Finite Group Databasebeta

← #14302972

Subgroups — #14302972

Order 1920.

628 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C3×#66262C211yesmaximal, characteristic, complemented
2#11755125C211yesmaximal, characteristic
3D5⋊#836C211yesmaximal, characteristic, complemented
4#11751724C212yesmaximal
5#11746518C212yesmaximal, complemented
6#10604671—31nomaximal, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C2×#6773C2211yescharacteristic
9D5⋊#208C2212yes—
10C3×Q8×AGL(1,𝔽5)—24no—
11C3×#4130—22no—
12C3×#4130C2212yes—
13#91836C411yescharacteristic, complemented
14C5×#944C411yescharacteristic, complemented
15#91411—22no—
16#91385C411yescharacteristic, complemented
17#91374C411yescharacteristic, complemented
18#91357C2211yescharacteristic
19#91348—22no—
20#91234—22no—
21#91158—24no—
22#71663—51nomaximal, Hall
23#66262D311yescharacteristic, complemented
24#66058—31no—
25D5⋊#101—31no—
26#65784—32no—
27#64863—32no—
28C2×C3×C5×Q8C2×C411yescharacteristic
29C2×D60C2×C411yescharacteristic
30C2×C5×Dic6C2×C411yescharacteristic
31C3×Q8×D5—22no—
32C3×Q8×D5—22no—
33C2×C3×C4×D5D412yes—
34C2×C3×C4×D5C2311yescharacteristic
35C2×C3×Dic10C2×C411yescharacteristic
36C2×C3×Dic10D412yes—
37C2×#1202—21no—
38C2×#1200—21no—
39#6773—22no—
40#6773—22no—
41#6762—22no—
42#6758—22no—
43D5⋊(C3⋊C8)—22no—
44C3×C22⋊AGL(1,𝔽5)—22no—
45C3×C4⋊AGL(1,𝔽5)—24no—
46C3×C4⋊AGL(1,𝔽5)—22no—
47C3×C4×AGL(1,𝔽5)—24no—
48C3×C4×AGL(1,𝔽5)D414yes—
49#6714—22no—
50C4×Dic15C2×C411yescharacteristic
51C5×Dic3⋊C4—22no—
52C4×C5×Dic3C2×C411yescharacteristic
53#6669—22no—
54#6668—21no—
55#6667—22no—
56#6665—21no—
57C3×#519—51no—
58#5816—51no—
59C2×#836—51no—
60#5005—52no—
61#4531—52no—
62C2×#713—31no—
63C2×Q8×D5C2×D311yescharacteristic
64Q8×AGL(1,𝔽5)—24no—
65#4130—22no—
66#4130C2×D312yes—
67D5⋊M16—32no—
68C5×C4⋊Q8—31no—
69#4093—31no—
70#4090—31no—
71#4042—62no—
72#4040—62no—
73#4021—31no—
74#4018—31no—
75#4010—64no—
76#4005—62no—
77#1912—151noSylow
78C22×D15—21no—
79C22×C3×D5C2×D411yescharacteristic
80C2×C3×AGL(1,𝔽5)—24no—
81C3×C5×Q8—22no—
82C2×C3×C4×C5C22⋊C412yes—
83C2×C3×C4×C5C22×C411yescharacteristic
84C2×Dic15—21no—
85D60—22no—
86C2×C5×Dic3—21no—
87C2×C5×Dic3—21no—
88C5×Dic6—22no—
89C2×C3×Dic5C22⋊C412yes—
90C2×C3×Dic5C2×D411yescharacteristic
91C3×C4×D5—22no—
92C3×C4×D5—22no—
93C3×C4×D5C2×D412yes—
94C3×C4×D5#4112yes—
95C3×Dic10—22no—
96C3×Dic10—24no—
97#1202—41no—
98#1202—41no—
99#1200—42no—
100C3⋊#170—42no—
101C22×C3×Q8—51no—
102C2×#235—51no—
103C3×C4×Q8—104no—
104C3×#117—52no—
105C3×#117—102no—
106#947—51no—
107#947—51no—
108#946—102no—
109#944—51no—
110#944AGL(1,𝔽5)11yescharacteristic, complemented
111#929—102no—
112#926—51no—
113C2×#208—52no—
114#836—102no—
115#805—104no—
116C2×C5×Q8—31no—
117C2×C5×Q8C4×D311yescharacteristic
118#713—62no—
119#713—62no—
120Q8×D5—22no—
