Finite Group Databasebeta

← C2×#11756722 #14514375

Subgroups — C2×#11756722 #14514375

Order 1920.

710 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 88 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×D5×SL(2,𝔽3)C211yesmaximal, characteristic
2#11756722C214yesmaximal, direct product factor
3C2×#91886C212yesmaximal, complemented
4C2×#65782—31nomaximal, Hall
5C22×C5×SL(2,𝔽3)C411yescharacteristic, complemented
6C2×#6763—41nomaximal
7C2×D5×SL(2,𝔽3)C412yescomplemented
8C2×D5×SL(2,𝔽3)C411yescharacteristic, complemented
9C2×D5×SL(2,𝔽3)C2211yescharacteristic
10C2×D5×SL(2,𝔽3)C2212yes—
11#91886C2214yescomplemented
12C2×#5420—51nomaximal, Hall
13C22×Q8×D5D311yescharacteristic
14C2×#4130—31no—
15C2×D5⋊M16—31no—
16#65782—34no—
17#65782—34no—
18C2×#4009—32no—
19C2×#4009—32no—
20C23×C3×D5—41no—
21C22×ΓL(1,𝔽16)—42no—
22C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
23C2×C5×SL(2,𝔽3)C2×C412yescomplemented
24#6763—44no—
25D5×SL(2,𝔽3)C2×C411yescharacteristic
26D5×SL(2,𝔽3)C2×C412yescomplemented
27D5×SL(2,𝔽3)C2311yescharacteristic
28C23×SL(2,𝔽3)—51no—
29#5420—54no—
30C2×#859—52no—
31C22×C5×Q8Dic311yescharacteristic, complemented
32C2×Q8×D5—31no—
33C2×Q8×D5—32no—
34C2×Q8×D5—31no—
35C2×Q8×D5—32no—
36C2×Q8×D5Dic312yescomplemented
37C2×Q8×D5Dic311yescharacteristic, complemented
38C2×Q8×D5C2×D311yescharacteristic
39C2×Q8×D5C2×D312yes—
40C22×C4×D5—31no—
41C22×Dic10—31no—
42C2×C22⋊AGL(1,𝔽5)—61no—
43C2×#704—61no—
44#4130—34no—
45#4130—62no—
46C2×C4⋊AGL(1,𝔽5)—32no—
47C2×C4×AGL(1,𝔽5)—61no—
48D5⋊M16—34no—
49D5⋊M16—62no—
50C2×#700—61no—
51C2×D5⋊C8—32no—
52#4009—38no—
53#4009—34no—
54#4009—34no—
55C2×#419—151noSylow
56C23×C3×C5—41no—
57C22×C3×D5—41no—
58C22×C3×D5—42no—
59C22×C3×D5—41no—
60C22×C3×D5—42no—
61C22×C3×D5—84no—
62C2×ΓL(1,𝔽16)—44no—
63C2×ΓL(1,𝔽16)—84no—
64C5×SL(2,𝔽3)C22×C411yescharacteristic
65C22×SL(2,𝔽3)—51no—
66C22×SL(2,𝔽3)—51no—
67C22×SL(2,𝔽3)—52no—
68C22×SL(2,𝔽3)—52no—
69C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, complemented
70C2×#217—201no—
71#859—54no—
72C23×D5S411yescharacteristic
73C22×AGL(1,𝔽5)—62no—
74C2×C5×Q8—31no—
75C2×C5×Q8—32no—
76C2×C5×Q8C2×Dic311yescharacteristic
77C2×C5×Q8C2×Dic312yescomplemented
78C22×C4×C5—31no—
79C22×Dic5—31no—
80Q8×D5—32no—
81Q8×D5—32no—
82Q8×D5—32no—
83Q8×D5—31no—
84Q8×D5—32no—
85Q8×D5—31no—
86Q8×D5—32no—
87Q8×D5—61no—
88Q8×D5—62no—
89Q8×D5—61no—
90Q8×D5C2×Dic311yescharacteristic
91Q8×D5C2×Dic312yescomplemented
92Q8×D5C22×D311yescharacteristic
93C2×C4×D5—31no—
94C2×C4×D5—32no—
95C2×C4×D5—32no—
96C2×C4×D5—31no—
97C2×C4×D5—31no—
98C2×C4×D5—32no—
99C2×C4×D5—31no—
100C2×C4×D5—32no—
101C2×Dic10—32no—
102C2×Dic10—31no—
103C2×Dic10—31no—
104C2×Dic10—32no—
105C2×Dic10—61no—
106C2×Dic10—62no—
107C22⋊AGL(1,𝔽5)—64no—
108#704—64no—
109C2×#170—62no—
110C4⋊AGL(1,𝔽5)—34no—
111C4⋊AGL(1,𝔽5)—62no—
112C4×AGL(1,𝔽5)—64no—
