Finite Group Databasebeta

← C5×#72123 #14316511

Subgroups — C5×#72123 #14316511

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#576C211yesmaximal, characteristic
2C5×#5195C211yesmaximal, characteristic, complemented
3C2×C5×#827C211yesmaximal, characteristic, complemented
4C5×#5143C211yesmaximal, characteristic, complemented
5C5×#4567C211yesmaximal, characteristic, complemented
6C5×#4563C211yesmaximal, characteristic, complemented
7C5×#4561C211yesmaximal, characteristic, complemented
8C5×#2194—31nomaximal, Hall
9C3×C5×H(5,𝔽2)—21no—
10C3×C5×AGL(1,ℤ/8)—22no—
11C3×C5×AGL(1,ℤ/8)—21no—
12C3×C5×#135—21no—
13C3×C5×#135C2211yescharacteristic
14C2×C3×C5×D8—21no—
15C2×C3×C5×D8C2211yescharacteristic
16C3×C5×#131C2211yescharacteristic
17C5×#951—21no—
18C2×C5×Dic12C2211yescharacteristic, complemented
19C5×#829—21no—
20C5×#827—21no—
21C5×#827—22no—
22C5×#824—21no—
23C5×#819C2211yescharacteristic, complemented
24C5×#812C2211yescharacteristic, complemented
25C2×C5×C3⋊C16C2211yescharacteristic, complemented
26#72123C511yesmaximal, characteristic, direct product factor, Hall
27C5×#576D311yescharacteristic
28C5×#510—31no—
29C2×C5×SD32—31no—
30C5×#473—31no—
31C5×#361—31no—
32C5×#359—31no—
33C5×#350—31no—
34C3×C5×#41—22no—
35C3×C5×#41—21no—
36C3×C5×#41—41no—
37C3×C5×#41D411yescharacteristic
38C2×C3×C5×D4—22no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—42no—
41C2×C3×C5×D4—41no—
42C2×C5×Dic6—21no—
43C3×C5×Q16D411yescharacteristic
44C3×C5×SD16—22no—
45C3×C5×SD16—21no—
46C3×C5×D8—22no—
47C3×C5×D8—21no—
48C3×C5×D8D411yescharacteristic
49C3×C5×D8D412yes—
50C3×C5×M16—21no—
51C3×C5×M16D411yescharacteristic
52C2×C3×C5×C8—21no—
53C2×C3×C5×C8C2311yescharacteristic
54C5×#216—41no—
55C5×#214—41no—
56C5×#208—21no—
57C5×Dic12—21no—
58C5×Dic12—21no—
59C5×C3⋊C16—21no—
60C5×C3⋊C16—21no—
61C3×#576C2×C511yescharacteristic
62#5195C2×C511yescharacteristic, complemented
63C2×#827C2×C511yescharacteristic, complemented
64#5143C2×C511yescharacteristic, complemented
65#4567C2×C511yescharacteristic, complemented
66#4563C2×C511yescharacteristic, complemented
67#4561C2×C511yescharacteristic, complemented
68C5×H(5,𝔽2)—21no—
69C5×#137—61no—
70C5×AGL(1,ℤ/8)—22no—
71C5×AGL(1,ℤ/8)—21no—
72C5×#135—21no—
73C5×#135C2×D311yescharacteristic
74C2×C5×Q16—31no—
75C2×C5×D8—21no—
76C2×C5×D8C2×D311yescharacteristic
77C5×#131C2×D311yescharacteristic
78C5×Q32—61no—
79C5×SD32—62no—
80C5×SD32—61no—
81C5×M32—31no—
82C2×C5×C16—31no—
83C5×#108—31no—
84C5×#101—61no—
85#2194—31noSylow
86C23×C3×C5—42no—
87C23×C3×C5—41no—
88C3×C5×Q8—21no—
89C3×C5×Q8D811yescharacteristic
90C3×C5×D4—21no—
91C3×C5×D4—21no—
92C3×C5×D4—22no—
93C3×C5×D4—42no—
94C3×C5×D4—42no—
95C3×C5×D4—41no—
96C3×C5×D4—41no—
97C3×C5×D4D811yescharacteristic
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—42no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C5×Dic3—41no—
103C5×Dic6—41no—
104C5×Dic6—41no—
105C3×C5×C8—21no—
106C3×C5×C8C2×D411yescharacteristic
107C3×C5×C8C2×D411yescharacteristic
108C5×C3⋊C8—41no—
109C3×H(5,𝔽2)—21no—
110C3×AGL(1,ℤ/8)—22no—
111C3×AGL(1,ℤ/8)—21no—
112C3×#135—21no—
113C3×#135C22×C511yescharacteristic
114C2×C3×D8—21no—
115C2×C3×D8C22×C511yescharacteristic
