Finite Group Databasebeta

← #38755357

Subgroups — #38755357

Order 25216.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#27161789C211yesmaximal, characteristic, complemented
2#27161433C212yesmaximal, complemented
3#27161429C211yesmaximal, characteristic, complemented
4#27161386C212yesmaximal, complemented
5#27161369C211yesmaximal, characteristic, complemented
6#27161291C212yesmaximal
7#27161281C211yesmaximal, characteristic, complemented
8Q8×#13096802C211yesmaximal, characteristic, complemented
9C4×#15691345C212yesmaximal, complemented
10C4×#15691343C211yesmaximal, characteristic
11#556×C197C211yesmaximal, characteristic, complemented
12C2×#15691345C2212yes—
13C2×#15691343C2211yescharacteristic
14#20263326C2211yescharacteristic
15#20263318C2212yes—
16#20263304—22no—
17#20263302—21no—
18C4×D788C2211yescharacteristic
19C4×D788C2212yes—
20C4⋊C4×D197—21no—
21#20263259C2211yescharacteristic
22#20263257C2212yes—
23C4⋊#13096802C2211yescharacteristic
24C4⋊#13096802C2212yes—
25#20263250C2212yes—
26#20263250C2212yes—
27#20263250C2212yes—
28#20263247C2212yes—
29#20263247C2211yescharacteristic
30#20263246C2211yescharacteristic
31C4×#13096802C2211yescharacteristic
32C4×#13096802C2212yes—
33C4⋊Q8×C197C2211yescharacteristic
34#126×C197—22no—
35#124×C197C2212yes—
36C22⋊Q8×C197—21no—
37C4×Q8×C197C2212yes—
38C4×Q8×C197C2211yescharacteristic
39C4×D4×C197C2211yescharacteristic
40#117×C197—21no—
41C2×Dic394—21no—
42C2×D788C2311yescharacteristic
43C2×C4×D197—21no—
44C2×C4×D197—22no—
45#15691345—24no—
46#15691343—22no—
47C2×#13096802C2311yescharacteristic
48C2×#13096802C2312yes—
49C2×#13096802C2311yescharacteristic
50C4⋊Dic197—21no—
51C4⋊Dic197C2312yes—
52C4⋊Dic197C2311yescharacteristic
53#15691337—22no—
54#15691337—22no—
55#15691335—22no—
56#15691335—21no—
57C2×Q8×C197D411yescharacteristic
58C2×Q8×C197C2312yes—
59C2×D4×C197C2311yescharacteristic
60C22×C4×C197—21no—
61C4⋊C4×C197—21no—
62C4⋊C4×C197—22no—
63C4⋊C4×C197—21no—
64C4⋊C4×C197D411yescharacteristic
65C4⋊C4×C197D412yes—
66C4⋊C4×C197C2311yescharacteristic
67C22⋊C4×C197—22no—
68C22⋊C4×C197—21no—
69C22⋊C4×C197—22no—
70C42×C197—22no—
71C42×C197C2311yescharacteristic
72C42×C197C2312yes—
73C22×D197—21no—
74Dic394—81no—
75D788—21no—
76D788#4112yes—
77C2×Dic197—21no—
78C2×Dic197—22no—
79C4×D197—42no—
80C4×D197—82no—
81#13096802—21no—
82#13096802—22no—
83#13096802—21no—
84C23×C197—21no—
85Q8×C197—22no—
86Q8×C197—22no—
87Q8×C197—22no—
88D4×C197—21no—
89D4×C197—21no—
90C2×C4×C197—21no—
91C2×C4×C197—22no—
92C2×C4×C197—42no—
93C2×C4×C197C2×D412yes—
94C2×C4×C197C2×D411yescharacteristic
95C2×C4×C197C2×D412yes—
96C2×C4×C197C2×D411yescharacteristic
97C2×C4×C197C2411yescharacteristic
98C22×C197—41no—
99C22×C197—41no—
100C22×C197—41no—
101C22×C197C22×D411yescharacteristic
102C2×D197—22no—
103C2×D197—41no—
104C2×D197—41no—
105C4×C197—22no—
106C4×C197—22no—
107C4×C197—21no—
108C4×C197—42no—
109C4×C197—41no—
110C4×C197#13512yes—
111C4×C197C2×#4111yescharacteristic
112C4×C1972−1+411yescharacteristic
