Finite Group Databasebeta

← C22×A4×#157 #13181828

Subgroups — C22×A4×#157 #13181828

Order 1728.

1030 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 96 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×A4×C3⋊D3C211yesmaximal, characteristic
2C2×A4×#157C216yesmaximal, direct product factor
3C24×#157C311yesmaximal, characteristic, complemented
4C22×C32×A4C411yescharacteristic, complemented
5C22×C3×#157—41nomaximal
6C2×A4×C3⋊D3C413yescomplemented
7C2×A4×C3⋊D3C2213yes—
8A4×#157C2214yesdirect product factor
9C24×C3⋊D3C2×C311yescharacteristic
10C23×#157—31no—
11C23×#157—31no—
12C23×#157—36no—
13C23×#157C2×C316yescomplemented
14C22×D3×A4—62no—
15C22×C3×C3⋊D3—41no—
16C2×C32×A4C2×C413yescomplemented
17C2×C3×#157—46no—
18A4×C3⋊D3C2×C413yescomplemented
19A4×C3⋊D3C2311yescharacteristic
20C22×C4×A4—91nomaximal
21C24×C32C3×C411yescharacteristic, complemented
22C23×C3⋊D3—31no—
23C23×C3⋊D3—33no—
24C23×C3⋊D3—31no—
25C23×C3⋊D3—33no—
26C23×C3⋊D3C3×C413yescomplemented
27C23×C3⋊D3C22×C313yes—
28C22×C3×A4—22no—
29C22×C3×A4—42no—
30C22×#157—36no—
31C22×#157—38no—
32C22×#157—31no—
33C22×#157—36no—
34C22×#157—36no—
35C22×#157—36no—
36C22×#157—312no—
37C22×#157A411yescharacteristic, direct product factor
38C22×#157C22×C314yescomplemented
39C2×D3×A4—66no—
40C2×D3×A4—66no—
41C22×C33—41no—
42C2×C3×C3⋊D3—43no—
43C2×C3×C3⋊D3—43no—
44C32×A4C22×C411yescharacteristic, complemented
45C3×#157—44no—
46C24×D3—62no—
47C23×A4—91no—
48C2×C4×A4—96no—
49C23×C32—33no—
50C23×C32—31no—
51C23×C32C2×C3×C413yescomplemented
52C22×C3⋊D3—32no—
53C22×C3⋊D3—36no—
54C22×C3⋊D3—36no—
55C22×C3⋊D3—31no—
56C22×C3⋊D3—33no—
57C22×C3⋊D3—33no—
58C22×C3⋊D3—33no—
59C22×C3⋊D3—33no—
60C22×C3⋊D3—33no—
61C22×C3⋊D3—33no—
62C22×C3⋊D3—36no—
63C22×C3⋊D3—33no—
64C22×C3⋊D3—33no—
65C22×C3⋊D3C2×C3×C413yescomplemented
66C22×C3⋊D3C2×A411yescharacteristic
67C22×C3⋊D3C23×C311yescharacteristic
68C22×C3×D3—242no—
69C2×C3×A4—26no—
70C2×C3×A4—46no—
71C2×#157—312no—
72C2×#157—36no—
73C2×#157—312no—
74C2×#157—34no—
75C2×#157—34no—
76C2×#157C2×A416yesdirect product factor
77D3×A4—66no—
78D3×A4—62no—
79C24×C4—91noSylow
80C2×C33—43no—
81C3×C3⋊D3—43no—
82C3×C3⋊D3—41no—
83C24×C3—22no—
84C23×D3—66no—
85C23×D3—66no—
86C23×D3—182no—
87C23×D3—186no—
88C23×D3—186no—
89C23×D3—182no—
90C22×A4—42no—
91C22×A4—93no—
92C22×A4—93no—
93C22×A4#15711yescharacteristic, direct product factor
94C22×C3×C4—361no—
95C4×A4—94no—
96C22×C32—33no—
97C22×C32—32no—
98C22×C32—33no—
99C22×C32—33no—
100C22×C32—82no—
101C22×C32—162no—
102C22×C32C4×A411yescharacteristic, complemented
103C22×C32C22×C3×C411yescharacteristic, complemented
104C2×C3⋊D3—36no—
105C2×C3⋊D3—36no—
106C2×C3⋊D3—33no—
107C2×C3⋊D3—33no—
108C2×C3⋊D3—33no—
109C2×C3⋊D3—36no—
110C2×C3⋊D3—33no—
111C2×C3⋊D3—33no—
112C2×C3⋊D3—31no—
113C2×C3⋊D3—33no—
114C2×C3⋊D3—31no—
115C2×C3⋊D3C4×A413yescomplemented
116C2×C3⋊D3C22×A413yes—
117C2×C3×D3—246no—
118C2×C3×D3—246no—
119C3×A4—22no—
120C3×A4—42no—
121#157—34no—
122#157C22×A414yesdirect product factor
123C25—91no—
124C23×C4—96no—
125C23×C4—271no—
126C23×C4—276no—
127C23×C4—271no—
128C33—41noSylow
129C23×C3—26no—
130C23×C3—62no—
131C23×C3—66no—
132C23×C3—361no—
133C22×D3—62no—
134C22×D3—66no—
135C22×D3—62no—
136C22×D3—186no—
137C22×D3—186no—
138C22×D3—186no—
139C22×D3—186no—
140C22×D3—182no—
141C22×D3—184no—
142C22×D3—1812no—
143C22×D3—1812no—
144C22×D3—186no—
145C22×D3—1812no—
146C22×D3—186no—
147C22×D3—186no—
148C22×D3—186no—
149C2×A4—46no—
