Finite Group Databasebeta

← #14313269

Subgroups — #14313269

Order 1920.

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

normal subgroups only

This group has 532 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×#65859C211yesmaximal, characteristic
2#11752420C211yesmaximal, characteristic, complemented
3#11752404C211yesmaximal, characteristic, complemented
4#11746735C214yesmaximal, complemented
5#10604382—31nomaximal, Hall
6C2×C3×#709C2211yescharacteristic
7C22×C3×D20C411yescharacteristic, complemented
8#91989—21no—
9D5⋊#208—21no—
10C3×C5×#117C411yescharacteristic, complemented
11C3×#4041—21no—
12C3×#4041—21no—
13C3×#4028C412yescomplemented
14C3×#4028C2212yes—
15C3×#4021—21no—
16C3×#4020—21no—
17#91242C2212yes—
18#91240C2212yes—
19#91234—21no—
20#91228—21no—
21#71859—51nomaximal, Hall
22#65859D311yescharacteristic
23#65761—31no—
24#65745—31no—
25#64864—34no—
26C23×C3×D5C2×C412yes—
27C22×C3×C4×C5C2×C411yescharacteristic
28C2×C3×C22⋊D5C2×C411yescharacteristic
29C2×C3×C22⋊D5C2311yescharacteristic
30C3×#709—41no—
31C3×#709—41no—
32C2×C3×D20—21no—
33C2×C3×D20—21no—
34C2×C3×D20—22no—
35C2×C3×D20—22no—
36C2×C3×D20—41no—
37C2×C3×D20—41no—
38C2×C3×D20D411yescharacteristic
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×Dic10D411yescharacteristic
42#6763—22no—
43#6762—22no—
44C4⋊ΓL(1,𝔽16)—41no—
45C4×ΓL(1,𝔽16)—41no—
46#6758—41no—
47D5⋊(C3⋊C8)—41no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C22⋊C4C2×C412yes—
50C3×C42×C5—21no—
51C3×#685—22no—
52C3×#685—22no—
53C3×#535—51no—
54#5240—51no—
55#5237—51no—
56#4536—54no—
57C2×#709C2×D311yescharacteristic
58C22×D20Dic311yescharacteristic, complemented
59#4130—61no—
60D5⋊M16—61no—
61C5×#117Dic311yescharacteristic, complemented
62#4041—21no—
63#4041—21no—
64#4028Dic312yescomplemented
65#4028C2×D312yes—
66#4021—21no—
67#4020—21no—
68#4013—32no—
69#4011—32no—
70#4005—61no—
71#3999—61no—
72#2122—151noSylow
73C23×C3×C5C22×C411yescharacteristic
74C22×C3×D5—22no—
75C22×C3×D5—42no—
76C22×C3×D5—82no—
77C22×C3×D5C22⋊C411yescharacteristic
78C22×C3×D5C2×D411yescharacteristic
79C2×ΓL(1,𝔽16)—42no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—22no—
83C2×C3×C4×C5C22⋊C411yescharacteristic
84C2×C3×C4×C5C22⋊C411yescharacteristic
85C3×C22⋊D5—41no—
86C3×C22⋊D5—41no—
87C2×C3×Dic5C22⋊C411yescharacteristic
88C2×C3×Dic5C2×D411yescharacteristic
89C3×D20—41no—
90C3×D20—42no—
91C3×D20—41no—
92C3×D20—41no—
93C3×D20—82no—
94C3×D20—82no—
95C3×C4×D5—21no—
96C3×C4×D5—21no—
97C3×C4×D5—21no—
98C3×C4×D5—21no—
99C3×Dic10—41no—
100C3⋊#170—42no—
101C2×C3×#41—51no—
102C22×C3×D4—51no—
103C3×#127—101no—
104C3×#124—101no—
105C3×#121—101no—
106C3×#121—101no—
107C3×#120—52no—
108C3×#120—52no—
109C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
110#926—101no—
111C2×#208—101no—
112#836—101no—
113#834—52no—
114#834—101no—
115#833—52no—
116C23×D5C2×Dic312yes—
117C22×C4×C5C2×Dic311yescharacteristic
118C2×C22⋊D5C2×Dic311yescharacteristic
119C2×C22⋊D5C22×D311yescharacteristic
120#709—41no—
121#709—41no—
122C2×D20—21no—
123C2×D20—21no—
