Finite Group Databasebeta

← #14316774

Subgroups — #14316774

Order 1920.

540 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#11756442C211yesmaximal, characteristic, complemented
2C3×C5×#455C211yesmaximal, characteristic, complemented
3#11751345C212yesmaximal, complemented
4#11746362C211yesmaximal, characteristic, complemented
5#11746296C212yesmaximal, complemented
6#10601784—31nomaximal, Hall
7C3×C5×H(5,𝔽2)C2211yescharacteristic
8C3×C5×#137—21no—
9C3×C5×#124C2211yescharacteristic
10#91836—21no—
11#91826—22no—
12#91826C2212yes—
13#91774—22no—
14#91771—22no—
15#91764—21no—
16C3×C5×#104—21no—
17C3×C5×#100C2211yescharacteristic
18#91118—22no—
19#91118C2212yes—
20#70991—51nomaximal, Hall
21#66017—31no—
22C5×#455D311yescharacteristic, complemented
23#65533—32no—
24#64761—31no—
25#64695—32no—
26C23×D15D412yescomplemented
27C3×C5×#41—21no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8D411yescharacteristic, complemented
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—41no—
32C2×C3×C5×D4C2311yescharacteristic
33C2×#1220—22no—
34C2×#1220—42no—
35C2×D60—21no—
36C2×D60—21no—
37C2×Dic30—21no—
38C3×C5×Q16—41no—
39C3×C5×SD16—41no—
40C3×C5×M16—21no—
41C3×C5×C22⋊C4—22no—
42C3×C42×C5D411yescharacteristic, complemented
43#6719—22no—
44#6719D412yescomplemented
45#6714—22no—
46#6714—22no—
47#6714—22no—
48#6714—42no—
49C4×Dic15—21no—
50#5683—51no—
51C3×#455D511yescharacteristic, complemented
52#5244—52no—
53#4541—51no—
54#4475—52no—
55C5×H(5,𝔽2)C2×D311yescharacteristic
56C5×#137—21no—
57C5×#124C2×D311yescharacteristic
58#4093—61no—
59#4083—32no—
60#4083—62no—
61#4031—62no—
62#4028—62no—
63#4021—61no—
64C5×#104—21no—
65C5×#100C2×D311yescharacteristic
66#3966—32no—
67#3966—62no—
68#2199—151noSylow
69C23×C3×C5—22no—
70C23×C3×C5C2×D412yes—
71C22×D15—22no—
72C22×D15—42no—
73C22×D15—82no—
74C3×C5×Q8—21no—
75C3×C5×Q8—41no—
76C3×C5×D4—21no—
77C3×C5×D4—41no—
78C3×C5×D4—42no—
79C3×C5×D4—41no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5C2×D411yescharacteristic
85#1220—82no—
86#1220—84no—
87C2×Dic15—22no—
88C2×Dic15—42no—
89D60—81no—
90D60—81no—
91Dic30—81no—
92C3×C5×C8—41no—
93C3×H(5,𝔽2)C2×D511yescharacteristic
94C3×#137—21no—
95C3×#124C2×D511yescharacteristic
96#947—101no—
97#937—52no—
98#937—102no—
99#885—102no—
100#882—102no—
101#875—101no—
102C3×#104—21no—
103C3×#100C2×D511yescharacteristic
104#834—52no—
105#834—102no—
106C23×D5—32no—
107C5×#41—21no—
108C5×#41—41no—
109C2×C5×Q8#6711yescharacteristic, complemented
110C2×C5×D4—22no—
111C2×C5×D4—41no—
112C2×C5×D4C22×D311yescharacteristic
113C2×C22⋊D5—62no—
114C2×C22⋊D5—122no—
115C2×D20—61no—
116C2×D20—61no—
