Finite Group Databasebeta

← C2×C5⋊#5931 #14514624

Subgroups — C2×C5⋊#5931 #14514624

Order 1920.

620 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 39 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×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C5⋊#5931C212yesmaximal, direct product factor
3C2×#65516—31nomaximal, Hall
4C22×C5⋊S4—41nomaximal
5C5×C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
6C2×C5⋊GL(2,𝔽3)—42nomaximal
7C2×#5931—51nomaximal, Hall
8C23×C5×Q8D311yescharacteristic, complemented
9C22×#688—32no—
10C2×#4076—31no—
11#65516—32no—
12#65516—34no—
13#65516—32no—
14C2×#3967—32no—
15C2×C22⋊#168—31no—
16C22×C5×A4—41no—
17C2×C5⋊S4—42no—
18C2×C5⋊S4—44no—
19C2×C5×SL(2,𝔽3)—42no—
20C2×C5×SL(2,𝔽3)—81no—
21C5⋊GL(2,𝔽3)—44no—
22C2×C22⋊SL(2,𝔽3)D511yescharacteristic, complemented
23#5931—52no—
24C22×C5×Q8—31no—
25C22×C5×Q8—32no—
26C22×C5×Q8—32no—
27C22×C5×Q8—61no—
28C22×C5×Q8—61no—
29C22×C5×Q8C2×D311yescharacteristic, complemented
30C23×C4×C5—31no—
31C22×C22⋊D5—61no—
32C22×D20—31no—
33C2×#688—34no—
34C2×#688—34no—
35C2×#688—62no—
36C2×#688—62no—
37C2×#688—62no—
38C2×#688—62no—
39#4076—32no—
40#4076—32no—
41#4076—61no—
42#4076—61no—
43C2×#685—61no—
44C2×C4⋊Dic5—31no—
45C22×#168—32no—
46#3967—34no—
47#3967—34no—
48C22⋊#168—32no—
49C22⋊#168—32no—
50C2×#448—151noSylow
51C22×D15—161no—
52C2×C5×A4—42no—
53C2×C5×A4—41no—
54C5⋊S4—44no—
55C5×SL(2,𝔽3)—42no—
56C5×SL(2,𝔽3)—81no—
57C22×S4—201no—
58C22⋊SL(2,𝔽3)C2×D511yescharacteristic, complemented
59C2×GL(2,𝔽3)—202no—
60C24×C5S411yescharacteristic
61C23×D5—61no—
62C2×C5×Q8—21no—
63C2×C5×Q8—32no—
64C2×C5×Q8—32no—
65C2×C5×Q8—32no—
66C2×C5×Q8—32no—
67C2×C5×Q8—32no—
68C2×C5×Q8—61no—
69C2×C5×Q8—61no—
70C2×C5×Q8—62no—
71C2×C5×Q8—62no—
72C2×C5×Q8—62no—
73C2×C5×Q8—61no—
74C2×C5×Q8—61no—
75C2×C5×Q8—61no—
76C2×C5×Q8—62no—
77C2×C5×Q8S412yescomplemented
78C22×C4×C5—31no—
79C22×C4×C5—31no—
80C22×C4×C5—31no—
81C22×C4×C5—31no—
82C22×C4×C5—31no—
83C22×C4×C5—31no—
84C22×C4×C5—62no—
85C22×C4×C5—62no—
86C2×C22⋊D5—62no—
87C2×C22⋊D5—62no—
88C2×C22⋊D5—122no—
89C2×C22⋊D5—122no—
90C22×Dic5—61no—
91C2×D20—32no—
92C2×D20—62no—
93C2×D20—62no—
94C2×D20—61no—
95#688—64no—
96#688—64no—
97#688—64no—
98#688—64no—
99#685—62no—
100#685—62no—
101C4⋊Dic5—32no—
102C4⋊Dic5—61no—
103C2×#168—34no—
104C2×#168—62no—
105C2×#168—62no—
106C23×Q8D1511yescharacteristic, complemented
107C22×SD16—152no—
108C2×#121—151no—