113#700—64no—
114D5⋊C8—34no—
115D5⋊C8—62no—
116C23×Q8—51no—
117C22×M16—151no—
118C2×#117—151no—
119#419—154no—
120#419—154no—
121C2×#103—152no—
122C2×#103—152no—
123C22×C3×C5—41no—
124C22×C3×C5—42no—
125C22×C3×C5—82no—
126C2×C3×D5—44no—
127C2×C3×D5—41no—
128C2×C3×D5—42no—
129C2×C3×D5—41no—
130C2×C3×D5—84no—
131C2×C3×D5—84no—
132C2×C3×D5—82no—
133ΓL(1,𝔽16)—84no—
134C24×C3—201no—
135C22×Dic3—202no—
136C2×SL(2,𝔽3)—51no—
137C2×SL(2,𝔽3)—51no—
138C2×SL(2,𝔽3)—52no—
139C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
140C2×SL(2,𝔽3)C2×AGL(1,𝔽5)12yescomplemented
141#217—204no—
142C23×C5SL(2,ℤ/4)11yescharacteristic
143C22×D5—24no—
144C22×D5SL(2,ℤ/4)12yes—
145C22×D5SL(2,ℤ/4)11yescharacteristic
146C22×D5C2×S412yes—
147C22×D5C2×S411yescharacteristic
148C2×AGL(1,𝔽5)—64no—
149C2×AGL(1,𝔽5)—124no—
150C5×Q8—31no—
151C5×Q8—32no—
152C5×Q8—61no—
153C5×Q8C22×Dic311yescharacteristic
154C2×C4×C5—31no—
155C2×C4×C5—32no—
156C2×C4×C5—32no—
157C2×C4×C5—31no—
158C2×Dic5—31no—
159C2×Dic5—32no—
160C2×Dic5—32no—
161C2×Dic5—31no—
162C4×D5—31no—
163C4×D5—32no—
164C4×D5—31no—
165C4×D5—32no—
166C4×D5—32no—
167C4×D5—32no—
168C4×D5—32no—
169C4×D5—31no—
170C4×D5—32no—
171C4×D5—31no—
172Dic10—31no—
173Dic10—32no—
174Dic10—31no—
175Dic10—61no—
176Dic10—62no—
177Dic10—62no—
178Dic10—61no—
179#170—64no—
180C22×Q8—51no—
181C22×Q8—51no—
182C22×Q8—52no—
183C22×Q8—52no—
184C22×Q8—151no—
185C22×Q8—152no—
186C22×Q8—151no—
187C22×Q8—151no—
188C22×Q8—152no—
189C22×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
190C23×C4—151no—
191C2×M16—154no—
192C2×M16—301no—
193C2×M16—302no—
194C2×M16—301no—
195C22×C8—152no—
196#117—154no—
197#117—302no—
198C2×C4⋊C4—152no—
199C2×C22⋊C4—301no—
200C2×C42—301no—
201#103—154no—
202#103—154no—
203#103—158no—
204C2×C3×C5—41no—
205C2×C3×C5—42no—
206C2×C3×C5—82no—
207C3×D5—42no—
208C3×D5—42no—
209C3×D5—82no—
210C23×C3—41no—
211C23×C3—201no—
212C23×C3—202no—
213C23×C3—202no—
214C23×C3—201no—
215C23×C3—404no—
216C2×Dic3—204no—
217C2×Dic3—404no—
218SL(2,𝔽3)C22×AGL(1,𝔽5)11yescharacteristic
219C22×C5—22no—
220C22×C5C2×SL(2,ℤ/4)12yes—
221C22×C5C2×SL(2,ℤ/4)11yescharacteristic
222C2×D5—22no—
223C2×D5—24no—
224C2×D5—24no—
225C2×D5#98314yes—
226C2×D5C2×SL(2,ℤ/4)12yes—
227C2×D5C2×SL(2,ℤ/4)11yescharacteristic
228C2×D5C22×S411yescharacteristic
229AGL(1,𝔽5)—124no—
230C4×C5—31no—
231C4×C5—31no—
232C4×C5—32no—
233Dic5—31no—
234Dic5—31no—
235Dic5—32no—
236C24—51no—
237C2×Q8—32no—
238C2×Q8—31no—
239C2×Q8—51no—
240C2×Q8—51no—
241C2×Q8—52no—
242C2×Q8—152no—
243C2×Q8—152no—
244C2×Q8—151no—
245C2×Q8—151no—
246C2×Q8—152no—
247C2×Q8—151no—
248C2×Q8—151no—
249C2×Q8—152no—
250C2×Q8—152no—
251C2×Q8—152no—
252C2×Q8—152no—
253C2×Q8—302no—
254C2×Q8—301no—
255C2×Q8—301no—
256C2×Q8—301no—
257C2×Q8—302no—
258C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
259C2×Q8C2×ΓL(1,𝔽16)12yescomplemented
260C22×C4—31no—
