Finite Group Databasebeta

← C3×#10598692 #14276911

Subgroups — C3×#10598692 #14276911

Order 1920.

452 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 92 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×#65764C211yesmaximal, characteristic
2C3×#65755C211yesmaximal, characteristic
3C3×C4×#685C211yesmaximal, characteristic
4#10598692C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C4×D5C2211yescharacteristic
6C2×C3×C22⋊AGL(1,𝔽5)—21no—
7C2×C3×#704—21no—
8C2×C3×C42×C5C411yescharacteristic, complemented
9C2×C3×#685—21no—
10C2×C3×C4×Dic5C411yescharacteristic, complemented
11C3×(C2×AGL(1,𝔽5))⋊C4—21no—
12C3×(C2×AGL(1,𝔽5))⋊C4C412yescomplemented
13C3×#4003—21no—
14C3×#4003C412yescomplemented
15C3×#3963—21no—
16C3×#1284—51nomaximal, Hall
17#65764C2×C311yescharacteristic
18#65755C2×C311yescharacteristic
19C4×#685C2×C311yescharacteristic
20C23×C3×D5D411yescharacteristic
21C22×C3×AGL(1,𝔽5)—22no—
22C22×C3×C4×C5—21no—
23C22×C3×C4×C5C2×C411yescharacteristic
24C22×C3×Dic5—21no—
25C22×C3×Dic5D411yescharacteristic
26C2×C3×C4×D5—41no—
27C2×C3×C4×D5—41no—
28C2×C3×C4×D5C2×C411yescharacteristic
29C2×C3×C4×D5C2×C411yescharacteristic
30C2×C3×C4×D5D411yescharacteristic
31C2×C3×C4×D5Q811yescharacteristic
32C3×C22⋊AGL(1,𝔽5)—41no—
33C3×C22⋊AGL(1,𝔽5)—41no—
34C3×#704—41no—
35C3×#704—41no—
36C2×C3×#170—22no—
37C3×C42×C5—41no—
38C3×#685—24no—
39C3×C4×Dic5—41no—
40C3×#407—51no—
41C3×#381—51no—
42C3×C4×C22⋊C4—51no—
43C22×C4×D5C22×C311yescharacteristic
44C2×C22⋊AGL(1,𝔽5)—21no—
45C2×#704—21no—
46C2×C42×C5C3×C411yescharacteristic, complemented
47C2×#685—21no—
48C2×C4×Dic5C3×C411yescharacteristic, complemented
49(C2×AGL(1,𝔽5))⋊C4—21no—
50(C2×AGL(1,𝔽5))⋊C4C3×C412yescomplemented
51#4003—21no—
52#4003C3×C412yescomplemented
53#3963—21no—
54#1284—51noSylow
55C23×C3×C5C22⋊C411yescharacteristic
56C22×C3×D5—21no—
57C22×C3×D5—21no—
58C22×C3×D5—22no—
59C22×C3×D5—41no—
60C22×C3×D5C22⋊C411yescharacteristic
61C22×C3×D5C4⋊C411yescharacteristic
62C2×C3×AGL(1,𝔽5)—42no—
63C2×C3×AGL(1,𝔽5)—42no—
64C2×C3×AGL(1,𝔽5)—42no—
65C2×C3×C4×C5—21no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5—22no—
68C2×C3×C4×C5—41no—
69C2×C3×C4×C5—41no—
70C2×C3×C4×C5C4211yescharacteristic
71C2×C3×C4×C5C4⋊C411yescharacteristic
72C2×C3×Dic5—21no—
73C2×C3×Dic5—21no—
74C2×C3×Dic5—22no—
75C2×C3×Dic5—41no—
76C2×C3×Dic5—41no—
77C2×C3×Dic5C22⋊C411yescharacteristic
78C2×C3×Dic5C4⋊C411yescharacteristic
79C3×C4×D5—41no—
80C3×C4×D5—41no—
81C3×C4×D5—41no—
82C3×C4×D5—41no—
83C3×#170—42no—
84C23×C3×C4—51no—
85C2×C3×M16—101no—
86C2×C3×C22⋊C4—101no—
87C2×C3×C22⋊C4—101no—
88C2×C3×C42—51no—
89C2×C3×C42AGL(1,𝔽5)11yescharacteristic, complemented
90C3×C22⋊C8—52no—
91C3×C22⋊C8—101no—
92C3×#95—52no—
93C3×#95—101no—
94C3×#95—101no—
95C23×D5C3×D411yescharacteristic
96C22×AGL(1,𝔽5)—22no—
97C22×C4×C5—21no—
98C22×C4×C5C2×C3×C411yescharacteristic
99C22×Dic5—21no—
100C22×Dic5C3×D411yescharacteristic
101C2×C4×D5—41no—
102C2×C4×D5—41no—
103C2×C4×D5C2×C3×C411yescharacteristic
104C2×C4×D5C2×C3×C411yescharacteristic
105C2×C4×D5C3×D411yescharacteristic
