Finite Group Databasebeta

← #28401345

Subgroups — #28401345

Order 14464.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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#20298935C212yesmaximal, complemented
3#20298932C212yesmaximal, complemented
4#20298839C211yesmaximal, characteristic, complemented
5#20298837C212yesmaximal, complemented
6#20298755C212yesmaximal, complemented
7#20298752C212yesmaximal, complemented
8C2×#16813534C211yesmaximal, characteristic
9#467×C113C211yesmaximal, characteristic, complemented
10C2×Q8×D113C2212yes—
11C22×C4×D113C2211yescharacteristic
12C2×Dic452C2211yescharacteristic, complemented
13C2×#14268637C2212yescomplemented
14C22×#11744751C2211yescharacteristic
15#16813602C2212yes—
16#16813601C2212yes—
17#16813600—22no—
18#16813599—22no—
19#16813598C2212yes—
20C4⋊C4×D113—22no—
21C4⋊C4×D113C2212yes—
22C22⋊C4×D113—22no—
23#16813556C2211yescharacteristic, complemented
24#16813555C2212yescomplemented
25#16813547C2212yescomplemented
26#16813546C2212yescomplemented
27#16813545C2212yescomplemented
28#16813534C2212yes—
29#16813534C2212yes—
30C2×Q16×C113C2211yescharacteristic, complemented
31C22×C8×C113C2211yescharacteristic
32C22⋊Q8×C113C2212yes—
33#107×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—
49C2×#11744751—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—
62C2×C8×C113—41no—
63C2×C8×C113C2311yescharacteristic
64C2×C8×C113C2311yescharacteristic
65C4⋊C4×C113—22no—
66C4⋊C4×C113C2312yes—
67C22⋊C4×C113—22no—
68C23×C113C2×D411yescharacteristic
69C22×D113—41no—
70C22×D113—41no—
71C22×D113Q1614yescomplemented
72C22×D113C2×D412yes—
73Q8×C113—22no—
74Q8×C113—22no—
75C2×C4×C113—21no—
76C2×C4×C113—21no—
77C2×C4×C113—22no—
78C2×C4×C113—22no—
79C2×C4×C113C2×D411yescharacteristic
80C2×C4×C113C2411yescharacteristic
81C2×Dic113—22no—
82C2×Dic113—22no—
83C2×Dic113—41no—
84C2×Dic113C2×D412yes—
85C4×D113—22no—
86C4×D113—22no—
87C4×D113—42no—
88C4×D113—44no—
89C4×D113—44no—
90Dic226—22no—
91Dic226—22no—
92Dic226—44no—
93C8×C113—21no—
94C8×C113—21no—
95#11744751—22no—
96C22×C113—21no—
97C22×C113—21no—
98C22×C113C2×Q1612yes—
99C22×C113C22×D411yescharacteristic
100C2×D113—22no—
101C2×D113—22no—
102C2×D113—44no—
103C2×D113C2×Q1614yes—
104C4×C113—21no—
105C4×C113—42no—
106C4×C113—42no—
107C4×C113H(5,𝔽2)11yescharacteristic
108C4×C113H(5,𝔽2)11yescharacteristic
109Dic113—22no—
110Dic113—42no—
111Dic113—42no—
112C2×C113—22no—
113C2×C113#53411yescharacteristic
114C2×C113C22×Q1611yescharacteristic
115C2×C113#57511yescharacteristic
116D113—24no—
117#3189—1131nomaximal, Sylow
118C113#318911yescharacteristic, complemented, Sylow
119C2×C22⋊Q8—1132no—
120#484—1132no—
121#484—1132no—
122#467—1131no—
123#467—1132no—
124#467D11311yescharacteristic, complemented
125#451—1132no—
126#451—1132no—
127C2×C22⋊C8—1131no—
128C22×Q8—1132no—
129C23×C4—1131no—
130C2×Q16—1132no—
131C2×Q16—1131no—
132C2×Q16C2×D11311yescharacteristic, complemented