113Dic197—41no—
114Dic197—42no—
115C2×C197—81no—
116C2×C197#54811yescharacteristic
117C2×C197C2×#13511yescharacteristic
118C2×C197#57711yescharacteristic
119D197—42no—
120C197#336011yescharacteristic, complemented, Sylow
121#3360—1971nomaximal, Sylow
122#556D19711yescharacteristic, complemented
123#554—1971no—
124#488—1972no—
125#487—1971no—
126#479—1972no—
127#476—1971no—
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—22no—
138#125—1972no—
139#124C2×D19712yes—
140C22⋊Q8—21no—
141C22⋊Q8—3941no—
142C22⋊Q8—3942no—
143C4×Q8C2×D19712yes—
144C4×Q8C2×D19711yescharacteristic
145C4×D4—1971no—
146C4×D4—1972no—
147C4×D4C2×D19711yescharacteristic
148#117—21no—
149C2×C4⋊C4—3941no—
150#107—1971no—
151#106—1972no—
152C4⋊C8—1971no—
153C4⋊C8—1972no—
154#103—1972no—
155#103—1972no—
156#103—1972no—
157#102—1971no—
158#102—1972no—
159#102—1971no—
160C4×C8—1971no—
161C4×C8—1972no—
162C2×Q8—3941no—
163C2×Q8C22⋊D19711yescharacteristic
164C2×Q8C22×D19712yes—
165C2×D4—1971no—
166C2×D4C22×D19711yescharacteristic
167C22×C4—21no—
168C22×C4—3941no—
169C22×C4—3942no—
170SD16—3944no—
171D8—3942no—
172C2×C8—1971no—
173C2×C8—1972no—
174C2×C8—1971no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—3942no—
179C4⋊C4—3942no—
180C4⋊C4—3941no—
181C4⋊C4—1972no—
182C4⋊C4—1971no—
183C4⋊C4C22⋊D19711yescharacteristic
184C4⋊C4C22⋊D19712yes—
185C4⋊C4C22×D19711yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—3941no—
190C22⋊C4—3942no—
191C42—22no—
192C42C22×D19711yescharacteristic
193C42C22×D19712yes—
194C23—21no—
195C23—3941no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—15761no—
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—3941no—
210C2×C4—3942no—
211C2×C4C2×C22⋊D19712yes—
212C2×C4C2×C22⋊D19712yes—
213C2×C4C2×C22⋊D19711yescharacteristic
214C2×C4C2×C22⋊D19711yescharacteristic
215C2×C4C23×D19711yescharacteristic
216C8—3941no—
217C8—3942no—
218C8—3941no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—7881no—
223C22—7881no—
224C22—3942no—
225C22C22×C22⋊D19711yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—7881no—
232C4—7882no—
233C4#2026334612yes—
234C4#41×D19711yescharacteristic
235C4#2026337311yescharacteristic
236C2—81no—
237C2—7882no—
238C2#2716173611yescharacteristic
239C2C2×#2026334611yescharacteristic
240C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#556×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691343C22
hypercenter#556D197
hypocenterC197#3360
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #27161789C2semidirect
2. #27161433C2semidirect
3. #27161429C2semidirect
4. #27161386C2semidirect
5. #27161369C2semidirect
7. #27161281C2semidirect
8. Q8×#13096802C2semidirect
9. C4×#15691345C2semidirect
11. #556×C197C2semidirect
120. C197#3360semidirect
122. #556D197semidirect

Maximal subgroups

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

subgroupindex
#271617892
#271614332
#271614292
#271613862
#271613692
#271612912
#271612812
Q8×#130968022
C4×#156913452
C4×#156913432
#556×C1972
#3360197