Finite Group Databasebeta

← #14313312

Subgroups — #14313312

Order 1920.

468 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 97 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×#65860C211yesmaximal, characteristic
2#11752420C211yesmaximal, characteristic
3#11752404C211yesmaximal, characteristic
4#11746736C214yesmaximal, complemented
5#10604383—31nomaximal, Hall
6C2×C3×#709C2211yescharacteristic
7C22×C3×Dic10C411yescharacteristic, complemented
8#91989—21no—
9D5⋊#208—21no—
10C3×C5×#117C411yescharacteristic, complemented
11C3×#4043—21no—
12C3×#4043—21no—
13C3×#4032C412yescomplemented
14C3×#4032C2212yes—
15C3×#4021—21no—
16C3×#4015—21no—
17#91242C2212yes—
18#91240C2212yes—
19#91234—21no—
20#91228—21no—
21#71870—51nomaximal, Hall
22#65860D311yescharacteristic
23#65761—31no—
24#65745—31no—
25#64865—34no—
26C22×C3×C4×C5C2×C411yescharacteristic
27C2×C3×C22⋊D5C2×C411yescharacteristic
28C2×C3×C22⋊D5C2311yescharacteristic
29C22×C3×Dic5C2×C412yes—
30C3×#709—41no—
31C3×#709—41no—
32C2×C3×D20D411yescharacteristic
33C2×C3×C4×D5D411yescharacteristic
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×Dic10—21no—
36C2×C3×Dic10—21no—
37C2×C3×Dic10—41no—
38C2×C3×Dic10—41no—
39C2×C3×Dic10D411yescharacteristic
40#6763—22no—
41#6762—22no—
42C4⋊ΓL(1,𝔽16)—41no—
43C4×ΓL(1,𝔽16)—41no—
44#6758—41no—
45D5⋊(C3⋊C8)—41no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C22⋊C4C2×C412yes—
48C3×C42×C5—21no—
49C3×#685—22no—
50C3×#685—22no—
51C3×C4⋊Dic5—22no—
52C3×C4⋊Dic5—22no—
53C3×#536—51no—
54#5240—51no—
55#5237—51no—
56#4539—54no—
57C2×#709C2×D311yescharacteristic
58C22×Dic10Dic311yescharacteristic, complemented
59#4130—61no—
60D5⋊M16—61no—
61C5×#117Dic311yescharacteristic, complemented
62#4043—21no—
63#4043—21no—
64#4032Dic312yescomplemented
65#4032C2×D312yes—
66#4021—21no—
67#4015—21no—
68#4013—32no—
69#4011—32no—
70#4005—61no—
71#3999—61no—
72#2123—151noSylow
73C23×C3×C5C22×C411yescharacteristic
74C22×C3×D5C22⋊C411yescharacteristic
75C22×C3×D5C2×D411yescharacteristic
76C2×ΓL(1,𝔽16)—42no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—21no—
79C2×C3×C4×C5—22no—
80C2×C3×C4×C5C22⋊C411yescharacteristic
81C2×C3×C4×C5C22⋊C411yescharacteristic
82C3×C22⋊D5—41no—
83C3×C22⋊D5—41no—
84C2×C3×Dic5—22no—
85C2×C3×Dic5—42no—
86C2×C3×Dic5C22⋊C411yescharacteristic
87C2×C3×Dic5C2×D411yescharacteristic
88C3×D20—41no—
89C3×C4×D5—21no—
90C3×C4×D5—21no—
91C3×C4×D5—21no—
92C3×C4×D5—21no—
93C3×Dic10—41no—
94C3×Dic10—42no—
95C3×Dic10—41no—
96C3×Dic10—41no—
97C3⋊#170—42no—
98C2×C3×#41—51no—
99C22×C3×Q8—51no—
100C3×C4⋊Q8—101no—
101C3×#124—101no—
102C3×#123—52no—
103C3×#123—52no—
104C3×C22⋊Q8—101no—
105C3×C22⋊Q8—101no—
106C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
107#926—101no—
108C2×#208—101no—
109#836—101no—
110#834—52no—
111#834—101no—
112#833—52no—
113C22×C4×C5C2×Dic311yescharacteristic
114C2×C22⋊D5C2×Dic311yescharacteristic