116C3×#131C22×C511yescharacteristic
117#951—21no—
118C2×Dic12C22×C511yescharacteristic, complemented
119#829—21no—
120#827—22no—
121#827—21no—
122#824—21no—
123#819C22×C511yescharacteristic, complemented
124#812C22×C511yescharacteristic, complemented
125C2×C3⋊C16C22×C511yescharacteristic, complemented
126C5×#41—22no—
127C5×#41—21no—
128C5×#41—41no—
129C5×#41#6711yescharacteristic
130C2×C5×Q8—61no—
131C2×C5×D4—22no—
132C2×C5×D4—21no—
133C2×C5×D4—42no—
134C2×C5×D4—41no—
135C5×Q16—61no—
136C5×Q16—61no—
137C5×Q16—121no—
138C5×Q16#6711yescharacteristic
139C5×SD16—22no—
140C5×SD16—21no—
141C5×SD16—121no—
142C5×D8—22no—
143C5×D8—21no—
144C5×D8#6711yescharacteristic
145C5×D8#6712yes—
146C5×M16—21no—
147C5×M16—61no—
148C5×M16#6711yescharacteristic
149C2×C5×C8—21no—
150C2×C5×C8C22×D311yescharacteristic
151C5×C16—61no—
152C5×C16—61no—
153#576C5×D311yescharacteristic
154#510—31no—
155C2×SD32—31no—
156#473—31no—
157#361—31no—
158#359—31no—
159#350—31no—
160C22×C3×C5—21no—
161C22×C3×C5—42no—
162C22×C3×C5—41no—
163C22×C3×C5—161no—
164C22×C3×C5—162no—
165C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
166C3×C4×C5—21no—
167C3×C4×C5—41no—
168C3×C4×C5#12011yescharacteristic
169C3×C4×C5C2×D811yescharacteristic
170C5×Dic3—81no—
171C3×#41—22no—
172C3×#41—21no—
173C3×#41—41no—
174C3×#41C5×D411yescharacteristic
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—42no—
178C2×C3×D4—41no—
179C2×Dic6—21no—
180C3×Q16C5×D411yescharacteristic
181C3×SD16—22no—
182C3×SD16—21no—
183C3×D8—22no—
184C3×D8—21no—
185C3×D8C5×D411yescharacteristic
186C3×D8C5×D412yes—
187C3×M16—21no—
188C3×M16C5×D411yescharacteristic
189C2×C3×C8—21no—
190C2×C3×C8C23×C511yescharacteristic
191#216—41no—
192#214—41no—
193#208—21no—
194Dic12—21no—
195Dic12—21no—
196C3⋊C16—21no—
197C3⋊C16—21no—
198C23×C5—42no—
199C23×C5—41no—
200C5×Q8—21no—
201C5×Q8—121no—
202C5×Q8—121no—
203C5×Q8#21311yescharacteristic
204C5×D4—21no—
205C5×D4—21no—
206C5×D4—22no—
207C5×D4—42no—
208C5×D4—42no—
209C5×D4—41no—
210C5×D4—41no—
211C5×D4#21311yescharacteristic
212C2×C4×C5—21no—
213C2×C4×C5—42no—
214C2×C4×C5—41no—
215C2×C4×C5—121no—
216C2×C4×C5C2×#6711yescharacteristic
217C5×C8—21no—
218C5×C8—121no—
219C5×C8C2×#6711yescharacteristic
220C5×C8C2×#6711yescharacteristic
221H(5,𝔽2)—21no—
222#137—61no—
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—21no—
225#135—21no—
226#135C2×C5×D311yescharacteristic
227C2×Q16—31no—
228C2×D8—21no—
229C2×D8C2×C5×D311yescharacteristic
230#131C2×C5×D311yescharacteristic
231Q32—61no—
232SD32—62no—
233SD32—61no—
234M32—31no—
235C2×C16—31no—
236#108—31no—
237#101—61no—
238C2×C3×C5—21no—
239C2×C3×C5—41no—
240C2×C3×C5—82no—
241C2×C3×C5—81no—
242C2×C3×C5#44711yescharacteristic
243C23×C3—42no—
244C23×C3—41no—
245C3×Q8—21no—
246C3×Q8C5×D811yescharacteristic
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—22no—
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4C5×D811yescharacteristic
255C2×C3×C4—21no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4C2×C5×D411yescharacteristic
259C2×Dic3—41no—
260Dic6—41no—
261Dic6—41no—
262C3×C8—21no—
263C3×C8C2×C5×D411yescharacteristic
264C3×C8C2×C5×D411yescharacteristic