121Q8×D5—22no—
122C2×D20—31no—
123C2×D20—61no—
124C2×C4×D5—61no—
125C2×C4×D5#6712yes—
126C2×C4×D5C22×D311yescharacteristic
127C2×Dic10C4×D311yescharacteristic
128C2×Dic10D1212yes—
129C22⋊AGL(1,𝔽5)—22no—
130#704—62no—
131C4⋊AGL(1,𝔽5)—24no—
132C4⋊AGL(1,𝔽5)—22no—
133C4×AGL(1,𝔽5)—24no—
134C4×AGL(1,𝔽5)#6714yes—
135#700—62no—
136D5⋊C8—62no—
137C5×C4⋊C4—62no—
138C42×C5—31no—
139#685—61no—
140#685—62no—
141#685—62no—
142#683—62no—
143#683—61no—
144C4×Dic5—31no—
145#536—151no—
146#519—51no—
147#421—152no—
148C2×#101—151no—
149H(3,ℤ/4)—152no—
150C22×C3×C5C2×C22⋊C411yescharacteristic
151C2×D15—42no—
152C2×D15—41no—
153C2×C3×D5—41no—
154C2×C3×D5#12011yescharacteristic
155C2×C3×D5#12311yescharacteristic
156C3×AGL(1,𝔽5)—44no—
157C3×C4×C5—22no—
158C3×C4×C5C4×D412yes—
159Dic15—41no—
160C3×Dic5—22no—
161C3×Dic5#12112yes—
162C5×Dic3—41no—
163C5×Dic3—41no—
164C2×C3×Q8—52no—
165C2×C3×Q8—51no—
166C2×C3×Q8—102no—
167C2×C3×Q8—102no—
168C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
169C22×C3×C4—51no—
170C22×C3×C4—52no—
171C2×#67—101no—
172#235—102no—
173#235—102no—
174C2×D12—51no—
175C2×C4×D3—101no—
176C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
177C3×C4⋊C4—102no—
178C3×C4⋊C4—104no—
179C3×C22⋊C4—102no—
180C3×C42—54no—
181C3×C42—104no—
182#217—101no—
183(C2×D3)⋊C4—102no—
184(C2×D3)⋊C4—102no—
185C4⋊Dic3—101no—
186Dic3⋊C4—22no—
187Dic3⋊C4—102no—
188C4×Dic3—51no—
189C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
190#208—102no—
191#208—102no—
192C2×C3⋊C8—102no—
193C22×D5—61no—
194C22×D5C2×#6711yescharacteristic
195C2×AGL(1,𝔽5)—24no—
196C5×Q8—22no—
197C5×Q8—62no—
198C2×C4×C5—61no—
199C2×C4×C5—61no—
200C2×C4×C5(C2×D3)⋊C412yes—
201C2×C4×C5C2×C4×D311yescharacteristic
202C2×Dic5—61no—
203C2×Dic5(C2×D3)⋊C412yes—
204C2×Dic5C2×D1211yescharacteristic
205D20—62no—
206D20—121no—
207D20—121no—
208C4×D5—22no—
209C4×D5—22no—
210C4×D5—122no—
211C4×D5#23512yes—
212C4×D5C2×#6712yes—
213Dic10—22no—
214Dic10—24no—
215#170—122no—
216C2×#41—151no—
217C22×Q8—51no—
218C2×M16—152no—
219C4⋊Q8—31no—
220C4⋊Q8—151no—
221#124—151no—
222#124—151no—
223#123—302no—
224C22⋊Q8—302no—
225C4×Q8—104no—
226#117—52no—
227#117—102no—
228#117—151no—
229#104—304no—
230#101—302no—
231C2×C3×C5—21no—
232C2×C3×C5#38611yescharacteristic
233D15—81no—
234C3×D5—21no—
235C3×D5—21no—
236C23×C3—51no—
237C22×D3—101no—
238C3×Q8—22no—
239C3×Q8—104no—
240C3×Q8—102no—
241C2×C3×C4—51no—
242C2×C3×C4—52no—
243C2×C3×C4—52no—
244C2×C3×C4—52no—
245C2×C3×C4—102no—
246C2×C3×C4—102no—
247C2×C3×C4—104no—
248C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
249C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
250#67—201no—
251#67—201no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—101no—
255D12—102no—
256C4×D3—202no—
257Dic6—22no—
258C3⋊C8—202no—
259C22×C5C2×(C2×D3)⋊C411yescharacteristic
