Finite Group Databasebeta

← #28401657

Subgroups — #28401657

Order 14464.

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

normal subgroups only

This group has 364 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#20299210C211yesmaximal, characteristic, complemented
2#20298993C211yesmaximal, characteristic, complemented
3#20298977C211yesmaximal, characteristic, complemented
4#20298966C211yesmaximal, characteristic, complemented
5#20298952C211yesmaximal, characteristic, complemented
6#20298817C212yesmaximal, complemented
7#20298805C212yesmaximal, complemented
8#20298760C212yesmaximal
9#20298754C212yesmaximal, complemented
10#20298659C211yesmaximal, characteristic, complemented
11#536×C113C211yesmaximal, characteristic
12C2×#14268637C2212yes—
13C2×#14268637C2212yes—
14C2×#14268637C2212yes—
15C2×#14268635C2212yes—
16#16813622C2211yescharacteristic
17#16813615—22no—
18#16813614C2211yescharacteristic
19#16813612—22no—
20#16813601C2212yes—
21C4×Dic226C2212yes—
22C4⋊#11744751C2212yes—
23#16813546C2212yes—
24#16813546C2212yes—
25#16813546C2212yes—
26#16813543C2212yes—
27C22⋊#11744751C2212yes—
28#16813533C2211yescharacteristic
29C4×#11744751C2211yescharacteristic
30C2×#41×C113C2211yescharacteristic
31C22×Q8×C113C2211yescharacteristic
32C4⋊Q8×C113C2211yescharacteristic
33C4⋊Q8×C113C2211yescharacteristic
34#124×C113C2211yescharacteristic
35#124×C113C2211yescharacteristic
36#123×C113—22no—
37C22⋊Q8×C113—22no—
38#117×C113C2211yescharacteristic
39C2×Dic226C2312yes—
40#14268637—42no—
41#14268637—42no—
42#14268637—42no—
43#14268635—42no—
44C2×#11744751C2312yes—
45C2×#11744751C2312yes—
46C4⋊Dic113—22no—
47C4⋊Dic113C2312yes—
48#14268626—22no—
49#14268626—22no—
50#14268626—22no—
51C22⋊Dic113—22no—
52C4×Dic113—22no—
53#41×C113—42no—
54C2×Q8×C113—21no—
55C2×Q8×C113—42no—
56C2×Q8×C113C2311yescharacteristic
57C2×Q8×C113C2311yescharacteristic
58C2×Q8×C113C2311yescharacteristic
59C2×D4×C113—21no—
60C2×D4×C113C2311yescharacteristic
61C22×C4×C113—21no—
62C22×C4×C113—21no—
63C22×C4×C113C2311yescharacteristic
64C4⋊C4×C113—22no—
65C4⋊C4×C113—22no—
66C4⋊C4×C113D412yes—
67C22⋊C4×C113—22no—
68C22⋊C4×C113—22no—
69C22⋊C4×C113D412yes—
70C42×C113C2311yescharacteristic
71C42×C113C2311yescharacteristic
72C23×C113—21no—
73C23×C113C2×D411yescharacteristic
74Q8×C113—22no—
75Q8×C113—22no—
76Q8×C113—22no—
77Q8×C113—42no—
78D4×C113—22no—
79D4×C113—42no—
80C2×C4×C113—22no—
81C2×C4×C113—21no—
82C2×C4×C113—21no—
83C2×C4×C113—21no—
84C2×C4×C113—42no—
85C2×C4×C113—42no—
86C2×C4×C113C2×D412yes—
87C2×C4×C113C2×D412yes—
88C2×C4×C113C2×D411yescharacteristic
89C2×C4×C113C2411yescharacteristic
90C2×Dic113—22no—
91C2×Dic113—22no—
92Dic226—22no—
93Dic226—22no—
94#11744751—22no—
95#11744751—22no—
96C22×C113—21no—
97C22×C113—22no—
98C22×C113—41no—
99C22×C113—42no—
100C22×C113C22×D411yescharacteristic
101C4×C113—21no—
102C4×C113—22no—
103C4×C113—22no—
104C4×C113—41no—
105C4×C113—41no—
106C4×C113—41no—
107C4×C113H(5,𝔽2)12yes—
