Finite Group Databasebeta

← #38754183

Subgroups — #38754183

Order 25216.

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

normal subgroups only

This group has 346 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#27161700C212yesmaximal, complemented
2#27161370C212yesmaximal, complemented
3#27161365C212yesmaximal, complemented
4#27161337C211yesmaximal, characteristic, complemented
5#27161332C211yesmaximal, characteristic
6#27161308C212yesmaximal, complemented
7#27161302C212yesmaximal, complemented
8#27161295C212yesmaximal, complemented
9#424×C197C211yesmaximal, characteristic, complemented
10C2×Dic788C2211yescharacteristic
11C2×#15691347C2212yes—
12C2×D1576C2211yescharacteristic
13#20263326C2212yes—
14#20263321—22no—
15#20263314C2212yes—
16#20263312—22no—
17#20263308—22no—
18#20263305C2212yes—
19#20263304—22no—
20#20263300C2212yes—
21C4×Dic394C2212yes—
22C4×D788C2212yes—
23#20263258C2211yescharacteristic
24#20263258C2212yes—
25#20263258C2211yescharacteristic
26#20263251C2212yes—
27#20263251C2212yes—
28#20263248C2212yes—
29#20263248C2212yes—
30C2×M16×C197C2211yescharacteristic
31C22×C8×C197C2211yescharacteristic
32#117×C197C2211yescharacteristic
33C4⋊C8×C197C2212yes—
34C4⋊C8×C197C2212yes—
35C2×Dic394—22no—
36C2×Dic394C2312yes—
37C2×D788C2312yes—
38C2×C22⋊D197—22no—
39C2×C4×D197—22no—
40Dic788—41no—
41#15691347—42no—
42D1576—41no—
43C4⋊Dic197—22no—
44C4⋊Dic197C2312yes—
45C4⋊Dic197C2312yes—
46#15691337—22no—
47#15691337—22no—
48C22⋊Dic197—22no—
49#15691335—22no—
50#15691335—22no—
51#15691335—22no—
52#15691335—22no—
53C4×Dic197—22no—
54C22×C4×C197C2311yescharacteristic
55M16×C197—41no—
56C2×C8×C197—41no—
57C2×C8×C197C2312yes—
58C2×C8×C197C2311yescharacteristic
59C2×C8×C197C2311yescharacteristic
60C4⋊C4×C197D412yes—
61C22⋊C4×C197D411yescharacteristic
62C22⋊C4×C197D411yescharacteristic
63C42×C197C2312yes—
64C22×D197—22no—
65Dic394—22no—
66Dic394—22no—
67Dic394—82no—
68D788—22no—
69D788—22no—
70C22⋊D197—82no—
71C2×Dic197—22no—
72C2×Dic197—22no—
73C2×Dic197—22no—
74C4×D197—82no—
75C23×C197C2×D411yescharacteristic
76C2×C4×C197—21no—
77C2×C4×C197—21no—
78C2×C4×C197C2×D412yes—
79C2×C4×C197C2×D412yes—
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2411yescharacteristic
82C8×C197—22no—
83C8×C197—21no—
84C8×C197—21no—
85C22×C197—21no—
86C22×C197—22no—
87C22×C197C22×D411yescharacteristic
88C2×D197—42no—
89C2×D197—42no—
90C2×D197—42no—
91C4×C197—21no—
92C4×C197—22no—
93C4×C197—22no—
94C4×C197H(5,𝔽2)11yescharacteristic
95C4×C1972−1+411yescharacteristic
96Dic197—42no—
97Dic197—42no—
98Dic197—42no—
99C2×C197—41no—
100C2×C197#53711yescharacteristic
101C2×C197#57712yes—
102D197—82no—
103C197#326311yescharacteristic, complemented, Sylow
104#3263—1971nomaximal, Sylow
105#542—1972no—
106#476—1972no—
107#475—1972no—
108#470—1971no—
109#469—1971no—
110#465—1972no—
111#464—1972no—
112#463—1972no—
113#424D19711yescharacteristic, complemented
114C2×Q16—1971no—
115C2×SD16—1972no—
116C2×D8—1971no—
117C2×M16C2×D19711yescharacteristic
118C22×C8C2×D19711yescharacteristic
119C4⋊Q8—1972no—
120#126—3942no—
121#124—3942no—
122#124—1972no—
123#123—3942no—
124C22⋊Q8—3942no—
125C22⋊Q8—1972no—
126#121—1972no—
127C4×Q8—1972no—
128C4×D4—1972no—
129#117C2×D19711yescharacteristic
130#106—1972no—
131#106—1971no—
132#106—1971no—
133C4⋊C8C2×D19712yes—
134C4⋊C8C2×D19712yes—
135#103—1972no—
136#103—1972no—
137#102—1972no—
138#102—1972no—
139C2×Q8—3942no—
140C2×Q8—1972no—
141C2×D4—3942no—
142C2×D4—1972no—
143C22×C4—3942no—
144C22×C4C22×D19711yescharacteristic
145Q16—7881no—
146SD16—7882no—
147D8—7881no—
148M16—41no—
149C2×C8—41no—
150C2×C8C22×D19711yescharacteristic
151C2×C8C22×D19712yes—
152C2×C8C22×D19711yescharacteristic
153C4⋊C4—3942no—
154C4⋊C4—3942no—
155C4⋊C4—3942no—
156C4⋊C4—1972no—
157C4⋊C4—1972no—
158C4⋊C4D78812yes—
159C22⋊C4—3942no—
160C22⋊C4—3942no—
161C22⋊C4—3942no—
162C22⋊C4—3942no—
163C22⋊C4—3942no—
164C22⋊C4D78811yescharacteristic
165C22⋊C4D78811yescharacteristic
166C42—3942no—
167C42C22×D19712yes—
168C23—3942no—
169C23C2×D78811yescharacteristic
170Q8—15762no—
171Q8—3942no—
172Q8—3942no—
173D4—15762no—
174D4—3942no—
175D4—3942no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—15762no—
179C2×C4—3942no—
180C2×C4—3942no—
181C2×C4—3942no—
182C2×C4C2×D78812yes—
183C2×C4C2×D78812yes—
184C2×C4C2×D78811yescharacteristic
185C2×C4C23×D19711yescharacteristic
186C8—22no—
187C8—21no—
188C8—21no—
189C22—22no—
190C22—21no—
191C22—7882no—
192C22—7882no—
193C22—7882no—
194C22C22×D78811yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—7882no—
199C4—7882no—
200C4—7882no—
201C4#2026337111yescharacteristic
202C4#2026337411yescharacteristic
203C2—41no—
204C2—15762no—
205C2#2716166011yescharacteristic
206C2#2716189112yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#424×C197C2
socleC22×C197C22×D4
prime radical#27161332C2
hypercenter#424D197
hypocenterC197#3263
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #27161700C2semidirect
2. #27161370C2semidirect
3. #27161365C2semidirect
4. #27161337C2semidirect
6. #27161308C2semidirect
7. #27161302C2semidirect
8. #27161295C2semidirect
9. #424×C197C2semidirect
103. C197#3263semidirect
113. #424D197semidirect

Maximal subgroups

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

subgroupindex
#271617002
#271613702
#271613652
#271613372
#271613322
#271613082
#271613022
#271612952
#424×C1972
#3263197