260C2×D5—41no—
261C2×D5—122no—
262C2×D5—121no—
263C2×D5#92911yescharacteristic
264C2×D5#95311yescharacteristic
265AGL(1,𝔽5)—44no—
266C4×C5—22no—
267C4×C5—121no—
268C4×C5—121no—
269C4×C5C4×#6712yes—
270Dic5—22no—
271Dic5—121no—
272Dic5#93012yes—
273#41—302no—
274#41—302no—
275C2×Q8—31no—
276C2×Q8—52no—
277C2×Q8—51no—
278C2×Q8—102no—
279C2×Q8—102no—
280C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
281C2×D4—151no—
282C2×D4—301no—
283C22×C4—51no—
284C22×C4—52no—
285C22×C4—301no—
286M16—302no—
287M16—302no—
288C2×C8—302no—
289C4⋊C4—62no—
290C4⋊C4—102no—
291C4⋊C4—104no—
292C4⋊C4—301no—
293C4⋊C4—302no—
294C22⋊C4—102no—
295C22⋊C4—302no—
296C22⋊C4—301no—
297C22⋊C4—302no—
298C42—31no—
299C42—54no—
300C42—104no—
301C42—151no—
302C3×C5#191211yescharacteristic, complemented, Hall
303C22×C3—51no—
304C22×C3—51no—
305C22×C3—201no—
306C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
307C2×D3—202no—
308C2×D3—201no—
309C3×C4—22no—
310C3×C4—52no—
311C3×C4—102no—
312C3×C4—204no—
313C3×C4D4×AGL(1,𝔽5)12yes—
314Dic3—41no—
315Dic3—41no—
316Dic3—201no—
317C2×C5—21no—
318C2×C5#521311yescharacteristic
319D5—21no—
320D5—21no—
321D5—241no—
322C23—51no—
323C23—301no—
324Q8—22no—
325Q8—62no—
326Q8—104no—
327Q8—102no—
328D4—302no—
329D4—601no—
330D4—601no—
331C2×C4—51no—
332C2×C4—52no—
333C2×C4—52no—
334C2×C4—52no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—102no—
338C2×C4—102no—
339C2×C4—104no—
340C2×C4—301no—
341C2×C4—602no—
342C2×C4(C2×D3)⋊AGL(1,𝔽5)12yes—
343C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
344C8—602no—
345C2×C3—21no—
346C2×C3—101no—
347C2×C3—101no—
348C2×C3#6577911yescharacteristic
349D3—401no—
350C5#7166311yescharacteristic, complemented, Sylow
351C22—51no—
352C22—51no—
353C22—201no—
354C22—602no—
355C22—601no—
356C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
357C4—22no—
358C4—52no—
359C4—102no—
360C4—121no—
361C4—121no—
362C4—204no—
363C4—601no—
364C4AGL(1,𝔽5)×#6712yes—
365C3#1060467111yescharacteristic, complemented, Sylow
366C2—21no—
367C2—101no—
368C2—101no—
369C2—1201no—
370C2#1175202211yescharacteristic
subgroupG/H
centerC2#11752022
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC2×C3×C5#386
prime radicalC2×#6773C22
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1912
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66262C2semidirect
3. D5⋊#836C2semidirect
5. #11746518C2semidirect
13. #91836C4semidirect
14. C5×#944C4semidirect
16. #91385C4semidirect
17. #91374C4semidirect
23. #66262D3semidirect
110. #944AGL(1,𝔽5)semidirect
302. C3×C5#1912semidirect
350. C5#71663semidirect
365. C3#10604671semidirect

Maximal subgroups

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

subgroupindex
C3×#662622
#117551252
D5⋊#8362
#117517242
#117465182
#106046713
#716635