Finite Group Databasebeta

← C5×#85359 #14511097

Subgroups — C5×#85359 #14511097

Order 1920.

518 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C2×C5×#983C211yesmaximal, characteristic
2C22×C5×#226C211yesmaximal, characteristic, complemented
3C5×#519⋊C3C211yesmaximal, characteristic, complemented
4C5×#5395C212yesmaximal, complemented
5C5×C4⋊#226C212yesmaximal, complemented
6C5×#3044—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C5×#983—22no—
9C2×C5×GL(2,𝔽3)C2211yescharacteristic
10C2×C5×#226—22no—
11C2×C5×#226C2212yes—
12C2×C5×#226C2211yescharacteristic
13C5×#886—41nomaximal
14C4×C5×SL(2,𝔽3)C2212yes—
15#85359C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×#137—31no—
17C22×C5×Q16—31no—
18C5×#536—31no—
19C5×#519D311yescharacteristic, complemented
20C5×#464—32no—
21C5×#464—32no—
22C5×#462—32no—
23C5×#462—32no—
24C5×#424—31no—
25C5×#419—31no—
26C5×#416—31no—
27C2×C5×#67—41no—
28C22×C5×Dic3—41no—
29C2×C5×SL(2,𝔽3)D412yes—
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31C5×GL(2,𝔽3)—21no—
32C5×#226—22no—
33C5×#226—21no—
34C3×C5×C22⋊C4—41no—
35C5×(C2×D3)⋊C4—42no—
36C5×C4⋊Dic3—42no—
37C2×#983C2×C511yescharacteristic
38C22×#226C2×C511yescharacteristic, complemented
39#519⋊C3C2×C511yescharacteristic, complemented
40#5395C2×C512yescomplemented
41C4⋊#226C2×C512yescomplemented
42C2×C5×#41—31no—
43C22×C5×Q8—31no—
44C22×C5×Q8C2×D311yescharacteristic
45C5×#137—62no—
46C5×#137—62no—
47C2×C5×Q16—31no—
48C2×C5×Q16—32no—
49C2×C5×Q16—31no—
50C2×C5×Q16—32no—
51C2×C5×Q16—62no—
52C2×C5×Q16—62no—
53C2×C5×SD16—31no—
54C2×C5×SD16—31no—
55C2×C5×M16—31no—
56C22×C5×C8—31no—
57C5×C4⋊Q8—32no—
58C5×#124—32no—
59C5×#123—61no—
60C5×#123—61no—
61C5×C22⋊Q8—62no—
62C4×C5×Q8—32no—
63C4×C5×Q8C2×D312yes—
64C5×#117—31no—
65C5×#117—31no—
66C5×C4⋊C8—32no—
67C5×C4⋊C8—32no—
68C5×#103—32no—
69C5×#103—32no—
70C5×#103—32no—
71C5×#102—32no—
72#3044—31noSylow
73C23×C3×C5—41no—
74C22×C5×D3—41no—
75C2×C3×C4×C5—42no—
76C5×#67—82no—
77C2×C5×Dic3—41no—
78C2×C5×Dic3—42no—
79C2×C5×Dic3—82no—
80C5×SL(2,𝔽3)C2×D411yescharacteristic
81C22×SL(2,𝔽3)C22×C511yescharacteristic
82#983—22no—
83C2×GL(2,𝔽3)C22×C511yescharacteristic
84C2×#226—22no—
85C2×#226C22×C511yescharacteristic
86C2×#226C22×C512yes—
87#886—41no—
88C4×SL(2,𝔽3)C22×C512yes—
89C5×#41—62no—
90C5×#41—62no—
91C2×C5×Q8—31no—
92C2×C5×Q8—32no—
93C2×C5×Q8—32no—
94C2×C5×Q8—31no—
95C2×C5×Q8—61no—
96C2×C5×Q8—62no—
97C2×C5×Q8—62no—
98C2×C5×Q8D1212yes—
