Finite Group Databasebeta

← C3×#10604556 #14284713

Subgroups — C3×#10604556 #14284713

Order 1920.

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

normal subgroups only

This group has 492 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×#65678C212yesmaximal, complemented
3C3×#64931C212yesmaximal, complemented
4C3×#64925C212yesmaximal, complemented
5#10604556C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#713C411yescharacteristic
7C2×C3×#711C411yescharacteristic
8C3×C5×#124C411yescharacteristic
9C3×#4090C2211yescharacteristic
10C3×#4086C2211yescharacteristic
11C3×#4030—22no—
12C3×#4027—22no—
13C3×#4021C411yescharacteristic
14C3×C42×D5C2211yescharacteristic
15C3×#4004C2214yes—
16C3×#4003—22no—
17C3×C4×#170—22no—
18C3×#1543—51nomaximal, Hall
19#66002C2×C311yescharacteristic
20#65678C2×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×C5×C22⋊C4—22no—
38C3×C42×C5C2×C411yescharacteristic
39C3×#685—22no—
40C3×#683—22no—
41C3×C4×Dic5D412yes—
42C3×C4×Dic5C2311yescharacteristic
43C3×#532—51no—
44C3×#431—52no—
45C3×#333—52no—
46C3×#331—52no—
47C2×#713C3×C411yescharacteristic
48C2×#711C3×C411yescharacteristic
49C5×#124C3×C411yescharacteristic
50#4090C22×C311yescharacteristic
51#4086C22×C311yescharacteristic
52#4030—22no—
53#4027—22no—
54#4021C3×C411yescharacteristic
55C42×D5C22×C311yescharacteristic
56#4004C22×C314yes—
57#4003—22no—
58C4×#170—22no—
59#1543—51noSylow
60C23×C3×C5—21no—
61C22×C3×D5—21no—
62C22×C3×D5C22⋊C411yescharacteristic
63C3×C5×Q8—41no—
64C3×C5×D4—41no—
65C2×C3×C4×C5—21no—
66C2×C3×C4×C5C22⋊C412yes—
67C2×C3×C4×C5C22×C411yescharacteristic
68C3×C22⋊D5—42no—
69C3×C22⋊D5—44no—
70C2×C3×Dic5—21no—
71C2×C3×Dic5—42no—
72C2×C3×Dic5C22⋊C411yescharacteristic
73C2×C3×Dic5C2×D412yes—
74C3×D20—41no—
75C3×D20—42no—
76C3×C4×D5—22no—
77C3×C4×D5—24no—
78C3×C4×D5—42no—
79C3×Dic10—41no—
80C3×#170—44no—
81C2×C3×#41—51no—
82C2×C3×#41—51no—
83C3×C4⋊Q8—51no—
84C3×#127—51no—
85C3×#124—51no—
86C3×#124AGL(1,𝔽5)11yescharacteristic
87C3×#121—102no—
88C3×C4×D4—102no—
89C2×C3×C42—51no—
90C3×C4⋊C8—54no—
91C3×C22⋊C8—102no—
92C3×C4×C8—102no—
93C2×C5×Q8C2×C3×C411yescharacteristic
94C2×C5×D4C2×C3×C411yescharacteristic
95C2×C22⋊D5—21no—
96C2×C22⋊D5—22no—
97C22×Dic5—21no—
98#713—42no—
99#711—42no—
100C2×D20—21no—
101C2×D20C2×C3×C411yescharacteristic
102C2×C4×D5—21no—
103C2×C4×D5C2×C3×C411yescharacteristic
104C2×C4×D5C3×D412yes—
105C2×Dic10C2×C3×C411yescharacteristic
106C2×#170—24no—
107C5×C22⋊C4—22no—
108C42×C5C2×C3×C411yescharacteristic
109#685—22no—
110#683—22no—
111C4×Dic5C3×D412yes—
112C4×Dic5C23×C311yescharacteristic
113#532—51no—
114#431—52no—
115#333—52no—
116#331—52no—
117C22×C3×C5—41no—
118C22×C3×C5—42no—
119C22×C3×C5C2×C22⋊C411yescharacteristic
120C2×C3×D5—21no—
121C2×C3×D5—22no—
122C2×C3×D5—41no—
123C2×C3×D5—42no—
124C3×C4×C5—21no—
125C3×C4×C5—22no—
126C3×C4×C5—41no—
127C3×Dic5—41no—
128C3×Dic5#10012yes—
129C3×Dic5#13514yes—
130C3×#41—202no—
131C3×#41—202no—
132C2×C3×Q8—51no—
133C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
134C2×C3×D4—51no—
135C2×C3×D4—101no—
136C2×C3×D4—102no—
137C2×C3×D4—101no—
138C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
139C22×C3×C4—52no—
140C22×C3×C4—51no—
141C22×C3×C4—101no—
142C22×C3×C4—101no—
143C2×C3×C8—104no—
144C3×C4⋊C4—102no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—102no—
147C3×C42—52no—
148C3×C42—51no—
149C3×C42C2×AGL(1,𝔽5)11yescharacteristic
150C23×C5—21no—
151C22×D5—21no—
152C22×D5C3×C22⋊C411yescharacteristic
153C5×Q8—41no—
154C5×D4—41no—
155C2×C4×C5—21no—
