Finite Group Databasebeta

← C22⋊C9×ΓL(1,𝔽25) #13147233

Subgroups — C22⋊C9×ΓL(1,𝔽25) #13147233

Order 1728.

318 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C4×D3×C22⋊C9C211yesmaximal, characteristic
2C3×C8×C22⋊C9C211yesmaximal, characteristic, complemented
3C3⋊C8×C22⋊C9C211yesmaximal, characteristic, complemented
4C22×C3×ΓL(1,𝔽25)C311yesmaximal, characteristic
5M16×C22⋊C9—31nomaximal
6C2×D3×C22⋊C9C411yescharacteristic
7C3×C4×C22⋊C9C2211yescharacteristic
8Dic3×C22⋊C9C411yescharacteristic
9C9×ΓL(1,𝔽25)—41nomaximal
10C22×C3×C4×D3C2×C311yescharacteristic
11C22×C32×C8C2×C311yescharacteristic
12C22×C3×C3⋊C8C2×C311yescharacteristic
13C2×C3×ΓL(1,𝔽25)—31no—
14C2×C3×ΓL(1,𝔽25)—31no—
15C2×C3×ΓL(1,𝔽25)—31no—
16C2×C3×ΓL(1,𝔽25)—31no—
17C2×C4×C22⋊C9—31no—
18C8×C22⋊C9—21no—
19C8×C22⋊C9—31no—
20C8×C22⋊C9D311yescharacteristic, complemented
21C2×C3×C22⋊C9C2×C411yescharacteristic
22D3×C22⋊C9—21no—
23C4×D3×C9—41no—
24C3×C8×C9—41no—
25C9×C3⋊C8—41no—
26C22×C3×M16—31no—
27C22×ΓL(1,𝔽25)C911yescharacteristic, complemented
28C23×C3×D3C3×C411yescharacteristic
29C22×C32×C4C22×C311yescharacteristic
30C22×C3×Dic3C3×C411yescharacteristic
31C2×C3×C4×D3—31no—
32C2×C3×C4×D3—31no—
33C2×C3×C4×D3—31no—
34C2×C3×C4×D3—31no—
35C22×C22⋊C9—31no—
36C2×C32×C8—31no—
37C2×C32×C8—31no—
38C2×C3×C3⋊C8—31no—
39C2×C3×C3⋊C8—31no—
40C3×ΓL(1,𝔽25)—32no—
41C3×ΓL(1,𝔽25)—31no—
42C3×ΓL(1,𝔽25)—31no—
43C3×ΓL(1,𝔽25)—31no—
44C3×ΓL(1,𝔽25)A411yescharacteristic
45C4×C22⋊C9—21no—
46C4×C22⋊C9—31no—
47C4×C22⋊C9C2×D311yescharacteristic
48C9×M16—121no—
49C2×D3×C9—41no—
50C3×C22⋊C9M1611yescharacteristic, complemented
51C3×C4×C9—41no—
52C9×Dic3—41no—
53C23×C3×C4—31no—
54C22×C4×D3C2×C911yescharacteristic
55C2×C3×M16—91no—
56C2×C3×M16—91no—
57C2×C3×M16—91no—
58C2×C3×M16—91no—
59C22×C3×C8—21no—
60C22×C3×C8—31no—
61C22×C3×C8C2×C911yescharacteristic, complemented
62C22×C3×C8C3×D311yescharacteristic
63C22×C3⋊C8C2×C911yescharacteristic, complemented
64C2×ΓL(1,𝔽25)—31no—
65C2×ΓL(1,𝔽25)—31no—
66C2×ΓL(1,𝔽25)—31no—
67C2×ΓL(1,𝔽25)—31no—
68C23×C32C2×C3×C411yescharacteristic
69C22×C3×D3—21no—
70C22×C3×D3—31no—
71C22×C3×D3—31no—
72C22×C3×D3—61no—
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C3×C4×D3—32no—
78C3×C4×D3—31no—
79C3×C4×D3—31no—
80C3×C4×D3—31no—
81C3×C4×D3C2×A411yescharacteristic
82C2×C22⋊C9—21no—
83C2×C22⋊C9—61no—
84C2×C22⋊C9C4×D311yescharacteristic
85C32×C8—31no—
86C32×C8C2×A411yescharacteristic
87C3×C3⋊C8—31no—
88C3×C3⋊C8C2×A411yescharacteristic
89C2×C4×C9—121no—
90C8×C9—41no—
91C8×C9—81no—
