Finite Group Databasebeta

← C3×#10604980 #14313309

Subgroups — C3×#10604980 #14313309

Order 1920.

468 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 108 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×#66150C211yesmaximal, characteristic
2C3×#65761C211yesmaximal, characteristic
3C3×#65745C211yesmaximal, characteristic
4C3×#64916C214yesmaximal, complemented
5#10604980C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×Q8C411yescharacteristic, complemented
7C2×C3×#709C2211yescharacteristic
8C3×#4130—21no—
9C3×D5⋊M16—21no—
10C3×#4093—21no—
11C3×#4092—21no—
12C3×#4092—21no—
13C3×#4090—21no—
14C3×#4075C412yescomplemented
15C3×#4075C2212yes—
16C3×#4072C411yescharacteristic, complemented
17C3×#4013C2212yes—
18C3×#4011C2212yes—
19C3×#4005—21no—
20C3×#3999—21no—
21C3×#2123—51nomaximal, Hall
22#66150C2×C311yescharacteristic
23#65761C2×C311yescharacteristic
24#65745C2×C311yescharacteristic
25#64916C2×C314yescomplemented
26C2×C3×C5×Q8—21no—
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8—41no—
29C2×C3×C5×Q8—41no—
30C22×C3×C4×C5C2×C411yescharacteristic
31C22×C3×C4×C5C2×C412yes—
32C2×C3×C22⋊D5C2×C411yescharacteristic
33C2×C3×C22⋊D5C2311yescharacteristic
34C3×#709—41no—
35C3×#709—41no—
36C2×C3×D20D411yescharacteristic
37C2×C3×C4×D5D411yescharacteristic
38C2×C3×C4×D5D411yescharacteristic
39C2×C3×Dic10D411yescharacteristic
40C3×C22⋊AGL(1,𝔽5)—22no—
41C3×#704—22no—
42C3×C4⋊AGL(1,𝔽5)—41no—
43C3×C4×AGL(1,𝔽5)—41no—
44C3×#700—41no—
45C3×D5⋊C8—41no—
46C3×C22⋊Dic5C2×C412yes—
47C3×#685—22no—
48C3×#685—22no—
49C3×C4⋊Dic5—21no—
50C3×#683—22no—
51C3×#683—22no—
52C3×C4×Dic5—21no—
53C3×#536—51no—
54C3×#413—51no—
55C3×#410—51no—
56C3×#352—54no—
57C22×C5×Q8C3×C411yescharacteristic, complemented
58C2×#709C22×C311yescharacteristic
59#4130—21no—
60D5⋊M16—21no—
61#4093—21no—
62#4092—21no—
63#4092—21no—
64#4090—21no—
65#4075C3×C412yescomplemented
66#4075C22×C312yes—
67#4072C3×C411yescharacteristic, complemented
68#4013C22×C312yes—
69#4011C22×C312yes—
70#4005—21no—
71#3999—21no—
72#2123—51noSylow
73C23×C3×C5C22×C411yescharacteristic
74C22×C3×D5C22⋊C411yescharacteristic
75C22×C3×D5C2×D411yescharacteristic
76C2×C3×AGL(1,𝔽5)—42no—
77C3×C5×Q8—42no—
78C3×C5×Q8—41no—
79C3×C5×Q8—41no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—22no—
83C2×C3×C4×C5—42no—
84C2×C3×C4×C5C22⋊C411yescharacteristic
85C2×C3×C4×C5C22⋊C411yescharacteristic
86C3×C22⋊D5—41no—
87C3×C22⋊D5—41no—
88C2×C3×Dic5—22no—
89C2×C3×Dic5C22⋊C411yescharacteristic
90C2×C3×Dic5C2×D411yescharacteristic
91C3×D20—41no—
92C3×C4×D5—21no—
93C3×C4×D5—21no—
94C3×C4×D5—21no—
95C3×C4×D5—21no—
96C3×Dic10—41no—
97C3×#170—42no—
98C2×C3×#41—51no—
99C22×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
100C2×C3×M16—101no—
101C3×C4⋊Q8—101no—
102C3×#124—101no—
103C3×#123—52no—
104C3×#123—52no—
105C3×C22⋊Q8—101no—
106C3×C22⋊Q8—101no—
107C3×#117—51no—
108C3×#117—101no—
109C3×#101—101no—
110C3×#100—52no—
111C3×#99—52no—
112C3×#99—101no—
113C2×C5×Q8—21no—
114C2×C5×Q8—21no—
