Finite Group Databasebeta

← #28401346

Subgroups — #28401346

Order 14464.

416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16813601C211yesmaximal, characteristic
2#20298933C212yesmaximal, complemented
3#20298931C212yesmaximal, complemented
4#20298837C214yesmaximal, complemented
5#20298754C212yesmaximal, complemented
6#20298753C212yesmaximal, complemented
7C2×C22⋊#11744751C211yesmaximal, characteristic
8C2×C22⋊Q8×C113C211yesmaximal, characteristic
9C22×Dic226C2211yescharacteristic
10C2×#14268637C2214yescomplemented
11C22×#11744751C2212yes—
12#16813601—21no—
13#16813601—21no—
14#16813601C2212yes—
15#16813601C2212yes—
16C2×C4⋊Dic113C2211yescharacteristic
17C2×#14268626—21no—
18C2×C22⋊Dic113—21no—
19#16813555C2214yescomplemented
20#16813546C2214yescomplemented
21#16813545C2214yescomplemented
22C22⋊#11744751C2212yes—
23C22⋊#11744751C2212yes—
24C22×Q8×C113C2211yescharacteristic
25C23×C4×C113C2211yescharacteristic
26C22⋊Q8×C113—21no—
27C22⋊Q8×C113—21no—
28C22⋊Q8×C113C2212yes—
29C22⋊Q8×C113C2212yes—
30C2×C4⋊C4×C113—21no—
31C2×C4⋊C4×C113C2211yescharacteristic
32C2×C22⋊C4×C113—21no—
33C2×Dic226—21no—
34C2×Dic226—42no—
35C2×Dic226C2312yes—
36C22×Dic113—21no—
37C22×Dic113—21no—
38#14268637—44no—
39C2×#11744751—42no—
40C2×#11744751C2314yes—
41C4⋊Dic113—21no—
42C4⋊Dic113C2312yes—
43#14268626—22no—
44#14268626—22no—
45C22⋊Dic113—22no—
46C22⋊Dic113—22no—
47C24×C113D411yescharacteristic
48C2×Q8×C113—21no—
49C2×Q8×C113—42no—
50C2×Q8×C113C2312yes—
51C22×C4×C113—21no—
52C22×C4×C113—21no—
53C22×C4×C113—42no—
54C22×C4×C113D411yescharacteristic
55C22×C4×C113D411yescharacteristic
56C22×C4×C113D411yescharacteristic
57C22×C4×C113C2311yescharacteristic
58C22×C4×C113C2311yescharacteristic
59C22×C4×C113C2311yescharacteristic
60C4⋊C4×C113—22no—
61C4⋊C4×C113—22no—
62C4⋊C4×C113—21no—
63C4⋊C4×C113C2312yes—
64C22⋊C4×C113—22no—
65C22⋊C4×C113—22no—
66C23×C113—41no—
67C23×C113—41no—
68C23×C113Q1614yescomplemented
69C23×C113C2×D411yescharacteristic
70C23×C113C2×D411yescharacteristic
71C23×C113C2×D411yescharacteristic
72Q8×C113—24no—
73Q8×C113—42no—
74C2×C4×C113—22no—
75C2×C4×C113—22no—
76C2×C4×C113—22no—
77C2×C4×C113—22no—
78C2×C4×C113—22no—
79C2×C4×C113—41no—
80C2×C4×C113—41no—
81C2×C4×C113—41no—
82C2×C4×C113—42no—
83C2×C4×C113—42no—
84C2×C4×C113C2×D411yescharacteristic
85C2×C4×C113C2×D411yescharacteristic
86C2×C4×C113C2×D411yescharacteristic
87C2×C4×C113C2411yescharacteristic
88C2×Dic113—22no—
89C2×Dic113—22no—
90C2×Dic113—42no—
91C2×Dic113—42no—
92Dic226—24no—
93Dic226—42no—
94#11744751—24no—
95C22×C113—21no—
96C22×C113—21no—
97C22×C113—21no—
98C22×C113—21no—
99C22×C113—21no—
100C22×C113—21no—
101C22×C113—44no—
102C22×C113C2×Q1612yes—
103C22×C113C2×Q1612yes—
104C22×C113C2×Q1612yes—
105C22×C113C22×D411yescharacteristic
106C4×C113—21no—
107C4×C113—21no—
108C4×C113—21no—
109C4×C113—42no—
110C4×C113—42no—
111C4×C113H(5,𝔽2)12yes—
112Dic113—42no—
113Dic113—42no—
114C2×C113—22no—
115C2×C113—22no—
116C2×C113—22no—
117C2×C113#53411yescharacteristic
118C2×C113C22×Q1611yescharacteristic
119C2×C113#57511yescharacteristic
120#3189—1131nomaximal, Sylow
