Subgroups — C₅×#33337 #11956057

Order 1280.

1128 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	C₅×#3407	C₂	1	2	yes	maximal, complemented
3	C₅×#2115	C₂	1	6	yes	maximal, complemented
4	C₂×C₅×#349	C₂	1	6	yes	maximal, complemented
5	C₂²×C₅×#41	C₄	1	1	yes	characteristic
6	C₅×#568	—	2	8	no	—
7	C₂×C₅×#131	C₄	1	3	yes	—
8	C₂×C₅×#131	C₄	1	1	yes	characteristic
9	C₂×C₅×#131	C₂²	1	3	yes	—
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	C₂²×C₅×M₁₆	C₄	1	3	yes	—
12	C₂²×C₅×M₁₆	C₂²	1	3	yes	—
13	C₅×#504	—	2	8	no	—
14	C₂×C₅×M₃₂	C₂²	1	12	yes	complemented
15	C₅×#349	C₂²	1	4	yes	—
16	C₅×#349	C₂²	1	12	yes	complemented
17	#33337	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₅×#41	—	2	4	no	—
19	C₂×C₅×#41	—	2	4	no	—
20	C₂×C₅×#41	C₈	1	1	yes	characteristic
21	C₂×C₅×#41	C₈	1	3	yes	—
22	C₂×C₅×#41	C₂×C₄	1	3	yes	—
23	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
25	C₂²×C₅×D₄	C₈	1	3	yes	—
26	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
27	C₅×#131	—	2	4	no	—
28	C₅×#131	—	2	12	no	—
29	C₅×#131	—	2	12	no	—
30	C₅×#131	D₄	1	8	yes	—
31	C₂×C₅×M₁₆	—	2	12	no	—
32	C₂×C₅×M₁₆	—	2	12	no	—
33	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
34	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
35	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
36	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
37	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₅×M₁₆	C₂³	1	3	yes	—
39	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
40	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
41	C₂²×C₅×C₈	C₂³	1	3	yes	—
42	C₅×M₃₂	—	2	12	no	—
43	C₅×M₃₂	C₂³	1	8	yes	—
44	C₂×C₅×C₁₆	—	2	12	no	—
45	C₂×#568	C₂×C₅	1	1	yes	characteristic
46	#3407	C₂×C₅	1	2	yes	complemented
47	#2115	C₂×C₅	1	6	yes	complemented
48	C₂×#349	C₂×C₅	1	6	yes	complemented
49	C₂⁴×C₅	C₂×C₈	1	3	yes	—
50	C₅×#41	—	2	4	no	—
51	C₅×#41	—	2	12	no	—
52	C₅×#41	—	2	6	no	—
53	C₅×#41	—	2	6	no	—
54	C₅×#41	C₂²⋊C₄	1	4	yes	—
55	C₅×#41	M₁₆	1	4	yes	—
56	C₂×C₅×Q₈	—	2	4	no	—
57	C₂×C₅×Q₈	C₂×C₈	1	3	yes	—
58	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
59	C₂×C₅×D₄	—	2	12	no	—
60	C₂×C₅×D₄	C₂×C₈	1	6	yes	—
61	C₂×C₅×D₄	C₂×C₈	1	3	yes	—
62	C₂×C₅×D₄	C₂×C₈	1	3	yes	—
63	C₂²×C₄×C₅	—	2	6	no	—
64	C₂²×C₄×C₅	—	2	6	no	—
65	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
66	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
67	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
71	C₅×M₁₆	—	2	6	no	—
72	C₅×M₁₆	—	2	6	no	—
73	C₅×M₁₆	—	2	6	no	—
74	C₅×M₁₆	—	2	2	no	—
75	C₅×M₁₆	—	2	6	no	—
76	C₅×M₁₆	C₂²⋊C₄	1	12	yes	—
77	C₂×C₅×C₈	—	2	6	no	—
78	C₂×C₅×C₈	—	2	12	no	—
79	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	12	yes	—
83	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
84	C₅×C₁₆	—	2	8	no	—
85	C₂²×#41	C₄×C₅	1	1	yes	characteristic
86	#568	—	2	8	no	—
87	C₂×#131	C₄×C₅	1	3	yes	—
88	C₂×#131	C₄×C₅	1	1	yes	characteristic
89	C₂×#131	C₂²×C₅	1	3	yes	—
90	C₂×#131	C₂²×C₅	1	1	yes	characteristic
91	C₂²×M₁₆	C₄×C₅	1	3	yes	—
92	C₂²×M₁₆	C₂²×C₅	1	3	yes	—
93	#504	—	2	8	no	—
94	C₂×M₃₂	C₂²×C₅	1	12	yes	complemented
95	#349	C₂²×C₅	1	4	yes	—
96	#349	C₂²×C₅	1	12	yes	complemented
97	C₂³×C₅	—	2	6	no	—
98	C₂³×C₅	—	8	3	no	—
99	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
100	C₂³×C₅	C₂²×C₈	1	3	yes	—
101	C₂³×C₅	C₂²×C₈	1	3	yes	—
102	C₅×Q₈	—	2	6	no	—
103	C₅×Q₈	C₂²⋊C₈	1	4	yes	—
104	C₅×D₄	—	2	12	no	—
105	C₅×D₄	—	2	6	no	—
106	C₅×D₄	C₂²⋊C₈	1	12	yes	—
107	C₂×C₄×C₅	—	2	6	no	—
108	C₂×C₄×C₅	—	2	3	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	3	no	—
112	C₂×C₄×C₅	—	2	6	no	—
113	C₂×C₄×C₅	—	2	6	no	—
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
115	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
117	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
118	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
119	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	2	3	no	—
122	C₅×C₈	C₂×C₂²⋊C₄	1	6	yes	—
123	C₅×C₈	C₂²×D₄	1	2	yes	—
124	C₂×#41	—	2	4	no	—
125	C₂×#41	—	2	4	no	—
126	C₂×#41	C₅×C₈	1	1	yes	characteristic
127	C₂×#41	C₅×C₈	1	3	yes	—
128	C₂×#41	C₂×C₄×C₅	1	3	yes	—
129	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
131	C₂²×D₄	C₅×C₈	1	3	yes	—
132	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
133	#131	—	2	4	no	—
134	#131	—	2	12	no	—
