Finite Group Databasebeta

← A4×C22⋊#157 #13180677

Subgroups — A4×C22⋊#157 #13180677

Order 1728.

734 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 42 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×#157C212yesmaximal, complemented
3C22×C22⋊#157C311yesmaximal, characteristic, complemented
4C22×C32×A4C411yescharacteristic, complemented
5C2×A4×C3⋊D3—22no—
6C2×A4×C3⋊D3C411yescharacteristic, complemented
7C2×A4×C3⋊D3C2211yescharacteristic
8A4×#157—22no—
9C3×C22⋊#157—41nomaximal
10C24×C3⋊D3C2×C311yescharacteristic
11C23×#157C2×C312yescomplemented
12C22×D3×A4—62no—
13C2×C22⋊#157—32no—
14C2×C22⋊#157—31no—
15C2×C22⋊#157—31no—
16C22×C3×C3⋊D3—41no—
17C2×C32×A4—21no—
18C2×C32×A4C2×C411yescharacteristic
19C2×C3×#157—42no—
20A4×C3⋊D3—21no—
21A4×C3⋊D3D412yes—
22A4×C22⋊C4—91nomaximal
23C24×C32C3×C411yescharacteristic, complemented
24C23×C3⋊D3—22no—
25C23×C3⋊D3—31no—
26C23×C3⋊D3—31no—
27C23×C3⋊D3—31no—
28C23×C3⋊D3—31no—
29C23×C3⋊D3—62no—
30C23×C3⋊D3C3×C411yescharacteristic, complemented
31C23×C3⋊D3C22×C311yescharacteristic
32C22×C3×A4—22no—
33C22×C3×A4—42no—
34C22×#157—22no—
35C22×#157—32no—
36C22×#157—32no—
37C22×#157—62no—
38C2×D3×A4—62no—
39C2×D3×A4—62no—
40C2×D3×A4—68no—
41C22⋊#157—32no—
42C22⋊#157—31no—
43C22⋊#157—32no—
44C22⋊#157A411yescharacteristic, direct product factor
45C22×C33—41no—
46C2×C3×C3⋊D3—41no—
47C2×C3×C3⋊D3—41no—
48C2×C3×C3⋊D3—82no—
49C32×A4C22⋊C411yescharacteristic, complemented
50C3×#157—82no—
51C24×D3—62no—
52C23×A4—91no—
53C2×C4×A4—92no—
54C23×C32—21no—
55C23×C32—31no—
56C23×C32—31no—
57C23×C32—61no—
58C23×C32C2×C3×C411yescharacteristic
59C22×C3⋊D3—21no—
60C22×C3⋊D3—31no—
61C22×C3⋊D3—32no—
62C22×C3⋊D3—31no—
63C22×C3⋊D3—32no—
64C22×C3⋊D3—31no—
65C22×C3⋊D3—31no—
66C22×C3⋊D3—31no—
67C22×C3⋊D3—31no—
68C22×C3⋊D3—31no—
69C22×C3⋊D3—62no—
70C22×C3⋊D3—62no—
71C22×C3⋊D3—62no—
72C22×C3⋊D3—62no—
73C22×C3⋊D3—62no—
74C22×C3⋊D3—62no—
75C22×C3⋊D3—62no—
76C22×C3⋊D3—62no—
77C22×C3⋊D3—61no—
78C22×C3⋊D3C3×D412yes—
79C22×C3⋊D3C2×A411yescharacteristic
80C22×C3×D3—242no—
81C2×C3×A4—24no—
82C2×C3×A4—22no—
83C2×C3×A4—44no—
84C2×C3×A4—42no—
85C2×#157—32no—
86C2×#157—62no—
87C2×#157—62no—
88C2×#157—62no—
89C2×#157—62no—
90C2×#157C2×A412yescomplemented
91D3×A4—64no—
92D3×A4—64no—
93C22×C22⋊C4—91noSylow
94C2×C33—41no—
95C2×C33—81no—
96C3×C3⋊D3—42no—
97C3×C3⋊D3—81no—
98C24×C3—22no—
99C23×D3—62no—
100C23×D3—62no—
101C23×D3—68no—
102C23×D3—182no—
103C23×D3—182no—
104C23×D3—182no—
105C23×D3—182no—
106C23×D3—188no—
107C22×A4—42no—
