Finite Group Databasebeta

← #28401500

Subgroups — #28401500

Order 14464.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#20299149C211yesmaximal, characteristic
2C4⋊Q8×D113C211yesmaximal, characteristic, complemented
3#20298972C211yesmaximal, characteristic, complemented
4#20298971C211yesmaximal, characteristic, complemented
5#20298950C211yesmaximal, characteristic, complemented
6#20298802C214yesmaximal, complemented
7#20298738C214yesmaximal, complemented
8#20298649C211yesmaximal, characteristic, complemented
9#487×C113C211yesmaximal, characteristic, complemented
10C2×#14268648—21no—
11C2×Q8×D113C2212yes—
12C2×#14268636C2214yes—
13C2×#14268634C2214yes—
14#16813622C2211yescharacteristic
15#16813621—21no—
16#16813621—21no—
17#16813620C2212yes—
18#16813619C2211yescharacteristic
19#16813611—21no—
20#16813597—21no—
21#16813597C2212yes—
22#16813584—21no—
23#16813584C2212yes—
24C4⋊C4×D113—21no—
25C4⋊C4×D113—21no—
26C42×D113C2211yescharacteristic
27#16813551C2212yes—
28#16813545C2214yes—
29#16813534C2212yes—
30C4×#11744751C2211yescharacteristic
31C4⋊Q8×C113C2211yescharacteristic
32C4⋊Q8×C113C2211yescharacteristic
33#103×C113C2214yes—
34C4×C8×C113C2211yescharacteristic
35#14268648—44no—
36C2×Dic226—22no—
37C2×Dic226C2312yes—
38Q8×D113—44no—
39C2×D452—22no—
40C2×D452—21no—
41C2×D452—21no—
42C2×D452C2312yes—
43C2×C4×D113—22no—
44C2×C4×D113—22no—
45C2×C4×D113D411yescharacteristic
46C2×C4×D113D411yescharacteristic
47C2×C4×D113C2311yescharacteristic
48#14268636—44no—
49#14268634—44no—
50C2×#11744751C2312yes—
51C4⋊Dic113—21no—
52C4⋊Dic113—21no—
53#14268626—22no—
54#14268626—22no—
55#14268624—22no—
56#14268624—22no—
57C4×Dic113D411yescharacteristic
58C4×Dic113D411yescharacteristic
59C4×Dic113C2311yescharacteristic
60C2×Q8×C113—21no—
61C2×Q8×C113C2312yes—
62C2×C8×C113C2312yes—
63C4⋊C4×C113—21no—
64C4⋊C4×C113—21no—
65C4⋊C4×C113—21no—
66C4⋊C4×C113C2312yes—
67C42×C113C2311yescharacteristic
68C22×D113—22no—
69C22×D113C2×D411yescharacteristic
70Q8×C113—24no—
71Q8×C113—42no—
72C2×C4×C113—22no—
73C2×C4×C113—22no—
74C2×C4×C113C2×D411yescharacteristic
75C2×C4×C113C2×D411yescharacteristic
76C2×C4×C113C2411yescharacteristic
77C2×Dic113—22no—
78C2×Dic113C2×D411yescharacteristic
79C2×Dic113C2×D411yescharacteristic
80C2×Dic113C2×D411yescharacteristic
81D452—24no—
82D452—44no—
83D452—42no—
84D452—42no—
85C4×D113—22no—
86C4×D113—22no—
87C4×D113—44no—
88C4×D113—44no—
89C4×D113SD1614yescomplemented
90Dic226—24no—
91Dic226—44no—
92C8×C113—22no—
93#11744751—22no—
94C22×C113C22×D411yescharacteristic
95C2×D113—21no—
96C2×D113—21no—
97C2×D113—42no—
98C2×D113—42no—
99C2×D113—42no—
100C2×D113C2×SD1612yes—
101C4×C113—42no—
102C4×C113—42no—
103C4×C113C2×SD1612yes—
104C4×C113#13712yes—
105C4×C113H(5,𝔽2)12yes—
106Dic113—21no—
107Dic113—21no—
108Dic113—42no—
109Dic113C2×SD1612yes—
110C2×C113#53511yescharacteristic
111C2×C113C22×SD1611yescharacteristic
112C2×C113C2×#13711yescharacteristic
