Finite Group Databasebeta

← #28401061

Subgroups — #28401061

Order 14464.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16813647C212yesmaximal, complemented
2C22×D904C211yesmaximal, characteristic, complemented
3#20299162C212yesmaximal, complemented
4#20298984C214yesmaximal, complemented
5#20298983C212yesmaximal, complemented
6#20298973C212yesmaximal, complemented
7#20298553C211yesmaximal, characteristic, complemented
8C2×#135×C113C211yesmaximal, characteristic, complemented
9C22×D452C2212yes—
10#16813647—28no—
11C2×#14268634C2214yescomplemented
12C2×D904—24no—
13C2×D904C2212yes—
14C2×D904C2212yes—
15C2×#14268632C2214yescomplemented
16C2×#14268631C2211yescharacteristic
17#16813618C2214yescomplemented
18#16813611C2214yescomplemented
19#16813596—24no—
20#16813590—24no—
21#16813581C2211yescharacteristic
22#16813532C2212yescomplemented
23C8×Dic113C2212yescomplemented
24C2×#41×C113C2212yes—
25#135×C113—24no—
26C2×Q16×C113C2211yescharacteristic, complemented
27C2×SD16×C113C2212yescomplemented
28C2×D8×C113C2211yescharacteristic, complemented
29C22×C8×C113C2211yescharacteristic
30C23×D113—22no—
31C2×D452—24no—
32C2×D452—24no—
33C2×D452—22no—
34C2×D452C2314yes—
35C2×C22⋊D113—28no—
36#14268634—28no—
37D904—24no—
38D904—24no—
39#14268632—28no—
40#14268631D414yescomplemented
41C2×#11744751C2312yes—
42C4⋊Dic113D412yes—
43C22⋊Dic113D412yes—
44#14268624—28no—
45C4×Dic113C2312yes—
46#41×C113—24no—
47#41×C113—24no—
48C2×Q8×C113C2312yes—
49C2×D4×C113—22no—
50C2×D4×C113C2312yes—
51C22×C4×C113—22no—
52C22×C4×C113C2311yescharacteristic
53Q16×C113—22no—
54SD16×C113—24no—
55D8×C113—22no—
56C2×C8×C113D414yescomplemented
57C2×C8×C113C2311yescharacteristic
58C2×C8×C113C2311yescharacteristic
59C23×C113—22no—
60C23×C113C2×D411yescharacteristic
61C22×D113—24no—
62C22×D113—44no—
63C22×D113—84no—
64Q8×C113—22no—
65Q8×C113—22no—
66D4×C113—22no—
67D4×C113—22no—
68D4×C113—44no—
69C2×C4×C113—22no—
70C2×C4×C113—44no—
71C2×C4×C113C2×D411yescharacteristic
72C2×C4×C113C2×D412yes—
73C2×C4×C113C2×D412yes—
74C2×C4×C113C2411yescharacteristic
75C22⋊D113—88no—
76C2×Dic113C2×D414yes—
77D452—24no—
78D452—24no—
79D452—44no—
80C8×C113C2×D412yes—
81C8×C113C2×D412yes—
82#11744751C2×D414yes—
83C22×C113—22no—
84C22×C113—42no—
85C22×C113—44no—
86C22×C113#12712yes—
87C22×C113C22×D411yescharacteristic
88C2×D113—44no—
89C2×D113—48no—
90C2×D113—88no—
91C4×C113—42no—
92C4×C113#12712yes—
93C4×C113C22×D411yescharacteristic
94C4×C113C22×D411yescharacteristic
95Dic113—24no—
96C2×C113—22no—
97C2×C113—82no—
98C2×C113C2×#12711yescharacteristic
99C2×C113#57612yes—
100D113—84no—
101#3155—1131nomaximal, Sylow
102C113#315511yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1132no—
104C2×#135D11311yescharacteristic, complemented
105C22×D8—1131no—
106#535—1132no—
107#496—1134no—
108#495—1132no—
109#493—1132no—