115C2×C22⋊D5C22×D311yescharacteristic
116C22×Dic5C2×Dic312yes—
117#709—41no—
118#709—41no—
119C2×D20#6711yescharacteristic
120C2×C4×D5#6711yescharacteristic
121C2×C4×D5#6711yescharacteristic
122C2×Dic10—21no—
123C2×Dic10—21no—
124C2×Dic10—41no—
125C2×Dic10—41no—
126C2×Dic10#6711yescharacteristic
127C22⋊AGL(1,𝔽5)—62no—
128#704—62no—
129C4⋊AGL(1,𝔽5)—121no—
130C4×AGL(1,𝔽5)—121no—
131#700—121no—
132D5⋊C8—121no—
133C5×C4⋊C4—21no—
134C5×C22⋊C4C2×Dic312yes—
135C42×C5—21no—
136#685—22no—
137#685—22no—
138C4⋊Dic5—22no—
139C4⋊Dic5—22no—
140#536—51no—
141#413—151no—
142#410—151no—
143#352—154no—
144C22×C3×C5—21no—
145C22×C3×C5—41no—
146C22×C3×C5C2×C22⋊C411yescharacteristic
147C2×C3×D5—21no—
148C2×C3×D5—21no—
149C2×C3×D5—41no—
150C2×C3×D5—41no—
151ΓL(1,𝔽16)—82no—
152C3×C4×C5—21no—
153C3×C4×C5—42no—
154C3×C4×C5#9911yescharacteristic
155C3×C4×C5#9911yescharacteristic
156C3×Dic5—21no—
157C3×Dic5—21no—
158C3×Dic5—42no—
159C3×#41—201no—
160C3×#41—201no—
161C2×C3×Q8—51no—
162C2×C3×Q8—101no—
163C2×C3×Q8—101no—
164C2×C3×Q8—201no—
165C2×C3×Q8—201no—
166C2×C3×D4—51no—
167C2×C3×D4—51no—
168C2×C3×D4—51no—
169C22×C3×C4—51no—
170C22×C3×C4—51no—
171C22×C3×C4—52no—
172C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
173C3×C4⋊C4—21no—
174C3×C4⋊C4—102no—
175C3×C4⋊C4—102no—
176C3×C22⋊C4—102no—
177C3×C22⋊C4—102no—
178C3×C22⋊C4C2×AGL(1,𝔽5)12yes—
179C3×C42—21no—
180#217—102no—
181C4⋊Dic3—201no—
182C4×Dic3—201no—
183#208—102no—
184#208—201no—
185C2×C3⋊C8—201no—
186C23×C5C22×Dic311yescharacteristic
187C22×D5#21711yescharacteristic
188C22×D5C2×#6711yescharacteristic
189C2×AGL(1,𝔽5)—122no—
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5—22no—
193C2×C4×C5#21711yescharacteristic
194C2×C4×C5#21711yescharacteristic
195C22⋊D5—41no—
196C22⋊D5—41no—
197C2×Dic5—22no—
198C2×Dic5—42no—
199C2×Dic5#21711yescharacteristic
200C2×Dic5C2×#6711yescharacteristic
201D20—41no—
202C4×D5—21no—
203C4×D5—21no—
204C4×D5—21no—
205C4×D5—21no—
206Dic10—41no—
207Dic10—42no—
208Dic10—41no—
209Dic10—41no—
210#170—122no—
211C2×#41—51no—
212C22×Q8—51no—
213C2×M16—301no—
214C4⋊Q8—101no—
215#124—101no—
216#123—52no—
217#123—52no—
218C22⋊Q8—101no—
219C22⋊Q8—101no—
220#117—301no—
221#117ΓL(1,𝔽16)11yescharacteristic, complemented
222#101—301no—
223#100—152no—
224#99—152no—
225#99—301no—
226C2×C3×C5—21no—
227C2×C3×C5—41no—
228C2×C3×C5C2×#9911yescharacteristic
229C3×D5—41no—
230C3×D5—41no—
231C23×C3—51no—
232C23×C3—51no—
233C23×C3C22×AGL(1,𝔽5)11yescharacteristic
234C3×Q8—201no—
235C3×Q8—202no—
236C3×Q8—201no—
237C3×Q8—201no—
238C3×D4—201no—
239C3×D4—201no—
240C3×D4—201no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—51no—
245C2×C3×C4—51no—
246C2×C3×C4—101no—
247C2×C3×C4—101no—
248C2×C3×C4—101no—
249C2×C3×C4—101no—
250C2×C3×C4—102no—
251C2×C3×C4—202no—
252C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
253C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
254C2×Dic3—202no—
255C3⋊C8—202no—