117C2×Dic10—61no—
118C5×Q16—41no—
119C5×SD16—41no—
120C5×M16—21no—
121C5×C22⋊C4—22no—
122C42×C5#6711yescharacteristic, complemented
123C22⋊Dic5—32no—
124C22⋊Dic5—62no—
125#685—62no—
126#685—62no—
127#685—62no—
128#685—122no—
129C4×Dic5—61no—
130H(3,𝔽4)—151no—
131B(4,𝔽2)—152no—
132#455D1511yescharacteristic, complemented
133#354—151no—
134AΣL(1,𝔽16)—152no—
135C22×C3×C5—22no—
136C22×C3×C5—42no—
137C22×C3×C5—41no—
138C22×C3×C5—82no—
139C22×C3×C5#12011yescharacteristic
140C2×D15—42no—
141C2×D15—42no—
142C2×D15—44no—
143C2×D15—42no—
144C2×D15—82no—
145C3×C4×C5—21no—
146C3×C4×C5—41no—
147C3×C4×C5—41no—
148C3×C4×C5—41no—
149Dic15—42no—
150Dic15—82no—
151C23×D3—52no—
152C3×#41—21no—
153C3×#41—41no—
154C2×C3×Q8C22⋊D511yescharacteristic, complemented
155C2×C3×D4—22no—
156C2×C3×D4—41no—
157C2×C3×D4C22×D511yescharacteristic
158C2×#67—102no—
159C2×#67—202no—
160C2×D12—101no—
161C2×D12—101no—
162C2×Dic6—101no—
163C3×Q16—41no—
164C3×SD16—41no—
165C3×M16—21no—
166C3×C22⋊C4—22no—
167C3×C42C22⋊D511yescharacteristic, complemented
168#217—52no—
169#217—102no—
170(C2×D3)⋊C4—102no—
171(C2×D3)⋊C4—102no—
172(C2×D3)⋊C4—102no—
173(C2×D3)⋊C4—202no—
174C4×Dic3—101no—
175C23×C5—22no—
176C23×C5D3×D412yes—
177C22×D5—62no—
178C22×D5—122no—
179C22×D5—242no—
180C5×Q8—21no—
181C5×Q8—41no—
182C5×D4—21no—
183C5×D4—41no—
184C5×D4—42no—
185C5×D4—41no—
186C2×C4×C5—22no—
187C2×C4×C5—21no—
188C2×C4×C5—21no—
189C2×C4×C5—41no—
190C2×C4×C5C2×#6711yescharacteristic
191C22⋊D5—242no—
192C22⋊D5—244no—
193C2×Dic5—62no—
194C2×Dic5—122no—
195D20—241no—
196D20—241no—
197Dic10—241no—
198C5×C8—41no—
199H(5,𝔽2)C2×D1511yescharacteristic
200#137—21no—
201#124—302no—
202#124—301no—
203#124—301no—
204#124C2×D1511yescharacteristic
205#120—152no—
206#120—302no—
207#120—302no—
208#104—21no—
209#100C2×D1511yescharacteristic
210#99—152no—
211#99—302no—
212C2×C3×C5—21no—
213C2×C3×C5—42no—
214C2×C3×C5—81no—
215C2×C3×C5B(4,𝔽2)11yescharacteristic
216D15—82no—
217C23×C3—22no—
218C23×C3D4×D512yes—
219C22×D3—102no—
220C22×D3—202no—
221C22×D3—402no—
222C3×Q8—21no—
223C3×Q8—41no—
224C3×D4—21no—
225C3×D4—41no—
226C3×D4—42no—
227C3×D4—41no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4C2×C22⋊D511yescharacteristic
233#67—402no—
234#67—404no—
235C2×Dic3—102no—
236C2×Dic3—202no—
237D12—401no—
238D12—401no—
239Dic6—401no—
240C3×C8—41no—
241C22×C5—22no—
242C22×C5—42no—
243C22×C5—41no—
244C22×C5—82no—
245C22×C5#93711yescharacteristic
246C2×D5—122no—
247C2×D5—122no—
248C2×D5—124no—
249C2×D5—122no—
250C2×D5—242no—
251C4×C5—21no—
252C4×C5—41no—
253C4×C5—41no—
254C4×C5—41no—
255Dic5—122no—
256Dic5—242no—
257C24—152no—
258#41—21no—
259#41—41no—
