Finite Group Databasebeta

← #27151669

Subgroups — #27151669

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
2C2×#15690769C211yesmaximal, characteristic, complemented
3#20261077C214yesmaximal, complemented
4#20261060C214yesmaximal, complemented
5#20260777C212yesmaximal, complemented
6#20260737C212yesmaximal, complemented
7#407×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690793—21no—
10#15690789C2212yes—
11#15690784C2212yes—
12#15690779—21no—
13C4×C22⋊D97—22no—
14#15690769—22no—
15#15690769C2214yes—
16#15690769C2211yescharacteristic
17C2×C4⋊Dic97C2212yes—
18C2×C22⋊Dic97—21no—
19C2×C4×Dic97—21no—
20#15690744C2214yescomplemented
21#15690742C2214yescomplemented
22#15690735C2214yescomplemented
23#15690732C2214yescomplemented
24C23×C4×C97C2211yescharacteristic
25C2×M16×C97C2212yes—
26C22⋊C8×C97C2214yes—
27#13095511—24no—
28C2×Dic194C2311yescharacteristic
29C2×D388C2311yescharacteristic
30C2×C22⋊D97—22no—
31C2×C22⋊D97—21no—
32C22×Dic97—22no—
33C2×C4×D97—21no—
34C4⋊Dic97—22no—
35C4⋊Dic97C2314yes—
36C4⋊Dic97C2312yes—
37#13095490—22no—
38C22⋊Dic97—24no—
39C22⋊Dic97—21no—
40C22⋊Dic97—22no—
41#13095488—22no—
42C4×Dic97—24no—
43C4×Dic97—21no—
44C24×C97D411yescharacteristic
45C22×C4×C97—24no—
46C22×C4×C97D411yescharacteristic
47C22×C4×C97D412yes—
48C22×C4×C97C2312yes—
49C22×C4×C97C2311yescharacteristic
50M16×C97—24no—
51C2×C8×C97C2314yes—
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×C97C2×D412yes—
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97#4114yes—
63C2×C4×C97C2411yescharacteristic
64C22⋊D97—44no—
65C22⋊D97—42no—
66C2×Dic97—24no—
67C2×Dic97—22no—
68C2×Dic97—21no—
69C2×Dic97—44no—
70D388—22no—
71C4×D97—42no—
72Dic194—22no—
73C8×C97—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#12312yes—
85C4×C97C2×#4112yes—
86Dic97—41no—
87Dic97—44no—
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×#117—971no—
98#485—974no—
99#482—974no—
100#428—972no—
101#417—972no—
102#407D9711yescharacteristic, complemented
103C2×#41—971no—
104C23×C4C2×D9711yescharacteristic
105C2×M16C2×D9712yes—
106#123—1941no—
107C22⋊Q8—972no—
108#121—972no—
109#120—1941no—
110C4×D4—1942no—
111#117—974no—
112#117—971no—
113#117—1942no—
114C2×C4⋊C4—972no—
115C2×C22⋊C4—1941no—
116C2×C42—1941no—
117#107—974no—
118#106—974no—
119#103—974no—
120#102—974no—
121C22⋊C8C2×D9714yes—
122C24D38811yescharacteristic
123#41—1944no—
124C2×Q8—971no—
125C2×D4—971no—
126C2×D4—1942no—
127C2×D4—1941no—
128C22×C4—24no—
129C22×C4—1942no—
130C22×C4—1941no—
131C22×C4D38811yescharacteristic
132C22×C4D38812yes—
133C22×C4C22×D9712yes—
134C22×C4C22×D9711yescharacteristic
135M16—24no—
136C2×C8C22×D9714yes—
137C4⋊C4—974no—
138C4⋊C4—972no—
139C4⋊C4—1942no—
140C4⋊C4—1942no—
141C22⋊C4—1944no—
142C22⋊C4—1941no—
143C22⋊C4—1942no—
144C22⋊C4—1942no—
145C42—1944no—
146C42—1941no—
147C23—22no—
148C23—42no—
149C23—1941no—
150C23C2×D38812yes—
151C23C2×D38811yescharacteristic
152Q8—1942no—
153D4—3884no—
154D4—3882no—
155D4—1942no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—3884no—
160C2×C4—3882no—
161C2×C4—1944no—
162C2×C4—1942no—
163C2×C4—1941no—
164C2×C4C2×D38812yes—
165C2×C4C2×D38811yescharacteristic
166C2×C4#1309551014yes—
167C2×C4C23×D9711yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—3881no—
175C22—3882no—
176C22#1569079412yes—
177C22C22×D38811yescharacteristic
178C4—22no—
179C4—3881no—
180C4—3884no—
181C4—3882no—
182C4#1569079412yes—
183C4C2×#1309551012yes—
184C2—22no—
185C2—42no—
186C2—7761no—
187C2C2×#1569079411yescharacteristic
188C2#2026153512yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#407×C97C2
socleC22×C97C22×D4
prime radical#20261246C2
hypercenter#407D97
hypocenterC97#3093
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#15690769C2semidirect
3. #20261077C2semidirect
4. #20261060C2semidirect
5. #20260777C2semidirect
6. #20260737C2semidirect
7. #407×C97C2semidirect
20. #15690744C22semidirect
21. #15690742C22semidirect
22. #15690735C22semidirect
23. #15690732C22semidirect
95. C97#3093semidirect
102. #407D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
C2×#156907692
#202610772
#202610602
#202607772
#202607372
#407×C972
#309397