109#448—152no—
110#448—154no—
111#448—152no—
112C2×#103—152no—
113C2×C22⋊C8—151no—
114C22×C3×C5—161no—
115C2×D15—162no—
116C2×D15—164no—
117C5×A4—41no—
118C22×A4—41no—
119C2×S4—202no—
120C2×S4—204no—
121C2×SL(2,𝔽3)—42no—
122C2×SL(2,𝔽3)—81no—
123GL(2,𝔽3)—204no—
124C23×C5—31no—
125C23×C5—31no—
126C23×C5—32no—
127C23×C5GL(2,𝔽3)12yescomplemented
128C23×C5C2×S411yescharacteristic
129C22×D5—62no—
130C22×D5—121no—
131C22×D5—122no—
132C22×D5—122no—
133C22×D5—121no—
134C5×Q8—21no—
135C5×Q8—32no—
136C5×Q8—32no—
137C5×Q8—32no—
138C5×Q8—61no—
139C5×Q8—62no—
140C5×Q8—61no—
141C5×Q8—62no—
142C5×Q8—61no—
143C5×Q8C2×S412yescomplemented
144C2×C4×C5—31no—
145C2×C4×C5—32no—
146C2×C4×C5—31no—
147C2×C4×C5—32no—
148C2×C4×C5—31no—
149C2×C4×C5—31no—
150C2×C4×C5—62no—
151C2×C4×C5—62no—
152C2×C4×C5—62no—
153C2×C4×C5—62no—
154C2×C4×C5—61no—
155C2×C4×C5—61no—
156C22⋊D5—124no—
157C22⋊D5—122no—
158C22⋊D5—122no—
159C2×Dic5—62no—
160C2×Dic5—122no—
161D20—64no—
162D20—122no—
163#168—64no—
164C22×Q8—31no—
165C22×Q8—32no—
166C22×Q8—32no—
167C22×Q8—61no—
168C22×Q8—61no—
169C22×Q8C2×D1511yescharacteristic, complemented
170C22×D4—151no—
171C22×D4—301no—
172C23×C4—31no—
173C2×SD16—154no—
174C2×SD16—154no—
175C2×SD16—302no—
176C2×SD16—302no—
177C2×SD16—302no—
178C2×SD16—302no—
179C22×C8—152no—
180#121—152no—
181#121—152no—
182#121—301no—
183#121—301no—
184C2×C4⋊C4—151no—
185C2×C22⋊C4—301no—
186#103—154no—
187#103—154no—
188C22⋊C8—152no—
189C22⋊C8—152no—
190C2×C3×C5—162no—
191C2×C3×C5—161no—
192D15—164no—
193C22×D3—801no—
194C2×A4—42no—
195C2×A4—41no—
196S4—204no—
197SL(2,𝔽3)—42no—
198SL(2,𝔽3)—81no—
199C22×C5—32no—
200C22×C5—31no—
201C22×C5—32no—
202C22×C5—32no—
203C22×C5—31no—
204C22×C5—31no—
205C22×C5—61no—
206C22×C5C2×GL(2,𝔽3)11yescharacteristic, complemented
207C22×C5#102011yescharacteristic
208C2×D5—122no—
209C2×D5—122no—
210C2×D5—122no—
211C2×D5—242no—
212C2×D5—242no—
213C4×C5—32no—
214C4×C5—32no—
215C4×C5—61no—
216C4×C5—61no—
217Dic5—122no—
218C24—301no—
219C24C5⋊S411yescharacteristic
220C2×Q8—21no—
221C2×Q8—32no—
222C2×Q8—32no—
223C2×Q8—32no—
224C2×Q8—32no—
225C2×Q8—32no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×Q8—62no—
229C2×Q8—62no—
230C2×Q8—62no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—62no—
235C2×Q8C5⋊S412yescomplemented
236C2×D4—152no—
237C2×D4—302no—
238C2×D4—302no—
239C2×D4—301no—
240C2×D4—302no—
241C2×D4—302no—
242C2×D4—602no—