124C2×D20—22no—
125C2×D20—22no—
126C2×D20—41no—
127C2×D20—41no—
128C2×D20#6711yescharacteristic
129C2×C4×D5#6711yescharacteristic
130C2×C4×D5#6711yescharacteristic
131C2×Dic10#6711yescharacteristic
132C22⋊AGL(1,𝔽5)—62no—
133#704—62no—
134C4⋊AGL(1,𝔽5)—121no—
135C4×AGL(1,𝔽5)—121no—
136#700—121no—
137D5⋊C8—121no—
138C5×C4⋊C4—21no—
139C5×C22⋊C4C2×Dic312yes—
140C42×C5—21no—
141#685—22no—
142#685—22no—
143#535—51no—
144#413—151no—
145#410—151no—
146AΣL(1,𝔽16)—154no—
147C22×C3×C5—21no—
148C22×C3×C5—41no—
149C22×C3×C5C2×C22⋊C411yescharacteristic
150C2×C3×D5—21no—
151C2×C3×D5—21no—
152C2×C3×D5—41no—
153C2×C3×D5—41no—
154C2×C3×D5—42no—
155C2×C3×D5—42no—
156C2×C3×D5—42no—
157C2×C3×D5—84no—
158ΓL(1,𝔽16)—82no—
159C3×C4×C5—21no—
160C3×C4×C5—42no—
161C3×C4×C5#9911yescharacteristic
162C3×C4×C5#9911yescharacteristic
163C3×Dic5—21no—
164C3×Dic5—21no—
165C24×C3—52no—
166C3×#41—201no—
167C3×#41—201no—
168C2×C3×Q8—51no—
169C2×C3×D4—51no—
170C2×C3×D4—51no—
171C2×C3×D4—51no—
172C2×C3×D4—102no—
173C2×C3×D4—102no—
174C2×C3×D4—101no—
175C2×C3×D4—101no—
176C2×C3×D4—201no—
177C2×C3×D4—201no—
178C22×C3×C4—51no—
179C22×C3×C4—51no—
180C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
181C3×C4⋊C4—21no—
182C3×C22⋊C4—102no—
183C3×C22⋊C4—102no—
184C3×C22⋊C4C2×AGL(1,𝔽5)12yes—
185C3×C42—21no—
186#217—102no—
187C4⋊Dic3—201no—
188C4×Dic3—201no—
189#208—102no—
190#208—201no—
191C2×C3⋊C8—201no—
192C23×C5C22×Dic311yescharacteristic
193C22×D5—22no—
194C22×D5—42no—
195C22×D5—82no—
196C22×D5#21711yescharacteristic
197C22×D5C2×#6711yescharacteristic
198C2×AGL(1,𝔽5)—122no—
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5—22no—
202C2×C4×C5#21711yescharacteristic
203C2×C4×C5#21711yescharacteristic
204C22⋊D5—41no—
205C22⋊D5—41no—
206C2×Dic5#21711yescharacteristic
207C2×Dic5C2×#6711yescharacteristic
208D20—41no—
209D20—42no—
210D20—41no—
211D20—41no—
212D20—82no—
213D20—82no—
214C4×D5—21no—
215C4×D5—21no—
216C4×D5—21no—
217C4×D5—21no—
218Dic10—41no—
219#170—122no—
220C2×#41—51no—
221C22×D4—51no—
222C2×M16—301no—
223#127—101no—
224#124—101no—
225#121—101no—
226#121—101no—
227#120—52no—
228#120—52no—
229#117—301no—
230#117ΓL(1,𝔽16)11yescharacteristic, complemented
231#101—301no—
232#100—152no—
233#99—152no—
234#99—301no—
235C2×C3×C5—21no—
236C2×C3×C5—41no—
237C2×C3×C5C2×#9911yescharacteristic
238C3×D5—41no—
239C3×D5—41no—
240C3×D5—82no—
241C23×C3—51no—
242C23×C3—51no—
243C23×C3—102no—
244C23×C3—202no—
245C23×C3—402no—
246C23×C3C22×AGL(1,𝔽5)11yescharacteristic
247C3×Q8—201no—
248C3×D4—201no—
249C3×D4—202no—
250C3×D4—201no—
251C3×D4—201no—
252C3×D4—201no—
253C3×D4—201no—
254C3×D4—402no—
255C3×D4—402no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—22no—
259C2×C3×C4—51no—
260C2×C3×C4—51no—
261C2×C3×C4—101no—
262C2×C3×C4—101no—
263C2×C3×C4—101no—
264C2×C3×C4—101no—
265C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
266C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
267C2×Dic3—202no—
268C3⋊C8—202no—
269C22×C5—21no—
270C22×C5—41no—
271C22×C5C2×#21711yescharacteristic
272C2×D5—21no—
273C2×D5—21no—
