Finite Group Databasebeta

← C22×#1572 #72252947

Subgroups — C22×#1572 #72252947

Order 5184.

2636 conjugacy classes of subgroups; only the 308 normal subgroups are shown.

normal subgroups only

This group has 2636 conjugacy classes of subgroups — too many to draw. Only its 308 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×C3⋊D3×#157C212yesmaximal
2C22×#10620198C211yesmaximal, characteristic
3C2×#1572C2112yesmaximal, direct product factor
4C22×C3⋊D32C2211yescharacteristic
5C22×#66472C412yescomplemented
6C22×C3⋊(C3⋊#157)C412yescomplemented
7C22×C32×#157C412yescomplemented
8C2×#10620198C416yescomplemented
9C2×#10620198C2216yes—
10C2×C3⋊D3×#157C4112yescomplemented
11C2×C3⋊D3×#157C22112yes—
12#1572C22116yesdirect product factor
13C22×D3×#157—64no—
14C22×C3⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
15C22×C32×C3⋊D3C2×C412yes—
16C2×C3⋊D32C2×C413yes—
17C2×C3⋊D32C2×C416yes—
18C2×C3⋊D32C2313yes—
19C2×#66472C2×C4112yescomplemented
20C2×C3⋊(C3⋊#157)C2×C4112yescomplemented
21C2×C32×#157C2×C4112yescomplemented
22#10620198C2×C4112yescomplemented
23#10620198C2314yes—
24C3⋊D3×#157C2×C4124yescomplemented
25C3⋊D3×#157C2318yes—
26C22×C4×#157—92nomaximal
27C22×D3×C3⋊D3—64no—
28C22×C3⋊#157—64no—
29C22×C3×#157—24no—
30C2×D3×#157—624no—
31C2×D3×#157—624no—
32C22×C34C4211yescharacteristic, complemented
33C2×C3⋊(C3⋊(C3⋊D3))C4213yescomplemented
34C2×C3⋊(C3⋊(C3⋊D3))C22×C413yes—
35C2×C32×C3⋊D3C4216yescomplemented
36C2×C32×C3⋊D3C22×C416yes—
37C3⋊D32C4216yescomplemented
38C3⋊D32C22×C413yes—
39C3⋊D32C22×C416yes—
40C3⋊D32C2411yescharacteristic
41#66472C22×C418yescomplemented
42C3⋊(C3⋊#157)C22×C418yescomplemented
43C32×#157C22×C418yescomplemented
44C23×#157—92no—
45C23×#157—92no—
46C22×C4×C3⋊D3—92no—
47C2×C4×#157—924no—
48C22×C3⋊(C3⋊D3)—64no—
49C22×C3⋊(C3⋊D3)—244no—
50C22×C3×C3⋊D3—24no—
51C22×C32×D3—64no—
52C2×D3×C3⋊D3—612no—
53C2×D3×C3⋊D3—612no—
54C2×D3×C3⋊D3—612no—
55C2×D3×C3⋊D3—612no—
56C2×C3⋊#157—624no—
57C2×C3×#157—224no—
58D3×#157—648no—
59D3×#157—616no—
60C2×C34C2×C4213yescomplemented
61C3⋊(C3⋊(C3⋊D3))C2×C4213yescomplemented
62C3⋊(C3⋊(C3⋊D3))C23×C411yescharacteristic
63C32×C3⋊D3C2×C4216yescomplemented
64C32×C3⋊D3C23×C412yes—
65C23×C3⋊D3—92no—
66C22×D32—364no—
67C22×#157—94no—
68C22×#157—912no—
69C22×#157—912no—
70C22×#157—92no—
71C22×#157—912no—
72C22×#157—912no—
73C22×#157—364no—
74C22×#157#15712yesdirect product factor
75C22×C32×C4—92no—
76C22×C3⋊Dic3—92no—
77C2×C4×C3⋊D3—912no—
78C2×C4×C3⋊D3—912no—
79C4×#157—932no—
80C22×C33—24no—
81C22×C33—84no—
82C2×C3⋊(C3⋊D3)—612no—
