Finite Group Databasebeta

← C5×#3518⋊C3 #14511356

Subgroups — C5×#3518⋊C3 #14511356

Order 1920.

530 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C5×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C5×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C5×#548⋊C3C212yesmaximal, complemented
4C5×#519⋊C3C212yesmaximal, complemented
5C4×C5×#41⋊C3C211yesmaximal, characteristic, complemented
6C5×#3518C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8C22×C5×SL(2,𝔽3)C2212yes—
9C3×C5×#124—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12C4×C5×SL(2,𝔽3)C2212yes—
13#3518⋊C3C511yesmaximal, characteristic, direct product factor, Hall
14C5×Q82C2×C311yescharacteristic, complemented
15C5×#556—32no—
16C5×#556—31no—
17C5×#555—31no—
18C5×D4×Q8C2×C311yescharacteristic
19C5×#548C2×C312yescomplemented
20C5×#542—32no—
21C5×#533—31no—
22C5×#529—31no—
23C5×#519C2×C312yescomplemented
24C4×C5×#41C2×C311yescharacteristic, complemented
25C2×C3×C5×Q8—41no—
26C2×C3×C5×D4—41no—
27C5×#41⋊C3—21no—
28C2×C5×SL(2,𝔽3)—22no—
29C2×C5×SL(2,𝔽3)C2311yescharacteristic
30C3×C5×C22⋊C4—42no—
31C3×C5×C22⋊C4—42no—
32C3×C42×C5—41no—
33Q8×SL(2,𝔽3)C2×C511yescharacteristic, complemented
34D4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
35#548⋊C3C2×C512yescomplemented
36#519⋊C3C2×C512yescomplemented
37C4×#41⋊C3C2×C511yescharacteristic, complemented
38C2×C5×#41C22×C311yescharacteristic
39C22×C5×Q8C22×C312yes—
40C5×C4⋊Q8—31no—
41C5×C4⋊Q8—31no—
42C5×C4⋊Q8—32no—
43C5×#126—34no—
44C5×#126—32no—
45C5×#125—31no—
46C5×#124—34no—
47C5×#124—32no—
48C5×#124A411yescharacteristic
49C5×#123—32no—
50C5×C22⋊Q8—32no—
51C5×C22⋊Q8—31no—
52C5×C22⋊Q8—32no—
53C5×C22⋊Q8—32no—
54C5×#121—32no—
55C5×#121—31no—
56C4×C5×Q8—32no—
57C4×C5×Q8—31no—
58C4×C5×Q8—32no—
59C4×C5×Q8—32no—
60C4×C5×Q8C22×C311yescharacteristic
61C4×C5×Q8C22×C311yescharacteristic
62C4×C5×Q8C22×C312yes—
63C4×C5×D4—31no—
64C4×C5×D4—31no—
65C4×C5×D4—31no—
66C4×C5×D4—32no—
67C5×#117—32no—
68C5×#117—32no—
69C5×#117—32no—
70C5×#117—31no—
71C2×C42×C5—31no—
72#3518C3×C511yescharacteristic, complemented, Sylow
73C23×C3×C5—42no—
74C3×C5×Q8—82no—
75C3×C5×D4—82no—
76C2×C3×C4×C5—41no—
77C2×C3×C4×C5—41no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—42no—
80C5×SL(2,𝔽3)#4111yescharacteristic
81C2×#41⋊C3C22×C511yescharacteristic
82C22×SL(2,𝔽3)C22×C512yes—
83C3×#124—41no—
84C4×SL(2,𝔽3)C22×C511yescharacteristic
85C4×SL(2,𝔽3)C22×C511yescharacteristic
86C4×SL(2,𝔽3)C22×C512yes—
87C5×#41—21no—
88C5×#41—61no—
89C2×C5×Q8—22no—
90C2×C5×Q8—32no—
91C2×C5×Q8—32no—
92C2×C5×Q8—62no—
93C2×C5×Q8C2×A411yescharacteristic
94C2×C5×Q8C23×C311yescharacteristic
95C2×C5×D4—31no—
96C2×C5×D4—32no—
97C2×C5×D4C2×A411yescharacteristic
98C22×C4×C5—32no—
99C22×C4×C5—31no—
100C22×C4×C5—31no—
101C22×C4×C5—31no—
102C5×C4⋊C4—31no—
103C5×C4⋊C4—31no—
104C5×C4⋊C4—32no—
105C5×C4⋊C4—32no—
106C5×C4⋊C4—31no—
107C5×C4⋊C4—31no—
108C5×C4⋊C4—32no—
109C5×C4⋊C4—32no—
110C5×C4⋊C4—32no—
111C5×C4⋊C4—32no—
112C5×C22⋊C4—32no—
113C5×C22⋊C4—34no—
114C5×C22⋊C4—32no—
115C5×C22⋊C4—32no—
116C5×C22⋊C4—32no—
117C5×C22⋊C4C2×A412yes—
118C5×C22⋊C4C2×A412yes—
119C42×C5—34no—
120C42×C5—31no—
121C42×C5—31no—
122C42×C5—31no—
123C42×C5—32no—
124C42×C5C2×A411yescharacteristic
125Q82C2×C3×C511yescharacteristic, complemented
126#556—32no—
127#556—31no—
128#555—31no—
129D4×Q8C2×C3×C511yescharacteristic
130#548C2×C3×C512yescomplemented
131#542—32no—
132#533—31no—
133#529—31no—
134#519C2×C3×C512yescomplemented
135C4×#41C2×C3×C511yescharacteristic, complemented
136C22×C3×C5—41no—
137C22×C3×C5—82no—
138C22×C3×C5—82no—
139C22×C3×C5—82no—
140C3×C4×C5—42no—
141C3×C4×C5—81no—
142C3×C4×C5—81no—
143C3×C4×C5—82no—
144C2×C3×Q8—41no—
145C2×C3×D4—41no—
146#41⋊C3—21no—
147C2×SL(2,𝔽3)—22no—
148C2×SL(2,𝔽3)C23×C511yescharacteristic
149C3×C22⋊C4—42no—
