Finite Group Databasebeta

← C2×#11751604 #14513235

Subgroups — C2×#11751604 #14513235

Order 1920.

610 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C22×C5⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C22×C5×#226C211yesmaximal, characteristic, complemented
3C2×Dic5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11751604C214yesmaximal, direct product factor
5C2×#65474—31nomaximal, Hall
6C22×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×C5⋊GL(2,𝔽3)—22no—
8C2×C5⋊GL(2,𝔽3)C2212yes—
9C2×C5×#226—22no—
10C2×C5×#226C2212yescomplemented
11C2×#6667—41nomaximal
12Dic5×SL(2,𝔽3)C2212yescomplemented
13C2×#5395—51nomaximal, Hall
14C22×C5×Q16—31no—
15C2×#4093—31no—
16C2×Q8×Dic5D311yescharacteristic, complemented
17C22×#688—31no—
18C2×#3967—31no—
19#65474—34no—
20#65474—34no—
21C2×#3953—31no—
22C2×#3946—31no—
23C23×D15—41no—
24C22×C5×Dic3—41no—
25C22×C3×Dic5—41no—
26C2×C5×SL(2,𝔽3)D412yes—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C5⋊GL(2,𝔽3)—22no—
29C5×#226—22no—
30#6667—44no—
31C22×GL(2,𝔽3)—51no—
32C22×#226D511yescharacteristic, complemented
33C2×C4×SL(2,𝔽3)—51no—
34#5395—54no—
35C22×C5×Q8—31no—
36C22×C5×Q8C2×D311yescharacteristic
37C22×D20—31no—
38C2×C5×Q16—32no—
39C2×C5×Q16—32no—
40C2×C5×Q16—62no—
41C2×C5×Q16—62no—
42C22×C5×C8—31no—
43#4093—34no—
44#4093—62no—
45Q8×Dic5—32no—
46Q8×Dic5C2×D312yescomplemented
47C2×#688—32no—
48C2×#688—32no—
49C2×#688—62no—
50C2×#688—62no—
51C2×#685—61no—
52C2×#685—61no—
53C2×C4⋊Dic5—31no—
54C2×C4×Dic5—31no—
55C22×#168—31no—
56#3967—34no—
57#3953—34no—
58#3946—34no—
59C2×#464—151noSylow
60C23×C3×C5—41no—
61C22×D15—42no—
62C22×D15—44no—
63C22×D15—82no—
64C22×D15—82no—
65C2×C5×Dic3—42no—
66C2×C5×Dic3—82no—
67C2×C3×Dic5—42no—
68C2×C3×Dic5—82no—
69C5×SL(2,𝔽3)C2×D411yescharacteristic
70C22×SL(2,𝔽3)C2×D511yescharacteristic
71C2×GL(2,𝔽3)—52no—
72C2×GL(2,𝔽3)—102no—
73C2×#226—22no—
74C2×#226C2×D512yescomplemented
75C2×(C2×D3)⋊C4—201no—
76C4×SL(2,𝔽3)—52no—
77C23×D5—61no—
78C2×C5×Q8—31no—
79C2×C5×Q8—32no—
80C2×C5×Q8—32no—
81C2×C5×Q8—61no—
82C2×C5×Q8—62no—
83C2×C5×Q8—62no—
84C2×C5×Q8D1212yes—
85C2×C5×Q8C22×D311yescharacteristic
86C22×C4×C5—31no—
87C22×C4×C5—61no—
88C22×Dic5—31no—
89C22×Dic5S411yescharacteristic
90C2×D20—32no—
91C2×D20—62no—
92C2×D20—62no—
93C2×D20—61no—
94C5×Q16—64no—
95C5×Q16—62no—
96C5×Q16—62no—
97C2×C5×C8—32no—
98C2×C5×C8—62no—
99#688—64no—
100#688—62no—
101#688—62no—
102#685—64no—
103#685—64no—
104C4⋊Dic5—32no—
105C4⋊Dic5—61no—
106C4×Dic5—32no—
107C4×Dic5—32no—
108C2×#168—32no—
109C2×#168—62no—
110C22×Q16—31no—
111C22×SD16—151no—
112C2×#124—151no—
113C2×C4×Q8—51no—
114#464—154no—
115#464—154no—
116C2×C4⋊C8—151no—
117C2×#103—151no—
118C2×#102—151no—
119C22×C3×C5—42no—
120C22×C3×C5—42no—
121C22×C3×C5—42no—
122C22×C3×C5—41no—
123C2×D15—44no—
124C2×D15—82no—
125C2×D15—82no—
126C2×D15—88no—
127C3×Dic5—82no—
128C5×Dic3—82no—
129C23×D3—201no—
130C22×C3×C4—201no—
131C22×Dic3—41no—
132C2×SL(2,𝔽3)C22⋊D512yes—
133C2×SL(2,𝔽3)C22×D511yescharacteristic
134GL(2,𝔽3)—102no—
135#226—22no—
136(C2×D3)⋊C4—204no—
137C23×C5C2×S411yescharacteristic
138C22×D5—62no—
139C22×D5—122no—
140C22×D5—122no—
141C22×D5—122no—
142C5×Q8—32no—
143C5×Q8—31no—
144C5×Q8—61no—
145C5×Q8—62no—
146C5×Q8—62no—
147C5×Q8—122no—
148C5×Q8C2×D1211yescharacteristic
149C2×C4×C5—32no—
150C2×C4×C5—32no—
151C2×C4×C5—31no—
152C2×C4×C5—31no—
153C2×C4×C5—62no—
154C2×C4×C5—122no—
155C2×Dic5—22no—
156C2×Dic5—32no—
157C2×Dic5—62no—
158C2×Dic5C2×S412yes—
159D20—62no—
160D20—62no—
161D20—122no—
162C5×C8—62no—