108C22×A4—91no—
109C22×A4—91no—
110C22×A4—182no—
111C22×A4#15711yescharacteristic, complemented
112C4×A4—182no—
113C3×C22⋊C4—361no—
114C22×C32—31no—
115C22×C32—31no—
116C22×C32—31no—
117C22×C32—61no—
118C22×C32—61no—
119C22×C32—61no—
120C22×C32—61no—
121C22×C32—82no—
122C22×C32—162no—
123C22×C32C3×C22⋊C411yescharacteristic, complemented
124C22×C32C4×A411yescharacteristic, complemented
125C2×C3⋊D3—22no—
126C2×C3⋊D3—32no—
127C2×C3⋊D3—31no—
128C2×C3⋊D3—32no—
129C2×C3⋊D3—32no—
130C2×C3⋊D3—32no—
131C2×C3⋊D3—31no—
132C2×C3⋊D3—62no—
133C2×C3⋊D3—61no—
134C2×C3⋊D3—61no—
135C2×C3⋊D3—62no—
136C2×C3⋊D3—64no—
137C2×C3⋊D3—62no—
138C2×C3⋊D3—61no—
139C2×C3⋊D3—61no—
140C2×C3⋊D3C4×A411yescharacteristic, complemented
141C2×C3⋊D3C22×A411yescharacteristic
142C2×C3×D3—242no—
143C2×C3×D3—242no—
144C2×C3×D3—248no—
145C3×A4—22no—
146C3×A4—42no—
147#157—22no—
148#157—62no—
149C25—91no—
150C23×C4—92no—
151C2×C22⋊C4—271no—
152C2×C22⋊C4—272no—
153C2×C22⋊C4—271no—
154C33—41noSylow
155C23×C3—24no—
156C23×C3—22no—
157C23×C3—62no—
158C23×C3—62no—
159C23×C3—64no—
160C23×C3—361no—
161C22×D3—62no—
162C22×D3—64no—
163C22×D3—64no—
164C22×D3—182no—
165C22×D3—182no—
166C22×D3—182no—
167C22×D3—182no—
168C22×D3—182no—
169C22×D3—184no—
170C22×D3—184no—
171C22×D3—182no—
172C22×D3—182no—
173C22×D3—188no—
174C22×D3—188no—
175C22×D3—188no—
176C22×D3—188no—
177C22×D3—188no—
178C22×D3—184no—
179C22×D3—188no—
180C22×D3—188no—
181C22×D3—188no—
182C2×A4—21no—
183C2×A4—44no—
184C2×A4—42no—
185C2×A4—92no—
186C2×A4—181no—
187C2×A4C2×#15711yescharacteristic
188C2×C3×C4—362no—
189C2×C32—21no—
190C2×C32—31no—
191C2×C32—31no—
192C2×C32—61no—
193C2×C32—82no—
194C2×C32—84no—
195C2×C32—164no—
196C2×C32—162no—
197C2×C32C2×C4×A411yescharacteristic
198C3⋊D3—21no—
199C3⋊D3—32no—
200C3⋊D3—61no—
201C3⋊D3D4×A412yes—
202C3×D3—244no—
203C3×D3—244no—
204C24—91no—
205C24—91no—
206C24—182no—
207C24—271no—
208C24—271no—
209C24—271no—
210C24—271no—
211C24—542no—
212C24C3×#15711yescharacteristic, complemented
213C22×C4—182no—
214C22×C4—272no—
215C22×C4—272no—
216C22×C4—542no—
217C22⋊C4—91no—
218C22⋊C4—272no—
219C22⋊C4—271no—
220C22⋊C4—272no—
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—41no—
224C22×C3—62no—
225C22×C3—62no—
226C22×C3—62no—
227C22×C3—64no—
228C22×C3—64no—
229C22×C3—64no—
230C22×C3—64no—
231C22×C3—162no—
232C22×C3—361no—
233C22×C3—361no—
234C22×C3—722no—
235C2×D3—62no—
236C2×D3—62no—
237C2×D3—68no—
238C2×D3—184no—
239C2×D3—184no—
240C2×D3—182no—
241C2×D3—182no—
242C2×D3—184no—