150C3×C22⋊C4—42no—
151C3×C42—41no—
152C23×C5—31no—
153C23×C5C22×A412yes—
154C5×Q8—22no—
155C5×Q8—31no—
156C5×Q8—62no—
157C5×Q8—62no—
158C5×Q8—122no—
159C5×Q8C3×#4111yescharacteristic
160C5×D4—22no—
161C5×D4—61no—
162C5×D4—61no—
163C5×D4—122no—
164C2×C4×C5—31no—
165C2×C4×C5—31no—
166C2×C4×C5—32no—
167C2×C4×C5—32no—
168C2×C4×C5—31no—
169C2×C4×C5—61no—
170C2×C4×C5—61no—
171C2×C4×C5—62no—
172C2×C4×C5—62no—
173C2×C4×C5—62no—
174C2×C4×C5—121no—
175C2×C4×C5C22×A411yescharacteristic
176C2×C4×C5C22×A411yescharacteristic
177C2×C4×C5C22×A412yes—
178C2×C4×C5C22×A411yescharacteristic
179C2×#41C22×C3×C511yescharacteristic
180C22×Q8C22×C3×C512yes—
181C4⋊Q8—31no—
182C4⋊Q8—31no—
183C4⋊Q8—32no—
184#126—34no—
185#126—32no—
186#125—31no—
187#124—34no—
188#124—32no—
189#124C5×A411yescharacteristic
190#123—32no—
191C22⋊Q8—32no—
192C22⋊Q8—31no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195#121—32no—
196#121—31no—
197C4×Q8—32no—
198C4×Q8—31no—
199C4×Q8—32no—
200C4×Q8—32no—
201C4×Q8C22×C3×C511yescharacteristic
202C4×Q8C22×C3×C511yescharacteristic
203C4×Q8C22×C3×C512yes—
204C4×D4—31no—
205C4×D4—31no—
206C4×D4—31no—
207C4×D4—32no—
208#117—32no—
209#117—32no—
210#117—32no—
211#117—31no—
212C2×C42—31no—
213C2×C3×C5—41no—
214C2×C3×C5—41no—
215C2×C3×C5—41no—
216C2×C3×C5—162no—
217C23×C3—42no—
218C3×Q8—82no—
219C3×D4—82no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224SL(2,𝔽3)C5×#4111yescharacteristic
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—61no—
229C22×C5—61no—
230C22×C5—61no—
231C22×C5C23×A411yescharacteristic
232C4×C5—21no—
233C4×C5—22no—
234C4×C5—21no—
235C4×C5—31no—
236C4×C5—31no—
237C4×C5—62no—
238C4×C5—61no—
239C4×C5—61no—
240C4×C5—62no—
241C4×C52−1+4⋊C312yes—
242#41—21no—
243#41—61no—
244C2×Q8—22no—
245C2×Q8—32no—
246C2×Q8—32no—
247C2×Q8—62no—
248C2×Q8C2×C5×A411yescharacteristic
249C2×Q8C23×C3×C511yescharacteristic
250C2×D4—31no—
251C2×D4—32no—
252C2×D4C2×C5×A411yescharacteristic
253C22×C4—32no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—31no—
257C4⋊C4—31no—
258C4⋊C4—31no—
259C4⋊C4—32no—
260C4⋊C4—32no—
261C4⋊C4—31no—
262C4⋊C4—31no—
263C4⋊C4—32no—
264C4⋊C4—32no—
265C4⋊C4—32no—
266C4⋊C4—32no—
267C22⋊C4—32no—
268C22⋊C4—34no—
269C22⋊C4—32no—
270C22⋊C4—32no—
271C22⋊C4—32no—
272C22⋊C4C2×C5×A412yes—
273C22⋊C4C2×C5×A412yes—
274C42—34no—
275C42—31no—
276C42—31no—
277C42—31no—
278C42—32no—
279C42C2×C5×A411yescharacteristic
280C3×C5—41noHall
281C22×C3—41no—
282C22×C3—82no—
283C22×C3—82no—
284C22×C3—82no—
285C3×C4—42no—
286C3×C4—81no—
287C3×C4—82no—
288C3×C4—81no—
289C2×C5—42no—
290C2×C5—121no—
291C2×C5A4×#4111yescharacteristic
292C2×C5C2×2−1+4⋊C311yescharacteristic
293C2×C5#594511yescharacteristic
294C23—31no—
295C23C22×C5×A412yes—
296Q8—22no—
297Q8—31no—
298Q8—62no—
299Q8—62no—
300Q8—122no—
301Q8C3×C5×#4111yescharacteristic
302D4—22no—
303D4—61no—
304D4—61no—
305D4—122no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4—31no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—121no—
317C2×C4C22×C5×A411yescharacteristic
318C2×C4C22×C5×A411yescharacteristic
319C2×C4C22×C5×A412yes—
320C2×C4C22×C5×A411yescharacteristic
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—162no—
325C5#3518⋊C311yescharacteristic, direct product factor, Sylow
326C22—22no—
327C22—22no—
328C22—22no—
329C22—61no—
330C22—61no—
331C22—61no—
332C22C23×C5×A411yescharacteristic
333C4—21no—
334C4—22no—
335C4—21no—
336C4—31no—
337C4—31no—
338C4—62no—
339C4—61no—
340C4—61no—
341C4—62no—
342C4C5×2−1+4⋊C312yes—
343C3—41noSylow
344C2—42no—
345C2—121no—
346C2C5×A4×#4111yescharacteristic
347C2C2×C5×2−1+4⋊C311yescharacteristic
348C2C5×#594511yescharacteristic
subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3518C3
hypercenterC5×#124A4
hypocenterQ8C3×C5×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×#124