110#405—1131no—
111C2×#41C2×D11312yes—
112C22×D4—1132no—
113AGL(1,ℤ/8)—2268no—
114#135—24no—
115C2×Q16C2×D11311yescharacteristic, complemented
116C2×SD16—1134no—
117C2×SD16C2×D11312yescomplemented
118C2×D8—1134no—
119C2×D8—1132no—
120C2×D8—1132no—
121C2×D8—2264no—
122C2×D8C2×D11311yescharacteristic, complemented
123C2×M16—1131no—
124C22×C8C2×D11311yescharacteristic
125#127—1134no—
126#124—1134no—
127#121—2264no—
128#120—2264no—
129#117—1131no—
130#97—1132no—
131C4×C8—1132no—
132C24—2262no—
133#41—24no—
134#41—24no—
135C2×Q8C22×D11312yes—
136C2×D4—22no—
137C2×D4—1134no—
138C2×D4—2268no—
139C2×D4—2262no—
140C2×D4—2264no—
141C2×D4—2264no—
142C2×D4C22×D11312yes—
143C22×C4—22no—
144C22×C4C22×D11311yescharacteristic
145Q16—22no—
146SD16—24no—
147SD16—2268no—
148D8—22no—
149D8—2268no—
150D8—2264no—
151D8—2264no—
152M16—1134no—
153C2×C8—1132no—
154C2×C8C22⋊D11314yescomplemented
155C2×C8C22×D11311yescharacteristic
156C2×C8C22×D11311yescharacteristic
157C4⋊C4—1132no—
158C22⋊C4—1132no—
159C22⋊C4—2268no—
160C42—1132no—
161C23—22no—
162C23—9044no—
163C23—4524no—
164C23—2264no—
165C23D4×D11311yescharacteristic
166Q8—22no—
167Q8—22no—
168D4—22no—
169D4—22no—
170D4—44no—
171D4—9048no—
172D4—4524no—
173D4—2264no—
174D4—2264no—
175C2×C4—22no—
176C2×C4—44no—
177C2×C4—1134no—
178C2×C4D4×D11311yescharacteristic
179C2×C4D4×D11312yes—
180C2×C4C2×C22⋊D11312yes—
181C2×C4C23×D11311yescharacteristic
182C8—1134no—
183C8C2×C22⋊D11312yes—
184C8C2×C22⋊D11312yes—
185C22—22no—
186C22—42no—
187C22—44no—
188C22—9048no—
189C22—4524no—
190C22—4528no—
191C22#1681361812yes—
192C22C2×D4×D11311yescharacteristic
193C4—42no—
194C4—2264no—
195C4#1681361812yes—
196C4C2×D4×D11311yescharacteristic
197C4C22×C22⋊D11311yescharacteristic
198C2—22no—
199C2—82no—
200C2—9044no—
201C2C2×#1681361811yescharacteristic
202C2#2029940012yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×#135×C113C2
socleC22×C113C22×D4
hypercenterC2×#135D113
hypocenterC113#3155
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16813647C2semidirect
2. C22×D904C2semidirect
3. #20299162C2semidirect
4. #20298984C2semidirect
5. #20298983C2semidirect
6. #20298973C2semidirect
7. #20298553C2semidirect
8. C2×#135×C113C2semidirect
11. C2×#14268634C22semidirect
15. C2×#14268632C22semidirect
17. #16813618C22semidirect
18. #16813611C22semidirect
22. #16813532C22semidirect
23. C8×Dic113C22semidirect
26. C2×Q16×C113C22semidirect
27. C2×SD16×C113C22semidirect
28. C2×D8×C113C22semidirect
40. #14268631D4semidirect
56. C2×C8×C113D4semidirect
102. C113#3155semidirect
104. C2×#135D113semidirect
115. C2×Q16C2×D113semidirect
117. C2×SD16C2×D113semidirect
122. C2×D8C2×D113semidirect
154. C2×C8C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136472
C22×D9042
#202991622
#202989842
#202989832
#202989732
#202985532
C2×#135×C1132
#3155113