106C2×C4×D5C3×Q811yescharacteristic
107C22⋊AGL(1,𝔽5)—41no—
108C22⋊AGL(1,𝔽5)—41no—
109#704—41no—
110#704—41no—
111C2×#170—22no—
112C42×C5—41no—
113#685—24no—
114C4×Dic5—41no—
115#407—51no—
116#381—51no—
117C4×C22⋊C4—51no—
118C22×C3×C5—21no—
119C22×C3×C5—21no—
120C22×C3×C5#9511yescharacteristic
121C22×C3×C5#9911yescharacteristic
122C22×C3×C5#9911yescharacteristic
123C2×C3×D5—21no—
124C2×C3×D5—21no—
125C2×C3×D5—21no—
126C2×C3×D5—21no—
127C2×C3×D5—21no—
128C2×C3×D5—42no—
129C2×C3×D5—42no—
130C2×C3×D5#10412yes—
131C3×AGL(1,𝔽5)—82no—
132C3×C4×C5—21no—
133C3×C4×C5—21no—
134C3×C4×C5—42no—
135C3×Dic5—21no—
136C3×Dic5—21no—
137C3×Dic5—42no—
138C24×C3—51no—
139C22×C3×C4—21no—
140C22×C3×C4—51no—
141C22×C3×C4—51no—
142C22×C3×C4—51no—
143C22×C3×C4—51no—
144C22×C3×C4—51no—
145C22×C3×C4—101no—
146C22×C3×C4—102no—
147C22×C3×C4—201no—
148C22×C3×C4—201no—
149C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
150C3×M16—201no—
151C3×M16—201no—
152C2×C3×C8—102no—
153C3×C22⋊C4—104no—
154C3×C22⋊C4—201no—
155C3×C22⋊C4—201no—
156C3×C42—41no—
157C3×C42—201no—
158C23×C5C3×C22⋊C411yescharacteristic
159C22×D5—21no—
160C22×D5—21no—
161C22×D5—22no—
162C22×D5—41no—
163C22×D5C3×C22⋊C411yescharacteristic
164C22×D5C3×C4⋊C411yescharacteristic
165C2×AGL(1,𝔽5)—42no—
166C2×AGL(1,𝔽5)—42no—
167C2×AGL(1,𝔽5)—42no—
168C2×C4×C5—21no—
169C2×C4×C5—21no—
170C2×C4×C5—22no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5C3×C4211yescharacteristic
174C2×C4×C5C3×C4⋊C411yescharacteristic
175C2×Dic5—21no—
176C2×Dic5—21no—
177C2×Dic5—22no—
178C2×Dic5—41no—
179C2×Dic5—41no—
180C2×Dic5C3×C22⋊C411yescharacteristic
181C2×Dic5C3×C4⋊C411yescharacteristic
182C4×D5—41no—
183C4×D5—41no—
184C4×D5—41no—
185C4×D5—41no—
186#170—42no—
187C23×C4—51no—
188C2×M16—101no—
189C2×C22⋊C4—101no—
190C2×C22⋊C4—101no—
191C2×C42—51no—
192C2×C42C3×AGL(1,𝔽5)11yescharacteristic, complemented
193C22⋊C8—52no—
194C22⋊C8—101no—
195#95—52no—
196#95—101no—
197#95—101no—
198C2×C3×C5—21no—
199C2×C3×C5—21no—
200C2×C3×C5H(3,ℤ/4)11yescharacteristic
201C2×C3×C5#33911yescharacteristic
202C2×C3×C5#34211yescharacteristic
203C3×D5—22no—
204C3×D5—41no—
205C23×C3—51no—
206C23×C3—51no—
207C23×C3—102no—
208C23×C3—101no—
209C23×C3—101no—
210C23×C3—201no—
211C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—51no—
218C2×C3×C4—51no—
219C2×C3×C4—101no—
220C2×C3×C4—101no—
221C2×C3×C4—102no—
222C2×C3×C4—201no—
223C2×C3×C4—201no—
224C2×C3×C4—201no—
225C2×C3×C4—201no—
226C2×C3×C4—201no—
227C2×C3×C4—201no—
228C2×C3×C4—202no—
229C2×C3×C4—202no—
230C2×C3×C4—202no—
231C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
232C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic
233C3×C8—202no—
234C22×C5—21no—
235C22×C5—21no—
236C22×C5C3×#9511yescharacteristic
237C22×C5C3×#9911yescharacteristic
238C22×C5C3×#9911yescharacteristic
239C2×D5—21no—
240C2×D5—21no—
241C2×D5—21no—
242C2×D5—21no—
243C2×D5—21no—
244C2×D5—42no—
245C2×D5—42no—
246C2×D5C3×#10412yes—
247AGL(1,𝔽5)—82no—
248C4×C5—21no—
249C4×C5—21no—
250C4×C5—42no—
251Dic5—21no—
252Dic5—21no—
253Dic5—42no—