256C22×C5—21no—
257C22×C5—41no—
258C22×C5C2×#21711yescharacteristic
259C2×D5—21no—
260C2×D5—21no—
261C2×D5—41no—
262C2×D5—41no—
263AGL(1,𝔽5)—242no—
264C4×C5—21no—
265C4×C5—42no—
266C4×C5#83411yescharacteristic
267C4×C5#83411yescharacteristic
268Dic5—21no—
269Dic5—21no—
270Dic5—42no—
271#41—201no—
272#41—201no—
273C2×Q8—51no—
274C2×Q8—101no—
275C2×Q8—101no—
276C2×Q8—201no—
277C2×Q8—201no—
278C2×D4—51no—
279C2×D4—51no—
280C2×D4—51no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—52no—
284C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
285M16—302no—
286M16—601no—
287C2×C8—601no—
288C4⋊C4—21no—
289C4⋊C4—102no—
290C4⋊C4—102no—
291C4⋊C4—601no—
292C22⋊C4—102no—
293C22⋊C4—102no—
294C22⋊C4—302no—
295C22⋊C4C2×ΓL(1,𝔽16)12yes—
296C42—21no—
297C42—601no—
298C3×C5#212311yescharacteristic, complemented, Hall
299C22×C3—21no—
300C22×C3—41no—
301C22×C3—101no—
302C22×C3—101no—
303C22×C3—201no—
304C22×C3—201no—
305C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—101no—
309C3×C4—101no—
310C3×C4—202no—
311C3×C4#399911yescharacteristic
312C3×C4#399911yescharacteristic
313Dic3—402no—
314C2×C5—21no—
315C2×C5—41no—
316C2×C5C2×#83411yescharacteristic
317D5—41no—
318D5—41no—
319C23—51no—
320C23—51no—
321C23C22×ΓL(1,𝔽16)11yescharacteristic
322Q8—201no—
323Q8—202no—
324Q8—201no—
325Q8—201no—
326D4—201no—
327D4—201no—
328D4—201no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—51no—
333C2×C4—51no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—102no—
339C2×C4—202no—
340C2×C4—602no—
341C2×C4#676311yescharacteristic
342C2×C4#676311yescharacteristic
343C8—602no—
344C2×C3—21no—
345C2×C3—41no—
346C2×C3—201no—
347C2×C3—201no—
348C2×C3C2×#399911yescharacteristic
349C5#7187011yescharacteristic, complemented, Sylow
350C22—21no—
351C22—41no—
352C22—101no—
353C22—101no—
354C22—201no—
355C22—201no—
356C22C2×#676311yescharacteristic
357C4—21no—
358C4—42no—
359C4—101no—
360C4—101no—
361C4—202no—
362C4—1202no—
363C4#9122811yescharacteristic
364C4#9122811yescharacteristic
365C3#1060438311yescharacteristic, complemented, Sylow
366C2—21no—
367C2—41no—
368C2—201no—
369C2—201no—
370C2C2×#9122811yescharacteristic
subgroupG/H
centerC2C2×#91228
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC3×C5×#117C4
socleC2×C3×C5C2×#99
prime radicalC2×C3×#709C22
cosocleC2×C3×C22⋊D5C23
hypercenter#117ΓL(1,𝔽16)
hypocenterC3×C5#2123
LCSG ▷ C23×C3×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #11746736C2semidirect
7. C22×C3×Dic10C4semidirect
10. C3×C5×#117C4semidirect
13. C3×#4032C4semidirect
58. C22×Dic10Dic3semidirect
61. C5×#117Dic3semidirect
64. #4032Dic3semidirect
106. C3×#117AGL(1,𝔽5)semidirect
221. #117ΓL(1,𝔽16)semidirect
298. C3×C5#2123semidirect
349. C5#71870semidirect
365. C3#10604383semidirect

Maximal subgroups

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

subgroupindex
C3×#658602
#117524202
#117524042
#117467362
#106043833
#718705