Finite Group Databasebeta

← C5×#72122 #14316385

Subgroups — C5×#72122 #14316385

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#576C211yesmaximal, characteristic, complemented
2C5×#5193C211yesmaximal, characteristic, complemented
3C2×C5×#826C211yesmaximal, characteristic, complemented
4C5×#5142C211yesmaximal, characteristic, complemented
5C5×#4567C211yesmaximal, characteristic, complemented
6C5×#4563C211yesmaximal, characteristic, complemented
7C5×#4561C211yesmaximal, characteristic, complemented
8C5×#2192—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×#949—21no—
18C2×C5×D24C2211yescharacteristic, complemented
19C5×#828—21no—
20C5×#826—21no—
21C5×#826—22no—
22C5×#823—21no—
23C5×#819C2211yescharacteristic, complemented
24C5×#812C2211yescharacteristic, complemented
25C2×C5×C3⋊C16C2211yescharacteristic, complemented
26#72122C511yesmaximal, characteristic, direct product factor, Hall
27C5×#576D311yescharacteristic, complemented
28C5×#509—31no—
29C2×C5×D16—31no—
30C5×#472—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×D12—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×#215—41no—
55C5×#213—41no—
56C5×#208—21no—
57C5×D24—21no—
58C5×D24—21no—
59C5×C3⋊C16—21no—
60C5×C3⋊C16—21no—
61C3×#576C2×C511yescharacteristic, complemented
62#5193C2×C511yescharacteristic, complemented
63C2×#826C2×C511yescharacteristic, complemented
64#5142C2×C511yescharacteristic, complemented
65#4567C2×C511yescharacteristic, complemented
66#4563C2×C511yescharacteristic, complemented
67#4561C2×C511yescharacteristic, complemented
68C5×H(5,𝔽2)—21no—
69C5×AGL(1,ℤ/8)—22no—
70C5×AGL(1,ℤ/8)—21no—
71C5×AGL(1,ℤ/8)—61no—
72C5×#135—21no—
73C5×#135C2×D311yescharacteristic
74C2×C5×D8—21no—
75C2×C5×D8—31no—
76C2×C5×D8C2×D311yescharacteristic
77C5×#131C2×D311yescharacteristic
78C5×SD32—61no—
79C5×D16—62no—
80C5×D16—61no—
81C5×M32—31no—
82C2×C5×C16—31no—
83C5×#108—31no—
84C5×#100—61no—
85#2192—31noSylow
86C23×C3×C5—42no—
87C23×C3×C5—41no—
88C22×C5×D3—41no—
89C3×C5×Q8—21no—
90C3×C5×Q8D811yescharacteristic
91C3×C5×D4—21no—
92C3×C5×D4—21no—
93C3×C5×D4—22no—
94C3×C5×D4—42no—
95C3×C5×D4—42no—
96C3×C5×D4—41no—
97C3×C5×D4—41no—
98C3×C5×D4D811yescharacteristic
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—42no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5C2×D411yescharacteristic
103C5×D12—41no—
104C5×D12—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#949—21no—
118C2×D24C22×C511yescharacteristic, complemented
119#828—21no—
120#826—22no—
121#826—21no—
122#823—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×D4—22no—
131C2×C5×D4—21no—
132C2×C5×D4—42no—
133C2×C5×D4—41no—
134C2×C5×D4—61no—
135C5×Q16#6711yescharacteristic
136C5×SD16—22no—
137C5×SD16—21no—
138C5×SD16—121no—
139C5×D8—22no—
140C5×D8—21no—
141C5×D8—61no—
142C5×D8—61no—
143C5×D8—121no—
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, complemented
154#509—31no—
155C2×D16—31no—
156#472—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
166C2×C5×D3—81no—
167C2×C5×D3—161no—
168C3×C4×C5—21no—
169C3×C4×C5—41no—
170C3×C4×C5#12011yescharacteristic
171C3×C4×C5C2×D811yescharacteristic
172C3×#41—22no—
173C3×#41—21no—
174C3×#41—41no—
175C3×#41C5×D411yescharacteristic
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—42no—
179C2×C3×D4—41no—
180C2×D12—21no—
181C3×Q16C5×D411yescharacteristic
182C3×SD16—22no—
183C3×SD16—21no—
184C3×D8—22no—
185C3×D8—21no—
186C3×D8C5×D411yescharacteristic
187C3×D8C5×D412yes—
188C3×M16—21no—
189C3×M16C5×D411yescharacteristic
190C2×C3×C8—21no—
191C2×C3×C8C23×C511yescharacteristic
192#215—41no—
193#213—41no—
194#208—21no—
195D24—21no—
196D24—21no—
197C3⋊C16—21no—
198C3⋊C16—21no—
199C23×C5—42no—
200C23×C5—41no—
201C23×C5—121no—
202C5×Q8—21no—
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—121no—
212C5×D4—121no—
213C5×D4#21311yescharacteristic
214C2×C4×C5—21no—
215C2×C4×C5—42no—
216C2×C4×C5—41no—
217C2×C4×C5C2×#6711yescharacteristic
218C5×C8—21no—
219C5×C8—121no—
220C5×C8C2×#6711yescharacteristic
221C5×C8C2×#6711yescharacteristic
222H(5,𝔽2)—21no—