265C3⋊C8—41no—
266C22×C5—21no—
267C22×C5—42no—
268C22×C5—41no—
269C22×C5—161no—
270C22×C5—162no—
271C22×C5#93211yescharacteristic
272C4×C5—21no—
273C4×C5—41no—
274C4×C5—241no—
275C4×C5C2×#21311yescharacteristic
276C4×C5#95311yescharacteristic
277#41—22no—
278#41—21no—
279#41—41no—
280#41C5×#6711yescharacteristic
281C2×Q8—61no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4—41no—
286Q16—61no—
287Q16—61no—
288Q16—121no—
289Q16C5×#6711yescharacteristic
290SD16—22no—
291SD16—21no—
292SD16—121no—
293D8—22no—
294D8—21no—
295D8C5×#6711yescharacteristic
296D8C5×#6712yes—
297M16—21no—
298M16—61no—
299M16C5×#6711yescharacteristic
300C2×C8—21no—
301C2×C8C22×C5×D311yescharacteristic
302C16—61no—
303C16—61no—
304C3×C5#219411yescharacteristic, complemented, Hall
305C22×C3—21no—
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—161no—
309C22×C3—162no—
310C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
311C3×C4—21no—
312C3×C4—41no—
313C3×C4C5×#12011yescharacteristic
314C3×C4C2×C5×D811yescharacteristic
315Dic3—81no—
316C2×C5—21no—
317C2×C5—41no—
318C2×C5—82no—
319C2×C5—81no—
320C2×C5#521711yescharacteristic
321C23—42no—
322C23—41no—
323Q8—21no—
324Q8—121no—
325Q8—121no—
326Q8C5×#21311yescharacteristic
327D4—21no—
328D4—21no—
329D4—22no—
330D4—42no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4C5×#21311yescharacteristic
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—121no—
339C2×C4C2×C5×#6711yescharacteristic
340C8—21no—
341C8—121no—
342C8C2×C5×#6711yescharacteristic
343C8C2×C5×#6711yescharacteristic
344C2×C3—21no—
345C2×C3—41no—
346C2×C3—82no—
347C2×C3—81no—
348C2×C3C5×#44711yescharacteristic
349C5#7212311yescharacteristic, direct product factor, Sylow
350C22—21no—
351C22—42no—
352C22—41no—
353C22—161no—
354C22—162no—
355C22C5×#93211yescharacteristic
356C4—21no—
357C4—41no—
358C4—241no—
359C4C2×C5×#21311yescharacteristic
360C4C5×#95311yescharacteristic
361C3C5×#219411yescharacteristic, complemented, Sylow
362C2—21no—
363C2—41no—
364C2—82no—
365C2—81no—
366C2C5×#521711yescharacteristic
subgroupG/H
centerC2×C5#5217
commutator, cosocleC2×C3×C8C23×C5
FrattiniC2×C8C22×C5×D3
Fitting, prime radicalC3×C5×#576C2
socleC2×C3×C5#447
hypercenterC5×#576D3
hypocenterC3C5×#2194
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#131 ◁ C5×#576

Complements

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

NH ≅ G/Ntype
2. C5×#5195C2semidirect
3. C2×C5×#827C2semidirect
4. C5×#5143C2semidirect
5. C5×#4567C2semidirect
6. C5×#4563C2semidirect
7. C5×#4561C2semidirect
18. C2×C5×Dic12C22semidirect
23. C5×#819C22semidirect
24. C5×#812C22semidirect
25. C2×C5×C3⋊C16C22semidirect
26. #72123C5direct (N × H)
62. #5195C2×C5semidirect
63. C2×#827C2×C5semidirect
64. #5143C2×C5semidirect
65. #4567C2×C5semidirect
66. #4563C2×C5semidirect
67. #4561C2×C5semidirect
118. C2×Dic12C22×C5semidirect
123. #819C22×C5semidirect
124. #812C22×C5semidirect
125. C2×C3⋊C16C22×C5semidirect
304. C3×C5#2194semidirect
349. C5#72123direct (N × H)
361. C3C5×#2194semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5762
C5×#51952
C2×C5×#8272
C5×#51432
C5×#45672
C5×#45632
C5×#45612
C5×#21943
#721235