Complements

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

NH ≅ G/Ntype
1. C5×Q8×SL(2,𝔽3)C2semidirect
2. C5×D4×SL(2,𝔽3)C2semidirect
3. C5×#548⋊C3C2semidirect
4. C5×#519⋊C3C2semidirect
5. C4×C5×#41⋊C3C2semidirect
6. C5×#3518C3semidirect
13. #3518⋊C3C5direct (N × H)
14. C5×Q82C2×C3semidirect
19. C5×#548C2×C3semidirect
23. C5×#519C2×C3semidirect
24. C4×C5×#41C2×C3semidirect
33. Q8×SL(2,𝔽3)C2×C5semidirect
34. D4×SL(2,𝔽3)C2×C5semidirect
35. #548⋊C3C2×C5semidirect
36. #519⋊C3C2×C5semidirect
37. C4×#41⋊C3C2×C5semidirect
72. #3518C3×C5semidirect
125. Q82C2×C3×C5semidirect
130. #548C2×C3×C5semidirect
134. #519C2×C3×C5semidirect
135. C4×#41C2×C3×C5semidirect
325. C5#3518⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×Q8×SL(2,𝔽3)2
C5×D4×SL(2,𝔽3)2
C5×#548⋊C32
C5×#519⋊C32
C4×C5×#41⋊C32
C5×#35183
C3×C5×#1244
#3518⋊C35