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—21no—
225AGL(1,ℤ/8)—61no—
226#135—21no—
227#135C2×C5×D311yescharacteristic
228C2×D8—21no—
229C2×D8—31no—
230C2×D8C2×C5×D311yescharacteristic
231#131C2×C5×D311yescharacteristic
232SD32—61no—
233D16—62no—
234D16—61no—
235M32—31no—
236C2×C16—31no—
237#108—31no—
238#100—61no—
239C2×C3×C5—21no—
240C2×C3×C5—41no—
241C2×C3×C5—82no—
242C2×C3×C5—81no—
243C2×C3×C5#44711yescharacteristic
244C5×D3—161no—
245C23×C3—42no—
246C23×C3—41no—
247C22×D3—41no—
248C3×Q8—21no—
249C3×Q8C5×D811yescharacteristic
250C3×D4—21no—
251C3×D4—21no—
252C3×D4—22no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4C5×D811yescharacteristic
258C2×C3×C4—21no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4C2×C5×D411yescharacteristic
262D12—41no—
263D12—41no—
264C3×C8—21no—
265C3×C8C2×C5×D411yescharacteristic
266C3×C8C2×C5×D411yescharacteristic
267C3⋊C8—41no—
268C22×C5—21no—
269C22×C5—42no—
270C22×C5—41no—
271C22×C5—161no—
272C22×C5—162no—
273C22×C5—241no—
274C22×C5—481no—
275C22×C5#93211yescharacteristic
276C4×C5—21no—
277C4×C5—41no—
278C4×C5C2×#21311yescharacteristic
279C4×C5#95311yescharacteristic
280#41—22no—
281#41—21no—
282#41—41no—
283#41C5×#6711yescharacteristic
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—61no—
289Q16C5×#6711yescharacteristic
290SD16—22no—
291SD16—21no—
292SD16—121no—
293D8—22no—
294D8—21no—
295D8—61no—
296D8—61no—
297D8—121no—
298D8C5×#6711yescharacteristic
299D8C5×#6712yes—
300M16—21no—
301M16—61no—
302M16C5×#6711yescharacteristic
303C2×C8—21no—
304C2×C8C22×C5×D311yescharacteristic
305C16—61no—
306C16—61no—
307C3×C5#219211yescharacteristic, complemented, Hall
308C22×C3—21no—
309C22×C3—42no—
310C22×C3—41no—
311C22×C3—161no—
312C22×C3—162no—
313C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
314C2×D3—81no—
315C2×D3—161no—
316C3×C4—21no—
317C3×C4—41no—
318C3×C4C5×#12011yescharacteristic
319C3×C4C2×C5×D811yescharacteristic
320C2×C5—21no—
321C2×C5—41no—
322C2×C5—82no—
323C2×C5—81no—
324C2×C5—481no—
325C2×C5#521711yescharacteristic
326C23—42no—
327C23—41no—
328C23—121no—
329Q8—21no—
330Q8C5×#21311yescharacteristic
331D4—21no—
332D4—21no—
333D4—22no—
334D4—42no—
335D4—42no—
336D4—41no—
337D4—41no—
338D4—121no—
339D4—121no—
340D4C5×#21311yescharacteristic
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4C2×C5×#6711yescharacteristic
345C8—21no—
346C8—121no—
347C8C2×C5×#6711yescharacteristic
348C8C2×C5×#6711yescharacteristic
349C2×C3—21no—
350C2×C3—41no—
351C2×C3—82no—
352C2×C3—81no—
353C2×C3C5×#44711yescharacteristic
354D3—161no—
355C5#7212211yescharacteristic, direct product factor, Sylow
356C22—21no—
357C22—42no—
358C22—41no—
359C22—161no—
360C22—162no—
361C22—241no—
362C22—481no—
363C22C5×#93211yescharacteristic
364C4—21no—
365C4—41no—
366C4C2×C5×#21311yescharacteristic
367C4C5×#95311yescharacteristic
368C3C5×#219211yescharacteristic, complemented, Sylow
369C2—21no—
370C2—41no—
371C2—82no—
372C2—81no—
373C2—481no—
374C2C5×#521711yescharacteristic
subgroupG/H
centerC2×C5#5217
commutator, cosocleC2×C3×C8C23×C5
FrattiniC2×C8C22×C5×D3
FittingC3×C5×#576C2
socleC2×C3×C5#447
hypercenterC5×#576D3
hypocenterC3C5×#2192
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
1. C3×C5×#576C2semidirect
2. C5×#5193C2semidirect
3. C2×C5×#826C2semidirect
4. C5×#5142C2semidirect
5. C5×#4567C2semidirect
6. C5×#4563C2semidirect
7. C5×#4561C2semidirect
18. C2×C5×D24C22semidirect
23. C5×#819C22semidirect
24. C5×#812C22semidirect
25. C2×C5×C3⋊C16C22semidirect
26. #72122C5direct (N × H)
27. C5×#576D3semidirect
61. C3×#576C2×C5semidirect
62. #5193C2×C5semidirect
63. C2×#826C2×C5semidirect
64. #5142C2×C5semidirect
65. #4567C2×C5semidirect
66. #4563C2×C5semidirect
67. #4561C2×C5semidirect
118. C2×D24C22×C5semidirect
123. #819C22×C5semidirect
124. #812C22×C5semidirect
125. C2×C3⋊C16C22×C5semidirect
153. #576C5×D3semidirect
307. C3×C5#2192semidirect
355. C5#72122direct (N × H)
368. C3C5×#2192semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5762
C5×#51932
C2×C5×#8262
C5×#51422
C5×#45672
C5×#45632
C5×#45612
C5×#21923
#721225