Finite Group Databasebeta

← A4×#1159 #13062166

Subgroups — A4×#1159 #13062166

Order 1344.

430 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C7×D4×A4C211yesmaximal, characteristic, complemented
2A4×C22⋊D7C212yesmaximal, complemented
3C2×A4×Dic7C212yesmaximal, complemented
4C4×A4×D7C211yesmaximal, characteristic, complemented
5A4×Dic14C211yesmaximal, characteristic, complemented
6C22×#1159C311yesmaximal, characteristic, complemented, Hall
7C22×C7×A4C2212yes—
8C2×A4×D7C2211yescharacteristic
9C3×#1159—41nomaximal
10C4×C7×A4C2211yescharacteristic
11A4×Dic7C2212yes—
12A4×Dic7C2211yescharacteristic
13C22×C7×D4C2×C311yescharacteristic, complemented
14C22×C22⋊D7C2×C312yescomplemented
15C23×Dic7C2×C312yescomplemented
16C2×#1159—31no—
17C2×#1159—31no—
18C2×#1159—31no—
19C2×#1159—31no—
20C2×#1159—32no—
21C2×#1159—32no—
22C22×C4×D7C2×C311yescharacteristic, complemented
23C22×Dic14C2×C311yescharacteristic, complemented
24A4×#41—71nomaximal, Hall
25C2×C7×A4—22no—
26C2×C7×A4C2311yescharacteristic
27A4×D7—21no—
28C3×C7×D4—41no—
29C3×C22⋊D7—42no—
30C2×C3×Dic7—42no—
31C3×C4×D7—41no—
32C3×Dic14—41no—
33C24×C7C22×C312yes—
34C23×D7C22×C311yescharacteristic
35C2×C7×D4—31no—
36C2×C7×D4—31no—
37C2×C7×D4—32no—
38C22×C4×C7C22×C311yescharacteristic
39C2×C22⋊D7—32no—
40C2×C22⋊D7—32no—
41C2×C22⋊D7—32no—
42C2×C22⋊D7—32no—
43C22×Dic7—32no—
44C22×Dic7—32no—
45C22×Dic7—32no—
46C22×Dic7—32no—
47C22×Dic7C22×C312yes—
48C22×Dic7C22×C311yescharacteristic
49#1159—34no—
50#1159—32no—
51#1159—32no—
52#1159—32no—
53#1159—34no—
54#1159—31no—
55#1159—32no—
56#1159—32no—
57#1159—31no—
58#1159—31no—
59#1159A411yescharacteristic, direct product factor
60C2×C4×D7—31no—
61C2×C4×D7—31no—
62C2×C4×D7—31no—
63C2×C4×D7—31no—
64C2×Dic14—31no—
65C2×Dic14—31no—
66C2×Dic14—32no—
67Q8×A4—71no—
68D4×A4—72no—
69D4×A4D711yescharacteristic, complemented
70C2×C4×A4—72no—
71C2×C4×A4—71no—
72C22×C3×C7—42no—
73C2×C3×D7—41no—
74C7×A4#4111yescharacteristic, complemented
75C3×C4×C7—41no—
76C3×Dic7—42no—
77C3×Dic7—41no—
78C22×#41—71noSylow
79C23×C7—22no—
80C23×C7—32no—
81C23×C7—32no—
82C23×C7—62no—
83C23×C7C23×C311yescharacteristic
84C22×D7—21no—
85C22×D7—31no—
86C22×D7—31no—
87C22×D7—61no—
88C7×D4—32no—
89C7×D4—32no—
90C7×D4—31no—
91C7×D4C2×A411yescharacteristic, complemented
92C2×C4×C7—31no—
93C2×C4×C7—31no—
94C22⋊D7—34no—
95C22⋊D7—32no—
96C22⋊D7—32no—
97C22⋊D7—32no—
98C22⋊D7C2×A412yescomplemented
99C2×Dic7—34no—
100C2×Dic7—32no—
101C2×Dic7—32no—
102C2×Dic7—31no—
103C2×Dic7—32no—
104C2×Dic7—32no—
105C2×Dic7—32no—
106C2×Dic7—31no—
107C2×Dic7C2×A412yescomplemented
108C4×D7—32no—
109C4×D7—31no—
110C4×D7—31no—
111C4×D7—31no—
112C4×D7C2×A411yescharacteristic, complemented
113Dic14—32no—
114Dic14—31no—
115Dic14—32no—
116Dic14C2×A411yescharacteristic, complemented
117C22×A4—71no—
118C22×A4C2×D712yes—
119C3×#41—281no—
120C4×A4—71no—
121C4×A4—72no—
122C4×A4C2×D711yescharacteristic
123C2×C3×C7—41no—
124C2×C3×C7—82no—
125C3×D7—81no—
126C2×#41—211no—
127C2×#41—211no—
128C2×#41—212no—
129C2×#41—211no—
130C2×#41—212no—
131C2×#41—211no—
132C22×Q8—71no—
133C22×D4—72no—
134C22×D4C3×D711yescharacteristic, complemented
135C23×C4—72no—
136C23×C4—71no—
137C22×C7—31no—
138C22×C7—31no—
139C22×C7—32no—
140C22×C7—62no—
141C22×C7—62no—
142C22×C7—62no—
143C22×C7—62no—
144C22×C7C3×#4111yescharacteristic, complemented
145C22×C7C22×A412yes—
146C2×D7—31no—
147C2×D7—61no—
148C2×D7—61no—