92C8×C9—121no—
93C22×M16—31noSylow
94C2×C3×C9—41no—
95D3×C9—81no—
96C24×C3—31no—
97C23×D3C4×C911yescharacteristic
98C22×C3×C4—21no—
99C22×C3×C4—31no—
100C22×C3×C4—91no—
101C22×C3×C4—91no—
102C22×C3×C4—91no—
103C22×C3×C4—91no—
104C22×C3×C4C22×C911yescharacteristic
105C22×C3×C4C2×C3×D311yescharacteristic
106C22×Dic3C4×C911yescharacteristic
107C2×C4×D3—31no—
108C2×C4×D3—31no—
109C2×C4×D3—31no—
110C2×C4×D3—31no—
111C3×M16—31no—
112C3×M16—91no—
113C3×M16—91no—
114C3×M16—92no—
115C3×M16—91no—
116C2×C3×C8—31no—
117C2×C3×C8—31no—
118C2×C3×C8—31no—
119C2×C3×C8—31no—
120C2×C3×C8—61no—
121C2×C3×C8—61no—
122C2×C3×C8—91no—
123C2×C3×C8—91no—
124C2×C3⋊C8—31no—
125C2×C3⋊C8—31no—
126ΓL(1,𝔽25)—32no—
127ΓL(1,𝔽25)—31no—
128ΓL(1,𝔽25)—31no—
129ΓL(1,𝔽25)—31no—
130ΓL(1,𝔽25)C22⋊C911yescharacteristic, direct product factor
131C22×C32—31no—
132C22×C32—31no—
133C22×C32C3×M1611yescharacteristic
134C2×C3×D3—31no—
135C2×C3×D3—61no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3C4×A411yescharacteristic
140C32×C4—31no—
141C32×C4C22×A411yescharacteristic
142C3×Dic3—31no—
143C3×Dic3C4×A411yescharacteristic
144C22×C9—121no—
145C22⋊C9—21no—
146C22⋊C9ΓL(1,𝔽25)11yescharacteristic, direct product factor
147C4×C9—41no—
148C4×C9—81no—
149C4×C9—121no—
150C23×C4—31no—
151C2×M16—91no—
152C2×M16—91no—
153C2×M16—91no—
154C2×M16—91no—
155C22×C8—31no—
156C22×C8D3×C911yescharacteristic, complemented
157C3×C9—41noSylow
158C23×C3—21no—
159C23×C3—61no—
160C23×C3—91no—
161C23×C3—91no—
162C23×C3—181no—
163C23×C3C2×C4×C911yescharacteristic
164C23×C3C3×C4×D311yescharacteristic
165C22×D3—21no—
166C22×D3—31no—
167C22×D3—31no—
168C22×D3—61no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—92no—
180C2×C3×C4—91no—
181C2×C3×C4—91no—
182C2×Dic3—31no—
183C2×Dic3—31no—
184C4×D3—32no—
185C4×D3—31no—
186C4×D3—31no—
187C4×D3—31no—
188C4×D3C2×C22⋊C911yescharacteristic
189C3×C8—21no—
190C3×C8—31no—
191C3×C8—31no—
192C3×C8—31no—
193C3×C8—61no—
194C3×C8—91no—
195C3×C8C2×C22⋊C911yescharacteristic, complemented
196C3×C8D3×A411yescharacteristic
197C3⋊C8—31no—
198C3⋊C8C2×C22⋊C911yescharacteristic, complemented
199C2×C32—31no—
200C2×C32—31no—
201C2×C32C2×C4×A411yescharacteristic
202C3×D3—21no—
203C3×D3—61no—
204C2×C9—41no—
205C2×C9—81no—
206C2×C9—241no—
207C24—31no—
208C22×C4—31no—
209C22×C4—91no—
210C22×C4—91no—
211C22×C4—91no—
212C22×C4—91no—
213C22×C4C2×D3×C911yescharacteristic
214M16—31no—
215M16—91no—
216M16—91no—
217M16—92no—
218M16—91no—
219C2×C8—31no—
220C2×C8—31no—
221C2×C8—91no—
