Finite Group Databasebeta

← C3×#10602739 #14316737

Subgroups — C3×#10602739 #14316737

Order 1920.

492 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 90 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×C5×#559C211yesmaximal, characteristic, complemented
2C3×#65366C211yesmaximal, characteristic
3C3×#65342C212yesmaximal, complemented
4C3×#64765C211yesmaximal, characteristic, complemented
5C3×#64759C212yesmaximal, complemented
6#10602739C311yesmaximal, characteristic, direct product factor, Hall
7C3×#4144C2211yescharacteristic
8C3×C5×#127C2211yescharacteristic
9C3×C5×#124—22no—
10C3×C5×#123—21no—
11C3×C5×#123C2212yes—
12C3×C4×C5×Q8—21no—
13C3×C4×C5×D4—22no—
14C3×#4078—21no—
15C3×#4075—22no—
16C3×#3966C2212yes—
17C3×#3965C2211yescharacteristic
18C3×#3938—22no—
19C3×#3937—21no—
20C3×#2198—51nomaximal, Hall
21C5×#559C2×C311yescharacteristic, complemented
22#65366C2×C311yescharacteristic
23#65342C2×C312yescomplemented
24#64765C2×C311yescharacteristic, complemented
25#64759C2×C312yescomplemented
26C2×C3×C5×Q8—21no—
27C2×C3×C5×D4—22no—
28C2×C3×C5×D4D411yescharacteristic
29C2×C3×C5×D4C2311yescharacteristic
30C22×C3×C4×C5—21no—
31C22×C3×C4×C5D412yes—
32C2×C3×C22⋊D5—22no—
33C3×#711—41no—
34C3×D4×D5—41no—
35C3×#709—21no—
36C3×C5×C4⋊C4—21no—
37C3×C5×C4⋊C4—22no—
38C3×C5×C22⋊C4—22no—
39C3×C5×C22⋊C4—21no—
40C3×C5×C22⋊C4—22no—
41C3×C5×C22⋊C4D412yes—
42C3×C42×C5—22no—
43C3×C42×C5D411yescharacteristic
44C3×C22⋊Dic5—22no—
45C3×#687—41no—
46C3×#686—41no—
47C3×#685—42no—
48C3×#683—42no—
49C3×#681—21no—
50C3×#559D511yescharacteristic, complemented
51C3×#458—52no—
52C3×#453—51no—
53C3×#353—51no—
54C3×#352—52no—
55#4144C22×C311yescharacteristic
56C5×#127C22×C311yescharacteristic
57C5×#124—22no—
58C5×#123—21no—
59C5×#123C22×C312yes—
60C4×C5×Q8—21no—
61C4×C5×D4—22no—
62#4078—21no—
63#4075—22no—
64#3966C22×C312yes—
65#3965C22×C311yescharacteristic
66#3938—22no—
67#3937—21no—
68#2198—51noSylow
69C23×C3×C5—21no—
70C23×C3×C5C2×D412yes—
71C22×C3×D5—22no—
72C3×C5×Q8—42no—
73C3×C5×D4—41no—
74C3×C5×D4—44no—
75C3×C5×D4—41no—
76C2×C3×C4×C5—21no—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5—42no—
80C2×C3×C4×C5—81no—
81C2×C3×C4×C5C2×D411yescharacteristic
82C3×C22⋊D5—42no—
83C3×C22⋊D5—41no—
84C2×C3×Dic5—22no—
85C2×C3×Dic5—42no—
86C3×D20—21no—
87C3×C4×D5—41no—
88C3×Dic10—21no—
89C3×#168—41no—
90C3×H(5,𝔽2)—51no—
91C3×AGL(1,ℤ/8)—101no—
92C3×#127C2×D511yescharacteristic
93C3×#124—22no—
94C3×#123—21no—
95C3×#123—102no—
96C3×#123C2×D512yes—
97C3×C4×Q8—21no—
98C3×C4×D4—22no—
99C3×#104—101no—
100C3×#100—51no—
101C3×#99—52no—
102C3×#99—102no—
103C2×C5×Q8—21no—
104C2×C5×D4—22no—
105C2×C5×D4C3×D411yescharacteristic
106C2×C5×D4C23×C311yescharacteristic
107C22×C4×C5—21no—
108C22×C4×C5C3×D412yes—
109C2×C22⋊D5—22no—
110#711—41no—
111D4×D5—41no—
112#709—21no—
113C5×C4⋊C4—21no—
114C5×C4⋊C4—22no—
115C5×C22⋊C4—22no—
116C5×C22⋊C4—21no—
117C5×C22⋊C4—22no—
118C5×C22⋊C4C3×D412yes—
119C42×C5—22no—
120C42×C5C3×D411yescharacteristic
121C22⋊Dic5—22no—
122#687—41no—
123#686—41no—
124#685—42no—
125#683—42no—
126#681—21no—
127#559C3×D511yescharacteristic, complemented
128#458—52no—
129#453—51no—
130#353—51no—
131#352—52no—
132C22×C3×C5—22no—
133C22×C3×C5—42no—
134C22×C3×C5—42no—
135C22×C3×C5—41no—
136C22×C3×C5#12011yescharacteristic
137C2×C3×D5—41no—
138C2×C3×D5—82no—
139C3×C4×C5—22no—
140C3×C4×C5—21no—
141C3×C4×C5—42no—
142C3×C4×C5—42no—
143C3×Dic5—41no—
144C3×Dic5—82no—
145C3×#41—101no—
146C3×#41—201no—
147C2×C3×Q8—21no—
148C2×C3×D4—22no—
149C2×C3×D4—102no—
150C2×C3×D4—201no—
151C2×C3×D4C22⋊D511yescharacteristic
152C2×C3×D4C22×D511yescharacteristic
153C22×C3×C4—21no—
154C22×C3×C4C22⋊D512yes—
155C3×SD16—201no—
156C3×D8—201no—
157C3×M16—101no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—202no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—102no—
165C3×C22⋊C4—202no—
166C3×C22⋊C4C22⋊D512yes—
167C3×C42—22no—
