Finite Group Databasebeta

← C2×#11756724 #14514378

Subgroups — C2×#11756724 #14514378

Order 1920.

626 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 80 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×D5×#41⋊C3C211yesmaximal, characteristic
2#11756724C214yesmaximal, direct product factor
3C2×#91885C212yesmaximal, complemented
4C2×#65795—31nomaximal, Hall
5C2×C5×#41⋊C3C411yescharacteristic, complemented
6C2×#6758—41nomaximal
7D5×#41⋊C3C412yescomplemented
8D5×#41⋊C3C2212yes—
9C2×D5×SL(2,𝔽3)C411yescharacteristic, complemented
10C2×C5⋊(#41⋊C3)C2211yescharacteristic
11#91885C2214yescomplemented
12C2×#5423—51nomaximal, Hall
13C2×D5×#41D311yescharacteristic
14C2×#4130—31no—
15C22×#700—31no—
16#65795—34no—
17#65795—34no—
18C2×#4010—32no—
19C2×#4008—32no—
20C2×C3×C4×D5—41no—
21C5×#41⋊C3C2×C412yescomplemented
22C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
23C2×C3⋊#170—42no—
24#6758—44no—
25D5×SL(2,𝔽3)C2×C412yescomplemented
26C5⋊(#41⋊C3)C2×C411yescharacteristic
27C5⋊(#41⋊C3)C2311yescharacteristic
28C22×#41⋊C3—51no—
29#5423—54no—
30C2×GU(2,𝔽3)—52no—
31C2×C5×#41Dic311yescharacteristic
32D5×#41—32no—
33D5×#41—32no—
34D5×#41Dic312yes—
35D5×#41C2×D312yes—
36C2×#713C2×D311yescharacteristic
37C2×Q8×D5Dic311yescharacteristic
38C2×#711—31no—
39C2×D4×D5—31no—
40C2×#709—31no—
41C22×C4×D5—31no—
42C2×C22⋊AGL(1,𝔽5)—61no—
43C22×#170—61no—
44#4130—34no—
45#4130—62no—
46C2×C4⋊AGL(1,𝔽5)—61no—
47C2×C4×AGL(1,𝔽5)—32no—
48C2×#700—34no—
49C2×#700—32no—
50C2×#700—62no—
51C2×#700—61no—
52#4010—34no—
53#4010—34no—
54#4008—38no—
55C2×#421—151noSylow
56C22×C3×D5—41no—
57C2×C3×C4×C5—41no—
58C2×C3×Dic5—41no—
59C3×C4×D5—42no—
60C3×C4×D5—42no—
61C5×SL(2,𝔽3)C22×C411yescharacteristic
62C3⋊#170—44no—
63C2×#41⋊C3—52no—
64C2×#41⋊C3—52no—
65C2×#41⋊C3—51no—
66C2×#41⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
67C22×SL(2,𝔽3)—51no—
68C2×#208—201no—
69GU(2,𝔽3)—54no—
70C23×D5—31no—
71C22×AGL(1,𝔽5)—62no—
72C5×#41—32no—
73C5×#41C2×Dic312yes—
74C2×C5×Q8C2×Dic311yescharacteristic
75C2×C5×D4—31no—
76C22×C4×C5—31no—
77C2×C22⋊D5—61no—
78C22×Dic5—31no—
79#713—31no—
80#713—31no—
81#713C2×Dic311yescharacteristic
82#713C22×D311yescharacteristic
83Q8×D5—32no—
84Q8×D5C2×Dic312yes—
85#711—32no—
86#711—32no—
87#711—61no—
88#711—61no—
89D4×D5—32no—
90D4×D5—32no—
91D4×D5—62no—
92#709—32no—
93#709—32no—
94#709—62no—
95C2×D20—31no—
96C2×C4×D5—31no—
97C2×C4×D5—31no—
98C2×C4×D5—31no—
99C2×C4×D5—32no—
100C2×C4×D5—32no—
101C2×C4×D5—32no—
102C2×C4×D5—32no—
103C2×C4×D5S411yescharacteristic
104C2×Dic10—31no—
105C22⋊AGL(1,𝔽5)—64no—
