Finite Group Databasebeta

← #38754159

Subgroups — #38754159

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#27161691C211yesmaximal, characteristic, complemented
2#27161684C211yesmaximal, characteristic
3#27161387C212yesmaximal, complemented
4#27161380C212yesmaximal, complemented
5#27161325C212yesmaximal, complemented
6#27161308C212yesmaximal, complemented
7#27161285C212yesmaximal, complemented
8#27161280C212yesmaximal, complemented
9#424×C197C211yesmaximal, characteristic, complemented
10C2×#15691347C2212yes—
11C2×#15691347C2212yes—
12#20263326C2211yescharacteristic
13#20263323C2211yescharacteristic
14#20263319—22no—
15#20263318C2211yescharacteristic
16#20263318C2211yescharacteristic
17#20263316—22no—
18#20263311—22no—
19#20263305C2212yes—
20#20263300C2212yes—
21#20263299—22no—
22C4×Dic394C2212yes—
23C4×D788C2212yes—
24#20263260C2212yes—
25#20263258C2212yes—
26#20263251C2212yes—
27#20263251C2212yes—
28#20263248C2212yes—
29#20263248C2212yes—
30C2×M16×C197C2211yescharacteristic
31C22×C8×C197C2211yescharacteristic
32#117×C197C2211yescharacteristic
33C4⋊C8×C197C2212yes—
34C4⋊C8×C197C2212yes—
35C2×Dic394C2312yes—
36C2×D788—22no—
37C2×D788C2312yes—
38C2×C22⋊D197—22no—
39C2×C4×D197—22no—
40#15691347—42no—
41#15691347—42no—
42C4⋊Dic197—22no—
43C4⋊Dic197C2312yes—
44C4⋊Dic197C2312yes—
45#15691337—22no—
46#15691337—22no—
47#15691337—22no—
48#15691337—22no—
49C22⋊Dic197—22no—
50#15691335—22no—
51#15691335—22no—
52C4×Dic197—22no—
53C22×C4×C197C2311yescharacteristic
54M16×C197—41no—
55C2×C8×C197—41no—
56C2×C8×C197C2312yes—
57C2×C8×C197C2312yes—
58C4⋊C4×C197D412yes—
59C22⋊C4×C197D412yes—
60C42×C197C2311yescharacteristic
61C42×C197C2311yescharacteristic
62C22×D197—22no—
63Dic394—22no—
64Dic394—22no—
65D788—22no—
66D788—22no—
67D788—82no—
68C22⋊D197—82no—
69C2×Dic197—22no—
70C2×Dic197—22no—
71C2×Dic197—22no—
72C4×D197—82no—
73C23×C197C2×D411yescharacteristic
74C2×C4×C197—21no—
75C2×C4×C197—21no—
76C2×C4×C197C2×D412yes—
77C2×C4×C197C2×D412yes—
78C2×C4×C197C2×D411yescharacteristic
79C2×C4×C197C2411yescharacteristic
80C8×C197—22no—
81C8×C197—22no—
82C22×C197—21no—
83C22×C197—22no—
84C22×C197C22×D411yescharacteristic
85C2×D197—42no—
86C2×D197—42no—
87C2×D197—42no—
88C4×C197—21no—
89C4×C197—22no—
90C4×C197—22no—
91C4×C197H(5,𝔽2)11yescharacteristic
92C4×C1972−1+411yescharacteristic
93Dic197—42no—
94Dic197—42no—
95Dic197—42no—
96C2×C197—41no—
97C2×C197#53711yescharacteristic
98C2×C197#57712yes—
99D197—82no—
100C197#326011yescharacteristic, complemented, Sylow
101#3260—1971nomaximal, Sylow
102#541—1971no—
103#540—1971no—
104#479—1972no—
105#478—1972no—
106#468—1972no—
107#465—1972no—
108#461—1972no—
109#460—1972no—
110#424D19711yescharacteristic, complemented
111C2×SD16—1972no—
112C2×SD16—1972no—
113C2×M16C2×D19711yescharacteristic
114C22×C8C2×D19711yescharacteristic
115C4⋊Q8—1971no—
116#127—1971no—
117#126—3942no—
118#125—3942no—
119#125—1971no—
120#125—1971no—
121#123—3942no—
122C22⋊Q8—1972no—
123#121—3942no—
124#121—1972no—
125C4×Q8—1972no—
126C4×D4—1972no—
127#117C2×D19711yescharacteristic
128#107—1972no—
129#106—1972no—
130C4⋊C8C2×D19712yes—
131C4⋊C8C2×D19712yes—
132#103—1972no—
133#103—1972no—
134#102—1972no—
135#102—1972no—
136C2×Q8—1972no—
137C2×D4—3942no—
138C2×D4—3942no—
139C2×D4—1972no—
140C22×C4—3942no—
141C22×C4C22×D19711yescharacteristic
142SD16—7882no—
143SD16—7882no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D19712yes—
147C2×C8C22×D19712yes—
148C4⋊C4—3942no—
149C4⋊C4—3942no—
150C4⋊C4—3942no—
151C4⋊C4—3942no—
152C4⋊C4—3942no—
153C4⋊C4—1972no—
154C4⋊C4—1972no—
155C4⋊C4D78812yes—
156C22⋊C4—3942no—
157C22⋊C4—3942no—
158C22⋊C4—3942no—
159C22⋊C4D78812yes—
160C42—3942no—
161C42C22×D19711yescharacteristic
162C42C22×D19711yescharacteristic
163C23—3942no—
164C23C2×D78811yescharacteristic
165Q8—3942no—
166Q8—3942no—
167D4—15762no—
168D4—15762no—
169D4—3942no—
170D4—3942no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—15762no—
174C2×C4—3942no—
175C2×C4—3942no—
176C2×C4—3942no—
177C2×C4C2×D78812yes—
178C2×C4C2×D78812yes—
179C2×C4C2×D78811yescharacteristic
180C2×C4C23×D19711yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—7882no—
186C22—7882no—
187C22—7882no—
188C22C22×D78811yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—7882no—
193C4—7882no—
194C4—7882no—
195C4#2026337111yescharacteristic
196C4#2026337411yescharacteristic
197C2—41no—
198C2—15762no—
199C2#2716166011yescharacteristic
200C2#2716189112yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#424×C197C2
socleC22×C197C22×D4
prime radical#27161684C2
hypercenter#424D197
hypocenterC197#3260
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. #27161691C2semidirect
3. #27161387C2semidirect
4. #27161380C2semidirect
5. #27161325C2semidirect
6. #27161308C2semidirect
7. #27161285C2semidirect
8. #27161280C2semidirect
9. #424×C197C2semidirect
100. C197#3260semidirect
110. #424D197semidirect

Maximal subgroups

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

subgroupindex
#271616912
#271616842
#271613872
#271613802
#271613252
#271613082
#271612852
#271612802
#424×C1972
#3260197