163#168—62no—
164C22×Q8—31no—
165C22×Q8D3×D511yescharacteristic
166C22×D4—151no—
167C2×Q16—32no—
168C2×Q16—32no—
169C2×Q16—62no—
170C2×Q16—62no—
171C2×SD16—152no—
172C2×SD16—152no—
173C2×SD16—302no—
174C2×SD16—302no—
175C22×C8—31no—
176C22×C8—151no—
177#124—154no—
178#124—302no—
179C4×Q8—52no—
180C4×Q8—152no—
181C2×C4⋊C4—151no—
182C2×C22⋊C4—301no—
183C2×C22⋊C4—301no—
184C2×C42—151no—
185C4⋊C8—154no—
186#103—154no—
187#102—154no—
188C2×C3×C5—44no—
189C2×C3×C5—41no—
190C2×C3×C5—41no—
191C2×C3×C5—41no—
192D15—84no—
193C23×C3—41no—
194C22×D3—202no—
195C22×D3—204no—
196C22×D3—402no—
197C22×D3—402no—
198C2×C3×C4—202no—
199C2×C3×C4—402no—
200C2×Dic3—42no—
201C2×Dic3—82no—
202SL(2,𝔽3)C2×C22⋊D511yescharacteristic
203C22×C5C4⋊S412yes—
204C22×C5#98312yes—
205C22×C5#98512yes—
206C22×C5C22×S411yescharacteristic
207C2×D5—122no—
208C2×D5—122no—
209C2×D5—122no—
210C2×D5—244no—
211C4×C5—31no—
212C4×C5—31no—
213C4×C5—32no—
214C4×C5—122no—
215Dic5—22no—
216Dic5—62no—
217C24—301no—
218C2×Q8—31no—
219C2×Q8—32no—
220C2×Q8—32no—
221C2×Q8—61no—
222C2×Q8—62no—
223C2×Q8—62no—
224C2×Q8#120312yes—
225C2×Q8C2×D3×D511yescharacteristic
226C2×D4—152no—
227C2×D4—302no—
228C2×D4—302no—
229C2×D4—301no—
230C22×C4—31no—
231C22×C4—51no—
232C22×C4—61no—
233C22×C4—151no—
234Q16—64no—
235Q16—62no—
236Q16—62no—
237SD16—304no—
238SD16—302no—
239SD16—302no—
240C2×C8—32no—
241C2×C8—62no—
242C2×C8—152no—
243C2×C8—302no—
244C4⋊C4—152no—
245C4⋊C4—301no—
246C22⋊C4—304no—
247C22⋊C4—304no—
248C42—152no—
249C42—152no—
250C3×C5—41noHall
251C22×C3—41no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—42no—
255C2×D3—204no—
256C2×D3—402no—
257C2×D3—402no—
258C2×D3—408no—
259C3×C4—402no—
260Dic3—82no—
261C2×C5#539514yescomplemented
262C2×C5C2×C4⋊S411yescharacteristic
263C2×C5C2×#98311yescharacteristic
264C2×C5C2×#98511yescharacteristic
265D5—242no—
266C23—302no—
267C23—602no—
268C23—602no—
269C23—602no—
270C23D5×S411yescharacteristic
271Q8—32no—
272Q8—31no—
273Q8—61no—
274Q8—62no—
275Q8—62no—
276Q8—122no—
277Q8C2×#120311yescharacteristic
278D4—302no—
279D4—302no—
280D4—602no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—32no—
285C2×C4—52no—
286C2×C4—62no—
287C2×C4—102no—
288C2×C4—122no—
289C2×C4—152no—
290C2×C4—302no—
291C8—62no—
292C8—302no—
293C2×C3—44no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297D3—404no—
298C5C2×#539511yescharacteristic, complemented, Sylow
299C22—602no—
300C22—602no—
301C22—602no—
302C22—1204no—
303C22#9189312yes—
304C22#9189712yes—
305C22#9190212yes—
306C22C2×D5×S411yescharacteristic
307C4—31no—
308C4—31no—
309C4—32no—
310C4—102no—
311C4—122no—
312C4—302no—
313C3—41noSylow
314C2—1202no—
315C2#1175160414yesdirect product factor
316C2C2×#9189311yescharacteristic
317C2C2×#9189711yescharacteristic
318C2C2×#9190211yescharacteristic
subgroupG/H
center, hypercenterC23D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FrattiniC22C2×D5×S4
FittingC22×C5×Q8C2×D3
socleC23×C5C2×S4
prime radicalC22×C5⋊GL(2,𝔽3)C2
hypocenterC5×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C5×#226C2semidirect
3. C2×Dic5×SL(2,𝔽3)C2semidirect
4. #11751604C2direct (N × H)
10. C2×C5×#226C22semidirect
12. Dic5×SL(2,𝔽3)C22semidirect
16. C2×Q8×Dic5D3semidirect
32. C22×#226D5semidirect
46. Q8×Dic5C2×D3semidirect
74. C2×#226C2×D5semidirect
261. C2×C5#5395semidirect
298. C5C2×#5395semidirect
315. C2#11751604direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5⋊GL(2,𝔽3)2
C22×C5×#2262
C2×Dic5×SL(2,𝔽3)2
#117516042
C2×#654743
C2×#66674
C2×#53955