Finite Group Databasebeta

← #27151667

Subgroups — #27151667

Order 12416.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#20261246C211yesmaximal, characteristic
2#20261076C214yesmaximal, complemented
3#20261059C214yesmaximal, complemented
4#20260775C212yesmaximal, complemented
5#20260734C212yesmaximal, complemented
6#20260699C211yesmaximal, characteristic, complemented
7C2×#117×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C2×#13095495C2212yes—
10#15690793—21no—
11#15690789C2212yes—
12#15690784C2212yes—
13#15690779—21no—
14C4×C22⋊D97—22no—
15#15690769C2211yescharacteristic
16#15690743C2214yescomplemented
17#15690741C2214yescomplemented
18#15690733C2214yescomplemented
19#15690730C2214yescomplemented
20C22⋊#11715895C2214yes—
21C23×C4×C97C2211yescharacteristic
22#117×C97—22no—
23#117×C97C2214yes—
24C2×C4⋊C4×C97C2212yes—
25C2×C22⋊C4×C97—21no—
26C2×C42×C97—21no—
27#13095511—24no—
28C2×Dic194C2311yescharacteristic
29C2×D388C2311yescharacteristic
30C2×C22⋊D97—22no—
31C2×C22⋊D97—21no—
32C2×C4×D97—21no—
33#13095495—24no—
34C2×#11715895C2314yes—
35C4⋊Dic97C2312yes—
36#13095490—22no—
37C22⋊Dic97—21no—
38C22⋊Dic97—22no—
39#13095488—22no—
40C4×Dic97—21no—
41C24×C97D411yescharacteristic
42C22×C4×C97—24no—
43C22×C4×C97—22no—
44C22×C4×C97D412yes—
45C22×C4×C97D411yescharacteristic
46C22×C4×C97C2312yes—
47C22×C4×C97C2311yescharacteristic
48C4⋊C4×C97—22no—
49C4⋊C4×C97C2314yes—
50C22⋊C4×C97—24no—
51C42×C97—24no—
52C23×C97—22no—
53C23×C97—42no—
54C23×C97C2×D412yes—
55C23×C97C2×D411yescharacteristic
56C22×D97—21no—
57C2×C4×C97—22no—
58C2×C4×C97—24no—
59C2×C4×C97—24no—
60C2×C4×C97—24no—
61C2×C4×C97—44no—
62C2×C4×C97C2×D412yes—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97#4114yes—
65C2×C4×C97C2411yescharacteristic
66C22⋊D97—44no—
67C22⋊D97—42no—
68C2×Dic97—22no—
69C2×Dic97—21no—
70D388—22no—
71C4×D97—42no—
72Dic194—22no—
73#11715895—24no—
74C22×C97—22no—
75C22×C97—22no—
76C22×C97—22no—
77C22×C97—22no—
78C22×C97—44no—
79C22×C97#12312yes—
80C22×C97C22×D411yescharacteristic
81C2×D97—41no—
82C2×D97—42no—
83C4×C97—22no—
84C4×C97—44no—
85C4×C97#12312yes—
86C4×C97C2×#4112yes—
87Dic97—41no—
88Dic97—42no—
89C2×C97—22no—
90C2×C97—42no—
91C2×C97C2×#12311yescharacteristic
92C2×C97#57512yes—
93D97—81no—
94#3093—971nomaximal, Sylow
95C97#309311yescharacteristic, complemented, Sylow
96#525—971no—
97C2×#117D9711yescharacteristic, complemented
98#485—974no—
99#482—974no—
100#428—972no—
101#417—972no—
102#407—971no—
103C2×#41—971no—
104C23×C4C2×D9711yescharacteristic
105C2×M16—972no—
106#123—1941no—
107C22⋊Q8—972no—
108#121—972no—
109#120—1941no—
110C4×D4—1942no—
111#117—22no—
112#117—971no—
113#117C2×D9714yes—
114C2×C4⋊C4C2×D9712yes—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117#107—974no—
118#106—974no—
119#103—974no—
120#102—974no—
121C22⋊C8—974no—
122C24C22⋊D9711yescharacteristic
123#41—1944no—
124C2×Q8—971no—
125C2×D4—971no—
126C2×D4—1942no—
127C2×D4—1941no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—1941no—
131C22×C4C22⋊D9711yescharacteristic
132C22×C4C22⋊D9712yes—
133C22×C4C22×D9712yes—
134C22×C4C22×D9711yescharacteristic
135M16—1944no—
136C2×C8—974no—
137C4⋊C4—22no—
138C4⋊C4—972no—
139C4⋊C4—1942no—
140C4⋊C4C22×D9714yes—
141C22⋊C4—24no—
142C22⋊C4—1941no—
143C22⋊C4—1942no—
144C22⋊C4—1942no—
145C42—24no—
146C42—1941no—
147C23—22no—
148C23—42no—
149C23—1941no—
150C23C2×C22⋊D9712yes—
151C23C2×C22⋊D9711yescharacteristic
152Q8—1942no—
153D4—3884no—
154D4—3882no—
155D4—1942no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—44no—
161C2×C4—3882no—
162C2×C4—1942no—
163C2×C4—1941no—
164C2×C4C2×C22⋊D9712yes—
165C2×C4C2×C22⋊D9711yescharacteristic
166C2×C4#1309551114yes—
167C2×C4C23×D9711yescharacteristic
168C8—1944no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—3881no—
175C22—3882no—
176C22#1569079312yes—
177C22C22×C22⋊D9711yescharacteristic
178C4—22no—
179C4—44no—
180C4—3881no—
181C4—3882no—
182C4#1569079312yes—
183C4C2×#1309551112yes—
184C2—22no—
185C2—42no—
186C2—7761no—
187C2C2×#1569079311yescharacteristic
188C2#2026153312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#117×C97C2
socleC22×C97C22×D4
prime radical#20261246C2
hypercenterC2×#117D97
hypocenterC97#3093
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20261076C2semidirect
3. #20261059C2semidirect
4. #20260775C2semidirect
5. #20260734C2semidirect
6. #20260699C2semidirect
7. C2×#117×C97C2semidirect
16. #15690743C22semidirect
17. #15690741C22semidirect
18. #15690733C22semidirect
19. #15690730C22semidirect
95. C97#3093semidirect
97. C2×#117D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
#202610762
#202610592
#202607752
#202607342
#202606992
C2×#117×C972
#309397