83C2×C3⋊(C3⋊D3)—612no—
84C2×C3⋊(C3⋊D3)—2412no—
85C2×C3⋊(C3⋊D3)—2412no—
86C2×C3×C3⋊D3—212no—
87C2×C3×C3⋊D3—212no—
88C2×C32×D3—612no—
89C2×C32×D3—612no—
90D3×C3⋊D3—624no—
91D3×C3⋊D3—612no—
92D3×C3⋊D3—612no—
93D3×C3⋊D3—612no—
94D3×C3⋊D3—64no—
95C3⋊#157—616no—
96C3×#157—216no—
97C22×C4×D3—544no—
98C34C22×C4211yescharacteristic, complemented, Sylow
99C23×C32—92no—
100C22×C3⋊D3—92no—
101C22×C3⋊D3—96no—
102C22×C3⋊D3—96no—
103C22×C3⋊D3—96no—
104C22×C3⋊D3—96no—
105C22×C3⋊D3—364no—
106C22×C3⋊D3—364no—
107C22×C3⋊D3—728no—
108C22×C3⋊D3—724no—
109C22×C3⋊D3C2×#15712yes—
110C22×C3×D3—128no—
111C2×D32—3612no—
112C2×D32—3624no—
113C2×D32—3612no—
114C2×#157—924no—
115C2×#157—98no—
116C2×#157—924no—
117C2×#157—912no—
118C2×#157—924no—
119C2×#157—98no—
120C2×#157—3624no—
121C2×#157C2×#157112yesdirect product factor
122C2×C32×C4—912no—
123C2×C3⋊Dic3—912no—
124C4×C3⋊D3—924no—
125C4×C3⋊D3—98no—
126C22×C42—811noSylow
127C2×C33—212no—
128C2×C33—812no—
129C3⋊(C3⋊D3)—612no—
130C3⋊(C3⋊D3)—64no—
131C3⋊(C3⋊D3)—2412no—
132C3⋊(C3⋊D3)—244no—
133C3×C3⋊D3—212no—
134C3×C3⋊D3—24no—
135C32×D3—612no—
136C32×D3—64no—
137C23×D3—544no—
138C22×C3×C4—184no—
139C22×Dic3—544no—
140C2×C4×D3—5424no—
141C2×C4×D3—5424no—
142C22×C32—44no—
143C22×C32—44no—
144C22×C32—88no—
145C22×C32—84no—
146C22×C32—96no—
147C22×C32—96no—
148C22×C32C4×#15712yescomplemented
149C2×C3⋊D3—96no—
150C2×C3⋊D3—96no—
151C2×C3⋊D3—96no—
152C2×C3⋊D3—96no—
153C2×C3⋊D3—912no—
154C2×C3⋊D3—92no—
155C2×C3⋊D3—96no—
156C2×C3⋊D3—3612no—
157C2×C3⋊D3—3612no—
158C2×C3⋊D3—3612no—
159C2×C3⋊D3—3612no—
160C2×C3⋊D3—7224no—
161C2×C3⋊D3—7224no—
162C2×C3⋊D3—7212no—
163C2×C3⋊D3—7212no—
164C2×C3⋊D3C4×#15716yescomplemented
165C2×C3⋊D3C22×#15716yes—
166C2×C3×D3—1224no—
167C2×C3×D3—1224no—
168D32—364no—
169D32—3624no—
170D32—3624no—
171D32—3612no—
172#157—916no—
173#157—98no—
174#157—3616no—
175#157C22×#15718yesdirect product factor
176C32×C4—98no—
177C3⋊Dic3—98no—
178C23×C4—812no—
179C23×C4—811no—
180C2×C42—8112no—
181C33—24no—
182C33—84no—
183C23×C3—184no—
184C22×D3—64no—
185C22×D3—5412no—
186C22×D3—5412no—
187C22×D3—5412no—
188C22×D3—5412no—
189C22×D3—544no—
190C22×D3—2164no—
191C2×C3×C4—1824no—
192C2×Dic3—5424no—
193C4×D3—5416no—
194C4×D3—5448no—
195C2×C32—412no—
196C2×C32—412no—
197C2×C32—824no—
198C2×C32—812no—
199C2×C32—92no—
200C2×C32—96no—
201C2×C32C2×C4×#15716yescomplemented
202C3⋊D3—92no—
203C3⋊D3—96no—
204C3⋊D3—3612no—
205C3⋊D3—364no—
206C3⋊D3—364no—
207C3⋊D3—3612no—
208C3⋊D3—7224no—
209C3⋊D3—728no—
210C3⋊D3—7212no—
211C3⋊D3—724no—
212C3⋊D3C2×C4×#15716yescomplemented
