Finite Group Databasebeta

← #27153171

Subgroups — #27153171

Order 12416.

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

normal subgroups only

This group has 370 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#20261340C211yesmaximal, characteristic, complemented
2#20261035C212yesmaximal, complemented
3#20261032C212yesmaximal, complemented
4#20260986C212yesmaximal, complemented
5#20260969C212yesmaximal, complemented
6#20260930C212yesmaximal, complemented
7#20260927C212yesmaximal, complemented
8#20260762C211yesmaximal, characteristic, complemented
9#540×C97C211yesmaximal, characteristic, complemented
10C2×#13095496C2212yes—
11C2×#13095496C2212yes—
12C2×#13095495C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14#15690807C2211yescharacteristic
15#15690802C2211yescharacteristic
16#15690795—22no—
17#15690784C2212yes—
18#15690783—22no—
19C4×D388C2212yes—
20#15690743C2212yes—
21#15690741C2212yes—
22C4⋊#11715895C2212yes—
23C4⋊#11715895C2212yes—
24#15690731C2212yes—
25#15690731C2212yes—
26#15690730C2212yes—
27#15690730C2212yes—
28#127×C97C2211yescharacteristic
29#125×C97C2211yescharacteristic
30#123×C97—22no—
31#121×C97—22no—
32#121×C97C2212yes—
33C4×D4×C97C2212yes—
34#117×C97C2211yescharacteristic
35C2×D388—22no—
36C2×D388C2312yes—
37C2×C22⋊D97—22no—
38C2×C4×D97—22no—
39#13095496—42no—
40#13095496—42no—
41#13095495—41no—
42C2×#11715895—41no—
43C2×#11715895C2312yes—
44C2×#11715895C2312yes—
45C4⋊Dic97C2312yes—
46#13095490—22no—
47#13095488—22no—
48#13095488—22no—
49C2×D4×C97—22no—
50C2×D4×C97—22no—
51C2×D4×C97C2312yes—
52C22×C4×C97—22no—
53C22×C4×C97C2311yescharacteristic
54C4⋊C4×C97—22no—
55C4⋊C4×C97D412yes—
56C4⋊C4×C97C2312yes—
57C22⋊C4×C97—22no—
58C22⋊C4×C97—22no—
59C22⋊C4×C97D412yes—
60C42×C97C2311yescharacteristic
61C42×C97C2311yescharacteristic
62C23×C97—22no—
63C23×C97C2×D411yescharacteristic
64C22×D97—22no—
65D4×C97—22no—
66D4×C97—22no—
67D4×C97—82no—
68D4×C97—82no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—22no—
72C2×C4×C97—82no—
73C2×C4×C97C2×D412yes—
74C2×C4×C97C2×D412yes—
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2411yescharacteristic
77C22⋊D97—82no—
78C2×Dic97—22no—
79D388—22no—
80D388—22no—
81D388—82no—
82C4×D97—82no—
83#11715895—22no—
84#11715895—22no—
85C22×C97—21no—
86C22×C97—22no—
87C22×C97—42no—
88C22×C97—42no—
89C22×C97—42no—
90C22×C97C22×D411yescharacteristic
91C2×D97—42no—
92C2×D97—42no—
93C2×D97—42no—
94C4×C97—21no—
95C4×C97—22no—
96C4×C97—22no—
97C4×C97—42no—
98C4×C97H(5,𝔽2)11yescharacteristic
99C4×C972−1+411yescharacteristic
100Dic97—42no—
101C2×C97—41no—
102C2×C97—82no—
103C2×C97#53711yescharacteristic
104C2×C97#57612yes—
105D97—82no—
106#3259—971nomaximal, Sylow
107C97#325911yescharacteristic, complemented, Sylow
108#540—971no—
109#540D9711yescharacteristic, complemented
110#478—972no—
111#478—972no—
112#469—972no—
113#466—972no—
114#459—972no—
115#459—972no—
116#424—971no—
117C2×D8—972no—
118C2×D8—972no—
119C2×M16—971no—
120C22×C8—971no—
121#127—971no—
122#127C2×D9711yescharacteristic
123#125—971no—
124#125C2×D9711yescharacteristic
125#123—22no—
126#123—1942no—
127#121—22no—
128#121—972no—
129#121—1942no—
130#121C2×D9712yes—
131C4×D4—972no—
132C4×D4C2×D9712yes—
133#117C2×D9711yescharacteristic
134#107—972no—
135#106—972no—
136C4⋊C8—972no—
137C4⋊C8—972no—
138#102—972no—
139#102—972no—
140#102—972no—
141#102—972no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—972no—
145C2×D4—1942no—
146C2×D4—1942no—
147C2×D4C22×D9712yes—
148C22×C4—22no—
149C22×C4—1942no—
150C22×C4C22×D9711yescharacteristic
151D8—3882no—
152D8—3882no—
153M16—3881no—
154C2×C8—972no—
155C2×C8—972no—
156C2×C8—3881no—
157C4⋊C4—22no—
158C4⋊C4—972no—
159C4⋊C4—1942no—
160C4⋊C4C22⋊D9712yes—
161C4⋊C4C22×D9712yes—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—1942no—
165C22⋊C4—1942no—
166C22⋊C4C22⋊D9712yes—
167C42C22×D9711yescharacteristic
168C42C22×D9711yescharacteristic
169C23—22no—
170C23—1942no—
171C23C2×C22⋊D9711yescharacteristic
172D4—22no—
173D4—22no—
174D4—82no—
175D4—82no—
176D4—7762no—
177D4—7762no—
178D4—1942no—
179D4—1942no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—82no—
184C2×C4—7762no—
185C2×C4—1942no—
186C2×C4C2×C22⋊D9712yes—
187C2×C4C2×C22⋊D9712yes—
188C2×C4C2×C22⋊D9711yescharacteristic
189C2×C4C23×D9711yescharacteristic
190C8—1942no—
191C8—1942no—
192C22—22no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—3882no—
198C22—3882no—
199C22—3882no—
200C22C22×C22⋊D9711yescharacteristic
201C4—22no—
202C4—22no—
203C4—21no—
204C4—42no—
205C4—3882no—
206C4#1569085611yescharacteristic
207C4#1569085711yescharacteristic
208C2—41no—
209C2—82no—
210C2—7762no—
211C2#2026131311yescharacteristic
212C2#2026153812yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#540×C97C2
socleC22×C97C22×D4
hypercenter#540D97
hypocenterC97#3259
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20261340C2semidirect
2. #20261035C2semidirect
3. #20261032C2semidirect
4. #20260986C2semidirect
5. #20260969C2semidirect
6. #20260930C2semidirect
7. #20260927C2semidirect
8. #20260762C2semidirect
9. #540×C97C2semidirect
107. C97#3259semidirect
109. #540D97semidirect

Maximal subgroups

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

subgroupindex
#202613402
#202610352
#202610322
#202609862
#202609692
#202609302
#202609272
#202607622
#540×C972
#325997