106C2×#170—64no—
107C2×#170—62no—
108C4⋊AGL(1,𝔽5)—64no—
109C4×AGL(1,𝔽5)—34no—
110C4×AGL(1,𝔽5)—62no—
111#700—34no—
112#700—62no—
113#700—64no—
114C22×#41—51no—
115C22×M16—151no—
116C2×#117—151no—
117#421—154no—
118#421—154no—
119C2×#104—152no—
120C2×#104—152no—
121C22×C3×C5—41no—
122C2×C3×D5—42no—
123C2×C3×D5—82no—
124C3×C4×C5—42no—
125C3×Dic5—41no—
126C3×Dic5—41no—
127C22×C3×C4—201no—
128#41⋊C3—51no—
129#41⋊C3—51no—
130#41⋊C3C2×AGL(1,𝔽5)12yescomplemented
131C2×SL(2,𝔽3)—52no—
132C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
133#208—204no—
134C2×C3⋊C8—202no—
135C23×C5—31no—
136C22×D5—31no—
137C22×D5—32no—
138C22×D5—32no—
139C22×D5—122no—
140C22×D5SL(2,ℤ/4)11yescharacteristic
141C2×AGL(1,𝔽5)—64no—
142C2×AGL(1,𝔽5)—124no—
143C5×Q8—31no—
144C5×Q8C22×Dic311yescharacteristic
145C5×D4—32no—
146C5×D4—61no—
147C2×C4×C5—31no—
148C2×C4×C5—32no—
149C2×C4×C5—32no—
150C2×C4×C5SL(2,ℤ/4)11yescharacteristic
151C22⋊D5—62no—
152C22⋊D5—62no—
153C2×Dic5—31no—
154C2×Dic5—32no—
155C2×Dic5—32no—
156C2×Dic5C2×S411yescharacteristic
157D20—31no—
158D20—31no—
159D20—61no—
160C4×D5—32no—
161C4×D5—32no—
162C4×D5—32no—
163C4×D5—32no—
164C4×D5—31no—
165C4×D5—31no—
166C4×D5—31no—
167C4×D5—31no—
168C4×D5SL(2,ℤ/4)12yes—
169C4×D5C2×S412yes—
170Dic10—32no—
171Dic10—61no—
172#170—64no—
173C2×#41—52no—
174C2×#41—52no—
175C2×#41—51no—
176C2×#41—151no—
177C2×#41—151no—
178C2×#41—152no—
179C2×#41—152no—
180C2×#41ΓL(1,𝔽16)11yescharacteristic
181C22×Q8—51no—
182C22×D4—151no—
183C23×C4—151no—
184C2×M16—152no—
185C2×M16—154no—
186C2×M16—301no—
187C2×M16—302no—
188C22×C8—301no—
189#117—154no—
190#117—302no—
191C2×C4⋊C4—301no—
192C2×C22⋊C4—301no—
193C2×C42—152no—
194#104—154no—
195#104—158no—
196#104—154no—
197C2×C3×C5—41no—
198C2×C3×C5—42no—
199C3×D5—82no—
200C23×C3—201no—
201C2×C3×C4—41no—
202C2×C3×C4—202no—
203C2×C3×C4—202no—
204C2×C3×C4—201no—
205SL(2,𝔽3)C22×AGL(1,𝔽5)11yescharacteristic
206C3⋊C8—204no—
207C22×C5—32no—
208C22×C5—62no—
209C22×C5C2×SL(2,ℤ/4)11yescharacteristic
210C2×D5—22no—
211C2×D5—31no—
212C2×D5—31no—
213C2×D5—62no—
214C2×D5—122no—
215C2×D5—122no—
216C2×D5C2×SL(2,ℤ/4)12yes—
217AGL(1,𝔽5)—124no—
218C4×C5—31no—
219C4×C5—31no—
220C4×C5C2×SL(2,ℤ/4)12yes—
221Dic5—32no—
222Dic5C2×SL(2,ℤ/4)11yescharacteristic
223Dic5C22×S411yescharacteristic
224C24—151no—
225#41—32no—
226#41—51no—
227#41—51no—
228#41—152no—
229#41—152no—
230#41—152no—
231#41—152no—
232#41—151no—
233#41—151no—
234#41—302no—
235#41—301no—
236#41—301no—
237#41C2×ΓL(1,𝔽16)12yes—
238C2×Q8—52no—
239C2×Q8—151no—
240C2×Q8—152no—
241C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