133C22×C8—1131no—
134C22×C8C2×D11311yescharacteristic
135C22⋊Q8—1132no—
136C22⋊Q8—1132no—
137C22⋊Q8—1132no—
138C22⋊Q8—2262no—
139C22⋊Q8—2262no—
140C22⋊Q8C2×D11312yes—
141C2×C4⋊C4—1132no—
142C2×C4⋊C4—2262no—
143C2×C22⋊C4—2262no—
144#107—1131no—
145#107—1132no—
146#107C2×D11311yescharacteristic, complemented
147#103—1132no—
148#103—1132no—
149#103—1132no—
150#103C2×D11312yescomplemented
151C22⋊C8—1132no—
152C22⋊C8—1132no—
153C24—1131no—
154C2×Q8—1132no—
155C2×Q8—4522no—
156C2×Q8—4522no—
157C2×Q8—2262no—
158C2×Q8C22×D11312yes—
159C22×C4—1131no—
160C22×C4—1132no—
161C22×C4—1132no—
162C22×C4—4521no—
163C22×C4—4521no—
164C22×C4—2262no—
165C22×C4—2262no—
166C22×C4C22×D11311yescharacteristic
167Q16—41no—
168Q16—4521no—
169Q16—4522no—
170C2×C8—41no—
171C2×C8—1132no—
172C2×C8—4521no—
173C2×C8C22×D11311yescharacteristic
174C2×C8C22×D11311yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—1132no—
177C4⋊C4—2262no—
178C4⋊C4—2262no—
179C4⋊C4—2262no—
180C4⋊C4—2262no—
181C4⋊C4C22×D11312yes—
182C22⋊C4—22no—
183C22⋊C4—2262no—
184C22⋊C4—2262no—
185C22⋊C4—2262no—
186C23—1134no—
187C23—1132no—
188C23—4521no—
189C23—4521no—
190C23D4×D11311yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—4524no—
194Q8—2262no—
195Q8—2262no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—1132no—
201C2×C4—4522no—
202C2×C4—4524no—
203C2×C4—4524no—
204C2×C4—4521no—
205C2×C4—2262no—
206C2×C4—2262no—
207C2×C4—2262no—
208C2×C4—2262no—
209C2×C4D4×D11311yescharacteristic
210C2×C4C23×D11311yescharacteristic
211C8—21no—
212C8—21no—
213C8—2262no—
214C22—21no—
215C22—21no—
216C22—1134no—
217C22—4524no—
218C22—2262no—
219C22—2262no—
220C22Q16×D11312yes—
221C22C2×D4×D11311yescharacteristic
222C4—21no—
223C4—42no—
224C4—42no—
225C4—4522no—
226C4—4522no—
227C4—2262no—
228C4#1681366711yescharacteristic
229C4#1681366811yescharacteristic
230C2—22no—
231C2—2264no—
232C2#2029915611yescharacteristic
233C2C2×Q16×D11311yescharacteristic
234C2#2029939511yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#467×C113C2
socleC22×C113C22×D4
prime radicalC23×D113D4
hypercenter#467D113
hypocenterC113#3189
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #20298935C2semidirect
3. #20298932C2semidirect
4. #20298839C2semidirect
5. #20298837C2semidirect
6. #20298755C2semidirect
7. #20298752C2semidirect
9. #467×C113C2semidirect
12. C2×Dic452C22semidirect
13. C2×#14268637C22semidirect
23. #16813556C22semidirect
24. #16813555C22semidirect
25. #16813547C22semidirect
26. #16813546C22semidirect
27. #16813545C22semidirect
30. C2×Q16×C113C22semidirect
33. #107×C113C22semidirect
34. #103×C113C22semidirect
71. C22×D113Q16semidirect
118. C113#3189semidirect
124. #467D113semidirect
132. C2×Q16C2×D113semidirect
146. #107C2×D113semidirect
150. #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
#202989352
#202989322
#202988392
#202988372
#202987552
#202987522
C2×#168135342
#467×C1132
#3189113