Finite Group Databasebeta

← #38754158

Subgroups — #38754158

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#27161383C212yesmaximal, complemented
3#27161379C212yesmaximal, complemented
4#27161323C212yesmaximal, complemented
5#27161306C212yesmaximal, complemented
6#27161284C212yesmaximal, complemented
7#27161276C212yesmaximal, complemented
8#27161106C211yesmaximal, characteristic, complemented
9#540×C197C211yesmaximal, characteristic
10C2×#15691346C2212yes—
11C2×#15691346C2212yes—
12C2×#15691342C2211yescharacteristic
13C22×#13096802C2211yescharacteristic
14#20263326C2211yescharacteristic
15#20263319—22no—
16#20263318C2211yescharacteristic
17#20263316—22no—
18#20263305C2212yes—
19C4×Dic394C2212yes—
20#20263259C2212yes—
21#20263257C2212yes—
22C4⋊#13096802C2212yes—
23C4⋊#13096802C2212yes—
24#20263249C2212yes—
25#20263249C2212yes—
26#20263247C2212yes—
27#20263247C2212yes—
28#127×C197C2211yescharacteristic
29#125×C197C2211yescharacteristic
30#123×C197—22no—
31#121×C197—22no—
32#121×C197C2212yes—
33C4×D4×C197C2212yes—
34#117×C197C2211yescharacteristic
35C2×Dic394C2312yes—
36#15691346—42no—
37#15691346—42no—
38#15691342—41no—
39C2×#13096802—41no—
40C2×#13096802C2312yes—
41C2×#13096802C2312yes—
42C4⋊Dic197—22no—
43C4⋊Dic197C2312yes—
44#15691337—22no—
45#15691337—22no—
46#15691337—22no—
47C22⋊Dic197—22no—
48C4×Dic197—22no—
49C2×D4×C197—22no—
50C2×D4×C197—22no—
51C2×D4×C197C2312yes—
52C22×C4×C197—22no—
53C22×C4×C197C2311yescharacteristic
54C4⋊C4×C197—22no—
55C4⋊C4×C197D412yes—
56C4⋊C4×C197C2312yes—
57C22⋊C4×C197—22no—
58C22⋊C4×C197—22no—
59C22⋊C4×C197D412yes—
60C42×C197C2311yescharacteristic
61C42×C197C2311yescharacteristic
62Dic394—22no—
63Dic394—22no—
64C2×Dic197—22no—
65C2×Dic197—22no—
66#13096802—22no—
67#13096802—22no—
68C23×C197—22no—
69C23×C197C2×D411yescharacteristic
70D4×C197—22no—
71D4×C197—22no—
72D4×C197—82no—
73D4×C197—82no—
74C2×C4×C197—21no—
75C2×C4×C197—21no—
76C2×C4×C197—22no—
77C2×C4×C197—82no—
78C2×C4×C197C2×D412yes—
79C2×C4×C197C2×D412yes—
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2411yescharacteristic
82C22×C197—21no—
83C22×C197—22no—
84C22×C197—42no—
85C22×C197—42no—
86C22×C197—42no—
87C22×C197C22×D411yescharacteristic
88C4×C197—21no—
89C4×C197—22no—
90C4×C197—22no—
91C4×C197—42no—
92C4×C197H(5,𝔽2)11yescharacteristic
93C4×C1972−1+411yescharacteristic
94Dic197—42no—
95Dic197—42no—
96C2×C197—41no—
97C2×C197—82no—
98C2×C197#53711yescharacteristic
99C2×C197#57712yes—
100C197#326011yescharacteristic, complemented, Sylow
101#3260—1971nomaximal, Sylow
102#541—1971no—
103#540D19711yescharacteristic
104#479—1972no—
105#478—1972no—
106#468—1972no—
107#465—1972no—
108#461—1972no—
109#460—1972no—
110#424—1971no—
111C2×SD16—1972no—
112C2×SD16—1972no—
113C2×M16—1971no—
114C22×C8—1971no—
115C4⋊Q8—1971no—
116#127C2×D19711yescharacteristic
117#126—3942no—
118#125—3942no—
119#125—1971no—
120#125C2×D19711yescharacteristic
121#123—22no—
122C22⋊Q8—1972no—
123#121—22no—
124#121C2×D19712yes—
125C4×Q8—1972no—
126C4×D4C2×D19712yes—
127#117C2×D19711yescharacteristic
128#107—1972no—
129#106—1972no—
130C4⋊C8—1972no—
131C4⋊C8—1972no—
132#103—1972no—
133#103—1972no—
134#102—1972no—
135#102—1972no—
136C2×Q8—1972no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4C22×D19712yes—
140C22×C4—22no—
141C22×C4C22×D19711yescharacteristic
142SD16—7882no—
143SD16—7882no—
144M16—7881no—
145C2×C8—7881no—
146C2×C8—1972no—
147C2×C8—1972no—
148C4⋊C4—22no—
149C4⋊C4—3942no—
150C4⋊C4—3942no—
151C4⋊C4—3942no—
152C4⋊C4—3942no—
153C4⋊C4—1972no—
154C4⋊C4C22⋊D19712yes—
155C4⋊C4C22×D19712yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—3942no—
159C22⋊C4C22⋊D19712yes—
160C42—3942no—
161C42C22×D19711yescharacteristic
162C42C22×D19711yescharacteristic
163C23—22no—
164C23C2×C22⋊D19711yescharacteristic
165Q8—3942no—
166Q8—3942no—
167D4—22no—
168D4—22no—
169D4—82no—
170D4—82no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—82no—
175C2×C4—3942no—
176C2×C4—3942no—
177C2×C4C2×C22⋊D19712yes—
178C2×C4C2×C22⋊D19712yes—
179C2×C4C2×C22⋊D19711yescharacteristic
180C2×C4C23×D19711yescharacteristic
181C8—3942no—
182C8—3942no—
183C22—22no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22C22×C22⋊D19711yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—7882no—
194C4—7882no—
195C4#2026337211yescharacteristic
196C4#2026337311yescharacteristic
197C2—41no—
198C2—82no—
199C2#2716165711yescharacteristic
200C2#2716189012yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radical#540×C197C2
socleC22×C197C22×D4
hypercenter#540D197
hypocenterC197#3260
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #27161691C2semidirect
2. #27161383C2semidirect
3. #27161379C2semidirect
4. #27161323C2semidirect
5. #27161306C2semidirect
6. #27161284C2semidirect
7. #27161276C2semidirect
8. #27161106C2semidirect
100. C197#3260semidirect

Maximal subgroups

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

subgroupindex
#271616912
#271613832
#271613792
#271613232
#271613062
#271612842
#271612762
#271611062
#540×C1972
#3260197