Subgroups — C₅×#33765 #11956485

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3058	C₂	1	2	yes	maximal, complemented
2	C₅×#2234	C₂	1	4	yes	maximal, complemented
3	C₅×#2218	C₂	1	2	yes	maximal, complemented
4	C₅×#2215	C₂	1	2	yes	maximal, complemented
5	C₅×#2184	C₂	1	4	yes	maximal, complemented
6	C₅×#2113	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×D₈	C₂²	1	2	yes	complemented
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	C₅×#535	—	2	2	no	—
10	C₂×C₅×D₁₆	C₂²	1	2	yes	—
11	C₂×C₅×D₁₆	C₂²	1	2	yes	—
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
14	C₅×#469	—	2	2	no	—
15	C₅×#469	—	2	2	no	—
16	C₅×#466	C₂²	1	4	yes	—
17	C₅×#466	C₂²	1	4	yes	complemented
18	C₅×#427	C₂²	1	2	yes	complemented
19	C₅×#418	—	2	2	no	—
20	C₅×#367	C₂²	1	2	yes	complemented
21	C₅×#366	C₂²	1	2	yes	complemented
22	C₅×#357	C₂²	1	4	yes	complemented
23	C₅×#357	C₂²	1	4	yes	complemented
24	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
25	#33765	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	D₄	1	1	yes	characteristic
27	C₂²×C₅×D₄	—	2	2	no	—
28	C₂×C₅×D₈	—	2	2	no	—
29	C₂×C₅×D₈	—	4	2	no	—
30	C₂×C₅×D₈	—	4	2	no	—
31	C₂×C₅×D₈	C₂³	1	4	yes	—
32	C₅×#131	—	4	1	no	—
33	C₅×#131	—	4	1	no	—
34	C₂×C₅×M₁₆	D₄	1	2	yes	—
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×#127	—	4	2	no	—
39	C₅×#124	—	4	2	no	—
40	C₅×#121	—	2	4	no	—
41	C₅×#121	—	2	4	no	—
42	C₅×#120	—	4	2	no	—
43	C₅×#120	—	4	2	no	—
44	C₅×#117	—	2	2	no	—
45	C₅×D₁₆	—	4	2	no	—
46	C₅×D₁₆	—	4	2	no	—
47	C₅×M₃₂	—	4	1	no	—
48	C₂×C₅×C₁₆	—	4	1	no	—
49	C₂×C₅×C₁₆	C₂³	1	2	yes	—
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₅×#107	C₂³	1	4	yes	—
52	C₅×#106	—	2	2	no	—
53	C₅×#102	—	2	4	no	—
54	C₅×#102	—	2	4	no	—
55	#3058	C₂×C₅	1	2	yes	complemented
56	#2234	C₂×C₅	1	4	yes	complemented
57	#2218	C₂×C₅	1	2	yes	complemented
58	#2215	C₂×C₅	1	2	yes	complemented
59	#2184	C₂×C₅	1	4	yes	complemented
60	#2113	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂⁴×C₅	—	4	2	no	—
62	C₅×#41	—	4	1	no	—
63	C₅×#41	—	4	1	no	—
64	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
65	C₂×C₅×D₄	—	2	4	no	—
66	C₂×C₅×D₄	—	4	4	no	—
67	C₂×C₅×D₄	—	4	4	no	—
68	C₂×C₅×D₄	—	4	4	no	—
69	C₂×C₅×D₄	—	4	4	no	—
70	C₂×C₅×D₄	—	4	2	no	—
71	C₂×C₅×D₄	D₈	1	1	yes	characteristic
72	C₂×C₅×D₄	D₈	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₅×D₈	—	2	4	no	—
76	C₅×D₈	—	2	4	no	—
77	C₅×D₈	—	4	4	no	—
78	C₅×M₁₆	—	4	2	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₂²⋊C₄	—	4	2	no	—
89	C₅×C₂²⋊C₄	—	4	4	no	—
90	C₅×C₂²⋊C₄	—	4	4	no	—
91	C₄²×C₅	—	4	2	no	—
92	C₅×C₁₆	—	2	2	no	—
93	C₅×C₁₆	—	2	2	no	—
94	C₂²×D₈	C₂²×C₅	1	2	yes	complemented
95	C₂×#131	C₂²×C₅	1	1	yes	characteristic
96	#535	—	2	2	no	—
97	C₂×D₁₆	C₂²×C₅	1	2	yes	—
98	C₂×D₁₆	C₂²×C₅	1	2	yes	—
99	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
100	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
101	#469	—	2	2	no	—
102	#469	—	2	2	no	—
103	#466	C₂²×C₅	1	4	yes	—
104	#466	C₂²×C₅	1	4	yes	complemented
105	#427	C₂²×C₅	1	2	yes	complemented
106	#418	—	2	2	no	—
107	#367	C₂²×C₅	1	2	yes	complemented
108	#366	C₂²×C₅	1	2	yes	complemented
109	#357	C₂²×C₅	1	4	yes	complemented
110	#357	C₂²×C₅	1	4	yes	complemented
111	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	complemented
112	C₂³×C₅	—	4	4	no	—
113	C₂³×C₅	—	8	4	no	—
114	C₂³×C₅	—	16	4	no	—
115	C₂³×C₅	C₂×D₈	1	2	yes	—
116	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
117	C₅×Q₈	—	4	1	no	—
118	C₅×D₄	—	4	2	no	—
119	C₅×D₄	—	4	1	no	—
120	C₅×D₄	—	4	4	no	—
121	C₅×D₄	—	4	4	no	—
122	C₅×D₄	—	8	4	no	—
123	C₅×D₄	—	16	4	no	—
124	C₅×D₄	—	16	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	2	2	no	—
135	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
136	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×#41	C₅×D₄	1	1	yes	characteristic