243C2×D3—184no—
244C2×D3—184no—
245C2×D3—184no—
246C2×D3—188no—
247C2×D3—188no—
248C2×D3—188no—
249C2×D3—184no—
250C2×D3—1816no—
251C2×D3—184no—
252A4—42no—
253A4C22⋊#15711yescharacteristic, direct product factor
254C3×C4—722no—
255C32—82no—
256C32—162no—
257C32A4×C22⋊C411yescharacteristic, complemented
258C23—21no—
259C23—31no—
260C23—31no—
261C23—61no—
262C23—91no—
263C23—92no—
264C23—181no—
265C23—271no—
266C23—271no—
267C23—271no—
268C23—271no—
269C23—272no—
270C23—271no—
271C23—272no—
272C23—271no—
273C23—271no—
274C23—541no—
275C23—542no—
276C23—542no—
277C23—542no—
278C23—542no—
279C23—542no—
280C23—542no—
281C23—542no—
282C23—542no—
283C23C2×C3×#15711yescharacteristic
284C2×C4—92no—
285C2×C4—272no—
286C2×C4—542no—
287C2×C4—542no—
288C2×C4—542no—
289C2×C4—542no—
290C2×C3—24no—
291C2×C3—22no—
292C2×C3—41no—
293C2×C3—62no—
294C2×C3—62no—
295C2×C3—64no—
296C2×C3—81no—
297C2×C3—164no—
298C2×C3—162no—
299C2×C3—362no—
300C2×C3—721no—
301D3—64no—
302D3—64no—
303D3—184no—
304D3—184no—
305C22—31no—
306C22—31no—
307C22—31no—
308C22—61no—
309C22—61no—
310C22—61no—
311C22—61no—
312C22—91no—
313C22—91no—
314C22—182no—
315C22—272no—
316C22—272no—
317C22—272no—
318C22—272no—
319C22—271no—
320C22—271no—
321C22—542no—
322C22—541no—
323C22—541no—
324C22—541no—
325C22—541no—
326C22—542no—
327C22—542no—
328C22—544no—
329C22C3×C22⋊#15711yescharacteristic, complemented
330C22A4×#15711yescharacteristic, complemented
331C4—182no—
332C4—542no—
333C3—22no—
334C3—41no—
335C3—162no—
336C2—21no—
337C2—31no—
338C2—31no—
339C2—61no—
340C2—92no—
341C2—181no—
342C2—272no—
343C2—541no—
344C2C2×A4×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×#157
commutator, socleC23×C32C2×C3×C4
FittingC24×C32C3×C4
prime radicalC22×A4×C3⋊D3C2
cosocleC23×C3⋊D3C22×C3
hypercenterC22A4×#157
hypocenterC22×C32C3×C22⋊C4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×A4×#157C2semidirect
3. C22×C22⋊#157C3semidirect
4. C22×C32×A4C4semidirect
6. C2×A4×C3⋊D3C4semidirect
11. C23×#157C2×C3semidirect
23. C24×C32C3×C4semidirect
30. C23×C3⋊D3C3×C4semidirect
44. C22⋊#157A4direct (N × H)
49. C32×A4C22⋊C4semidirect
90. C2×#157C2×A4semidirect
111. C22×A4#157semidirect
123. C22×C32C3×C22⋊C4semidirect
124. C22×C32C4×A4semidirect
140. C2×C3⋊D3C4×A4semidirect
212. C24C3×#157semidirect
253. A4C22⋊#157direct (N × H)
257. C32A4×C22⋊C4semidirect
329. C22C3×C22⋊#157semidirect
330. C22A4×#157semidirect

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
C22×C22⋊#1573
C3×C22⋊#1574
A4×C22⋊C49