243C2×D4—602no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—301no—
253SD16—304no—
254SD16—304no—
255SD16—304no—
256SD16—304no—
257C2×C8—154no—
258C2×C8—302no—
259C2×C8—302no—
260C4⋊C4—152no—
261C4⋊C4—301no—
262C22⋊C4—302no—
263C22⋊C4—302no—
264C3×C5—161noHall
265C22×C3—161no—
266C2×D3—802no—
267C2×D3—804no—
268A4—41no—
269C2×C5—31no—
270C2×C5—31no—
271C2×C5—32no—
272C2×C5#593112yescomplemented
273C2×C5C2×#102011yescharacteristic
274D5—242no—
275C23—31no—
276C23—31no—
277C23—32no—
278C23—302no—
279C23—601no—
280C23—602no—
281C23—602no—
282C23—601no—
283C23C5⋊GL(2,𝔽3)12yescomplemented
284C23C2×C5⋊S411yescharacteristic
285Q8—21no—
286Q8—32no—
287Q8—32no—
288Q8—32no—
289Q8—61no—
290Q8—62no—
291Q8—61no—
292Q8—62no—
293Q8—61no—
294Q8C2×C5⋊S412yescomplemented
295D4—304no—
296D4—602no—
297D4—604no—
298D4—602no—
299D4—602no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—302no—
313C2×C4—602no—
314C8—304no—
315C2×C3—162no—
316C2×C3—161no—
317D3—804no—
318C5C2×#593111yescharacteristic, complemented, Sylow
319C22—32no—
320C22—31no—
321C22—32no—
322C22—32no—
323C22—31no—
324C22—31no—
325C22—61no—
326C22—602no—
327C22—602no—
328C22—602no—
329C22—1202no—
330C22—1202no—
331C22C2×C5⋊GL(2,𝔽3)11yescharacteristic, complemented
332C22C5⋊#102011yescharacteristic
333C4—32no—
334C4—32no—
335C4—61no—
336C4—61no—
337C4—602no—
338C3—161noSylow
339C2—31no—
340C2—31no—
341C2—32no—
342C2—1202no—
343C2C5⋊#593112yesdirect product factor
344C2C2×C5⋊#102011yescharacteristic
subgroupG/H
center, hypercenterC22C5⋊#1020
commutator, cosocle, hypocenterC5×C22⋊SL(2,𝔽3)C22
FrattiniC2C2×C5⋊#1020
FittingC23×C5×Q8D3
socleC24×C5S4
LCSG ▷ C5×C22⋊SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C5×C22⋊SL(2,𝔽3)C2semidirect
2. C5⋊#5931C2direct (N × H)
5. C5×C22⋊SL(2,𝔽3)C22semidirect
8. C23×C5×Q8D3semidirect
22. C2×C22⋊SL(2,𝔽3)D5semidirect
29. C22×C5×Q8C2×D3semidirect
58. C22⋊SL(2,𝔽3)C2×D5semidirect
77. C2×C5×Q8S4semidirect
106. C23×Q8D15semidirect
127. C23×C5GL(2,𝔽3)semidirect
143. C5×Q8C2×S4semidirect
169. C22×Q8C2×D15semidirect
206. C22×C5C2×GL(2,𝔽3)semidirect
235. C2×Q8C5⋊S4semidirect
272. C2×C5#5931semidirect
283. C23C5⋊GL(2,𝔽3)semidirect
294. Q8C2×C5⋊S4semidirect
318. C5C2×#5931semidirect
331. C22C2×C5⋊GL(2,𝔽3)semidirect
343. C2C5⋊#5931direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊SL(2,𝔽3)2
C5⋊#59312
C2×#655163
C22×C5⋊S44
C2×C5⋊GL(2,𝔽3)4
C2×#59315