138	C₂²×D₄	—	2	2	no	—
139	C₂×D₈	—	2	2	no	—
140	C₂×D₈	—	4	2	no	—
141	C₂×D₈	—	4	2	no	—
142	C₂×D₈	C₂³×C₅	1	4	yes	—
143	#131	—	4	1	no	—
144	#131	—	4	1	no	—
145	C₂×M₁₆	C₅×D₄	1	2	yes	—
146	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₅	1	2	yes	—
148	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
149	#127	—	4	2	no	—
150	#124	—	4	2	no	—
151	#121	—	2	4	no	—
152	#121	—	2	4	no	—
153	#120	—	4	2	no	—
154	#120	—	4	2	no	—
155	#117	—	2	2	no	—
156	D₁₆	—	4	2	no	—
157	D₁₆	—	4	2	no	—
158	M₃₂	—	4	1	no	—
159	C₂×C₁₆	—	4	1	no	—
160	C₂×C₁₆	C₂³×C₅	1	2	yes	—
161	C₂×C₁₆	C₂³×C₅	1	2	yes	—
162	#107	C₂³×C₅	1	4	yes	—
163	#106	—	2	2	no	—
164	#102	—	2	4	no	—
165	#102	—	2	4	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	4	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	8	4	no	—
171	C₂²×C₅	—	8	4	no	—
172	C₂²×C₅	—	8	4	no	—
173	C₂²×C₅	—	16	8	no	—
174	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	8	4	no	—
178	C₄×C₅	#534	1	1	yes	characteristic
179	C₄×C₅	#575	1	1	yes	characteristic
180	C₂⁴	—	4	2	no	—
181	#41	—	4	1	no	—
182	#41	—	4	1	no	—
183	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	—	4	4	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	C₅×D₈	1	1	yes	characteristic
191	C₂×D₄	C₅×D₈	1	2	yes	—
192	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
193	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	D₈	—	2	4	no	—
195	D₈	—	2	4	no	—
196	D₈	—	4	4	no	—
197	M₁₆	—	4	1	no	—
198	M₁₆	—	4	2	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	—	4	2	no	—
211	C₁₆	—	2	2	no	—
212	C₁₆	—	2	2	no	—
213	C₂×C₅	—	4	1	no	—
214	C₂×C₅	—	4	2	no	—
215	C₂×C₅	—	16	4	no	—
216	C₂×C₅	#3186	1	1	yes	characteristic
217	C₂×C₅	#3415	1	2	yes	—
218	C₂³	—	4	4	no	—
219	C₂³	—	8	4	no	—
220	C₂³	—	16	4	no	—
221	C₂³	C₂×C₅×D₈	1	2	yes	—
222	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
223	Q₈	—	4	1	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	1	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	4	no	—
228	D₄	—	8	4	no	—
229	D₄	—	16	4	no	—
230	D₄	—	16	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
237	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
242	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
243	C₅	#33765	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	16	8	no	—
252	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	8	4	no	—
256	C₄	C₅×#534	1	1	yes	characteristic
257	C₄	C₅×#575	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	4	1	no	—
260	C₂	—	16	4	no	—
261	C₂	C₅×#3186	1	1	yes	characteristic
262	C₂	C₅×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3058	C₂	semidirect
2. C₅×#2234	C₂	semidirect
3. C₅×#2218	C₂	semidirect
4. C₅×#2215	C₂	semidirect
5. C₅×#2184	C₂	semidirect
6. C₅×#2113	C₂	semidirect
7. C₂²×C₅×D₈	C₂²	semidirect
12. C₂×C₅×M₃₂	C₂²	semidirect
13. C₂²×C₅×C₁₆	C₂²	semidirect
17. C₅×#466	C₂²	semidirect
18. C₅×#427	C₂²	semidirect
20. C₅×#367	C₂²	semidirect
21. C₅×#366	C₂²	semidirect
22. C₅×#357	C₂²	semidirect
23. C₅×#357	C₂²	semidirect
24. C₅×C₂²⋊C₁₆	C₂²	semidirect
25. #33765	C₅	direct (N × H)
55. #3058	C₂×C₅	semidirect
56. #2234	C₂×C₅	semidirect
57. #2218	C₂×C₅	semidirect
58. #2215	C₂×C₅	semidirect
59. #2184	C₂×C₅	semidirect
60. #2113	C₂×C₅	semidirect
94. C₂²×D₈	C₂²×C₅	semidirect
99. C₂×M₃₂	C₂²×C₅	semidirect
100. C₂²×C₁₆	C₂²×C₅	semidirect
104. #466	C₂²×C₅	semidirect
105. #427	C₂²×C₅	semidirect
107. #367	C₂²×C₅	semidirect
108. #366	C₂²×C₅	semidirect
109. #357	C₂²×C₅	semidirect
110. #357	C₂²×C₅	semidirect
111. C₂²⋊C₁₆	C₂²×C₅	semidirect
243. C₅	#33765	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3058	2
C₅×#2234	2
C₅×#2218	2
C₅×#2215	2
C₅×#2184	2
C₅×#2113	2
#33765	5

Subgroups — C5×#33765 #11956485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33765 #11956485