113D113—22no—
114D113—82no—
115#3206—1131nomaximal, Sylow
116C113#320611yescharacteristic, complemented, Sylow
117#532—1131no—
118C2×C4⋊Q8—1131no—
119#492—1131no—
120#492—1131no—
121#487—1131no—
122#487D11311yescharacteristic, complemented
123#461—1134no—
124#448—1134no—
125#431—1131no—
126C2×#41—2261no—
127C22×Q8—1132no—
128C2×SD16—1134no—
129C2×SD16—1134no—
130C4⋊Q8—1132no—
131C4⋊Q8—1131no—
132C4⋊Q8—2261no—
133C4⋊Q8—2261no—
134C4⋊Q8C2×D11311yescharacteristic
135C4⋊Q8C2×D11311yescharacteristic
136#127—1131no—
137#124—2261no—
138#121—1132no—
139#121—2261no—
140C4×D4—1132no—
141C4×D4—2261no—
142C2×C4⋊C4—2261no—
143C2×C4⋊C4—2261no—
144C2×C42—1131no—
145C4⋊C8—1132no—
146#103—1134no—
147#103C2×D11314yes—
148C22⋊C8—1132no—
149C4×C8—1131no—
150C4×C8C2×D11311yescharacteristic
151#41—4524no—
152C2×Q8—21no—
153C2×Q8—1132no—
154C2×Q8—4524no—
155C2×Q8—2262no—
156C2×Q8C22×D11312yes—
157C2×D4—1132no—
158C2×D4—2262no—
159C2×D4—2261no—
160C2×D4—2261no—
161C22×C4—1131no—
162C22×C4—1131no—
163C22×C4—1131no—
164C22×C4—2262no—
165C22×C4—2262no—
166SD16—4524no—
167SD16—4524no—
168C2×C8—1132no—
169C2×C8C22×D11312yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—2262no—
174C4⋊C4—2262no—
175C4⋊C4—2261no—
176C4⋊C4—2261no—
177C4⋊C4C22×D11312yes—
178C22⋊C4—2262no—
179C22⋊C4—2262no—
180C42—1131no—
181C42—1131no—
182C42—1131no—
183C42C22×D11311yescharacteristic
184C23—1131no—
185C23—2262no—
186Q8—24no—
187Q8—42no—
188Q8—4524no—
189Q8—2264no—
190D4—4524no—
191D4—4522no—
192D4—4522no—
193D4—2264no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—1131no—
197C2×C4—1134no—
198C2×C4—1131no—
199C2×C4—1131no—
200C2×C4—4524no—
201C2×C4—4524no—
202C2×C4—2262no—
203C2×C4—2262no—
204C2×C4—2262no—
205C2×C4D4×D11311yescharacteristic
206C2×C4D4×D11311yescharacteristic
207C2×C4C23×D11311yescharacteristic
208C8—22no—
209C8—2262no—
210C22—1132no—
211C22—4522no—
212C22—4522no—
213C22—4522no—
214C22—2261no—
215C22—2261no—
216C22C2×D4×D11311yescharacteristic
217C4—42no—
218C4—42no—
219C4—1132no—
220C4—4522no—
221C4—2261no—
222C4—2261no—
223C4SD16×D11312yes—
224C4#1681365412yes—
225C4#1681366812yes—
226C2—9042no—
227C2—2262no—
228C2#2029916611yescharacteristic
229C2C2×SD16×D11311yescharacteristic
230C2C2×#1681365411yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#487×C113C2
socleC22×C113C22×D4
prime radical#20299149C2
hypercenter#487D113
hypocenterC113#3206
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D113C2semidirect
3. #20298972C2semidirect
4. #20298971C2semidirect
5. #20298950C2semidirect
6. #20298802C2semidirect
7. #20298738C2semidirect
8. #20298649C2semidirect
9. #487×C113C2semidirect
89. C4×D113SD16semidirect
116. C113#3206semidirect
122. #487D113semidirect

Maximal subgroups

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

subgroupindex
#202991492
C4⋊Q8×D1132
#202989722
#202989712
#202989502
#202988022
#202987382
#202986492
#487×C1132
#3206113