Finite Group Databasebeta

← C3×#10604558 #14285424

Subgroups — C3×#10604558 #14285424

Order 1920.

508 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C3×#66002C211yesmaximal, characteristic
2C3×#65682C212yesmaximal, complemented
3C3×#64931C212yesmaximal, complemented
4C3×#64925C212yesmaximal, complemented
5#10604558C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#713C411yescharacteristic
7C2×C3×#711C411yescharacteristic
8C2×C3×#700—22no—
9C3×C5×#124C411yescharacteristic
10C3×#4090C2211yescharacteristic
11C3×#4086C2211yescharacteristic
12C3×#4030—22no—
13C3×#4027—22no—
14C3×#4021C411yescharacteristic
15C3×C42×D5C2211yescharacteristic
16C3×#4004—22no—
17C3×#4004C2214yes—
18C3×#1560—51nomaximal, Hall
19#66002C2×C311yescharacteristic
20#65682C2×C312yescomplemented
21#64931C2×C312yescomplemented
22#64925C2×C312yescomplemented
23C2×C3×C5×Q8C2×C411yescharacteristic
24C2×C3×C5×D4C2×C411yescharacteristic
25C2×C3×C22⋊D5—21no—
26C2×C3×C22⋊D5—22no—
27C22×C3×Dic5—21no—
28C3×#713—42no—
29C3×#711—42no—
30C2×C3×D20—21no—
31C2×C3×D20C2×C411yescharacteristic
32C2×C3×C4×D5—21no—
33C2×C3×C4×D5C2×C411yescharacteristic
34C2×C3×C4×D5D412yes—
35C2×C3×Dic10C2×C411yescharacteristic
36C2×C3×#170—24no—
37C3×#700—44no—
38C3×C5×C22⋊C4—22no—
39C3×C42×C5C2×C411yescharacteristic
40C3×#685—22no—
41C3×#683—22no—
42C3×C4×Dic5D412yes—
43C3×C4×Dic5C2311yescharacteristic
44C3×#532—51no—
45C3×#423—52no—
46C3×#333—52no—
47C3×#331—52no—
48C2×#713C3×C411yescharacteristic
49C2×#711C3×C411yescharacteristic
50C2×#700—22no—
51C5×#124C3×C411yescharacteristic
52#4090C22×C311yescharacteristic
53#4086C22×C311yescharacteristic
54#4030—22no—
55#4027—22no—
56#4021C3×C411yescharacteristic
57C42×D5C22×C311yescharacteristic
58#4004—22no—
59#4004C22×C314yes—
60#1560—51noSylow
61C23×C3×C5—21no—
62C22×C3×D5—21no—
63C22×C3×D5C22⋊C411yescharacteristic
64C3×C5×Q8—41no—
65C3×C5×D4—41no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5C22⋊C412yes—
68C2×C3×C4×C5C22×C411yescharacteristic
69C3×C22⋊D5—42no—
70C3×C22⋊D5—42no—
71C3×C22⋊D5—42no—
72C2×C3×Dic5—21no—
73C2×C3×Dic5—42no—
74C2×C3×Dic5C22⋊C411yescharacteristic
75C2×C3×Dic5C2×D412yes—
76C3×D20—41no—
77C3×D20—42no—
78C3×C4×D5—22no—
79C3×C4×D5—22no—
80C3×C4×D5—22no—
81C3×C4×D5—42no—
82C3×Dic10—41no—
83C3×#170—44no—
84C2×C3×#41—51no—
85C2×C3×#41—51no—
86C2×C3×M16—102no—
87C3×C4⋊Q8—51no—
88C3×#127—51no—
89C3×#124—51no—
90C3×#124AGL(1,𝔽5)11yescharacteristic
91C3×#121—102no—
92C3×C4×D4—102no—
93C2×C3×C42—51no—
94C3×C4⋊C8—54no—
95C3×C4⋊C8—102no—
96C2×C5×Q8C2×C3×C411yescharacteristic
97C2×C5×D4C2×C3×C411yescharacteristic
98C2×C22⋊D5—21no—
99C2×C22⋊D5—22no—
100C22×Dic5—21no—
101#713—42no—
102#711—42no—
103C2×D20—21no—
104C2×D20C2×C3×C411yescharacteristic
105C2×C4×D5—21no—
106C2×C4×D5C2×C3×C411yescharacteristic
107C2×C4×D5C3×D412yes—
108C2×Dic10C2×C3×C411yescharacteristic
109C2×#170—24no—
110#700—44no—
111C5×C22⋊C4—22no—
112C42×C5C2×C3×C411yescharacteristic
113#685—22no—
114#683—22no—
115C4×Dic5C3×D412yes—
116C4×Dic5C23×C311yescharacteristic
117#532—51no—
118#423—52no—
119#333—52no—
120#331—52no—
121C22×C3×C5—41no—
122C22×C3×C5—42no—
123C22×C3×C5C2×C22⋊C411yescharacteristic
124C2×C3×D5—21no—
125C2×C3×D5—22no—
126C2×C3×D5—41no—
127C2×C3×D5—42no—
128C3×C4×C5—21no—
129C3×C4×C5—22no—
130C3×C4×C5—41no—
131C3×Dic5—41no—
132C3×Dic5#10012yes—
133C3×Dic5AGL(1,ℤ/8)12yes—
134C3×Dic5#13712yes—
135C3×#41—202no—
136C3×#41—202no—
137C2×C3×Q8—51no—
138C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
139C2×C3×D4—51no—
140C2×C3×D4—101no—
141C2×C3×D4—102no—
142C2×C3×D4—101no—
143C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
144C22×C3×C4—52no—
145C22×C3×C4—51no—
146C22×C3×C4—101no—
147C22×C3×C4—101no—
148C3×M16—204no—
149C2×C3×C8—104no—
150C3×C4⋊C4—102no—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—102no—
153C3×C42—52no—
154C3×C42—51no—
155C3×C42C2×AGL(1,𝔽5)11yescharacteristic
156C23×C5—21no—
157C22×D5—21no—