168C3×C42C22⋊D511yescharacteristic
169C23×C5—21no—
170C23×C5C2×C3×D412yes—
171C22×D5—22no—
172C5×Q8—42no—
173C5×D4—41no—
174C5×D4—41no—
175C5×D4—44no—
176C2×C4×C5—21no—
177C2×C4×C5—22no—
178C2×C4×C5—22no—
179C2×C4×C5—42no—
180C2×C4×C5—81no—
181C2×C4×C5C2×C3×D411yescharacteristic
182C22⋊D5—42no—
183C22⋊D5—41no—
184C2×Dic5—22no—
185C2×Dic5—42no—
186D20—21no—
187C4×D5—41no—
188Dic10—21no—
189#168—41no—
190H(5,𝔽2)—51no—
191AGL(1,ℤ/8)—101no—
192#127C2×C3×D511yescharacteristic
193#124—22no—
194#123—21no—
195#123—102no—
196#123C2×C3×D512yes—
197C4×Q8—21no—
198C4×D4—22no—
199#104—101no—
200#100—51no—
201#99—52no—
202#99—102no—
203C2×C3×C5—21no—
204C2×C3×C5—42no—
205C2×C3×C5—81no—
206C2×C3×C5B(4,𝔽2)11yescharacteristic
207C3×D5—81no—
208C23×C3—21no—
209C23×C3—102no—
210C23×C3C2×C22⋊D512yes—
211C3×Q8—42no—
212C3×Q8—101no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—44no—
216C3×D4—101no—
217C3×D4—202no—
218C3×D4—201no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—42no—
223C2×C3×C4—81no—
224C2×C3×C4—102no—
225C2×C3×C4—201no—
226C2×C3×C4—202no—
227C2×C3×C4C2×C22⋊D511yescharacteristic
228C3×C8—201no—
229C22×C5—22no—
230C22×C5—42no—
231C22×C5—42no—
232C22×C5—41no—
233C22×C5C3×#12011yescharacteristic
234C2×D5—41no—
235C2×D5—82no—
236C4×C5—22no—
237C4×C5—21no—
238C4×C5—42no—
239C4×C5—42no—
240Dic5—41no—
241Dic5—82no—
242#41—101no—
243#41—201no—
244C2×Q8—21no—
245C2×D4—22no—
246C2×D4—102no—
247C2×D4—201no—
248C2×D4C3×C22⋊D511yescharacteristic
249C2×D4C22×C3×D511yescharacteristic
250C22×C4—21no—
251C22×C4C3×C22⋊D512yes—
252SD16—201no—
253D8—201no—
254M16—101no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—202no—
258C22⋊C4—22no—
259C22⋊C4—21no—
260C22⋊C4—22no—
261C22⋊C4—102no—
262C22⋊C4—202no—
263C22⋊C4C3×C22⋊D512yes—
264C42—22no—
265C42C3×C22⋊D511yescharacteristic
266C3×C5#219811yescharacteristic, complemented, Hall
267C22×C3—22no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—41no—
271C22×C3—201no—
272C22×C3—402no—
273C22×C3#409911yescharacteristic
274C3×C4—22no—
275C3×C4—21no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—201no—
279C3×C4—402no—
280C2×C5—21no—
281C2×C5—42no—
282C2×C5—81no—
283C2×C5C3×B(4,𝔽2)11yescharacteristic
284D5—81no—
285C23—21no—
286C23—102no—
287C23C2×C3×C22⋊D512yes—
288Q8—42no—
289Q8—101no—
290D4—41no—
291D4—41no—
292D4—44no—
293D4—101no—
294D4—202no—
295D4—201no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—102no—
302C2×C4—201no—
303C2×C4—202no—
304C2×C4C2×C3×C22⋊D511yescharacteristic
305C8—201no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—81no—
309C2×C3—401no—
310C2×C3#6553311yescharacteristic
311C5C3×#219811yescharacteristic, complemented, Sylow
312C22—22no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—201no—
317C22—402no—
318C22C3×#409911yescharacteristic
319C4—22no—
320C4—21no—
321C4—42no—
322C4—42no—
323C4—201no—
324C4—402no—
325C3#1060273911yescharacteristic, direct product factor, Sylow
326C2—21no—
327C2—42no—
328C2—81no—
329C2—401no—
330C2C3×#6553311yescharacteristic
subgroupG/H
centerC2×C3#65533
commutator, cosocleC2×C5×D4C23×C3
FrattiniC2×D4C22×C3×D5
FittingC3×C5×#559C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC3×#65366C2
hypercenterC3×#559D5
hypocenterC5C3×#2198
LCSG ▷ C2×C5×D4 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#559

Complements

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

NH ≅ G/Ntype
1. C3×C5×#559C2semidirect
3. C3×#65342C2semidirect
4. C3×#64765C2semidirect
5. C3×#64759C2semidirect
6. #10602739C3direct (N × H)
21. C5×#559C2×C3semidirect
23. #65342C2×C3semidirect
24. #64765C2×C3semidirect
25. #64759C2×C3semidirect
50. C3×#559D5semidirect
127. #559C3×D5semidirect
266. C3×C5#2198semidirect
311. C5C3×#2198semidirect
325. C3#10602739direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5592
C3×#653662
C3×#653422
C3×#647652
C3×#647592
#106027393
C3×#21985