135	#131	—	2	12	no	—
136	#131	C₅×D₄	1	8	yes	—
137	C₂×M₁₆	—	2	12	no	—
138	C₂×M₁₆	—	2	12	no	—
139	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
140	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
141	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
142	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
143	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
144	C₂×M₁₆	C₂³×C₅	1	3	yes	—
145	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
146	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
147	C₂²×C₈	C₂³×C₅	1	3	yes	—
148	M₃₂	—	2	12	no	—
149	M₃₂	C₂³×C₅	1	8	yes	—
150	C₂×C₁₆	—	2	12	no	—
151	C₂²×C₅	—	2	1	no	—
152	C₂²×C₅	—	2	3	no	—
153	C₂²×C₅	—	4	3	no	—
154	C₂²×C₅	—	4	1	no	—
155	C₂²×C₅	—	4	3	no	—
156	C₂²×C₅	—	8	6	no	—
157	C₂²×C₅	C₂×C₂²⋊C₈	1	6	yes	—
158	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	3	no	—
161	C₄×C₅	C₂×C₂²⋊C₈	1	6	yes	—
162	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
164	C₂⁴	C₂×C₅×C₈	1	3	yes	—
165	#41	—	2	4	no	—
166	#41	—	2	12	no	—
167	#41	—	2	6	no	—
168	#41	—	2	6	no	—
169	#41	C₅×C₂²⋊C₄	1	4	yes	—
170	#41	C₅×M₁₆	1	4	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₂×C₅×C₈	1	3	yes	—
173	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
174	C₂×D₄	—	2	12	no	—
175	C₂×D₄	C₂×C₅×C₈	1	6	yes	—
176	C₂×D₄	C₂×C₅×C₈	1	3	yes	—
177	C₂×D₄	C₂×C₅×C₈	1	3	yes	—
178	C₂²×C₄	—	2	6	no	—
179	C₂²×C₄	—	2	6	no	—
180	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
181	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
182	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
183	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
184	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
186	M₁₆	—	2	6	no	—
187	M₁₆	—	2	6	no	—
188	M₁₆	—	2	6	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	6	no	—
191	M₁₆	C₅×C₂²⋊C₄	1	12	yes	—
192	C₂×C₈	—	2	6	no	—
193	C₂×C₈	—	2	12	no	—
194	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
195	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
196	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
197	C₂×C₈	C₂×C₅×D₄	1	12	yes	—
198	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
199	C₁₆	—	2	8	no	—
200	C₂×C₅	—	2	1	no	—
201	C₂×C₅	—	2	6	no	—
202	C₂×C₅	—	4	3	no	—
203	C₂×C₅	—	4	1	no	—
204	C₂×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic
205	C₂³	—	2	6	no	—
206	C₂³	—	8	3	no	—
207	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
208	C₂³	C₂²×C₅×C₈	1	3	yes	—
209	C₂³	C₂²×C₅×C₈	1	3	yes	—
210	Q₈	—	2	6	no	—
211	Q₈	C₅×C₂²⋊C₈	1	4	yes	—
212	D₄	—	2	12	no	—
213	D₄	—	2	6	no	—
214	D₄	C₅×C₂²⋊C₈	1	12	yes	—
215	C₂×C₄	—	2	6	no	—
216	C₂×C₄	—	2	6	no	—
217	C₂×C₄	—	2	6	no	—
218	C₂×C₄	—	2	3	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	3	no	—
222	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
223	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
225	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
226	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
227	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
228	C₈	—	2	1	no	—
229	C₈	—	2	3	no	—
230	C₈	C₂×C₅×C₂²⋊C₄	1	6	yes	—
231	C₈	C₂²×C₅×D₄	1	2	yes	—
232	C₅	#33337	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²	—	2	1	no	—
234	C₂²	—	2	3	no	—
235	C₂²	—	4	3	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	3	no	—
238	C₂²	—	8	6	no	—
239	C₂²	C₂×C₅×C₂²⋊C₈	1	6	yes	—
240	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	3	no	—
243	C₄	C₂×C₅×C₂²⋊C₈	1	6	yes	—
244	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	6	no	—
248	C₂	—	4	1	no	—
249	C₂	—	4	3	no	—
250	C₂	C₂²×C₅×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂²×C₂²⋊C₈
prime radical	C₂²×C₅×#41	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#3407	C₂	semidirect
3. C₅×#2115	C₂	semidirect
4. C₂×C₅×#349	C₂	semidirect
14. C₂×C₅×M₃₂	C₂²	semidirect
16. C₅×#349	C₂²	semidirect
17. #33337	C₅	direct (N × H)
46. #3407	C₂×C₅	semidirect
47. #2115	C₂×C₅	semidirect
48. C₂×#349	C₂×C₅	semidirect
94. C₂×M₃₂	C₂²×C₅	semidirect
96. #349	C₂²×C₅	semidirect
232. C₅	#33337	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×#568	2
C₅×#3407	2
C₅×#2115	2
C₂×C₅×#349	2
#33337	5

Subgroups — C5×#33337 #11956057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33337 #11956057