121C113#318911yescharacteristic, complemented, Sylow
122C2×C22⋊Q8—1131no—
123C2×C22⋊Q8D11311yescharacteristic
124#484—1132no—
125#484—1132no—
126#467—1134no—
127#451—1132no—
128#451—1132no—
129C2×C22⋊C8—1131no—
130C22×Q8—1131no—
131C22×Q8C2×D11311yescharacteristic
132C23×C4C2×D11311yescharacteristic
133C2×Q16—1134no—
134C22×C8—1132no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—1132no—
138C22⋊Q8—1132no—
139C22⋊Q8—2261no—
140C22⋊Q8—2261no—
141C22⋊Q8C2×D11312yes—
142C22⋊Q8C2×D11312yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—1131no—
145C2×C4⋊C4—2261no—
146C2×C4⋊C4C2×D11311yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—2261no—
149#107—1134no—
150#103—1134no—
151#103—1134no—
152C22⋊C8—1132no—
153C22⋊C8—1132no—
154C24C22⋊D11311yescharacteristic
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—1132no—
158C2×Q8—4522no—
159C2×Q8—2261no—
160C2×Q8C22×D11312yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—2261no—
165C22×C4—2261no—
166C22×C4C22⋊D11311yescharacteristic
167C22×C4C22⋊D11311yescharacteristic
168C22×C4C22⋊D11311yescharacteristic
169C22×C4C22×D11311yescharacteristic
170C22×C4C22×D11311yescharacteristic
171C22×C4C22×D11311yescharacteristic
172Q16—4524no—
173C2×C8—1134no—
174C2×C8—4522no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—1132no—
179C4⋊C4—2261no—
180C4⋊C4—2262no—
181C4⋊C4—2262no—
182C4⋊C4C22×D11312yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—2262no—
186C22⋊C4—2262no—
187C23—41no—
188C23—41no—
189C23#1426863714yescomplemented
190C23C2×C22⋊D11311yescharacteristic
191C23C2×C22⋊D11311yescharacteristic
192C23C2×C22⋊D11311yescharacteristic
193Q8—24no—
194Q8—42no—
195Q8—4522no—
196Q8—2264no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—4522no—
208C2×C4—4522no—
209C2×C4—2262no—
210C2×C4—2262no—
211C2×C4C2×C22⋊D11311yescharacteristic
212C2×C4C2×C22⋊D11311yescharacteristic
213C2×C4C2×C22⋊D11311yescharacteristic
214C2×C4C23×D11311yescharacteristic
215C8—2264no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—44no—
223C22C2×#1426863712yes—
224C22C2×#1426863712yes—
225C22C2×#1426863712yes—
226C22C22×C22⋊D11311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—4522no—
233C4—4522no—
234C4#1681366712yes—
235C2—22no—
236C2—22no—
237C2—22no—
238C2#2029915711yescharacteristic
239C2C22×#1426863711yescharacteristic
240C2#2029939611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×C22⋊Q8×C113C2
socleC22×C113C22×D4
prime radicalC24×C113D4
hypercenterC2×C22⋊Q8D113
hypocenterC113#3189
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20298933C2semidirect
3. #20298931C2semidirect
4. #20298837C2semidirect
5. #20298754C2semidirect
6. #20298753C2semidirect
10. C2×#14268637C22semidirect
19. #16813555C22semidirect
20. #16813546C22semidirect
21. #16813545C22semidirect
68. C23×C113Q16semidirect
121. C113#3189semidirect
189. C23#14268637semidirect

Maximal subgroups

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

subgroupindex
C2×#168136012
#202989332
#202989312
#202988372
#202987542
#202987532
C2×C22⋊#117447512
C2×C22⋊Q8×C1132
#3189113