Subgroups — C₅×#33630 #11956350

Order 1280.

542 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
3	C₅×#2229	C₂	1	4	yes	maximal, complemented
4	C₅×#2228	C₂	1	4	yes	maximal, complemented
5	C₅×#2159	C₂	1	2	yes	maximal, complemented
6	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2139	C₂	1	2	yes	maximal
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂×C₅×#125	—	2	1	no	—
10	C₄×C₅×#41	—	2	1	no	—
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₅×#499	—	2	2	no	—
13	C₅×#499	C₂²	1	4	yes	—
14	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
15	C₄×C₅×SD₁₆	—	2	2	no	—
16	C₄×C₅×D₈	C₂²	1	2	yes	—
17	C₅×#427	C₂²	1	1	yes	characteristic
18	C₂×C₅×#107	C₂²	1	2	yes	—
19	C₂×C₅×#106	—	2	1	no	—
20	C₅×#416	—	2	1	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#367	C₂²	1	4	yes	—
23	C₅×#366	C₂²	1	4	yes	—
24	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₅×#358	C₂²	1	4	yes	—
26	C₅×#357	C₂²	1	4	yes	—
27	#33630	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	—	2	1	no	—
29	C₅×#135	—	2	2	no	—
30	C₅×#135	Q₈	1	4	yes	—
31	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
32	C₂×C₅×SD₁₆	—	2	1	no	—
33	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#125	—	2	4	no	—
37	C₅×#125	—	4	1	no	—
38	C₅×#125	—	4	1	no	—
39	C₄×C₅×Q₈	—	2	2	no	—
40	C₄×C₅×D₄	—	2	2	no	—
41	C₄×C₅×D₄	—	4	2	no	—
42	C₅×#117	—	2	1	no	—
43	C₅×#117	—	4	1	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	—	4	2	no	—
46	C₂×C₄²×C₅	—	4	1	no	—
47	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₅×M₃₂	—	2	4	no	—
49	C₂×C₅×C₁₆	C₂³	1	4	yes	—
50	C₅×#107	—	2	2	no	—
51	C₅×#107	C₂³	1	4	yes	—
52	C₅×#107	C₂³	1	2	yes	—
53	C₅×#106	—	2	4	no	—
54	C₅×#106	—	2	1	no	—
55	C₅×#103	—	2	2	no	—
56	C₅×#102	—	2	2	no	—
57	C₄×C₅×C₈	D₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	2	yes	—
59	C₂×#499	C₂×C₅	1	1	yes	characteristic, complemented
60	C₄×#135	C₂×C₅	1	1	yes	characteristic
61	#2229	C₂×C₅	1	4	yes	complemented
62	#2228	C₂×C₅	1	4	yes	complemented
63	#2159	C₂×C₅	1	2	yes	complemented
64	#2150	C₂×C₅	1	1	yes	characteristic, complemented
65	#2139	C₂×C₅	1	2	yes	—
66	C₅×#41	—	2	4	no	—
67	C₅×#41	—	4	2	no	—
68	C₂×C₅×Q₈	—	2	1	no	—
69	C₂×C₅×D₄	—	2	1	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂²×C₄×C₅	—	4	1	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	D₈	1	2	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₅×Q₁₆	—	2	1	no	—
77	C₅×Q₁₆	C₂×Q₈	1	2	yes	—
78	C₅×SD₁₆	—	2	4	no	—
79	C₅×D₈	—	2	1	no	—
80	C₅×D₈	C₂×Q₈	1	2	yes	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	4	2	no	—
90	C₅×C₄⋊C₄	—	4	4	no	—
91	C₅×C₄⋊C₄	—	4	4	no	—
92	C₅×C₄⋊C₄	—	4	2	no	—
93	C₅×C₂²⋊C₄	—	4	1	no	—
94	C₅×C₂²⋊C₄	—	4	2	no	—
95	C₄²×C₅	—	4	1	no	—
96	C₄²×C₅	—	4	2	no	—
97	C₄²×C₅	D₈	1	1	yes	characteristic
98	C₄²×C₅	D₈	1	1	yes	characteristic
99	C₄²×C₅	C₂×D₄	1	2	yes	—
100	C₅×C₁₆	—	2	4	no	—
101	C₂×#135	C₂²×C₅	1	1	yes	characteristic
102	C₂×#125	—	2	1	no	—
103	C₄×#41	—	2	1	no	—
104	C₂×M₃₂	C₂²×C₅	1	2	yes	—
105	#499	—	2	2	no	—
106	#499	C₂²×C₅	1	4	yes	—
107	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
108	C₄×SD₁₆	—	2	2	no	—
109	C₄×D₈	C₂²×C₅	1	2	yes	—
110	#427	C₂²×C₅	1	1	yes	characteristic
111	C₂×#107	C₂²×C₅	1	2	yes	—
112	C₂×#106	—	2	1	no	—
113	#416	—	2	1	no	—
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	#367	C₂²×C₅	1	4	yes	—
116	#366	C₂²×C₅	1	4	yes	—
117	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
118	#358	C₂²×C₅	1	4	yes	—
119	#357	C₂²×C₅	1	4	yes	—
120	C₂³×C₅	—	4	1	no	—
121	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
122	C₅×Q₈	—	2	2	no	—
123	C₅×Q₈	—	4	1	no	—
124	C₅×D₄	—	2	2	no	—
125	C₅×D₄	—	4	4	no	—
126	C₅×D₄	—	4	1	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	8	4	no	—
133	C₂×C₄×C₅	—	8	4	no	—
