Finite Group Databasebeta

← #28401394

Subgroups — #28401394

Order 14464.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22⋊Q8×D113C212yesmaximal
2#20298930C212yesmaximal, complemented
3#20298927C212yesmaximal, complemented
4#20298856C211yesmaximal, characteristic, complemented
5#20298854C212yesmaximal, complemented
6#20298755C212yesmaximal, complemented
7#20298752C212yesmaximal, complemented
8#20298561C211yesmaximal, characteristic
9#470×C113C211yesmaximal, characteristic, complemented
10C2×Q8×D113C2212yes—
11C22×C4×D113C2211yescharacteristic
12C2×Dic452C2211yescharacteristic, complemented
13C2×#14268637C2212yescomplemented
14C2×#14268631C2211yescharacteristic
15#16813602C2212yes—
16#16813601C2212yes—
17#16813600—22no—
18#16813599—22no—
19#16813598C2212yes—
20C4⋊C4×D113—22no—
21C4⋊C4×D113C2212yes—
22C22⋊C4×D113—22no—
23#16813554C2211yescharacteristic, complemented
24#16813553C2212yescomplemented
25#16813547C2212yescomplemented
26#16813546C2212yescomplemented
27#16813545C2212yescomplemented
28#16813534C2212yes—
29#16813534C2212yes—
30C2×Q16×C113C2211yescharacteristic, complemented
31C2×M16×C113C2211yescharacteristic
32C22⋊Q8×C113C2212yes—
33#106×C113C2211yescharacteristic, complemented
34#103×C113C2212yescomplemented
35C23×D113D411yescharacteristic
36C2×Dic226—22no—
37C2×Dic226C2312yes—
38Q8×D113—42no—
39Q8×D113—42no—
40C22×Dic113D411yescharacteristic
41C2×C4×D113—22no—
42C2×C4×D113—22no—
43C2×C4×D113—41no—
44C2×C4×D113—41no—
45C2×C4×D113D412yes—
46C2×C4×D113C2312yes—
47Dic452—41no—
48#14268637—42no—
49#14268631—41no—
50C2×#11744751C2312yes—
51C4⋊Dic113—22no—
52C4⋊Dic113C2312yes—
53#14268626—22no—
54#14268626—22no—
55#14268626—22no—
56C22⋊Dic113—22no—
57#14268624—22no—
58#14268624—22no—
59C2×Q8×C113C2312yes—
60C22×C4×C113C2311yescharacteristic
61Q16×C113—41no—
62M16×C113—41no—
63C2×C8×C113C2311yescharacteristic
64C2×C8×C113C2311yescharacteristic
65C4⋊C4×C113—22no—
66C4⋊C4×C113C2312yes—
67C22⋊C4×C113—22no—
68C23×C113C2×D411yescharacteristic
69C22×D113—24no—
70C22×D113—41no—
71C22×D113C2×D412yes—
72Q8×C113—22no—
73Q8×C113—22no—
74C2×C4×C113—21no—
75C2×C4×C113—21no—
76C2×C4×C113—22no—
77C2×C4×C113—22no—
78C2×C4×C113C2×D411yescharacteristic
79C2×C4×C113C2411yescharacteristic
80C2×Dic113—22no—
81C2×Dic113—22no—
82C2×Dic113—41no—
83C2×Dic113C2×D412yes—
84C4×D113—22no—
85C4×D113—22no—
86C4×D113—42no—
87C4×D113—44no—
88C4×D113—44no—
89Dic226—22no—
90Dic226—22no—
91Dic226—44no—
92C8×C113—21no—
93C8×C113—21no—
94#11744751—22no—
95C22×C113—21no—
96C22×C113—22no—
97C22×C113C22×D411yescharacteristic
98C2×D113—22no—
99C2×D113—22no—
100C2×D113—44no—
101C2×D113#13714yes—
102C4×C113—21no—
103C4×C113—42no—
104C4×C113—42no—
105C4×C113H(5,𝔽2)11yescharacteristic
106C4×C113H(5,𝔽2)11yescharacteristic
107Dic113—22no—
108Dic113—42no—
109Dic113—42no—
110C2×C113—41no—
111C2×C113#53411yescharacteristic
112C2×C113C2×#13712yes—
113D113—24no—
114#3195—1131nomaximal, Sylow
115C113#319511yescharacteristic, complemented, Sylow
116C2×C22⋊Q8—1132no—
117#483—1132no—
118#483—1132no—
119#470—1131no—