158C22×D5C3×C22⋊C411yescharacteristic
159C5×Q8—41no—
160C5×D4—41no—
161C2×C4×C5—21no—
162C2×C4×C5C3×C22⋊C412yes—
163C2×C4×C5C22×C3×C411yescharacteristic
164C22⋊D5—42no—
165C22⋊D5—42no—
166C22⋊D5—42no—
167C2×Dic5—21no—
168C2×Dic5—42no—
169C2×Dic5C3×C22⋊C411yescharacteristic
170C2×Dic5C2×C3×D412yes—
171D20—41no—
172D20—42no—
173C4×D5—22no—
174C4×D5—22no—
175C4×D5—22no—
176C4×D5—42no—
177Dic10—41no—
178#170—44no—
179C2×#41—51no—
180C2×#41—51no—
181C2×M16—102no—
182C4⋊Q8—51no—
183#127—51no—
184#124—51no—
185#124C3×AGL(1,𝔽5)11yescharacteristic
186#121—102no—
187C4×D4—102no—
188C2×C42—51no—
189C4⋊C8—54no—
190C4⋊C8—102no—
191C2×C3×C5—81no—
192C2×C3×C5C2×#10011yescharacteristic
193C2×C3×C5#41712yes—
194C3×D5—41no—
195C3×D5—81no—
196C23×C3—21no—
197C23×C3—51no—
198C23×C3—101no—
199C3×Q8—41no—
200C3×Q8—201no—
201C3×D4—41no—
202C3×D4—201no—
203C3×D4—202no—
204C3×D4—202no—
205C3×D4—202no—
206C3×D4—202no—
207C2×C3×C4—21no—
208C2×C3×C4—51no—
209C2×C3×C4—52no—
210C2×C3×C4—102no—
211C2×C3×C4—102no—
212C2×C3×C4—102no—
213C2×C3×C4—101no—
214C2×C3×C4—202no—
215C2×C3×C4—202no—
216C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
217C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
218C3×C8—204no—
219C22×C5—41no—
220C22×C5—42no—
221C22×C5C2×C3×C22⋊C411yescharacteristic
222C2×D5—21no—
223C2×D5—22no—
224C2×D5—41no—
225C2×D5—42no—
226C4×C5—21no—
227C4×C5—22no—
228C4×C5—41no—
229Dic5—41no—
230Dic5C3×#10012yes—
231Dic5C3×AGL(1,ℤ/8)12yes—
232Dic5C3×#13712yes—
233#41—202no—
234#41—202no—
235C2×Q8—51no—
236C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
237C2×D4—51no—
238C2×D4—101no—
239C2×D4—102no—
240C2×D4—101no—
241C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
242C22×C4—52no—
243C22×C4—51no—
244C22×C4—101no—
245C22×C4—101no—
246M16—204no—
247C2×C8—104no—
248C4⋊C4—102no—
249C22⋊C4—22no—
250C22⋊C4—102no—
251C42—52no—
252C42—51no—
253C42C2×C3×AGL(1,𝔽5)11yescharacteristic
254C3×C5#156011yescharacteristic, complemented, Hall
255C22×C3—42no—
256C22×C3—41no—
257C22×C3—102no—
258C22×C3—101no—
259C22×C3—202no—
260C22×C3—201no—
261C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
262C3×C4—22no—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—52no—
266C3×C4—52no—
267C3×C4—52no—
268C3×C4—201no—
269C2×C5—81no—
270C2×C5C2×C3×#10011yescharacteristic
271C2×C5C3×#41712yes—
272D5—41no—
273D5—81no—
274C23—21no—
275C23—51no—
276C23—101no—
277Q8—41no—
278Q8—201no—
279D4—41no—
280D4—201no—
281D4—202no—
282D4—202no—
283D4—202no—
284D4—202no—
285C2×C4—21no—
286C2×C4—52no—
287C2×C4—51no—
288C2×C4—102no—
289C2×C4—102no—
290C2×C4—102no—
291C2×C4—101no—
292C2×C4—202no—
293C2×C4—202no—
294C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
295C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
296C8—204no—
297C2×C3—81no—
298C2×C3—201no—
299C2×C3—401no—
300C2×C3D5⋊#10011yescharacteristic
301C2×C3#6579312yes—
302C5C3×#156011yescharacteristic, complemented, Sylow
303C22—41no—
304C22—42no—
305C22—101no—
306C22—102no—
307C22—201no—
308C22—202no—
309C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
310C4—21no—
311C4—22no—
312C4—41no—
313C4—52no—
314C4—52no—
315C4—52no—
316C4—201no—
317C3#1060455811yescharacteristic, direct product factor, Sylow
318C2—81no—
319C2—201no—
320C2—401no—
321C2C3×D5⋊#10011yescharacteristic
322C2C3×#6579312yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#124C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66002C2
cosocleC4×Dic5C23×C3
hypercenterC3×#124AGL(1,𝔽5)
hypocenterC5C3×#1560
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C3×#65682C2semidirect
3. C3×#64931C2semidirect
4. C3×#64925C2semidirect
5. #10604558C3direct (N × H)
20. #65682C2×C3semidirect
21. #64931C2×C3semidirect
22. #64925C2×C3semidirect
254. C3×C5#1560semidirect
302. C5C3×#1560semidirect
317. C3#10604558direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660022
C3×#656822
C3×#649312
C3×#649252
#106045583
C3×#15605