Finite Group Databasebeta

← C3×#10598928 #14288393

Subgroups — C3×#10598928 #14288393

Order 1920.

508 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 106 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×#65844C211yesmaximal, characteristic
2C3×#65272C211yesmaximal, characteristic, complemented
3C3×#65025C212yesmaximal, complemented
4C3×C5×#324C211yesmaximal, characteristic, complemented
5C3×#64690C212yesmaximal, complemented
6#10598928C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#709—21no—
8C2×C3×C5×C4⋊C4C2211yescharacteristic
9C3×#4076—21no—
10C3×#4076C2212yes—
11C2×C3×C4⋊Dic5C2211yescharacteristic
12C2×C3×#683—21no—
13C3×#4043—22no—
14C3×#4041—22no—
15C3×C5×C22⋊C8C2212yes—
16C3×#3963—21no—
17C3×#3963C2211yescharacteristic
18C3×#3953—22no—
19C3×#3950—22no—
20C3×#1610—51nomaximal, Hall
21#65844C2×C311yescharacteristic
22#65272C2×C311yescharacteristic, complemented
23#65025C2×C312yescomplemented
24C5×#324C2×C311yescharacteristic, complemented
25#64690C2×C312yescomplemented
26C22×C3×C4×C5D411yescharacteristic
27C22×C3×C4×C5D411yescharacteristic
28C22×C3×C4×C5C2311yescharacteristic
29C2×C3×C22⋊D5—22no—
30C22×C3×Dic5—21no—
31C22×C3×Dic5—21no—
32C3×#709—42no—
33C3×#709—42no—
34C2×C3×D20—21no—
35C2×C3×D20—21no—
36C2×C3×D20D412yes—
37C2×C3×C4×D5—22no—
38C2×C3×Dic10—21no—
39C2×C3×C5×C8—22no—
40C3×C5×C4⋊C4—22no—
41C3×#685—22no—
42C3×#685—22no—
43C3×C4⋊Dic5—21no—
44C3×C4⋊Dic5—22no—
45C3×C4⋊Dic5D412yes—
46C3×#683—42no—
47C3×#537—51no—
48C3×#480—52no—
49C3×#398—51no—
50C3×#327—52no—
51C3×#324D511yescharacteristic, complemented
52C2×#709—21no—
53C2×C5×C4⋊C4C22×C311yescharacteristic
54#4076—21no—
55#4076C22×C312yes—
56C2×C4⋊Dic5C22×C311yescharacteristic
57C2×#683—21no—
58#4043—22no—
59#4041—22no—
60C5×C22⋊C8C22×C312yes—
61#3963—21no—
62#3963C22×C311yescharacteristic
63#3953—22no—
64#3950—22no—
65#1610—51noSylow
66C23×C3×C5C2×D411yescharacteristic
67C22×C3×D5—22no—
68C2×C3×C4×C5—21no—
69C2×C3×C4×C5—21no—
70C2×C3×C4×C5—21no—
71C2×C3×C4×C5—41no—
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5D812yes—
74C2×C3×C4×C5C2×D412yes—
75C3×C22⋊D5—42no—
76C3×C22⋊D5—42no—
77C2×C3×Dic5—22no—
78C2×C3×Dic5—22no—
79C2×C3×Dic5—41no—
80C2×C3×Dic5—41no—
81C2×C3×Dic5—41no—
82C2×C3×Dic5—41no—
83C2×C3×Dic5—41no—
84C3×D20—42no—
85C3×D20—42no—
86C3×D20—81no—
87C3×C4×D5—44no—
88C3×Dic10—42no—
89C3×C5×C8—42no—
90C2×C3×#41—101no—
91C3×C22⋊Q8—102no—
92C3×#121—52no—
93C3×#121—102no—
94C3×#121—101no—
95C2×C3×C4⋊C4—51no—
96C2×C3×C4⋊C4—101no—
97C2×C3×C4⋊C4C2×D511yescharacteristic
98C3×#107—102no—
99C3×#102—102no—
100C3×C22⋊C8C2×D512yes—
101C3×#95—51no—
102C3×#95—101no—
103C22×C4×C5C3×D411yescharacteristic
104C22×C4×C5C3×D411yescharacteristic
105C22×C4×C5C23×C311yescharacteristic