274C2×D5—41no—
275C2×D5—41no—
276C2×D5—42no—
277C2×D5—42no—
278C2×D5—42no—
279C2×D5—84no—
280AGL(1,𝔽5)—242no—
281C4×C5—21no—
282C4×C5—42no—
283C4×C5#83411yescharacteristic
284C4×C5#83411yescharacteristic
285Dic5—21no—
286Dic5—21no—
287C24—52no—
288#41—201no—
289#41—201no—
290C2×Q8—51no—
291C2×D4—51no—
292C2×D4—51no—
293C2×D4—51no—
294C2×D4—102no—
295C2×D4—102no—
296C2×D4—101no—
297C2×D4—101no—
298C2×D4—201no—
299C2×D4—201no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
303M16—302no—
304M16—601no—
305C2×C8—601no—
306C4⋊C4—21no—
307C4⋊C4—601no—
308C22⋊C4—102no—
309C22⋊C4—102no—
310C22⋊C4—302no—
311C22⋊C4C2×ΓL(1,𝔽16)12yes—
312C42—21no—
313C42—601no—
314C3×C5#212211yescharacteristic, complemented, Hall
315C22×C3—21no—
316C22×C3—41no—
317C22×C3—101no—
318C22×C3—101no—
319C22×C3—201no—
320C22×C3—201no—
321C22×C3—202no—
322C22×C3—202no—
323C22×C3—202no—
324C22×C3—404no—
325C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
326C3×C4—21no—
327C3×C4—42no—
328C3×C4—101no—
329C3×C4—101no—
330C3×C4#399911yescharacteristic
331C3×C4#399911yescharacteristic
332Dic3—402no—
333C2×C5—21no—
334C2×C5—41no—
335C2×C5C2×#83411yescharacteristic
336D5—41no—
337D5—41no—
338D5—82no—
339C23—51no—
340C23—51no—
341C23—102no—
342C23—202no—
343C23—402no—
344C23C22×ΓL(1,𝔽16)11yescharacteristic
345Q8—201no—
346D4—201no—
347D4—202no—
348D4—201no—
349D4—201no—
350D4—201no—
351D4—201no—
352D4—402no—
353D4—402no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—51no—
358C2×C4—51no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—602no—
364C2×C4#676311yescharacteristic
365C2×C4#676311yescharacteristic
366C8—602no—
367C2×C3—21no—
368C2×C3—41no—
369C2×C3—201no—
370C2×C3—201no—
371C2×C3—402no—
372C2×C3C2×#399911yescharacteristic
373C5#7185911yescharacteristic, complemented, Sylow
374C22—21no—
375C22—41no—
376C22—101no—
377C22—101no—
378C22—201no—
379C22—201no—
380C22—202no—
381C22—202no—
382C22—202no—
383C22—404no—
384C22C2×#676311yescharacteristic
385C4—21no—
386C4—42no—
387C4—101no—
388C4—101no—
389C4—1202no—
390C4#9122811yescharacteristic
391C4#9122811yescharacteristic
392C3#1060438211yescharacteristic, complemented, Sylow
393C2—21no—
394C2—41no—
395C2—201no—
396C2—201no—
397C2—402no—
398C2C2×#9122811yescharacteristic
subgroupG/H
centerC2C2×#91228
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC3×C5×#117C4
socleC2×C3×C5C2×#99
prime radicalC3×#65859C2
cosocleC2×C3×C22⋊D5C23
hypercenter#117ΓL(1,𝔽16)
hypocenterC3×C5#2122
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
2. #11752420C2semidirect
3. #11752404C2semidirect
4. #11746735C2semidirect
7. C22×C3×D20C4semidirect
10. C3×C5×#117C4semidirect
13. C3×#4028C4semidirect
58. C22×D20Dic3semidirect
61. C5×#117Dic3semidirect
64. #4028Dic3semidirect
109. C3×#117AGL(1,𝔽5)semidirect
230. #117ΓL(1,𝔽16)semidirect
314. C3×C5#2122semidirect
373. C5#71859semidirect
392. C3#10604382semidirect

Maximal subgroups

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

subgroupindex
C3×#658592
#117524202
#117524042
#117467352
#106043823
#718595