99C2×C5×Q8C22×D311yescharacteristic
100C2×C5×D4—31no—
101C2×C5×D4—61no—
102C22×C4×C5—31no—
103C22×C4×C5—61no—
104C22×C4×C5—61no—
105C5×Q16—64no—
106C5×Q16—62no—
107C5×Q16—62no—
108C5×Q16—61no—
109C5×Q16—62no—
110C5×Q16—61no—
111C5×SD16—62no—
112C5×SD16—61no—
113C5×SD16—61no—
114C5×M16—62no—
115C2×C5×C8—32no—
116C2×C5×C8—31no—
117C2×C5×C8—31no—
118C2×C5×C8—62no—
119C5×C4⋊C4—32no—
120C5×C4⋊C4—32no—
121C5×C4⋊C4—62no—
122C5×C4⋊C4—62no—
123C5×C22⋊C4—31no—
124C5×C22⋊C4—62no—
125C5×C22⋊C4—62no—
126C5×C22⋊C4S411yescharacteristic
127C42×C5—32no—
128C42×C5—61no—
129C2×#137—31no—
130C22×Q16—31no—
131#536—31no—
132#519C5×D311yescharacteristic, complemented
133#464—32no—
134#464—32no—
135#462—32no—
136#462—32no—
137#424—31no—
138#419—31no—
139#416—31no—
140C22×C3×C5—42no—
141C22×C3×C5—41no—
142C22×C3×C5—82no—
143C2×C5×D3—81no—
144C2×C5×D3—82no—
145C3×C4×C5—82no—
146C5×Dic3—82no—
147C5×Dic3—81no—
148C2×#67—41no—
149C22×Dic3—41no—
150C2×SL(2,𝔽3)C5×D412yes—
151C2×SL(2,𝔽3)C23×C511yescharacteristic
152GL(2,𝔽3)—21no—
153#226—22no—
154#226—21no—
155C3×C22⋊C4—41no—
156(C2×D3)⋊C4—42no—
157C4⋊Dic3—42no—
158C23×C5—61no—
159C23×C5C2×S411yescharacteristic
160C5×Q8—32no—
161C5×Q8—31no—
162C5×Q8—61no—
163C5×Q8—62no—
164C5×Q8—61no—
165C5×Q8—62no—
166C5×Q8—61no—
167C5×Q8—122no—
168C5×Q8C2×D1211yescharacteristic
169C5×D4—61no—
170C5×D4—61no—
171C5×D4—122no—
172C2×C4×C5—32no—
173C2×C4×C5—32no—
174C2×C4×C5—31no—
175C2×C4×C5—31no—
176C2×C4×C5—32no—
177C2×C4×C5—62no—
178C2×C4×C5—61no—
179C2×C4×C5—122no—
180C2×C4×C5—122no—
181C2×C4×C5C2×S412yes—
182C5×C8—62no—
183C5×C8—61no—
184C5×C8—61no—
185C2×#41—31no—
186C22×Q8—31no—
187C22×Q8C2×C5×D311yescharacteristic
188#137—62no—
189#137—62no—
190C2×Q16—32no—
191C2×Q16—31no—
192C2×Q16—32no—
193C2×Q16—31no—
194C2×Q16—62no—
195C2×Q16—62no—
196C2×SD16—31no—
197C2×SD16—31no—
198C2×M16—31no—
199C22×C8—31no—
200C4⋊Q8—32no—
201#124—32no—
202#123—61no—
203#123—61no—
204C22⋊Q8—62no—
205C4×Q8—32no—
206C4×Q8C2×C5×D312yes—
207#117—31no—
208#117—31no—
209C4⋊C8—32no—
210C4⋊C8—32no—
211#103—32no—
212#103—32no—
213#103—32no—
214#102—32no—
215C2×C3×C5—41no—
216C2×C3×C5—42no—
217C2×C3×C5—82no—
218C5×D3—161no—
219C23×C3—41no—
220C22×D3—41no—
221C2×C3×C4—42no—
222#67—82no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225C2×Dic3—82no—
226SL(2,𝔽3)C2×C5×D411yescharacteristic
227C22×C5—22no—
228C22×C5—121no—
229C22×C5—122no—
230C22×C5C4⋊S412yes—
231C22×C5C22×S411yescharacteristic