106C2×C22⋊D5—22no—
107C22×Dic5—21no—
108C22×Dic5—21no—
109#709—42no—
110#709—42no—
111C2×D20—21no—
112C2×D20—21no—
113C2×D20C3×D412yes—
114C2×C4×D5—22no—
115C2×Dic10—21no—
116C2×C5×C8—22no—
117C5×C4⋊C4—22no—
118#685—22no—
119#685—22no—
120C4⋊Dic5—21no—
121C4⋊Dic5—22no—
122C4⋊Dic5C3×D412yes—
123#683—42no—
124#537—51no—
125#480—52no—
126#398—51no—
127#327—52no—
128#324C3×D511yescharacteristic, complemented
129C22×C3×C5—21no—
130C22×C3×C5—21no—
131C22×C3×C5#12011yescharacteristic
132C22×C3×C5C2×D811yescharacteristic
133C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
134C2×C3×D5—42no—
135C2×C3×D5—42no—
136C2×C3×D5—42no—
137C3×C4×C5—22no—
138C3×C4×C5—22no—
139C3×C4×C5—41no—
140C3×Dic5—42no—
141C3×Dic5—42no—
142C3×Dic5—81no—
143C3×#41—202no—
144C3×#41—202no—
145C2×C3×Q8—101no—
146C2×C3×D4—52no—
147C2×C3×D4—101no—
148C2×C3×D4—101no—
149C2×C3×D4—102no—
150C22×C3×C4—102no—
151C22×C3×C4—101no—
152C22×C3×C4—101no—
153C22×C3×C4D2011yescharacteristic
154C22×C3×C4D2011yescharacteristic
155C22×C3×C4C22×D511yescharacteristic
156C2×C3×C8—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—52no—
159C3×C4⋊C4—101no—
160C3×C4⋊C4—102no—
161C3×C4⋊C4—202no—
162C3×C22⋊C4—102no—
163C3×C22⋊C4—102no—
164C23×C5C2×C3×D411yescharacteristic
165C22×D5—22no—
166C2×C4×C5—21no—
167C2×C4×C5—21no—
168C2×C4×C5—21no—
169C2×C4×C5—41no—
170C2×C4×C5—41no—
171C2×C4×C5C3×D812yes—
172C2×C4×C5C2×C3×D412yes—
173C22⋊D5—42no—
174C22⋊D5—42no—
175C2×Dic5—22no—
176C2×Dic5—22no—
177C2×Dic5—41no—
178C2×Dic5—41no—
179C2×Dic5—41no—
180C2×Dic5—41no—
181C2×Dic5—41no—
182D20—42no—
183D20—42no—
184D20—81no—
185C4×D5—44no—
186Dic10—42no—
187C5×C8—42no—
188C2×#41—101no—
189C22⋊Q8—102no—
190#121—52no—
191#121—102no—
192#121—101no—
193C2×C4⋊C4—51no—
194C2×C4⋊C4—101no—
195C2×C4⋊C4C2×C3×D511yescharacteristic
196#107—102no—
197#102—102no—
198C22⋊C8C2×C3×D512yes—
199#95—51no—
200#95—101no—
201C2×C3×C5—21no—
202C2×C3×C5—21no—
203C2×C3×C5#44711yescharacteristic
204C2×C3×C5#45611yescharacteristic
205C2×C3×C5#45811yescharacteristic
206C3×D5—82no—
207C23×C3—102no—
208C23×C3C2×D2011yescharacteristic
209C3×Q8—202no—
210C3×D4—202no—
211C3×D4—202no—
212C3×D4—202no—
213C3×D4—202no—
214C3×D4—401no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—102no—
221C2×C3×C4—102no—
222C2×C3×C4—204no—
223C2×C3×C4—201no—
224C2×C3×C4—201no—
225C2×C3×C4—201no—
226C2×C3×C4—201no—
227C2×C3×C4—201no—
228C2×C3×C4D4012yes—
229C2×C3×C4D4×D512yes—
230C3×C8—42no—
231C22×C5—21no—
232C22×C5—21no—
233C22×C5C3×#12011yescharacteristic
234C22×C5C2×C3×D811yescharacteristic
235C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
236C2×D5—42no—
237C2×D5—42no—
238C2×D5—42no—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—41no—
242Dic5—42no—
243Dic5—42no—
244Dic5—81no—
