114Dic197—41no—
115C2×C197—81no—
116C2×C197#54811yescharacteristic
117C2×C197C2×#13511yescharacteristic
118C2×C197#57711yescharacteristic
119D197—42no—
120C197#336011yescharacteristic, complemented, Sylow
121#3360—1971nomaximal, Sylow
122#556—1971no—
123#554—1971no—
124#488—1972no—
125#487—1971no—
126#479—1972no—
127#476D19711yescharacteristic, complemented
128#463—1972no—
129#461—1971no—
130C8×Q8—1971no—
131C4×SD16—1972no—
132C4×D8—1971no—
133C2×SD16—1972no—
134C2×D8—1971no—
135C4⋊Q8—1971no—
136C4⋊Q8C2×D19711yescharacteristic
137#126—3942no—
138#125—1972no—
139#124—1972no—
140C22⋊Q8—3941no—
141C22⋊Q8—3941no—
142C22⋊Q8—3942no—
143C4×Q8—1972no—
144C4×Q8—1971no—
145C4×D4—1972no—
146C4×D4—1971no—
147C4×D4C2×D19711yescharacteristic
148#117—3941no—
149C2×C4⋊C4—3941no—
150#107—1971no—
151#106C2×D19712yes—
152C4⋊C8—1972no—
153C4⋊C8C2×D19711yescharacteristic
154#103—1972no—
155#103—1972no—
156#103—1972no—
157#102—1971no—
158#102—1971no—
159#102C2×D19712yes—
160C4×C8—1972no—
161C4×C8—1971no—
162C2×Q8—21no—
163C2×Q8—1972no—
164C2×Q8—1971no—
165C2×D4—1971no—
166C2×D4C22×D19711yescharacteristic
167C22×C4—21no—
168C22×C4—3942no—
169C22×C4—3941no—
170SD16—3944no—
171D8—3942no—
172C2×C8—1971no—
173C2×C8—1971no—
174C2×C8C22×D19712yes—
175C4⋊C4—21no—
176C4⋊C4—3942no—
177C4⋊C4—3942no—
178C4⋊C4—3942no—
179C4⋊C4—3941no—
180C4⋊C4—3941no—
181C4⋊C4—1971no—
182C4⋊C4—1972no—
183C4⋊C4—1971no—
184C4⋊C4C22×D19711yescharacteristic
185C4⋊C4C22×D19712yes—
186C22⋊C4—21no—
187C22⋊C4—3942no—
188C22⋊C4—3942no—
189C22⋊C4—3942no—
190C22⋊C4—3941no—
191C42—3942no—
192C42—1972no—
193C42C22×D19711yescharacteristic
194C23—21no—
195C23—3941no—
196Q8—81no—
197Q8—3942no—
198Q8—3942no—
199Q8—3942no—
200D4—21no—
201D4—21no—
202D4—3941no—
203D4—1972no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—7882no—
208C2×C4—15762no—
209C2×C4—3942no—
210C2×C4—3941no—
211C2×C4—1972no—
212C2×C4—1972no—
213C2×C4D4×D19711yescharacteristic
214C2×C4D4×D19711yescharacteristic
215C2×C4C23×D19711yescharacteristic
216C8—22no—
217C8—3941no—
218C8—3941no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—7881no—
223C22—7881no—
224C22—3942no—
225C22C2×D4×D19711yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—7881no—
230C4—7882no—
231C4—3942no—
232C4—3942no—
233C4#2026334712yes—
234C4#41×D19711yescharacteristic
235C4#2026337411yescharacteristic
236C2—81no—
237C2—7882no—
238C2#2716173911yescharacteristic
239C2C2×#2026334711yescharacteristic
240C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#476×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691343C22
hypercenter#476D197
hypocenterC197#3360
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #27161799C2semidirect
2. #27161790C2semidirect
3. #27161432C2semidirect
4. #27161431C2semidirect
5. #27161389C2semidirect
7. #27161285C2semidirect
8. #27161203C2semidirect
9. #27161166C2semidirect
11. #476×C197C2semidirect
120. C197#3360semidirect
127. #476D197semidirect

Maximal subgroups

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

subgroupindex
#271617992
#271617902
#271614322
#271614312
#271613892
#271612932
#271612852
#271612032
#271611662
C4×#156913432
#476×C1972
#3360197