Subgroups — C₅×#12602 #11935322

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#349	C₂	1	2	yes	maximal
3	C₄×C₅×M₃₂	C₂	1	2	yes	maximal, complemented
4	C₅×#1377	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₅×#117	C₄	1	1	yes	characteristic
8	C₂×C₅×M₃₂	C₂²	1	4	yes	—
9	C₅×#405	—	2	2	no	—
10	C₅×#405	—	2	2	no	—
11	C₂×C₅×#97	C₄	1	1	yes	characteristic
12	C₂×C₅×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#349	C₄	1	8	yes	complemented
16	C₅×#346	—	2	2	no	—
17	C₄×C₅×C₁₆	—	2	2	no	—
18	#12602	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₅×C₈	—	2	1	no	—
24	C₂²×C₅×C₈	—	2	1	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	C₈	1	4	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₅×#117	—	2	2	no	—
32	C₅×#117	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
34	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₅×M₃₂	—	2	4	no	—
38	C₅×M₃₂	C₂×C₄	1	8	yes	—
39	C₂×C₅×C₁₆	—	2	4	no	—
40	C₅×#97	—	2	2	no	—
41	C₅×#97	—	2	2	no	—
42	C₄×C₅×C₈	—	2	1	no	—
43	C₄×C₅×C₈	—	2	1	no	—
44	C₄×C₅×C₈	D₄	1	2	yes	—
45	C₄×C₅×C₈	D₄	1	2	yes	—
46	C₂×#405	C₂×C₅	1	1	yes	characteristic
47	C₂×#349	C₂×C₅	1	2	yes	—
48	C₄×M₃₂	C₂×C₅	1	2	yes	complemented
49	#1377	C₂×C₅	1	2	yes	complemented
50	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	2	1	no	—
53	C₂²×C₄×C₅	—	2	1	no	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	C₄²	1	2	yes	—
56	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₅×M₁₆	—	2	8	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	C₄²	1	2	yes	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
72	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
73	C₂×C₅×C₈	M₁₆	1	2	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₅×C₈	#41	1	2	yes	—
77	C₅×C₄⋊C₄	—	2	4	no	—
78	C₅×C₂²⋊C₄	—	2	4	no	—
79	C₄²×C₅	—	2	1	no	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅	M₁₆	1	2	yes	—
83	C₅×C₁₆	—	2	8	no	—
84	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
85	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
86	C₂×#117	C₄×C₅	1	1	yes	characteristic
87	C₂×M₃₂	C₂²×C₅	1	4	yes	—
88	#405	—	2	2	no	—
89	#405	—	2	2	no	—
90	C₂×#97	C₄×C₅	1	1	yes	characteristic
91	C₂×#97	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	#349	C₄×C₅	1	8	yes	complemented
95	#346	—	2	2	no	—
96	C₄×C₁₆	—	2	2	no	—
97	C₂³×C₅	—	2	1	no	—
98	C₂³×C₅	—	2	1	no	—
99	C₂³×C₅	—	4	2	no	—
100	C₂³×C₅	C₄×C₈	1	2	yes	—
101	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	4	2	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
112	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	#117	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₅×C₈	—	2	2	no	—
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	C₂²⋊C₈	1	2	yes	—
122	C₅×C₈	C₂²⋊C₈	1	2	yes	—
123	C₅×C₈	C₄×D₄	1	4	yes	—
124	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	C₅×C₈	1	4	yes	—
132	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
134	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
136	#117	—	2	2	no	—
137	#117	—	2	2	no	—
138	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
139	C₂×C₂²⋊C₄	C₅×C₈	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
142	M₃₂	—	2	4	no	—
143	M₃₂	C₂×C₄×C₅	1	8	yes	—
144	C₂×C₁₆	—	2	4	no	—
145	#97	—	2	2	no	—
146	#97	—	2	2	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	C₅×D₄	1	2	yes	—
150	C₄×C₈	C₅×D₄	1	2	yes	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	4	2	no	—
154	C₂²×C₅	—	4	2	no	—
155	C₂²×C₅	—	4	2	no	—
156	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
157	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₅	#431	1	1	yes	characteristic
159	C₄×C₅	—	2	4	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
164	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₄×C₅	#431	1	1	yes	characteristic
166	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	C₄²×C₅	1	2	yes	—
172	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
173	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
175	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
177	M₁₆	—	2	8	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₄²×C₅	1	2	yes	—
185	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
188	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
189	C₂×C₈	C₅×M₁₆	1	2	yes	—
190	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
191	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
192	C₂×C₈	C₅×#41	1	2	yes	—
193	C₄⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
198	C₄²	C₅×M₁₆	1	2	yes	—
199	C₁₆	—	2	8	no	—
200	C₂×C₅	—	2	1	no	—
201	C₂×C₅	—	2	1	no	—
202	C₂×C₅	—	4	2	no	—
203	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
204	C₂×C₅	#2115	1	2	yes	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	4	2	no	—
208	C₂³	C₄×C₅×C₈	1	2	yes	—
209	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
220	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
221	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₅×#117	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
226	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	C₅×C₂²⋊C₈	1	2	yes	—
230	C₈	C₅×C₂²⋊C₈	1	2	yes	—
231	C₈	C₄×C₅×D₄	1	4	yes	—
232	C₅	#12602	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂×C₄×C₅×C₈	1	1	yes	characteristic
239	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
240	C₂²	C₅×#431	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	—	4	2	no	—
244	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₄	C₄×C₅×M₁₆	1	1	yes	characteristic
246	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
247	C₄	C₅×#431	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	C₄×C₅×C₂²⋊C₈	1	1	yes	characteristic
252	C₂	C₅×#2115	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×M₃₂	C₂	semidirect
4. C₅×#1377	C₂	semidirect
15. C₅×#349	C₄	semidirect
18. #12602	C₅	direct (N × H)
48. C₄×M₃₂	C₂×C₅	semidirect
49. #1377	C₂×C₅	semidirect
94. #349	C₄×C₅	semidirect
232. C₅	#12602	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×#405	2
C₂×C₅×#349	2
C₄×C₅×M₃₂	2
C₅×#1377	2
#12602	5

Subgroups — C5×#12602 #11935322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12602 #11935322