156C2×C4×C5C3×C22⋊C412yes—
157C2×C4×C5C22×C3×C411yescharacteristic
158C22⋊D5—42no—
159C22⋊D5—44no—
160C2×Dic5—21no—
161C2×Dic5—42no—
162C2×Dic5C3×C22⋊C411yescharacteristic
163C2×Dic5C2×C3×D412yes—
164D20—41no—
165D20—42no—
166C4×D5—22no—
167C4×D5—24no—
168C4×D5—42no—
169Dic10—41no—
170#170—44no—
171C2×#41—51no—
172C2×#41—51no—
173C4⋊Q8—51no—
174#127—51no—
175#124—51no—
176#124C3×AGL(1,𝔽5)11yescharacteristic
177#121—102no—
178C4×D4—102no—
179C2×C42—51no—
180C4⋊C8—54no—
181C22⋊C8—102no—
182C4×C8—102no—
183C2×C3×C5—81no—
184C2×C3×C5C2×#10011yescharacteristic
185C2×C3×C5#41612yes—
186C3×D5—41no—
187C3×D5—81no—
188C23×C3—21no—
189C23×C3—51no—
190C23×C3—101no—
191C3×Q8—41no—
192C3×Q8—201no—
193C3×D4—41no—
194C3×D4—201no—
195C3×D4—202no—
196C3×D4—204no—
197C3×D4—202no—
198C2×C3×C4—21no—
199C2×C3×C4—51no—
200C2×C3×C4—52no—
201C2×C3×C4—104no—
202C2×C3×C4—102no—
203C2×C3×C4—101no—
204C2×C3×C4—202no—
205C2×C3×C4—202no—
206C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
207C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
208C3×C8—204no—
209C22×C5—41no—
210C22×C5—42no—
211C22×C5C2×C3×C22⋊C411yescharacteristic
212C2×D5—21no—
213C2×D5—22no—
214C2×D5—41no—
215C2×D5—42no—
216C4×C5—21no—
217C4×C5—22no—
218C4×C5—41no—
219Dic5—41no—
220Dic5C3×#10012yes—
221Dic5C3×#13514yes—
222#41—202no—
223#41—202no—
224C2×Q8—51no—
225C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
226C2×D4—51no—
227C2×D4—101no—
228C2×D4—102no—
229C2×D4—101no—
230C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
231C22×C4—52no—
232C22×C4—51no—
233C22×C4—101no—
234C22×C4—101no—
235C2×C8—104no—
236C4⋊C4—102no—
237C22⋊C4—22no—
238C22⋊C4—102no—
239C42—52no—
240C42—51no—
241C42C2×C3×AGL(1,𝔽5)11yescharacteristic
242C3×C5#154311yescharacteristic, complemented, Hall
243C22×C3—42no—
244C22×C3—41no—
245C22×C3—102no—
246C22×C3—101no—
247C22×C3—202no—
248C22×C3—201no—
249C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
250C3×C4—22no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4—54no—
254C3×C4—52no—
255C3×C4—201no—
256C2×C5—81no—
257C2×C5C2×C3×#10011yescharacteristic
258C2×C5C3×#41612yes—
259D5—41no—
260D5—81no—
261C23—21no—
262C23—51no—
263C23—101no—
264Q8—41no—
265Q8—201no—
266D4—41no—
267D4—201no—
268D4—202no—
269D4—204no—
270D4—202no—
271C2×C4—21no—
272C2×C4—52no—
273C2×C4—51no—
274C2×C4—102no—
275C2×C4—104no—
276C2×C4—101no—
277C2×C4—202no—
278C2×C4—202no—
279C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
280C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
281C8—204no—
282C2×C3—81no—
283C2×C3—201no—
284C2×C3—401no—
285C2×C3D5⋊#10011yescharacteristic
286C2×C3#6579412yes—
287C5C3×#154311yescharacteristic, complemented, Sylow
288C22—41no—
289C22—42no—
290C22—101no—
291C22—102no—
292C22—201no—
293C22—202no—
294C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
295C4—21no—
296C4—22no—
297C4—41no—
298C4—52no—
299C4—54no—
300C4—201no—
301C3#1060455611yescharacteristic, direct product factor, Sylow
302C2—81no—
303C2—201no—
304C2—401no—
305C2C3×D5⋊#10011yescharacteristic
306C2C3×#6579412yes—
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×#1543
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×#65678C2semidirect
3. C3×#64931C2semidirect
4. C3×#64925C2semidirect
5. #10604556C3direct (N × H)
20. #65678C2×C3semidirect
21. #64931C2×C3semidirect
22. #64925C2×C3semidirect
242. C3×C5#1543semidirect
287. C5C3×#1543semidirect
301. C3#10604556direct (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×#656782
C3×#649312
C3×#649252
#106045563
C3×#15435