254C24—51no—
255C22×C4—21no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—101no—
262C22×C4—102no—
263C22×C4—201no—
264C22×C4—201no—
265C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
266M16—201no—
267M16—201no—
268C2×C8—102no—
269C22⋊C4—104no—
270C22⋊C4—201no—
271C22⋊C4—201no—
272C42—41no—
273C42—201no—
274C3×C5#128411yescharacteristic, complemented, Hall
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—52no—
278C22×C3—101no—
279C22×C3—101no—
280C22×C3—101no—
281C22×C3—101no—
282C22×C3—101no—
283C22×C3—202no—
284C22×C3—202no—
285C22×C3#399911yescharacteristic
286C22×C3(C2×AGL(1,𝔽5))⋊C411yescharacteristic
287C22×C3#401111yescharacteristic
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—42no—
291C3×C4—101no—
292C3×C4—101no—
293C3×C4—202no—
294C3×C4—402no—
295C2×C5—21no—
296C2×C5—21no—
297C2×C5C3×H(3,ℤ/4)11yescharacteristic
298C2×C5C3×#33911yescharacteristic
299C2×C5C3×#34211yescharacteristic
300D5—22no—
301D5—41no—
302C23—51no—
303C23—51no—
304C23—102no—
305C23—101no—
306C23—101no—
307C23—201no—
308C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—51no—
315C2×C4—51no—
316C2×C4—101no—
317C2×C4—101no—
318C2×C4—102no—
319C2×C4—201no—
320C2×C4—201no—
321C2×C4—201no—
322C2×C4—201no—
323C2×C4—201no—
324C2×C4—201no—
325C2×C4—202no—
326C2×C4—202no—
327C2×C4—202no—
328C2×C4C3×C4×AGL(1,𝔽5)11yescharacteristic
329C2×C4C3×C4⋊AGL(1,𝔽5)11yescharacteristic
330C8—202no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—102no—
334C2×C3—201no—
335C2×C3#6486211yescharacteristic
336C2×C3#6486311yescharacteristic
337C2×C3#6491711yescharacteristic
338C5C3×#128411yescharacteristic, complemented, Sylow
339C22—21no—
340C22—21no—
341C22—52no—
342C22—101no—
343C22—101no—
344C22—101no—
345C22—101no—
346C22—101no—
347C22—202no—
348C22—202no—
349C22C3×#399911yescharacteristic
350C22C3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
351C22C3×#401111yescharacteristic
352C4—21no—
353C4—21no—
354C4—42no—
355C4—101no—
356C4—101no—
357C4—202no—
358C4—402no—
359C3#1059869211yescharacteristic, direct product factor, Sylow
360C2—21no—
361C2—21no—
362C2—102no—
363C2—201no—
364C2C3×#6486211yescharacteristic
365C2C3×#6486311yescharacteristic
366C2C3×#6491711yescharacteristic
subgroupG/H
centerC22×C3(C2×AGL(1,𝔽5))⋊C4
commutatorC2×C4×C5C3×C42
FrattiniC22×C4C2×C3×AGL(1,𝔽5)
FittingC2×C3×C42×C5C4
socleC22×C3×C5#95
prime radicalC23×C3×D5D4
cosocleC22×C4×D5C22×C3
hypercenterC2×C3×C42AGL(1,𝔽5)
hypocenterC5C3×#1284
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
4. #10598692C3direct (N × H)
8. C2×C3×C42×C5C4semidirect
10. C2×C3×C4×Dic5C4semidirect
12. C3×(C2×AGL(1,𝔽5))⋊C4C4semidirect
14. C3×#4003C4semidirect
46. C2×C42×C5C3×C4semidirect
48. C2×C4×Dic5C3×C4semidirect
50. (C2×AGL(1,𝔽5))⋊C4C3×C4semidirect
52. #4003C3×C4semidirect
89. C2×C3×C42AGL(1,𝔽5)semidirect
192. C2×C42C3×AGL(1,𝔽5)semidirect
274. C3×C5#1284semidirect
338. C5C3×#1284semidirect
359. C3#10598692direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#657642
C3×#657552
C3×C4×#6852
#105986923
C3×#12845