115C2×C5×Q8—41no—
116C2×C5×Q8—41no—
117C22×C4×C5C2×C3×C411yescharacteristic
118C22×C4×C5C2×C3×C412yes—
119C2×C22⋊D5C2×C3×C411yescharacteristic
120C2×C22⋊D5C23×C311yescharacteristic
121#709—41no—
122#709—41no—
123C2×D20C3×D411yescharacteristic
124C2×C4×D5C3×D411yescharacteristic
125C2×C4×D5C3×D411yescharacteristic
126C2×Dic10C3×D411yescharacteristic
127C22⋊AGL(1,𝔽5)—22no—
128#704—22no—
129C4⋊AGL(1,𝔽5)—41no—
130C4×AGL(1,𝔽5)—41no—
131#700—41no—
132D5⋊C8—41no—
133C22⋊Dic5C2×C3×C412yes—
134#685—22no—
135#685—22no—
136C4⋊Dic5—21no—
137#683—22no—
138#683—22no—
139C4×Dic5—21no—
140#536—51no—
141#413—51no—
142#410—51no—
143#352—54no—
144C22×C3×C5—21no—
145C22×C3×C5—41no—
146C22×C3×C5C2×C22⋊C411yescharacteristic
147C2×C3×D5—21no—
148C2×C3×D5—21no—
149C2×C3×D5—41no—
150C2×C3×D5—41no—
151C3×AGL(1,𝔽5)—82no—
152C3×C4×C5—21no—
153C3×C4×C5—42no—
154C3×C4×C5#9911yescharacteristic
155C3×C4×C5#9911yescharacteristic
156C3×Dic5—21no—
157C3×Dic5—21no—
158C3×Dic5—42no—
159C3×#41—201no—
160C3×#41—201no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—41no—
164C2×C3×Q8—41no—
165C2×C3×Q8—51no—
166C2×C3×D4—51no—
167C2×C3×D4—51no—
168C2×C3×D4—51no—
169C22×C3×C4—51no—
170C22×C3×C4—51no—
171C22×C3×C4C2×AGL(1,𝔽5)12yes—
172C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
173C3×M16—102no—
174C3×M16—201no—
175C2×C3×C8—201no—
176C3×C4⋊C4—102no—
177C3×C4⋊C4—102no—
178C3×C4⋊C4—101no—
179C3×C4⋊C4—201no—
180C3×C22⋊C4—52no—
181C3×C22⋊C4—102no—
182C3×C22⋊C4—102no—
183C3×C22⋊C4—102no—
184C3×C42—101no—
185C3×C42—201no—
186C23×C5C22×C3×C411yescharacteristic
187C22×D5C3×C22⋊C411yescharacteristic
188C22×D5C2×C3×D411yescharacteristic
189C2×AGL(1,𝔽5)—42no—
190C5×Q8—42no—
191C5×Q8—41no—
192C5×Q8—41no—
193C2×C4×C5—21no—
194C2×C4×C5—21no—
195C2×C4×C5—22no—
196C2×C4×C5—42no—
197C2×C4×C5C3×C22⋊C411yescharacteristic
198C2×C4×C5C3×C22⋊C411yescharacteristic
199C22⋊D5—41no—
200C22⋊D5—41no—
201C2×Dic5—22no—
202C2×Dic5C3×C22⋊C411yescharacteristic
203C2×Dic5C2×C3×D411yescharacteristic
204D20—41no—
205C4×D5—21no—
206C4×D5—21no—
207C4×D5—21no—
208C4×D5—21no—
209Dic10—41no—
210#170—42no—
211C2×#41—51no—
212C22×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
213C2×M16—101no—
214C4⋊Q8—101no—
215#124—101no—
216#123—52no—
217#123—52no—
218C22⋊Q8—101no—
219C22⋊Q8—101no—
220#117—51no—
221#117—101no—
222#101—101no—
223#100—52no—
224#99—52no—
225#99—101no—
226C2×C3×C5—21no—
227C2×C3×C5—41no—
228C2×C3×C5C2×#9911yescharacteristic
229C3×D5—41no—
230C3×D5—41no—
231C23×C3—51no—
232C23×C3—51no—
233C23×C3C22×AGL(1,𝔽5)11yescharacteristic
234C3×Q8—42no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—201no—
238C3×D4—201no—
239C3×D4—201no—
240C3×D4—201no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—42no—
245C2×C3×C4—51no—
246C2×C3×C4—51no—
247C2×C3×C4—101no—
248C2×C3×C4—101no—
249C2×C3×C4—101no—
250C2×C3×C4—101no—
251C2×C3×C4—102no—
252C2×C3×C4—202no—
253C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
254C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
