Finite Group Databasebeta

← #38754174

Subgroups — #38754174

Order 25216.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#27161702C212yesmaximal, complemented
2#27161372C212yesmaximal, complemented
3#27161358C212yesmaximal, complemented
4#27161327C212yesmaximal, complemented
5#27161315C211yesmaximal, characteristic
6#27161303C212yesmaximal, complemented
7#27161297C212yesmaximal, complemented
8#27161105C211yesmaximal, characteristic, complemented
9#467×C197C211yesmaximal, characteristic, complemented
10C2×#15691345C2212yes—
11C2×D1576C2211yescharacteristic
12C2×#15691342C2211yescharacteristic
13#20263325C2212yes—
14#20263321—22no—
15#20263315C2212yes—
16#20263312—22no—
17#20263308—22no—
18#20263306—22no—
19#20263300C2212yes—
20Q8×Dic197C2212yes—
21#20263288C2212yes—
22#20263285C2211yescharacteristic
23#20263260C2211yescharacteristic
24#20263259C2212yes—
25#20263255C2212yes—
26#20263255C2212yes—
27#20263250C2212yes—
28#20263248C2212yes—
29#20263246C2212yes—
30C2×Q16×C197C2211yescharacteristic
31C22×C8×C197C2211yescharacteristic
32C22⋊Q8×C197C2212yes—
33#107×C197C2211yescharacteristic
34#103×C197C2212yes—
35C2×Dic394—22no—
36C2×D788C2312yes—
37C2×C22⋊D197—22no—
38C2×C4×D197—22no—
39#15691345—42no—
40D1576—41no—
41#15691342—41no—
42C2×#13096802C2312yes—
43C4⋊Dic197—22no—
44C4⋊Dic197D411yescharacteristic
45C4⋊Dic197D411yescharacteristic
46C4⋊Dic197C2312yes—
47#15691337—22no—
48C22⋊Dic197D412yes—
49#15691335—22no—
50#15691335—22no—
51#15691335—22no—
52#15691335—22no—
53C4×Dic197—22no—
54C4×Dic197C2312yes—
55C2×Q8×C197C2312yes—
56C22×C4×C197C2311yescharacteristic
57Q16×C197—41no—
58C2×C8×C197—41no—
59C2×C8×C197C2311yescharacteristic
60C2×C8×C197C2311yescharacteristic
61C4⋊C4×C197—22no—
62C4⋊C4×C197C2312yes—
63C22⋊C4×C197—22no—
64C22×D197—22no—
65Dic394—82no—
66D788—22no—
67D788—22no—
68C22⋊D197—82no—
69C2×Dic197—22no—
70C2×Dic197C2×D412yes—
71C2×Dic197C2×D412yes—
72C4×D197—82no—
73#13096802—22no—
74C23×C197C2×D411yescharacteristic
75Q8×C197—22no—
76Q8×C197—22no—
77C2×C4×C197—21no—
78C2×C4×C197—21no—
79C2×C4×C197—22no—
80C2×C4×C197—22no—
81C2×C4×C197C2×D411yescharacteristic
82C2×C4×C197C2411yescharacteristic
83C8×C197—21no—
84C8×C197—21no—
85C22×C197—21no—
86C22×C197—21no—
87C22×C197—21no—
88C22×C197C22×D411yescharacteristic
89C2×D197—42no—
90C2×D197—42no—
91C2×D197—42no—
92C4×C197—21no—
93C4×C197—42no—
94C4×C197—42no—
95C4×C197H(5,𝔽2)11yescharacteristic
96C4×C1972−1+411yescharacteristic
97Dic197—22no—
98Dic197—22no—
99Dic197—42no—
100C2×C197—41no—
101C2×C197#53711yescharacteristic
102C2×C197#57611yescharacteristic
103C2×C197#57811yescharacteristic
104D197—82no—
105C197#326211yescharacteristic, complemented, Sylow
106#3262—1971nomaximal, Sylow
107#542—1972no—
108#477—1972no—
109#474—1972no—
110#468—1972no—
111#467D19711yescharacteristic, complemented
112#466—1971no—
113#464—1972no—
114#463—1972no—
115#424—1971no—
116C2×Q16C2×D19711yescharacteristic
117C2×SD16—1972no—
118C2×D8—1971no—
119C2×M16—1971no—
120C22×C8C2×D19711yescharacteristic
121C4⋊Q8—1972no—
122#126—3942no—
123#124—3942no—
124#124—1972no—
125#123—3942no—
126C22⋊Q8—3942no—
127C22⋊Q8C2×D19712yes—
128#121—1972no—
129C4×Q8—1972no—
130C4×D4—1972no—
131#117—1971no—
132#107—1971no—
133#107—1972no—
134#107C2×D19711yescharacteristic
135C4⋊C8—1972no—
136C4⋊C8—1972no—
137#103—1972no—
138#103C2×D19712yes—
139#102—1972no—
140#102—1972no—
141C2×Q8—3942no—
142C2×Q8C22×D19712yes—
143C2×D4—3942no—
144C2×D4—1972no—
145C22×C4—3942no—
146C22×C4C22×D19711yescharacteristic
147Q16—41no—
148SD16—7882no—
149D8—7881no—
150M16—7881no—
151C2×C8—41no—
152C2×C8—1972no—
153C2×C8C22×D19711yescharacteristic
154C2×C8C22×D19711yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—3942no—
157C4⋊C4—3942no—
158C4⋊C4—1972no—
159C4⋊C4—1971no—
160C4⋊C4—1971no—
161C4⋊C4C22×D19712yes—
162C22⋊C4—22no—
163C22⋊C4—3942no—
164C22⋊C4—3942no—
165C22⋊C4—3942no—
166C22⋊C4—3942no—
167C22⋊C4—1972no—
168C42—3942no—
169C42—1972no—
170C23—3942no—
171C23D4×D19711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—15762no—
175D4—15762no—
176D4—3942no—
177D4—3942no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—15762no—
183C2×C4—3942no—
184C2×C4—1972no—
185C2×C4—1972no—
186C2×C4D4×D19711yescharacteristic
187C2×C4C23×D19711yescharacteristic
188C8—21no—
189C8—21no—
190C8—3942no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—7882no—
195C22—7882no—
196C22—7882no—
197C22C2×D4×D19711yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—7882no—
202C4—3942no—
203C4—3942no—
204C4#2026337111yescharacteristic
205C4#2026337311yescharacteristic
206C2—41no—
207C2—15762no—
208C2#2716165611yescharacteristic
209C2#2716188111yescharacteristic
210C2#2716189211yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#467×C197C2
socleC22×C197C22×D4
prime radical#27161315C2
hypercenter#467D197
hypocenterC197#3262
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #27161702C2semidirect
2. #27161372C2semidirect
3. #27161358C2semidirect
4. #27161327C2semidirect
6. #27161303C2semidirect
7. #27161297C2semidirect
8. #27161105C2semidirect
9. #467×C197C2semidirect
105. C197#3262semidirect
111. #467D197semidirect

Maximal subgroups

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

subgroupindex
#271617022
#271613722
#271613582
#271613272
#271613152
#271613032
#271612972
#271611052
#467×C1972
#3262197