260C2×Q8—301no—
261C2×Q8#122011yescharacteristic, complemented
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—301no—
265C2×D4—301no—
266C2×D4—302no—
267C2×D4—602no—
268C2×D4C22×D1511yescharacteristic
269Q16—41no—
270SD16—41no—
271M16—21no—
272C22⋊C4—22no—
273C22⋊C4—152no—
274C22⋊C4—302no—
275C22⋊C4—302no—
276C22⋊C4—302no—
277C22⋊C4—302no—
278C22⋊C4—602no—
279C42—301no—
280C42#122011yescharacteristic, complemented
281C3×C5#219911yescharacteristic, complemented, Hall
282C22×C3—22no—
283C22×C3—42no—
284C22×C3—41no—
285C22×C3—82no—
286C22×C3#408311yescharacteristic
287C2×D3—202no—
288C2×D3—202no—
289C2×D3—204no—
290C2×D3—202no—
291C2×D3—402no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296Dic3—202no—
297Dic3—402no—
298C2×C5—21no—
299C2×C5—42no—
300C2×C5—81no—
301C2×C5#503211yescharacteristic
302D5—242no—
303C23—22no—
304C23—302no—
305C23—602no—
306C23—1202no—
307C23D4×D1512yes—
308Q8—21no—
309Q8—41no—
310Q8—1201no—
311D4—21no—
312D4—41no—
313D4—42no—
314D4—41no—
315D4—1201no—
316D4—1201no—
317D4—1202no—
318D4—1204no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—302no—
324C2×C4—602no—
325C2×C4C2×#122011yescharacteristic
326C8—41no—
327C2×C3—21no—
328C2×C3—42no—
329C2×C3—81no—
330C2×C3#6532111yescharacteristic
331D3—402no—
332C5#7099111yescharacteristic, complemented, Sylow
333C22—22no—
334C22—42no—
335C22—41no—
336C22—82no—
337C22—602no—
338C22—602no—
339C22—604no—
340C22—602no—
341C22—1202no—
342C22#9182611yescharacteristic
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—602no—
348C4—1202no—
349C3#1060178411yescharacteristic, complemented, Sylow
350C2—21no—
351C2—42no—
352C2—81no—
353C2—1202no—
354C2#1175113311yescharacteristic
subgroupG/H
centerC2#11751133
commutator, cosocleC2×C3×C5×D4C23
FrattiniC2×D4C22×D15
FittingC3×C5×#455C2
socleC2×C3×C5B(4,𝔽2)
hypercenter#455D15
hypocenterC3×C5#2199
LCSG ▷ C2×C3×C5×D4 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #11756442C2semidirect
2. C3×C5×#455C2semidirect
3. #11751345C2semidirect
4. #11746362C2semidirect
5. #11746296C2semidirect
22. C5×#455D3semidirect
26. C23×D15D4semidirect
29. C2×C3×C5×Q8D4semidirect
42. C3×C42×C5D4semidirect
44. #6719D4semidirect
51. C3×#455D5semidirect
109. C2×C5×Q8#67semidirect
122. C42×C5#67semidirect
132. #455D15semidirect
154. C2×C3×Q8C22⋊D5semidirect
167. C3×C42C22⋊D5semidirect
261. C2×Q8#1220semidirect
280. C42#1220semidirect
281. C3×C5#2199semidirect
332. C5#70991semidirect
349. C3#10601784semidirect

Maximal subgroups

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

subgroupindex
#117564422
C3×C5×#4552
#117513452
#117463622
#117462962
#106017843
#709915