Finite Group Databasebeta

← #27152109

Subgroups — #27152109

Order 12416.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#20261291C211yesmaximal, characteristic, complemented
2#127×D97C211yesmaximal, characteristic
3#20261101C211yesmaximal, characteristic, complemented
4#20261100C211yesmaximal, characteristic, complemented
5#20261099C211yesmaximal, characteristic, complemented
6#20261080C214yesmaximal, complemented
7#20260738C214yesmaximal, complemented
8#20260687C211yesmaximal, characteristic, complemented
9#490×C97C211yesmaximal, characteristic, complemented
10C2×D4×D97—22no—
11C2×D4×D97C2212yes—
12C2×#13095494C2212yes—
13#15690809C2212yescomplemented
14#15690807C2211yescharacteristic
15#15690806—22no—
16#15690806C2212yes—
17#15690802C2211yescharacteristic, complemented
18#15690788—22no—
19#15690775—22no—
20#15690768C2212yes—
21C42×D97C2211yescharacteristic
22#15690732C2214yescomplemented
23#15690731C2214yescomplemented
24#15690730C2214yescomplemented
25#15690721C2211yescharacteristic, complemented
26C8×Dic97C2212yescomplemented
27#127×C97C2211yescharacteristic
28#125×C97C2211yescharacteristic, complemented
29#102×C97C2214yescomplemented
30#97×C97C2211yescharacteristic, complemented
31C23×D97—22no—
32C2×Dic194—22no—
33C2×D388—22no—
34C2×D388C2312yes—
35C2×C22⋊D97—22no—
36C2×C22⋊D97—24no—
37D4×D97—28no—
38D4×D97—48no—
39C2×C4×D97D412yes—
40C2×C4×D97C2311yescharacteristic
41#13095494—24no—
42C2×#11715895C2312yes—
43C4⋊Dic97C2312yes—
44#13095490—22no—
45#13095490—22no—
46#13095488—22no—
47C4×Dic97—22no—
48C4×Dic97D412yes—
49C4×Dic97C2311yescharacteristic
50C2×D4×C97—22no—
51C2×D4×C97C2312yes—
52C2×C8×C97C2312yes—
53C4⋊C4×C97—22no—
54C4⋊C4×C97C2312yes—
55C42×C97C2311yescharacteristic
56C23×C97—22no—
57C22×D97—22no—
58C22×D97—48no—
59C22×D97—44no—
60C22×D97C2×D411yescharacteristic
61D4×C97—24no—
62D4×C97—44no—
63C2×C4×C97—22no—
64C2×C4×C97C2×D412yes—
65C2×C4×C97C2411yescharacteristic
66C22⋊D97—44no—
67C22⋊D97—48no—
68C2×Dic97—22no—
69C2×Dic97C2×D411yescharacteristic
70C2×Dic97C2×D412yes—
71D388—24no—
72D388—44no—
73C4×D97—24no—
74C4×D97#4114yes—
75Dic194—44no—
76C8×C97—22no—
77#11715895—22no—
78C22×C97—42no—
79C22×C97—44no—
80C22×C97C22×D411yescharacteristic
81C2×D97—22no—
82C2×D97—42no—
83C2×D97—44no—
84C2×D97—88no—
85C2×D97#12412yes—
86C4×C97—22no—
87C4×C97—42no—
88C4×C97C2×#4112yes—
89Dic97—42no—
90Dic97#12412yes—
91Dic97AGL(1,ℤ/8)14yes—
92C2×C97—82no—
93C2×C97C2×#12411yescharacteristic
94C2×C97C2×AGL(1,ℤ/8)12yes—
95D97—22no—
96D97—82no—
97#3138—971nomaximal, Sylow
98C97#313811yescharacteristic, complemented, Sylow
99#533—971no—
100C2×#127—971no—
101#490—971no—
102#490—971no—
103#490—971no—
104#490D9711yescharacteristic, complemented
105#486—974no—
106#418—974no—
107C4×M16—971no—
108C22×D4—972no—
109C22×D4—1942no—
110C2×M16—972no—
111C4⋊Q8—972no—
112#127—972no—
113#127—971no—
114#127—1942no—
115#127C2×D9711yescharacteristic
116#125—971no—
117#125C2×D9711yescharacteristic, complemented
118C22⋊Q8—1942no—
119C4×Q8—1942no—
120#117—972no—
121C2×C42—971no—
122#102—974no—
123#102—974no—
124#102—974no—
125#102C2×D9714yescomplemented
126#97—971no—
127#97C2×D9711yescharacteristic, complemented
128C4×C8—972no—
129C24—1942no—
130C2×Q8—1942no—
131C2×D4—22no—
132C2×D4—972no—
133C2×D4—3888no—
134C2×D4—1948no—
135C2×D4—1944no—
136C2×D4—1942no—
137C2×D4—1942no—
138C2×D4C22×D9712yes—
139C22×C4—971no—
140C22×C4—972no—
141M16—1944no—
142C2×C8—972no—
143C2×C8C22×D9712yes—
144C4⋊C4—22no—
145C4⋊C4—972no—
146C4⋊C4—1942no—
147C4⋊C4—1942no—
148C4⋊C4C22×D9712yes—
149C22⋊C4—1942no—
150C42—972no—
151C42—971no—
152C42—1942no—
153C42C22×D9711yescharacteristic
154C23—22no—
155C23—971no—
156C23—3884no—
157C23—3888no—
158C23—1942no—
159Q8—3884no—
160D4—24no—
161D4—44no—
162D4—3884no—
163D4—3888no—
164D4—3884no—
165D4—1944no—
166C2×C4—22no—
167C2×C4—971no—
168C2×C4—972no—
169C2×C4—974no—
170C2×C4—1944no—
171C2×C4—1942no—
172C2×C4D4×D9712yes—
173C2×C4C23×D9711yescharacteristic
174C8—22no—
175C8—1942no—
176C22—42no—
177C22—44no—
178C22—7768no—
179C22—972no—
180C22—3882no—
181C22—3884no—
182C22—1942no—
183C22C2×D4×D9711yescharacteristic
184C4—22no—
185C4—42no—
186C4—972no—
187C4—974no—
188C4—3882no—
189C4#41×D9712yes—
190C2—82no—
191C2—7762no—
192C2—1942no—
193C2#124×D9711yescharacteristic
194C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#490×C97C2
socleC22×C97C22×D4
prime radical#127×D97C2
hypercenter#490D97
hypocenterC97#3138
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #20261291C2semidirect
3. #20261101C2semidirect
4. #20261100C2semidirect
5. #20261099C2semidirect
6. #20261080C2semidirect
7. #20260738C2semidirect
8. #20260687C2semidirect
9. #490×C97C2semidirect
13. #15690809C22semidirect
17. #15690802C22semidirect
22. #15690732C22semidirect
23. #15690731C22semidirect
24. #15690730C22semidirect
25. #15690721C22semidirect
26. C8×Dic97C22semidirect
28. #125×C97C22semidirect
29. #102×C97C22semidirect
30. #97×C97C22semidirect
98. C97#3138semidirect
104. #490D97semidirect
117. #125C2×D97semidirect
125. #102C2×D97semidirect
127. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612912
#127×D972
#202611012
#202611002
#202610992
#202610802
#202607382
#202606872
#490×C972
#313897