Subgroups — C₂×C₅×#1307 #11930837

Order 1280.

570 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×#432	C₂	1	2	yes	maximal
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₅×#1307	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×M₁₆	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₅×#432	—	2	2	no	—
7	C₅×#432	—	2	2	no	—
8	C₅×#432	C₄	1	4	yes	—
9	C₅×#432	C₂²	1	4	yes	—
10	C₅×#423	—	2	2	no	—
11	C₅×#423	C₄	1	2	yes	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	—	2	2	no	—
14	C₂×C₅×C₄⋊C₈	—	2	2	no	—
15	C₂×C₅×C₂²⋊C₈	C₄	1	4	yes	—
16	C₂×C₅×#97	—	2	2	no	—
17	C₂×#1307	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	—	4	4	no	—
22	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₅×C₈	—	2	4	no	—
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊C₈	—	2	4	no	—
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×C₄⋊C₈	—	2	4	no	—
37	C₅×C₄⋊C₈	—	2	4	no	—
38	C₅×C₂²⋊C₈	—	2	4	no	—
39	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
40	C₅×#97	—	2	4	no	—
41	C₅×#97	—	2	4	no	—
42	C₂×#432	C₂×C₅	1	2	yes	—
43	C₂×#423	C₂×C₅	1	1	yes	characteristic
44	#1307	C₂×C₅	1	4	yes	direct product factor
45	C₂⁴×C₅	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	—	4	4	no	—
49	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₄²	1	2	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
61	C₅×M₁₆	—	4	4	no	—
62	C₅×M₁₆	—	4	4	no	—
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	—	2	8	no	—
65	C₂×C₅×C₈	—	4	8	no	—
66	C₂×C₅×C₈	—	4	4	no	—
67	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅	C₂×D₄	1	2	yes	—
76	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂²×M₁₆	C₄×C₅	1	2	yes	—
78	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
79	#432	—	2	2	no	—
80	#432	—	2	2	no	—
81	#432	C₄×C₅	1	4	yes	—
82	#432	C₂²×C₅	1	4	yes	—
83	#423	—	2	2	no	—
84	#423	C₄×C₅	1	2	yes	—
85	#423	C₂²×C₅	1	2	yes	—
86	C₂×C₄⋊C₈	—	2	2	no	—
87	C₂×C₄⋊C₈	—	2	2	no	—
88	C₂×C₂²⋊C₈	C₄×C₅	1	4	yes	—
89	C₂×#97	—	2	2	no	—
90	C₂³×C₅	—	2	2	no	—
91	C₂³×C₅	—	2	4	no	—
92	C₂³×C₅	#95	1	1	yes	characteristic
93	C₂³×C₅	#95	1	1	yes	characteristic
94	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	#95	1	1	yes	characteristic
109	C₂×C₄×C₅	#95	1	2	yes	—
110	C₂×C₄×C₅	#95	1	2	yes	—
111	C₂×C₄×C₅	#95	1	1	yes	characteristic
112	C₂×C₄×C₅	#100	1	2	yes	—
113	C₂×C₄×C₅	#101	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
118	C₅×C₈	—	4	8	no	—
119	C₅×C₈	—	4	4	no	—
120	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	—	4	4	no	—
124	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
130	C₂×C₄²	C₅×D₄	1	2	yes	—
131	C₂×C₄²	C₅×D₄	1	2	yes	—
132	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
133	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
134	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	2	4	no	—
139	C₄⋊C₈	—	2	4	no	—
140	C₂²⋊C₈	—	2	4	no	—
141	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
142	#97	—	2	4	no	—
143	#97	—	2	4	no	—
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	4	4	no	—
149	C₂²×C₅	#343	1	2	yes	—
150	C₂²×C₅	#344	1	4	yes	—
151	C₂²×C₅	C₂×#95	1	1	yes	characteristic
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	1	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	2	2	no	—
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	#343	1	2	yes	—
159	C₄×C₅	C₂×#100	1	1	yes	characteristic
160	C₄×C₅	C₂×#101	1	1	yes	characteristic
161	C₂⁴	C₄²×C₅	1	1	yes	characteristic
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
166	C₂²×C₄	C₄²×C₅	1	2	yes	—
167	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
168	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
170	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
173	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
175	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
177	M₁₆	—	4	4	no	—
178	M₁₆	—	4	4	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	8	no	—
182	C₂×C₈	—	4	4	no	—
183	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
184	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
185	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
186	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
187	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
188	C₄²	C₅×C₄⋊C₄	1	2	yes	—
189	C₄²	C₅×C₄⋊C₄	1	2	yes	—
190	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
191	C₄²	C₂×C₅×D₄	1	2	yes	—
192	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
193	C₂×C₅	—	4	1	no	—
194	C₂×C₅	—	4	1	no	—
195	C₂×C₅	#1307	1	4	yes	direct product factor
196	C₂×C₅	C₂×#343	1	1	yes	characteristic
197	C₂×C₅	C₂×#344	1	2	yes	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	4	no	—
200	C₂³	C₅×#95	1	1	yes	characteristic
201	C₂³	C₅×#95	1	1	yes	characteristic
202	C₂³	C₂×C₄²×C₅	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₅×#95	1	1	yes	characteristic
217	C₂×C₄	C₅×#95	1	2	yes	—
218	C₂×C₄	C₅×#95	1	2	yes	—
219	C₂×C₄	C₅×#95	1	1	yes	characteristic
220	C₂×C₄	C₅×#100	1	2	yes	—
221	C₂×C₄	C₅×#101	1	2	yes	—
222	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
223	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
226	C₈	—	4	8	no	—
227	C₈	—	4	4	no	—
228	C₅	C₂×#1307	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	4	no	—
234	C₂²	C₅×#343	1	2	yes	—
235	C₂²	C₅×#344	1	4	yes	—
236	C₂²	C₂×C₅×#95	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	C₅×#343	1	2	yes	—
244	C₄	C₂×C₅×#100	1	1	yes	characteristic
245	C₄	C₂×C₅×#101	1	1	yes	characteristic
246	C₂	—	4	1	no	—
247	C₂	—	4	1	no	—
248	C₂	C₅×#1307	1	4	yes	direct product factor
249	C₂	C₂×C₅×#343	1	1	yes	characteristic
250	C₂	C₂×C₅×#344	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	#95
commutator	C₂³	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂⁴×C₅	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1307	C₂	direct (N × H)
17. C₂×#1307	C₅	direct (N × H)
44. #1307	C₂×C₅	direct (N × H)
195. C₂×C₅	#1307	direct (N × H)
228. C₅	C₂×#1307	direct (N × H)
248. C₂	C₅×#1307	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×#432	2
C₂×C₅×#423	2
C₅×#1307	2
C₂×#1307	5

Subgroups — C2×C5×#1307 #11930837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1307 #11930837