134	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅	#137	1	2	yes	—
139	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₅×C₈	—	2	2	no	—
141	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
142	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
143	C₅×C₈	C₂×#41	1	1	yes	characteristic
144	C₂×#41	—	2	1	no	—
145	#135	—	2	2	no	—
146	#135	C₅×Q₈	1	4	yes	—
147	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
148	C₂×SD₁₆	—	2	1	no	—
149	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
150	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
152	#125	—	2	4	no	—
153	#125	—	4	1	no	—
154	#125	—	4	1	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	—	4	2	no	—
158	#117	—	2	1	no	—
159	#117	—	4	1	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	4	2	no	—
162	C₂×C₄²	—	4	1	no	—
163	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
164	M₃₂	—	2	4	no	—
165	C₂×C₁₆	C₂³×C₅	1	4	yes	—
166	#107	—	2	2	no	—
167	#107	C₂³×C₅	1	4	yes	—
168	#107	C₂³×C₅	1	2	yes	—
169	#106	—	2	4	no	—
170	#106	—	2	1	no	—
171	#103	—	2	2	no	—
172	#102	—	2	2	no	—
173	C₄×C₈	C₅×D₄	1	2	yes	—
174	C₄×C₈	C₂³×C₅	1	2	yes	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	4	2	no	—
177	C₂²×C₅	—	8	2	no	—
178	C₂²×C₅	#474	1	2	yes	—
179	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	4	2	no	—
183	C₄×C₅	—	8	2	no	—
184	C₄×C₅	—	8	4	no	—
185	C₄×C₅	#474	1	2	yes	—
186	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₄×C₅	C₂×#137	1	1	yes	characteristic
188	#41	—	2	4	no	—
189	#41	—	4	2	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₅×D₈	1	2	yes	—
197	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	Q₁₆	—	2	1	no	—
199	Q₁₆	C₂×C₅×Q₈	1	2	yes	—
200	SD₁₆	—	2	4	no	—
201	D₈	—	2	1	no	—
202	D₈	C₂×C₅×Q₈	1	2	yes	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
207	C₂×C₈	C₅×#41	1	2	yes	—
208	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	1	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	2	no	—
219	C₄²	C₅×D₈	1	1	yes	characteristic
220	C₄²	C₅×D₈	1	1	yes	characteristic
221	C₄²	C₂×C₅×D₄	1	2	yes	—
222	C₁₆	—	2	4	no	—
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	—	8	2	no	—
225	C₂×C₅	C₂×#474	1	1	yes	characteristic
226	C₂×C₅	#3414	1	2	yes	—
227	C₂³	—	4	1	no	—
228	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	4	1	no	—
231	D₄	—	2	2	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
242	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
243	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
244	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
245	C₂×C₄	C₅×#137	1	2	yes	—
246	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
249	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
250	C₈	C₂×C₅×#41	1	1	yes	characteristic
251	C₅	#33630	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	C₅×#474	1	2	yes	—
256	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	8	4	no	—
262	C₄	C₅×#474	1	2	yes	—
263	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
264	C₄	C₂×C₅×#137	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	8	2	no	—
267	C₂	C₂×C₅×#474	1	1	yes	characteristic
268	C₂	C₅×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#499	C₂	semidirect
3. C₅×#2229	C₂	semidirect
4. C₅×#2228	C₂	semidirect
5. C₅×#2159	C₂	semidirect
6. C₅×#2150	C₂	semidirect
27. #33630	C₅	direct (N × H)
59. C₂×#499	C₂×C₅	semidirect
61. #2229	C₂×C₅	semidirect
62. #2228	C₂×C₅	semidirect
63. #2159	C₂×C₅	semidirect
64. #2150	C₂×C₅	semidirect
251. C₅	#33630	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#499	2
C₄×C₅×#135	2
C₅×#2229	2
C₅×#2228	2
C₅×#2159	2
C₅×#2150	2
C₅×#2139	2
#33630	5

Subgroups — C5×#33630 #11956350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33630 #11956350