213C3⋊D3C23×#15712yes—
214C3×D3—1224no—
215C3×D3—128no—
216C24—811no—
217C22×C4—92no—
218C22×C4—8112no—
219C22×C4—8112no—
220C22×C4—812no—
221C22×C4—812no—
222C22×C4—816no—
223C22×C4—816no—
224C42—8116no—
225C22×C3—24no—
226C22×C3—84no—
227C22×C3—1812no—
228C22×C3—1812no—
229C2×D3—612no—
230C2×D3—612no—
231C2×D3—544no—
232C2×D3—5412no—
233C2×D3—5412no—
234C2×D3—5412no—
235C2×D3—5424no—
236C2×D3—5412no—
237C2×D3—5412no—
238C2×D3—21612no—
239C2×D3—21612no—
240C3×C4—1816no—
241Dic3—5416no—
242C32—44no—
243C32—44no—
244C32—88no—
245C32—84no—
246C32C22×C4×#15712yescomplemented
247C23—92no—
248C23—813no—
249C23—816no—
250C23—813no—
251C23—811no—
252C2×C4—912no—
253C2×C4—818no—
254C2×C4—8124no—
255C2×C4—8112no—
256C2×C4—814no—
257C2×C4—8112no—
258C2×C4—8112no—
259C2×C3—212no—
260C2×C3—812no—
261C2×C3—184no—
262C2×C3—1812no—
263D3—612no—
264D3—64no—
265D3—544no—
266D3—5412no—
267D3—2164no—
268D3—21612no—
269C22—96no—
270C22—96no—
271C22—811no—
272C22—816no—
273C22—813no—
274C22—816no—
275C22—813no—
276C22—813no—
277C22#157211yescharacteristic, direct product factor
278C4—98no—
279C4—818no—
280C4—818no—
281C3—24no—
282C3—84no—
283C2—92no—
284C2—96no—
285C2—811no—
286C2—813no—
287C2C2×#157213yesdirect product factor
subgroupG/H
center, hypercenterC22#1572
commutator, hypocenterC34C22×C42
FrattiniC1G
Fitting, socleC22×C34C42
prime radicalC22×C3⋊D32C22
cosocleC3⋊D32C24
LCSG ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. C2×#1572C2direct (N × H)
5. C22×#66472C4semidirect
6. C22×C3⋊(C3⋊#157)C4semidirect
7. C22×C32×#157C4semidirect
8. C2×#10620198C4semidirect
10. C2×C3⋊D3×#157C4semidirect
12. #1572C22direct (N × H)
19. C2×#66472C2×C4semidirect
20. C2×C3⋊(C3⋊#157)C2×C4semidirect
21. C2×C32×#157C2×C4semidirect
22. #10620198C2×C4semidirect
24. C3⋊D3×#157C2×C4semidirect
32. C22×C34C42semidirect
33. C2×C3⋊(C3⋊(C3⋊D3))C42semidirect
35. C2×C32×C3⋊D3C42semidirect
37. C3⋊D32C42semidirect
41. #66472C22×C4semidirect
42. C3⋊(C3⋊#157)C22×C4semidirect
43. C32×#157C22×C4semidirect
60. C2×C34C2×C42semidirect
61. C3⋊(C3⋊(C3⋊D3))C2×C42semidirect
63. C32×C3⋊D3C2×C42semidirect
74. C22×#157#157direct (N × H)
98. C34C22×C42semidirect
121. C2×#157C2×#157direct (N × H)
148. C22×C32C4×#157semidirect
164. C2×C3⋊D3C4×#157semidirect
175. #157C22×#157direct (N × H)
201. C2×C32C2×C4×#157semidirect
212. C3⋊D3C2×C4×#157semidirect
246. C32C22×C4×#157semidirect
277. C22#1572direct (N × H)
287. C2C2×#1572direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊D3×#1572
C22×#106201982
C2×#15722
C22×C4×#1579