120#470—1132no—
121#470D11311yescharacteristic, complemented
122#451—1132no—
123#451—1132no—
124#407—1131no—
125C22×Q8—1132no—
126C23×C4—1131no—
127C2×Q16—1132no—
128C2×Q16—1131no—
129C2×Q16C2×D11311yescharacteristic, complemented
130C2×M16—1131no—
131C2×M16C2×D11311yescharacteristic
132C22⋊Q8—1132no—
133C22⋊Q8—1132no—
134C22⋊Q8—1132no—
135C22⋊Q8—2262no—
136C22⋊Q8—2262no—
137C22⋊Q8C2×D11312yes—
138C2×C4⋊C4—1132no—
139C2×C4⋊C4—2262no—
140C2×C22⋊C4—2262no—
141#106—1131no—
142#106—1132no—
143#106C2×D11311yescharacteristic, complemented
144#103—1132no—
145#103—1132no—
146#103—1132no—
147#103C2×D11312yescomplemented
148C22⋊C8—1132no—
149C22⋊C8—1132no—
150C24—1131no—
151C2×Q8—1132no—
152C2×Q8—4522no—
153C2×Q8—4522no—
154C2×Q8—2262no—
155C2×Q8C22×D11312yes—
156C22×C4—1131no—
157C22×C4—1132no—
158C22×C4—1132no—
159C22×C4—4521no—
160C22×C4—4521no—
161C22×C4—2262no—
162C22×C4—2262no—
163C22×C4C22×D11311yescharacteristic
164Q16—41no—
165Q16—4521no—
166Q16—4522no—
167M16—41no—
168M16—4521no—
169C2×C8—1132no—
170C2×C8C22×D11311yescharacteristic
171C2×C8C22×D11311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—1132no—
174C4⋊C4—2262no—
175C4⋊C4—2262no—
176C4⋊C4—2262no—
177C4⋊C4—2262no—
178C4⋊C4C22×D11312yes—
179C22⋊C4—22no—
180C22⋊C4—2262no—
181C22⋊C4—2262no—
182C22⋊C4—2262no—
183C23—1132no—
184C23—4521no—
185C23—2264no—
186C23D4×D11311yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—4524no—
190Q8—2262no—
191Q8—2262no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—1132no—
197C2×C4—4522no—
198C2×C4—4524no—
199C2×C4—4524no—
200C2×C4—4521no—
201C2×C4—2262no—
202C2×C4—2262no—
203C2×C4—2262no—
204C2×C4—2262no—
205C2×C4D4×D11311yescharacteristic
206C2×C4C23×D11311yescharacteristic
207C8—21no—
208C8—21no—
209C8—2262no—
210C22—22no—
211C22—21no—
212C22—1134no—
213C22—4524no—
214C22—2262no—
215C22—2262no—
216C22C2×D4×D11311yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—4522no—
221C4—4522no—
222C4—2262no—
223C4#1681366711yescharacteristic
224C4#1681366811yescharacteristic
225C2—41no—
226C2—2264no—
227C2#2029915611yescharacteristic
228C2#137×D11312yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#470×C113C2
socleC22×C113C22×D4
prime radicalC23×D113D4
hypercenter#470D113
hypocenterC113#3195
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #20298930C2semidirect
3. #20298927C2semidirect
4. #20298856C2semidirect
5. #20298854C2semidirect
6. #20298755C2semidirect
7. #20298752C2semidirect
9. #470×C113C2semidirect
12. C2×Dic452C22semidirect
13. C2×#14268637C22semidirect
23. #16813554C22semidirect
24. #16813553C22semidirect
25. #16813547C22semidirect
26. #16813546C22semidirect
27. #16813545C22semidirect
30. C2×Q16×C113C22semidirect
33. #106×C113C22semidirect
34. #103×C113C22semidirect
115. C113#3195semidirect
121. #470D113semidirect
129. C2×Q16C2×D113semidirect
143. #106C2×D113semidirect
147. #103C2×D113semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1132
#202989302
#202989272
#202988562
#202988542
#202987552
#202987522
#202985612
#470×C1132
#3195113