242C2×D4—31no—
243C2×D4—151no—
244C2×D4—152no—
245C2×D4—152no—
246C2×D4—302no—
247C2×D4—301no—
248C22×C4—31no—
249C22×C4—51no—
250C22×C4—151no—
251C22×C4—152no—
252C22×C4—152no—
253C22×C4—152no—
254C22×C4—152no—
255C22×C4—151no—
256C22×C4—151no—
257C22×C4—151no—
258C22×C4—302no—
259M16—154no—
260M16—304no—
261M16—302no—
262C2×C8—302no—
263C2×C8—304no—
264C4⋊C4—304no—
265C22⋊C4—304no—
266C42—154no—
267C42—302no—
268C3×C5—41noHall
269C22×C3—41no—
270C22×C3—202no—
271C22×C3—402no—
272C3×C4—42no—
273C3×C4—201no—
274C3×C4—201no—
275C2×C5—62no—
276C2×C5#542312yescomplemented
277C2×C5C22×SL(2,ℤ/4)11yescharacteristic
278D5—22no—
279D5—61no—
280D5—61no—
281C23—31no—
282C23—51no—
283C23—152no—
284C23—152no—
285C23—151no—
286C23—602no—
287Q8—31no—
288Q8—152no—
289Q8—301no—
290Q8C22×ΓL(1,𝔽16)11yescharacteristic
291D4—32no—
292D4—61no—
293D4—151no—
294D4—151no—
295D4—301no—
296D4—302no—
297D4—302no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—51no—
302C2×C4—52no—
303C2×C4—52no—
304C2×C4—151no—
305C2×C4—151no—
306C2×C4—151no—
307C2×C4—151no—
308C2×C4—152no—
309C2×C4—152no—
310C2×C4—152no—
311C2×C4—152no—
312C2×C4—151no—
313C2×C4—152no—
314C2×C4—152no—
315C2×C4—304no—
316C2×C4—604no—
317C2×C4A4⋊AGL(1,𝔽5)11yescharacteristic
318C8—304no—
319C2×C3—41no—
320C2×C3—42no—
321C2×C3—402no—
322C5C2×#542311yescharacteristic, complemented, Sylow
323C22—32no—
324C22—52no—
325C22—62no—
326C22—102no—
327C22—151no—
328C22—151no—
329C22—302no—
330C22—602no—
331C22—602no—
332C22C2×A4⋊AGL(1,𝔽5)11yescharacteristic
333C4—31no—
334C4—31no—
335C4—51no—
336C4—51no—
337C4—152no—
338C4—604no—
339C4C2×A4⋊AGL(1,𝔽5)12yes—
340C3—41noSylow
341C2—62no—
342C2—102no—
343C2—301no—
344C2—301no—
345C2#1175672412yesdirect product factor
346C2C22×A4⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22C2×A4⋊AGL(1,𝔽5)
commutator, hypocenterC5×SL(2,𝔽3)C22×C4
FrattiniC2C22×A4⋊AGL(1,𝔽5)
FittingC2×C5×#41Dic3
socleC22×C5C2×SL(2,ℤ/4)
prime radicalC2×D5×#41⋊C3C2
cosocleC5⋊(#41⋊C3)C23
hypercenterC2×C4A4⋊AGL(1,𝔽5)
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11756724C2direct (N × H)
3. C2×#91885C2semidirect
5. C2×C5×#41⋊C3C4semidirect
7. D5×#41⋊C3C4semidirect
9. C2×D5×SL(2,𝔽3)C4semidirect
11. #91885C22semidirect
21. C5×#41⋊C3C2×C4semidirect
25. D5×SL(2,𝔽3)C2×C4semidirect
66. C2×#41⋊C3AGL(1,𝔽5)semidirect
130. #41⋊C3C2×AGL(1,𝔽5)semidirect
276. C2×C5#5423semidirect
322. C5C2×#5423semidirect
345. C2#11756724direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×#41⋊C32
#117567242
C2×#918852
C2×#657953
C2×#67584
C2×#54235