150C2×A4—91no—
151C2×A4—93no—
152C2×A4C2×#15713yesdirect product factor
153C2×C3×C4—366no—
154C2×C32—33no—
155C2×C32—31no—
156C2×C32—86no—
157C2×C32—166no—
158C2×C32C2×C4×A413yescomplemented
159C3⋊D3—31no—
160C3⋊D3—33no—
161C3⋊D3C2×C4×A413yescomplemented
162C3⋊D3C23×A411yescharacteristic
163C3×D3—246no—
164C3×D3—242no—
165C24—93no—
166C24—93no—
167C24—271no—
168C24—273no—
169C24—273no—
170C24—271no—
171C24C3×#15711yescharacteristic, complemented
172C22×C4—91no—
173C22×C4—94no—
174C22×C4—276no—
175C22×C4—276no—
176C22×C4—271no—
177C22×C4—278no—
178C22×C4—276no—
179C22×C4—2712no—
180C22×C4—276no—
181C22×C3—22no—
182C22×C3—22no—
183C22×C3—41no—
184C22×C3—66no—
185C22×C3—66no—
186C22×C3—66no—
187C22×C3—64no—
188C22×C3—162no—
189C22×C3—363no—
190C22×C3—363no—
191C2×D3—66no—
192C2×D3—66no—
193C2×D3—182no—
194C2×D3—186no—
195C2×D3—186no—
196C2×D3—186no—
197C2×D3—1812no—
198C2×D3—186no—
199C2×D3—186no—
200C2×D3—1812no—
201C2×D3—1812no—
202C2×D3—186no—
203C2×D3—182no—
204A4—42no—
205A4C22×#15711yescharacteristic, direct product factor
206C3×C4—364no—
207C32—82no—
208C32—162no—
209C32C22×C4×A411yescharacteristic, complemented
210C23—33no—
211C23—31no—
212C23—91no—
213C23—91no—
214C23—93no—
215C23—273no—
216C23—273no—
217C23—273no—
218C23—273no—
219C23—273no—
220C23—273no—
221C23—276no—
222C23—273no—
223C23—273no—
224C23—272no—
225C23—276no—
226C23—276no—
227C23—271no—
228C23C2×C3×#15713yescomplemented
229C2×C4—96no—
230C2×C4—274no—
231C2×C4—2712no—
232C2×C4—276no—
233C2×C4—2712no—
234C2×C4—274no—
235C2×C3—26no—
236C2×C3—43no—
237C2×C3—62no—
238C2×C3—66no—
239C2×C3—166no—
240C2×C3—361no—
241C2×C3—363no—
242D3—66no—
243D3—62no—
244D3—182no—
245D3—186no—
246C22—33no—
247C22—32no—
248C22—33no—
249C22—33no—
250C22—93no—
251C22—93no—
252C22—271no—
253C22—273no—
254C22—273no—
255C22—273no—
256C22—276no—
257C22—271no—
258C22—273no—
259C22—276no—
260C22—276no—
261C22—273no—
262C22—273no—
263C22A4×#15711yescharacteristic, direct product factor
264C22C22×C3×#15711yescharacteristic, complemented
265C4—94no—
266C4—274no—
267C3—22no—
268C3—41no—
269C3—162no—
270C2—31no—
271C2—33no—
272C2—91no—
273C2—93no—
274C2—271no—
275C2—273no—
276C2C2×A4×#15713yesdirect product factor
subgroupG/H
center, hypercenterC22A4×#157
commutator, hypocenterC22×C32C22×C3×C4
FrattiniC1G
Fitting, socleC24×C32C3×C4
prime radicalC22×A4×C3⋊D3C2
cosocleC22×C3⋊D3C23×C3
LCSG ▷ C22×C32
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×A4×#157C2direct (N × H)
3. C24×#157C3semidirect
4. C22×C32×A4C4semidirect
6. C2×A4×C3⋊D3C4semidirect
8. A4×#157C22direct (N × H)
13. C23×#157C2×C3semidirect
16. C2×C32×A4C2×C4semidirect
18. A4×C3⋊D3C2×C4semidirect
21. C24×C32C3×C4semidirect
26. C23×C3⋊D3C3×C4semidirect
37. C22×#157A4direct (N × H)
38. C22×#157C22×C3semidirect
44. C32×A4C22×C4semidirect
51. C23×C32C2×C3×C4semidirect
65. C22×C3⋊D3C2×C3×C4semidirect
76. C2×#157C2×A4direct (N × H)
93. C22×A4#157direct (N × H)
102. C22×C32C4×A4semidirect
103. C22×C32C22×C3×C4semidirect
115. C2×C3⋊D3C4×A4semidirect
122. #157C22×A4direct (N × H)
152. C2×A4C2×#157direct (N × H)
158. C2×C32C2×C4×A4semidirect
161. C3⋊D3C2×C4×A4semidirect
171. C24C3×#157semidirect
205. A4C22×#157direct (N × H)
209. C32C22×C4×A4semidirect
228. C23C2×C3×#157semidirect
263. C22A4×#157direct (N × H)
264. C22C22×C3×#157semidirect
276. C2C2×A4×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C22×A4×C3⋊D32
C2×A4×#1572
C24×#1573
C22×C3×#1574
C22×C4×A49