108Dic113—42no—
109Dic113—42no—
110C2×C113—41no—
111C2×C113—81no—
112C2×C113#53511yescharacteristic
113C2×C113#57812yes—
114#3223—1131nomaximal, Sylow
115C113#322311yescharacteristic, complemented, Sylow
116#541—1131no—
117#536D11311yescharacteristic
118#497—1131no—
119#494—1131no—
120#491—1131no—
121#488—1131no—
122#464—1132no—
123#462—1132no—
124#452—1132no—
125#451—1132no—
126#433—1131no—
127C2×#41C2×D11311yescharacteristic
128C22×Q8C2×D11311yescharacteristic
129C2×Q16—1132no—
130C2×Q16—1132no—
131C2×Q16—1132no—
132C2×SD16—1132no—
133C4⋊Q8—1131no—
134C4⋊Q8C2×D11311yescharacteristic
135C4⋊Q8C2×D11311yescharacteristic
136#126—2262no—
137#125—1131no—
138#125—2262no—
139#124C2×D11311yescharacteristic
140#124C2×D11311yescharacteristic
141#123—22no—
142C22⋊Q8—22no—
143C22⋊Q8—1132no—
144C4×Q8—1132no—
145#117C2×D11311yescharacteristic
146C4⋊C8—1132no—
147#103—1132no—
148#103—1132no—
149#103—1132no—
150#102—1132no—
151C22⋊C8—1132no—
152#97—1131no—
153C4×C8—1131no—
154#41—42no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—1132no—
158C2×Q8C22×D11311yescharacteristic
159C2×Q8C22×D11311yescharacteristic
160C2×Q8C22×D11311yescharacteristic
161C2×D4—21no—
162C2×D4C22×D11311yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D11311yescharacteristic
166Q16—4522no—
167Q16—4522no—
168Q16—4522no—
169SD16—4522no—
170C2×C8—1132no—
171C2×C8—1132no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—1132no—
175C4⋊C4—2262no—
176C4⋊C4—2262no—
177C4⋊C4—2262no—
178C4⋊C4—2262no—
179C4⋊C4C22⋊D11312yes—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—2262no—
183C22⋊C4C22⋊D11312yes—
184C42—2262no—
185C42C22×D11311yescharacteristic
186C42C22×D11311yescharacteristic
187C23—21no—
188C23C2×C22⋊D11311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—2262no—
194Q8—2262no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—2262no—
204C2×C4—2262no—
205C2×C4C2×C22⋊D11312yes—
206C2×C4C2×C22⋊D11312yes—
207C2×C4C2×C22⋊D11311yescharacteristic
208C2×C4C23×D11311yescharacteristic
209C8—2262no—
210C8—2262no—
211C22—22no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22C22×C22⋊D11311yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—4522no—
223C4—4522no—
224C4#1681366712yes—
225C2—41no—
226C2—81no—
227C2#2029916511yescharacteristic
228C2#2029941312yes—
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#536×C113C2
socleC22×C113C22×D4
prime radicalC2×#41×C113C22
hypercenter#536D113
hypocenterC113#3223
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #20299210C2semidirect
2. #20298993C2semidirect
3. #20298977C2semidirect
4. #20298966C2semidirect
5. #20298952C2semidirect
6. #20298817C2semidirect
7. #20298805C2semidirect
9. #20298754C2semidirect
10. #20298659C2semidirect
115. C113#3223semidirect

Maximal subgroups

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

subgroupindex
#202992102
#202989932
#202989772
#202989662
#202989522
#202988172
#202988052
#202987602
#202987542
#202986592
#536×C1132
#3223113