Finite Group Databasebeta

← #38755365

Subgroups — #38755365

Order 25216.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#27161803C211yesmaximal, characteristic, complemented
2#27161786C211yesmaximal, characteristic, complemented
3#27161435C212yesmaximal, complemented
4#27161429C211yesmaximal, characteristic, complemented
5#27161384C212yesmaximal, complemented
6#27161370C211yesmaximal, characteristic, complemented
7#27161296C212yesmaximal
8#27161203C211yesmaximal, characteristic, complemented
9SD16×Dic197C212yesmaximal, complemented
10C4×#15691343C211yesmaximal, characteristic
11#461×C197C211yesmaximal, characteristic, complemented
12C2×#15691343C2211yescharacteristic
13#20263326C2211yescharacteristic
14#20263321—22no—
15#20263317C2212yes—
16#20263315C2212yes—
17#20263306—21no—
18#20263305—21no—
19#20263303—22no—
20Q8×Dic197C2212yes—
21C4×Dic394C2211yescharacteristic
22C4×D788C2211yescharacteristic
23D4×Dic197C2212yes—
24#20263284—21no—
25C2×C4⋊Dic197—21no—
26#20263259C2211yescharacteristic
27#20263258C2212yes—
28#20263255C2212yes—
29#20263250C2212yes—
30#20263250C2212yes—
31#20263248C2212yes—
32#20263247C2211yescharacteristic
33#20263246C2211yescharacteristic
34C4×#13096802C2211yescharacteristic
35C8×Dic197C2212yes—
36C2×SD16×C197C2212yes—
37C4⋊Q8×C197C2211yescharacteristic
38C4×D4×C197C2211yescharacteristic
39C4⋊C8×C197C2211yescharacteristic
40#103×C197C2212yes—
41C2×Dic394—21no—
42C2×Dic394D411yescharacteristic
43C2×D788C2311yescharacteristic
44C22×Dic197—22no—
45C2×C4×D197—21no—
46#15691343—22no—
47C2×#13096802C2311yescharacteristic
48C2×#13096802C2311yescharacteristic
49C4⋊Dic197—22no—
50C4⋊Dic197—22no—
51C4⋊Dic197—21no—
52C4⋊Dic197D411yescharacteristic
53C4⋊Dic197C2311yescharacteristic
54C4⋊Dic197C2312yes—
55#15691337—22no—
56#15691337D412yes—
57C22⋊Dic197—22no—
58#15691335—22no—
59#15691335—22no—
60#15691335—21no—
61C4×Dic197—22no—
62C4×Dic197C2312yes—
63C2×Q8×C197C2312yes—
64C2×D4×C197C2311yescharacteristic
65C22×C4×C197—21no—
66SD16×C197—24no—
67C2×C8×C197C2312yes—
68C4⋊C4×C197—21no—
69C4⋊C4×C197—21no—
70C4⋊C4×C197C2311yescharacteristic
71C22⋊C4×C197—21no—
72C42×C197C2311yescharacteristic
73C22×D197—21no—
74Dic394—22no—
75Dic394—81no—
76D788—21no—
77D788—21no—
78C2×Dic197—21no—
79C2×Dic197—22no—
80C2×Dic197—82no—
81C2×Dic197C2×D412yes—
82C2×Dic197C2×D412yes—
83C4×D197—42no—
84#13096802—21no—
85#13096802—21no—
86C23×C197—21no—
87Q8×C197—22no—
88Q8×C197—22no—
89D4×C197—21no—
90D4×C197#4112yes—
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—22no—
99C22×C197—41no—
100C22×C197—41no—
101C22×C197C22×D411yescharacteristic
102C2×D197—41no—
103C2×D197—41no—
104C2×D197—41no—
105C4×C197—21no—
106C4×C197—42no—
107C4×C197—41no—
108C4×C197#13512yes—
109C4×C197C2×#4111yescharacteristic
110C4×C1972−1+411yescharacteristic
111Dic197—22no—
112Dic197—22no—
113Dic197—41no—
114Dic197—42no—
115C2×C197—42no—
116C2×C197#54811yescharacteristic
117C2×C197C2×#13511yescharacteristic
118C2×C197#57711yescharacteristic
119D197—81no—
120C197#336011yescharacteristic, complemented, Sylow
121#3360—1971nomaximal, Sylow
122#556—1971no—
123#554—1971no—
124#488—1972no—
125#487—1971no—
126#479—1972no—
127#476—1971no—
128#463—1972no—
129#461D19711yescharacteristic, complemented
130C8×Q8—1971no—
131C4×SD16—1972no—
132C4×D8—1971no—
133C2×SD16C2×D19712yes—
134C2×D8—1971no—
135C4⋊Q8—1971no—
136C4⋊Q8C2×D19711yescharacteristic
137#126—3942no—
138#125—1972no—
139#124—1972no—
140C22⋊Q8—3942no—
141C22⋊Q8—3941no—
142C22⋊Q8—3941no—
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#106—1972no—
152C4⋊C8—1972no—
153C4⋊C8C2×D19711yescharacteristic
154#103—1972no—
155#103—1972no—
156#103C2×D19712yes—
157#102—1971no—
158#102—1972no—
159#102—1971no—
160C4×C8—1972no—
161C4×C8—1971no—
162C2×Q8—3941no—
163C2×Q8—1971no—
164C2×Q8C22×D19712yes—
165C2×D4—1971no—
166C2×D4C22×D19711yescharacteristic
167C22×C4—21no—
168C22×C4—3941no—
169C22×C4—3942no—
170SD16—24no—
171D8—3942no—
172C2×C8—1971no—
173C2×C8—1971no—
174C2×C8C22×D19712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—3942no—
178C4⋊C4—3942no—
179C4⋊C4—3942no—
180C4⋊C4—3941no—
181C4⋊C4—1972no—
182C4⋊C4—1971no—
183C4⋊C4—1972no—
184C4⋊C4—1971no—
185C4⋊C4C22×D19711yescharacteristic
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—22no—
197Q8—22no—
198Q8—15761no—
199Q8—3942no—
200D4—21no—
201D4—3941no—
202D4—3941no—
203D4#1569135712yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—7882no—
208C2×C4—15762no—
209C2×C4—3941no—
210C2×C4—3942no—
211C2×C4—1972no—
212C2×C4—1972no—
213C2×C4D4×D19711yescharacteristic
214C2×C4D4×D19711yescharacteristic
215C2×C4C23×D19711yescharacteristic
216C8—22no—
217C8—3941no—
218C8—3941no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—7881no—
223C22—7881no—
224C22—7881no—
225C22C2×D4×D19711yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—7881no—
230C4—7882no—
231C4—3942no—
232C4—3942no—
233C4#2026334712yes—
234C4C2×#1569135711yescharacteristic
235C4#2026337311yescharacteristic
236C2—42no—
237C2—15761no—
238C2#2716173511yescharacteristic
239C2C2×#2026334711yescharacteristic
240C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#461×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691343C22
hypercenter#461D197
hypocenterC197#3360
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #27161803C2semidirect
2. #27161786C2semidirect
3. #27161435C2semidirect
4. #27161429C2semidirect
5. #27161384C2semidirect
6. #27161370C2semidirect
8. #27161203C2semidirect
9. SD16×Dic197C2semidirect
11. #461×C197C2semidirect
120. C197#3360semidirect
129. #461D197semidirect

Maximal subgroups

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

subgroupindex
#271618032
#271617862
#271614352
#271614292
#271613842
#271613702
#271612962
#271612032
SD16×Dic1972
C4×#156913432
#461×C1972
#3360197