261C22×C4—151no—
262C22×C4—151no—
263C22×C4—152no—
264C22×C4—152no—
265C22×C4—152no—
266C22×C4—151no—
267C22×C4—152no—
268C22×C4—151no—
269C22×C4—151no—
270C22×C4—302no—
271M16—304no—
272M16—304no—
273C2×C8—154no—
274C2×C8—302no—
275C2×C8—302no—
276C4⋊C4—154no—
277C4⋊C4—302no—
278C22⋊C4—304no—
279C42—304no—
280C3×C5—41noHall
281C22×C3—42no—
282C22×C3—41no—
283C22×C3—82no—
284C22×C3—201no—
285C22×C3—204no—
286C22×C3—201no—
287C22×C3—202no—
288C22×C3—402no—
289C22×C3—404no—
290C22×C3—404no—
291Dic3—404no—
292C2×C5—22no—
293C2×C5#542012yescomplemented
294C2×C5C22×SL(2,ℤ/4)11yescharacteristic
295D5—22no—
296D5#542012yescomplemented
297D5C2×#98312yes—
298C23—51no—
299C23—52no—
300C23—52no—
301C23—51no—
302C23—104no—
303C23A4⋊AGL(1,𝔽5)11yescharacteristic
304Q8—31no—
305Q8—32no—
306Q8—61no—
307Q8—151no—
308Q8—151no—
309Q8—152no—
310Q8—302no—
311Q8—301no—
312Q8—302no—
313Q8—301no—
314Q8C22×ΓL(1,𝔽16)11yescharacteristic
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—152no—
320C2×C4—151no—
321C2×C4—151no—
322C2×C4—151no—
323C2×C4—152no—
324C2×C4—151no—
325C2×C4—152no—
326C2×C4—152no—
327C2×C4—152no—
328C2×C4—152no—
329C2×C4—152no—
330C2×C4—152no—
331C2×C4—151no—
332C2×C4—151no—
333C2×C4—304no—
334C2×C4—604no—
335C8—304no—
336C2×C3—41no—
337C2×C3—42no—
338C2×C3—82no—
339C2×C3—202no—
340C2×C3—202no—
341C2×C3—402no—
342C5C2×#542011yescharacteristic, complemented, Sylow
343C22—22no—
344C22—51no—
345C22—51no—
346C22—54no—
347C22—52no—
348C22—104no—
349C22—104no—
350C22—102no—
351C22C2×A4⋊AGL(1,𝔽5)12yes—
352C22C2×A4⋊AGL(1,𝔽5)11yescharacteristic
353C4—31no—
354C4—32no—
355C4—31no—
356C4—151no—
357C4—152no—
358C4—151no—
359C4—604no—
360C3—41noSylow
361C2—22no—
362C2—52no—
363C2—52no—
364C2—102no—
365C2#1175672212yesdirect product factor
366C2C22×A4⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22C2×A4⋊AGL(1,𝔽5)
commutator, hypocenterC5×SL(2,𝔽3)C22×C4
FrattiniC2C22×A4⋊AGL(1,𝔽5)
FittingC22×C5×Q8Dic3
socleC22×C5C2×SL(2,ℤ/4)
prime radicalC22×D5×SL(2,𝔽3)C2
cosocleD5×SL(2,𝔽3)C23
hypercenterC23A4⋊AGL(1,𝔽5)
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11756722C2direct (N × H)
3. C2×#91886C2semidirect
5. C22×C5×SL(2,𝔽3)C4semidirect
7. C2×D5×SL(2,𝔽3)C4semidirect
8. C2×D5×SL(2,𝔽3)C4semidirect
11. #91886C22semidirect
23. C2×C5×SL(2,𝔽3)C2×C4semidirect
26. D5×SL(2,𝔽3)C2×C4semidirect
31. C22×C5×Q8Dic3semidirect
36. C2×Q8×D5Dic3semidirect
37. C2×Q8×D5Dic3semidirect
69. C22×SL(2,𝔽3)AGL(1,𝔽5)semidirect
77. C2×C5×Q8C2×Dic3semidirect
91. Q8×D5C2×Dic3semidirect
140. C2×SL(2,𝔽3)C2×AGL(1,𝔽5)semidirect
189. C22×Q8ΓL(1,𝔽16)semidirect
259. C2×Q8C2×ΓL(1,𝔽16)semidirect
293. C2×C5#5420semidirect
296. D5#5420semidirect
342. C5C2×#5420semidirect
365. C2#11756722direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D5×SL(2,𝔽3)2
#117567222
C2×#918862
C2×#657823
C2×#67634
C2×#54205