149C2×D7—61no—
150C2×D7—61no—
151C2×D7C22×A411yescharacteristic
152C4×C7—31no—
153C4×C7C22×A411yescharacteristic
154Dic7—32no—
155Dic7—31no—
156Dic7C22×A412yes—
157Dic7C22×A411yescharacteristic
158C2×A4—22no—
159C2×A4—141no—
160C2×A4C22×D711yescharacteristic
161C3×Q8—281no—
162C3×D4—41no—
163C3×D4—282no—
164C2×C3×C4—281no—
165C2×C3×C4—282no—
166C3×C7—41noHall
167C24—71no—
168C24C2×C3×D712yes—
169#41—71no—
170#41—212no—
171#41—211no—
172#41—212no—
173#41—211no—
174#41—212no—
175#41—212no—
176#41—214no—
177#41—214no—
178#41—212no—
179#41—211no—
180C2×Q8—211no—
181C2×Q8—211no—
182C2×Q8—212no—
183C2×D4—31no—
184C2×D4—31no—
185C2×D4—32no—
186C2×D4—212no—
187C2×D4—212no—
188C2×D4—212no—
189C2×D4—212no—
190C22×C4—71no—
191C22×C4—72no—
192C22×C4—211no—
193C22×C4—211no—
194C22×C4—211no—
195C22×C4—211no—
196C22×C4—212no—
197C22×C4—212no—
198C22×C4—212no—
199C22×C4—212no—
200C22×C4C2×C3×D711yescharacteristic
201C2×C7—22no—
202C2×C7—31no—
203C2×C7—31no—
204C2×C7—62no—
205C2×C7C23×A411yescharacteristic
206D7—21no—
207D7—61no—
208C22×C3—42no—
209C22×C3—281no—
210A4#115911yescharacteristic, direct product factor
211C3×C4—41no—
212C3×C4—281no—
213C3×C4—282no—
214C23—22no—
215C23—32no—
216C23—32no—
217C23—62no—
218C23—141no—
219C23—211no—
220C23—211no—
221C23—421no—
222C23C22×C3×D711yescharacteristic
223Q8—71no—
224Q8—212no—
225Q8—212no—
226Q8—211no—
227D4—32no—
228D4—32no—
229D4—31no—
230D4—72no—
231D4—212no—
232D4—212no—
233D4—214no—
234D4—212no—
235D4A4×D711yescharacteristic, complemented
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—72no—
239C2×C4—71no—
240C2×C4—211no—
241C2×C4—211no—
242C2×C4—212no—
243C2×C4—211no—
244C2×C4—212no—
245C2×C4—212no—
246C2×C4—212no—
247C2×C4—214no—
248C2×C4—212no—
249C2×C4—211no—
250C2×C4—211no—
251C2×C4—212no—
252C7A4×#4111yescharacteristic, complemented, Sylow
253C2×C3—41no—
254C2×C3—82no—
255C2×C3—561no—
256C22—31no—
257C22—31no—
258C22—32no—
259C22—62no—
260C22—62no—
261C22—62no—
262C22—62no—
263C22—71no—
264C22—211no—
265C22—421no—
266C22—421no—
267C22—421no—
268C22—421no—
269C22C3×#115911yescharacteristic, complemented
270C22C2×A4×D712yes—
271C4—31no—
272C4—71no—
273C4—72no—
274C4—211no—
275C4—212no—
276C4C2×A4×D711yescharacteristic
277C3—41noSylow
278C2—22no—
279C2—31no—
280C2—31no—
281C2—62no—
282C2—141no—
283C2—421no—
284C2C22×A4×D711yescharacteristic
subgroupG/H
center, FrattiniC2C22×A4×D7
commutator, socle, cosocleC23×C7C23×C3
FittingC22×C7×D4C2×C3
hypercenterD4A4×D7
hypocenterC22×C7C3×#41
LCSG ▷ C23×C7 ▷ C22×C7
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C7×D4×A4C2semidirect
2. A4×C22⋊D7C2semidirect
3. C2×A4×Dic7C2semidirect
4. C4×A4×D7C2semidirect
5. A4×Dic14C2semidirect
6. C22×#1159C3semidirect
13. C22×C7×D4C2×C3semidirect
14. C22×C22⋊D7C2×C3semidirect
15. C23×Dic7C2×C3semidirect
22. C22×C4×D7C2×C3semidirect
23. C22×Dic14C2×C3semidirect
59. #1159A4direct (N × H)
69. D4×A4D7semidirect
74. C7×A4#41semidirect
91. C7×D4C2×A4semidirect
98. C22⋊D7C2×A4semidirect
107. C2×Dic7C2×A4semidirect
112. C4×D7C2×A4semidirect
116. Dic14C2×A4semidirect
134. C22×D4C3×D7semidirect
144. C22×C7C3×#41semidirect
210. A4#1159direct (N × H)
235. D4A4×D7semidirect
252. C7A4×#41semidirect
269. C22C3×#1159semidirect

Maximal subgroups

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

subgroupindex
C7×D4×A42
A4×C22⋊D72
C2×A4×Dic72
C4×A4×D72
A4×Dic142
C22×#11593
C3×#11594
A4×#417