245#41—202no—
246#41—202no—
247C2×Q8—101no—
248C2×D4—52no—
249C2×D4—101no—
250C2×D4—101no—
251C2×D4—102no—
252C22×C4—102no—
253C22×C4—101no—
254C22×C4—101no—
255C22×C4C3×D2011yescharacteristic
256C22×C4C3×D2011yescharacteristic
257C22×C4C22×C3×D511yescharacteristic
258C2×C8—22no—
259C4⋊C4—22no—
260C4⋊C4—52no—
261C4⋊C4—101no—
262C4⋊C4—102no—
263C4⋊C4—202no—
264C22⋊C4—102no—
265C22⋊C4—102no—
266C3×C5#161011yescharacteristic, complemented, Hall
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—202no—
270C22×C3—202no—
271C22×C3—202no—
272C22×C3#402811yescharacteristic
273C22×C3C2×D4011yescharacteristic
274C22×C3#405411yescharacteristic
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—41no—
278C3×C4—202no—
279C3×C4—202no—
280C3×C4—401no—
281C2×C5—21no—
282C2×C5—21no—
283C2×C5C3×#44711yescharacteristic
284C2×C5C3×#45611yescharacteristic
285C2×C5C3×#45811yescharacteristic
286D5—82no—
287C23—102no—
288C23C2×C3×D2011yescharacteristic
289Q8—202no—
290D4—202no—
291D4—202no—
292D4—202no—
293D4—202no—
294D4—401no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—102no—
301C2×C4—102no—
302C2×C4—204no—
303C2×C4—201no—
304C2×C4—201no—
305C2×C4—201no—
306C2×C4—201no—
307C2×C4—201no—
308C2×C4C3×D4012yes—
309C2×C4C3×D4×D512yes—
310C8—42no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—402no—
314C2×C3#6502111yescharacteristic
315C2×C3#6503111yescharacteristic
316C2×C3#6504111yescharacteristic
317C5C3×#161011yescharacteristic, complemented, Sylow
318C22—21no—
319C22—21no—
320C22—202no—
321C22—202no—
322C22—202no—
323C22C3×#402811yescharacteristic
324C22C2×C3×D4011yescharacteristic
325C22C3×#405411yescharacteristic
326C4—22no—
327C4—22no—
328C4—41no—
329C4—202no—
330C4—202no—
331C4—401no—
332C3#1059892811yescharacteristic, direct product factor, Sylow
333C2—21no—
334C2—21no—
335C2—402no—
336C2C3×#6502111yescharacteristic
337C2C3×#6503111yescharacteristic
338C2C3×#6504111yescharacteristic
subgroupG/H
centerC22×C3#4028
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#324C2
socleC22×C3×C5#120
prime radicalC3×#65844C2
hypercenterC3×#324D5
hypocenterC5C3×#1610
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#324

Complements

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

NH ≅ G/Ntype
2. C3×#65272C2semidirect
3. C3×#65025C2semidirect
4. C3×C5×#324C2semidirect
5. C3×#64690C2semidirect
6. #10598928C3direct (N × H)
22. #65272C2×C3semidirect
23. #65025C2×C3semidirect
24. C5×#324C2×C3semidirect
25. #64690C2×C3semidirect
51. C3×#324D5semidirect
128. #324C3×D5semidirect
266. C3×C5#1610semidirect
317. C5C3×#1610semidirect
332. C3#10598928direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658442
C3×#652722
C3×#650252
C3×C5×#3242
C3×#646902
#105989283
C3×#16105