232C4×C5—22no—
233C4×C5—31no—
234C4×C5—32no—
235C4×C5—31no—
236C4×C5—62no—
237C4×C5—121no—
238C4×C5—122no—
239#41—62no—
240#41—62no—
241C2×Q8—31no—
242C2×Q8—32no—
243C2×Q8—32no—
244C2×Q8—31no—
245C2×Q8—61no—
246C2×Q8—62no—
247C2×Q8—62no—
248C2×Q8C5×D1212yes—
249C2×Q8C22×C5×D311yescharacteristic
250C2×D4—31no—
251C2×D4—61no—
252C22×C4—31no—
253C22×C4—61no—
254C22×C4—61no—
255Q16—64no—
256Q16—62no—
257Q16—62no—
258Q16—61no—
259Q16—62no—
260Q16—61no—
261SD16—62no—
262SD16—61no—
263SD16—61no—
264M16—62no—
265C2×C8—32no—
266C2×C8—31no—
267C2×C8—31no—
268C2×C8—62no—
269C4⋊C4—32no—
270C4⋊C4—32no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C22⋊C4—31no—
274C22⋊C4—62no—
275C22⋊C4—62no—
276C22⋊C4C5×S411yescharacteristic
277C42—32no—
278C42—61no—
279C3×C5—41noHall
280C22×C3—42no—
281C22×C3—41no—
282C22×C3—82no—
283C2×D3—81no—
284C2×D3—82no—
285C3×C4—82no—
286Dic3—82no—
287Dic3—81no—
288C2×C5—22no—
289C2×C5—241no—
290C2×C5C2×C4⋊S411yescharacteristic
291C2×C5#592712yes—
292C23—61no—
293C23C2×C5×S411yescharacteristic
294Q8—32no—
295Q8—31no—
296Q8—61no—
297Q8—62no—
298Q8—61no—
299Q8—62no—
300Q8—61no—
301Q8—122no—
302Q8C2×C5×D1211yescharacteristic
303D4—61no—
304D4—61no—
305D4—122no—
306C2×C4—32no—
307C2×C4—32no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—32no—
311C2×C4—62no—
312C2×C4—61no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4C2×C5×S412yes—
316C8—62no—
317C8—61no—
318C8—61no—
319C2×C3—41no—
320C2×C3—42no—
321C2×C3—82no—
322D3—161no—
323C5#8535911yescharacteristic, direct product factor, Sylow
324C22—22no—
325C22—121no—
326C22—122no—
327C22C5×C4⋊S412yes—
328C22C22×C5×S411yescharacteristic
329C4—22no—
330C4—31no—
331C4—32no—
332C4—31no—
333C4—62no—
334C4—121no—
335C4—122no—
336C3—41noSylow
337C2—22no—
338C2—241no—
339C2C2×C5×C4⋊S411yescharacteristic
340C2C5×#592712yes—
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×#519D3
prime radicalC2×C5×#983C2
hypercenterC5×C22⋊C4S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C5×#226C2semidirect
3. C5×#519⋊C3C2semidirect
4. C5×#5395C2semidirect
5. C5×C4⋊#226C2semidirect
15. #85359C5direct (N × H)
19. C5×#519D3semidirect
38. C22×#226C2×C5semidirect
39. #519⋊C3C2×C5semidirect
40. #5395C2×C5semidirect
41. C4⋊#226C2×C5semidirect
132. #519C5×D3semidirect
323. C5#85359direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#9832
C22×C5×#2262
C5×#519⋊C32
C5×#53952
C5×C4⋊#2262
C5×#30443
C5×#8864
#853595