Finite Group Databasebeta

← #27152138

Subgroups — #27152138

Order 12416.

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

normal subgroups only

This group has 392 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#20261286C211yesmaximal, characteristic
2#125×D97C211yesmaximal, characteristic, complemented
3#20261103C211yesmaximal, characteristic, complemented
4#20261101C211yesmaximal, characteristic, complemented
5#20261100C211yesmaximal, characteristic, complemented
6#20261091C214yesmaximal, complemented
7#20260735C214yesmaximal, complemented
8#20260687C211yesmaximal, characteristic, complemented
9#491×C97C211yesmaximal, characteristic, complemented
10C2×#13095512C2212yes—
11C2×#13095494C2212yes—
12#15690807C2211yescharacteristic
13#15690802C2211yescharacteristic
14#15690801C2212yes—
15#15690800—22no—
16#15690799C2212yes—
17#15690785—22no—
18#15690772—22no—
19C4⋊C4×D97—22no—
20C4⋊C4×D97C2212yes—
21C42×D97C2211yescharacteristic
22#15690734C2214yes—
23#15690732C2214yes—
24#15690730C2214yes—
25#15690721C2211yescharacteristic
26C8×Dic97C2212yes—
27C4⋊Q8×C97C2211yescharacteristic
28#125×C97C2211yescharacteristic
29#103×C97C2214yes—
30#97×C97C2211yescharacteristic
31#13095512—28no—
32C2×D388—22no—
33C2×D388—22no—
34C2×D388C2312yes—
35C2×C4×D97—22no—
36C2×C4×D97—22no—
37C2×C4×D97D412yes—
38C2×C4×D97C2311yescharacteristic
39#13095494—24no—
40C2×#11715895C2312yes—
41C4⋊Dic97—22no—
42C4⋊Dic97C2312yes—
43#13095490—22no—
44#13095490—22no—
45#13095490—22no—
46#13095488—24no—
47C4×Dic97D412yes—
48C4×Dic97C2311yescharacteristic
49C2×Q8×C97C2312yes—
50C2×C8×C97C2312yes—
51C4⋊C4×C97—22no—
52C4⋊C4×C97—22no—
53C4⋊C4×C97C2312yes—
54C42×C97C2311yescharacteristic
55C22×D97—22no—
56C22×D97C2×D411yescharacteristic
57Q8×C97—24no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97C2×D412yes—
61C2×C4×C97C2411yescharacteristic
62C2×Dic97—22no—
63C2×Dic97C2×D412yes—
64C2×Dic97C2×D411yescharacteristic
65D388—24no—
66D388—44no—
67D388—44no—
68C4×D97—42no—
69C4×D97—44no—
70C4×D97—44no—
71C4×D97#4114yes—
72C8×C97—22no—
73#11715895—22no—
74C22×C97C22×D411yescharacteristic
75C2×D97—22no—
76C2×D97—42no—
77C2×D97—44no—
78C2×D97#12412yes—
79C4×C97—22no—
80C4×C97—42no—
81C4×C97—42no—
82C4×C97C2×#4112yes—
83Dic97—22no—
84Dic97—42no—
85Dic97#12412yes—
86C2×C97C2×#12411yescharacteristic
87C2×C97#57512yes—
88D97—22no—
89D97—82no—
90#3142—971nomaximal, Sylow
91C97#314211yescharacteristic, complemented, Sylow
92#532—971no—
93C2×#125—971no—
94#491—971no—
95#491D9711yescharacteristic, complemented
96#490—971no—
97#490—971no—
98#488—974no—
99#417—974no—
100C4×M16—971no—
101C2×#41—972no—
102C2×M16—972no—
103C4⋊Q8C2×D9711yescharacteristic
104#127—971no—
105#125—972no—
106#125—971no—
107#125—1942no—
108#125C2×D9711yescharacteristic
109#124—972no—
110#121—1942no—
111C4×D4—1942no—
112C2×C4⋊C4—972no—
113C2×C4⋊C4—1942no—
114C2×C42—971no—
115#103—974no—
116#103C2×D9714yes—
117#102—974no—
118#102—974no—
119#97—971no—
120#97C2×D9711yescharacteristic
121C4×C8—972no—
122#41—1948no—
123C2×Q8C22×D9712yes—
124C2×D4—972no—
125C2×D4—1942no—
126C2×D4—1942no—
127C22×C4—971no—
128C22×C4—972no—
129C22×C4—1942no—
130C22×C4—1942no—
131M16—1944no—
132C2×C8—972no—
133C2×C8C22×D9712yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—972no—
137C4⋊C4—1942no—
138C4⋊C4—1942no—
139C4⋊C4—1942no—
140C4⋊C4—1942no—
141C4⋊C4C22×D9712yes—
142C22⋊C4—1944no—
143C42—972no—
144C42—971no—
145C42C22×D9711yescharacteristic
146C23—971no—
147C23—1942no—
148Q8—24no—
149D4—3884no—
150D4—3884no—
151D4—1944no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—971no—
155C2×C4—974no—
156C2×C4—972no—
157C2×C4—3882no—
158C2×C4—3884no—
159C2×C4—3884no—
160C2×C4—1942no—
161C2×C4D4×D9712yes—
162C2×C4C23×D9711yescharacteristic
163C8—22no—
164C8—1942no—
165C22—972no—
166C22—3882no—
167C22—3884no—
168C22—1942no—
169C22C2×D4×D9711yescharacteristic
170C4—22no—
171C4—42no—
172C4—42no—
173C4—972no—
174C4—3882no—
175C4—1942no—
176C4#41×D9712yes—
177C2—7762no—
178C2—1942no—
179C2#124×D9711yescharacteristic
180C2#2026153412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#491×C97C2
socleC22×C97C22×D4
prime radical#20261286C2
hypercenter#491D97
hypocenterC97#3142
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #125×D97C2semidirect
3. #20261103C2semidirect
4. #20261101C2semidirect
5. #20261100C2semidirect
6. #20261091C2semidirect
7. #20260735C2semidirect
8. #20260687C2semidirect
9. #491×C97C2semidirect
91. C97#3142semidirect
95. #491D97semidirect

Maximal subgroups

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

subgroupindex
#202612862
#125×D972
#202611032
#202611012
#202611002
#202610912
#202607352
#202606872
#491×C972
#314297