Finite Group Databasebeta

← #38754182

Subgroups — #38754182

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#27161696C212yesmaximal, complemented
2#27161367C212yesmaximal, complemented
3#27161363C212yesmaximal, complemented
4#27161337C211yesmaximal, characteristic, complemented
5#27161306C212yesmaximal, complemented
6#27161300C212yesmaximal, complemented
7#27161292C212yesmaximal, complemented
8#27161107C211yesmaximal, characteristic, complemented
9#469×C197C211yesmaximal, characteristic
10C2×Dic788C2211yescharacteristic
11C2×#15691346C2212yes—
12C22×#13096802C2211yescharacteristic
13#20263325C2212yes—
14#20263319—22no—
15#20263313—22no—
16#20263313C2212yes—
17#20263307—22no—
18#20263305C2212yes—
19#20263303—22no—
20#20263291C2212yes—
21D4×Dic197C2212yes—
22#20263285C2211yescharacteristic
23#20263258C2211yescharacteristic
24#20263257C2212yes—
25#20263255C2212yes—
26#20263255C2212yes—
27#20263251C2212yes—
28#20263249C2212yes—
29#20263247C2212yes—
30C2×D8×C197C2211yescharacteristic
31C2×M16×C197C2211yescharacteristic
32#121×C197C2212yes—
33#106×C197C2211yescharacteristic
34#102×C197C2212yes—
35C2×Dic394—22no—
36C2×Dic394C2312yes—
37C22×Dic197—22no—
38Dic788—41no—
39#15691346—42no—
40C2×#13096802—41no—
41C2×#13096802C2312yes—
42C4⋊Dic197D412yes—
43C4⋊Dic197C2312yes—
44#15691337—22no—
45#15691337—22no—
46#15691337—22no—
47C22⋊Dic197—22no—
48C22⋊Dic197—22no—
49C22⋊Dic197—22no—
50C22⋊Dic197—22no—
51C22⋊Dic197D411yescharacteristic
52C22⋊Dic197D411yescharacteristic
53C4×Dic197—22no—
54C4×Dic197C2312yes—
55C2×D4×C197—22no—
56C2×D4×C197C2312yes—
57C22×C4×C197C2311yescharacteristic
58D8×C197—41no—
59M16×C197—41no—
60C2×C8×C197C2311yescharacteristic
61C2×C8×C197C2311yescharacteristic
62C4⋊C4×C197C2312yes—
63C22⋊C4×C197—22no—
64Dic394—22no—
65Dic394—22no—
66Dic394—82no—
67C2×Dic197—22no—
68C2×Dic197—22no—
69C2×Dic197—82no—
70C2×Dic197C2×D412yes—
71C2×Dic197C2×D412yes—
72#13096802—22no—
73C23×C197—22no—
74C23×C197C2×D411yescharacteristic
75D4×C197—22no—
76D4×C197—22no—
77D4×C197—82no—
78C2×C4×C197—21no—
79C2×C4×C197—21no—
80C2×C4×C197—22no—
81C2×C4×C197C2×D411yescharacteristic
82C2×C4×C197C2411yescharacteristic
83C8×C197—21no—
84C8×C197—21no—
85C22×C197—21no—
86C22×C197—22no—
87C22×C197—42no—
88C22×C197—42no—
89C22×C197—42no—
90C22×C197C22×D411yescharacteristic
91C4×C197—21no—
92C4×C197—42no—
93C4×C197H(5,𝔽2)11yescharacteristic
94C4×C1972−1+411yescharacteristic
95Dic197—22no—
96Dic197—22no—
97Dic197—42no—
98Dic197—42no—
99C2×C197—41no—
100C2×C197—82no—
101C2×C197#53711yescharacteristic
102C2×C197#57712yes—
103C197#326311yescharacteristic, complemented, Sylow
104#3263—1971nomaximal, Sylow
105#542—1972no—
106#476—1972no—
107#475—1972no—
108#470—1971no—
109#469D19711yescharacteristic
110#465—1972no—
111#464—1972no—
112#463—1972no—
113#424—1971no—
114C2×Q16—1971no—
115C2×SD16—1972no—
116C2×D8C2×D19711yescharacteristic
117C2×M16C2×D19711yescharacteristic
118C22×C8—1971no—
119C4⋊Q8—1972no—
120#126—3942no—
121#124—3942no—
122#124—1972no—
123#123—3942no—
124C22⋊Q8—3942no—
125C22⋊Q8—1972no—
126#121C2×D19712yes—
127C4×Q8—1972no—
128C4×D4—1972no—
129#117—1971no—
130#106—1971no—
131#106—1972no—
132#106C2×D19711yescharacteristic
133C4⋊C8—1972no—
134C4⋊C8—1972no—
135#103—1972no—
136#103—1972no—
137#102—1972no—
138#102C2×D19712yes—
139C2×Q8—3942no—
140C2×Q8—1972no—
141C2×D4—22no—
142C2×D4C22×D19712yes—
143C22×C4—3942no—
144C22×C4C22×D19711yescharacteristic
145Q16—7881no—
146SD16—7882no—
147D8—41no—
148M16—41no—
149C2×C8—7881no—
150C2×C8—1972no—
151C2×C8C22×D19711yescharacteristic
152C2×C8C22×D19711yescharacteristic
153C4⋊C4—3942no—
154C4⋊C4—3942no—
155C4⋊C4—3942no—
156C4⋊C4—1972no—
157C4⋊C4—1972no—
158C4⋊C4C22×D19712yes—
159C22⋊C4—22no—
160C22⋊C4—3942no—
161C22⋊C4—3942no—
162C22⋊C4—3942no—
163C22⋊C4—3942no—
164C22⋊C4—1971no—
165C22⋊C4—1971no—
166C42—3942no—
167C42—1972no—
168C23—22no—
169C23D4×D19711yescharacteristic
170Q8—15762no—
171Q8—3942no—
172Q8—3942no—
173D4—22no—
174D4—22no—
175D4—82no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—15762no—
180C2×C4—3942no—
181C2×C4—3942no—
182C2×C4—1972no—
183C2×C4—1972no—
184C2×C4D4×D19711yescharacteristic
185C2×C4C23×D19711yescharacteristic
186C8—21no—
187C8—21no—
188C8—3942no—
189C22—22no—
190C22—21no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22C2×D4×D19711yescharacteristic
195C4—21no—
196C4—42no—
197C4—7882no—
198C4—7882no—
199C4—3942no—
200C4—3942no—
201C4#2026337111yescharacteristic
202C4#2026337311yescharacteristic
203C2—41no—
204C2—82no—
205C2#2716165611yescharacteristic
206C2#2716188912yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radical#469×C197C2
socleC22×C197C22×D4
hypercenter#469D197
hypocenterC197#3263
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #27161696C2semidirect
2. #27161367C2semidirect
3. #27161363C2semidirect
4. #27161337C2semidirect
5. #27161306C2semidirect
6. #27161300C2semidirect
7. #27161292C2semidirect
8. #27161107C2semidirect
103. C197#3263semidirect

Maximal subgroups

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

subgroupindex
#271616962
#271613672
#271613632
#271613372
#271613062
#271613002
#271612922
#271611072
#469×C1972
#3263197