222C2×C8—91no—
223C22×C3—21no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—31no—
229C22×C3—61no—
230C22×C3—61no—
231C22×C3—91no—
232C22×C3—181no—
233C22×C3—181no—
234C22×C3—181no—
235C22×C3—181no—
236C22×C3C9×M1611yescharacteristic, complemented
237C22×C3C3×ΓL(1,𝔽25)11yescharacteristic
238C2×D3—31no—
239C2×D3—61no—
240C2×D3—61no—
241C2×D3—61no—
242C2×D3—61no—
243C2×D3C4×C22⋊C911yescharacteristic
244C3×C4—21no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—61no—
249C3×C4—91no—
250C3×C4C22×C22⋊C911yescharacteristic
251C3×C4C2×D3×A411yescharacteristic
252Dic3—31no—
253Dic3C4×C22⋊C911yescharacteristic
254C32A4×M1611yescharacteristic
255C9—41no—
256C9—81no—
257C23—61no—
258C23—91no—
259C23—91no—
260C23—181no—
261C23C4×D3×C911yescharacteristic
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—92no—
269C2×C4—91no—
270C2×C4—91no—
271C8—31no—
272C8—31no—
273C8—91no—
274C8D3×C22⋊C911yescharacteristic, complemented
275C2×C3—21no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—61no—
281C2×C3—61no—
282C2×C3—61no—
283C2×C3—181no—
284C2×C3C2×C4×C22⋊C911yescharacteristic
285C2×C3C4×D3×A411yescharacteristic
286D3—21no—
287D3—61no—
288C22—31no—
289C22—31no—
290C22—31no—
291C22—91no—
292C22—181no—
293C22—181no—
294C22—181no—
295C22—181no—
296C22C9×ΓL(1,𝔽25)11yescharacteristic, complemented
297C4—31no—
298C4—31no—
299C4—91no—
300C4C2×D3×C22⋊C911yescharacteristic
301C3—21no—
302C3M16×C22⋊C911yescharacteristic, complemented
303C3A4×ΓL(1,𝔽25)11yescharacteristic
304C2—31no—
305C2—31no—
306C2—61no—
307C2—181no—
308C2C4×D3×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC3×C4C2×D3×A4
commutatorC23×C3C2×C4×C9
FittingC22×C32×C8C2×C3
socleC23×C32C2×C3×C4
prime radicalC23×C3×D3C3×C4
cosocleC22×C32×C4C22×C3
hypercenterC3×C8D3×A4
hypocenterC22×C3C9×M16
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×C8×C22⋊C9C2semidirect
3. C3⋊C8×C22⋊C9C2semidirect
20. C8×C22⋊C9D3semidirect
27. C22×ΓL(1,𝔽25)C9semidirect
50. C3×C22⋊C9M16semidirect
61. C22×C3×C8C2×C9semidirect
63. C22×C3⋊C8C2×C9semidirect
130. ΓL(1,𝔽25)C22⋊C9direct (N × H)
146. C22⋊C9ΓL(1,𝔽25)direct (N × H)
156. C22×C8D3×C9semidirect
195. C3×C8C2×C22⋊C9semidirect
198. C3⋊C8C2×C22⋊C9semidirect
236. C22×C3C9×M16semidirect
274. C8D3×C22⋊C9semidirect
296. C22C9×ΓL(1,𝔽25)semidirect
302. C3M16×C22⋊C9semidirect

Maximal subgroups

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

subgroupindex
C4×D3×C22⋊C92
C3×C8×C22⋊C92
C3⋊C8×C22⋊C92
C22×C3×ΓL(1,𝔽25)3
M16×C22⋊C93
C9×ΓL(1,𝔽25)4