255C3×C8—202no—
256C22×C5—21no—
257C22×C5—41no—
258C22×C5C2×C3×C22⋊C411yescharacteristic
259C2×D5—21no—
260C2×D5—21no—
261C2×D5—41no—
262C2×D5—41no—
263AGL(1,𝔽5)—82no—
264C4×C5—21no—
265C4×C5—42no—
266C4×C5C3×#9911yescharacteristic
267C4×C5C3×#9911yescharacteristic
268Dic5—21no—
269Dic5—21no—
270Dic5—42no—
271#41—201no—
272#41—201no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—51no—
278C2×D4—51no—
279C2×D4—51no—
280C2×D4—51no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
284C22×C4C2×C3×AGL(1,𝔽5)12yes—
285M16—102no—
286M16—201no—
287C2×C8—201no—
288C4⋊C4—102no—
289C4⋊C4—102no—
290C4⋊C4—101no—
291C4⋊C4—201no—
292C22⋊C4—52no—
293C22⋊C4—102no—
294C22⋊C4—102no—
295C22⋊C4—102no—
296C42—101no—
297C42—201no—
298C3×C5#212311yescharacteristic, complemented, Hall
299C22×C3—21no—
300C22×C3—41no—
301C22×C3—101no—
302C22×C3—101no—
303C22×C3—201no—
304C22×C3—201no—
305C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—101no—
309C3×C4—101no—
310C3×C4—202no—
311C3×C4—402no—
312C3×C4#401111yescharacteristic
313C3×C4#401111yescharacteristic
314C2×C5—21no—
315C2×C5—41no—
316C2×C5C2×C3×#9911yescharacteristic
317D5—41no—
318D5—41no—
319C23—51no—
320C23—51no—
321C23C22×C3×AGL(1,𝔽5)11yescharacteristic
322Q8—42no—
323Q8—41no—
324Q8—41no—
325Q8—201no—
326D4—201no—
327D4—201no—
328D4—201no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—51no—
334C2×C4—51no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—102no—
340C2×C4—202no—
341C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
342C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
343C8—202no—
344C2×C3—21no—
345C2×C3—41no—
346C2×C3—201no—
347C2×C3—201no—
348C2×C3C2×#401111yescharacteristic
349C5C3×#212311yescharacteristic, complemented, Sylow
350C22—21no—
351C22—41no—
352C22—101no—
353C22—101no—
354C22—201no—
355C22—201no—
356C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
357C4—21no—
358C4—42no—
359C4—101no—
360C4—101no—
361C4—202no—
362C4—402no—
363C4C3×#401111yescharacteristic
364C4C3×#401111yescharacteristic
365C3#1060498011yescharacteristic, direct product factor, Sylow
366C2—21no—
367C2—41no—
368C2—201no—
369C2—201no—
370C2C2×C3×#401111yescharacteristic
subgroupG/H
centerC2×C3C2×#4011
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×Q8C4
socleC2×C3×C5C2×#99
prime radicalC2×C3×#709C22
cosocleC2×C22⋊D5C23×C3
hypercenterC22×C3×Q8AGL(1,𝔽5)
hypocenterC5C3×#2123
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
4. C3×#64916C2semidirect
5. #10604980C3direct (N × H)
6. C22×C3×C5×Q8C4semidirect
14. C3×#4075C4semidirect
16. C3×#4072C4semidirect
25. #64916C2×C3semidirect
57. C22×C5×Q8C3×C4semidirect
65. #4075C3×C4semidirect
67. #4072C3×C4semidirect
99. C22×C3×Q8AGL(1,𝔽5)semidirect
212. C22×Q8C3×AGL(1,𝔽5)semidirect
298. C3×C5#2123semidirect
349. C5C3×#2123semidirect
365. C3#10604980direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661502
C3×#657612
C3×#657452
C3×#649162
#106049803
C3×#21235