Subgroups — C₂×C₅×#2657 #11957342

Order 1280.

5142 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#127	C₂	1	3	yes	maximal, complemented
2	C₂²×C₅×#124	C₂	1	3	yes	maximal, complemented
3	C₅×#2657	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#392	C₂	1	8	yes	maximal, complemented
5	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×D₄	C₂²	1	3	yes	—
8	C₂³×C₅×D₄	C₂²	1	12	yes	—
9	C₂×C₅×#127	—	2	24	no	—
10	C₂×C₅×#127	C₂²	1	24	yes	complemented
11	C₂×C₅×#124	—	2	24	no	—
12	C₂×C₅×#124	C₂²	1	24	yes	complemented
13	C₂²×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
14	C₂²×C₄²×C₅	C₂²	1	3	yes	—
15	C₅×#392	C₂²	1	64	yes	complemented
16	C₅×#376	C₂²	1	8	yes	complemented
17	C₂×C₅×#95	C₂²	1	4	yes	—
18	C₂×#2657	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂⁵×C₅	C₂³	1	6	yes	—
20	C₂²×C₅×Q₈	—	2	12	no	—
21	C₂²×C₅×Q₈	C₂³	1	4	yes	—
22	C₂²×C₅×D₄	—	2	12	no	—
23	C₂²×C₅×D₄	—	2	48	no	—
24	C₂²×C₅×D₄	—	2	24	no	—
25	C₂²×C₅×D₄	—	4	48	no	—
26	C₂²×C₅×D₄	C₂³	1	12	yes	—
27	C₂²×C₅×D₄	C₂³	1	48	yes	—
28	C₂³×C₄×C₅	C₂³	1	2	yes	—
29	C₂³×C₄×C₅	C₂³	1	6	yes	—
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₅×#127	—	2	96	no	—
32	C₅×#124	—	2	96	no	—
33	C₂×C₅×C₂²⋊C₄	—	2	48	no	—
34	C₂×C₅×C₂²⋊C₄	C₂³	1	48	yes	—
35	C₂×C₄²×C₅	D₄	1	24	yes	complemented
36	C₂×C₄²×C₅	C₂³	1	12	yes	—
37	C₅×#95	C₂³	1	16	yes	—
38	C₂²×#127	C₂×C₅	1	3	yes	complemented
39	C₂²×#124	C₂×C₅	1	3	yes	complemented
40	#2657	C₂×C₅	1	16	yes	direct product factor
41	C₂×#392	C₂×C₅	1	8	yes	complemented
42	C₂×#376	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂⁴×C₅	—	2	12	no	—
44	C₂⁴×C₅	—	2	24	no	—
45	C₂⁴×C₅	—	2	48	no	—
46	C₂⁴×C₅	C₂⁴	1	12	yes	—
47	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
48	C₂×C₅×Q₈	—	2	24	no	—
49	C₂×C₅×Q₈	—	4	16	no	—
50	C₂×C₅×D₄	—	2	24	no	—
51	C₂×C₅×D₄	—	2	96	no	—
52	C₂×C₅×D₄	—	2	48	no	—
53	C₂×C₅×D₄	—	4	96	no	—
54	C₂×C₅×D₄	—	4	192	no	—
55	C₂×C₅×D₄	—	4	48	no	—
56	C₂²×C₄×C₅	—	2	24	no	—
57	C₂²×C₄×C₅	—	2	12	no	—
58	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
61	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
62	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
63	C₅×C₂²⋊C₄	—	2	96	no	—
64	C₄²×C₅	C₂×D₄	1	48	yes	complemented
65	C₂³×Q₈	C₂²×C₅	1	1	yes	characteristic
66	C₂³×D₄	C₂²×C₅	1	3	yes	—
67	C₂³×D₄	C₂²×C₅	1	12	yes	—
68	C₂×#127	—	2	24	no	—
69	C₂×#127	C₂²×C₅	1	24	yes	complemented
70	C₂×#124	—	2	24	no	—
71	C₂×#124	C₂²×C₅	1	24	yes	complemented
72	C₂²×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
73	C₂²×C₄²	C₂²×C₅	1	3	yes	—
74	#392	C₂²×C₅	1	64	yes	complemented
75	#376	C₂²×C₅	1	8	yes	complemented
76	C₂×#95	C₂²×C₅	1	4	yes	—
77	C₂³×C₅	—	2	12	no	—
78	C₂³×C₅	—	2	24	no	—
79	C₂³×C₅	—	2	48	no	—
80	C₂³×C₅	—	4	24	no	—
81	C₂³×C₅	—	4	96	no	—
82	C₂³×C₅	—	4	48	no	—
83	C₂³×C₅	C₂²×D₄	1	6	yes	—
84	C₂³×C₅	H(5,𝔽₂)	1	8	yes	—
85	C₂³×C₅	C₂⁵	1	1	yes	characteristic
86	C₅×Q₈	—	4	16	no	—
87	C₅×D₄	—	4	192	no	—
88	C₅×D₄	—	4	48	no	—
89	C₂×C₄×C₅	—	2	48	no	—
90	C₂×C₄×C₅	—	2	24	no	—
91	C₂×C₄×C₅	—	2	12	no	—
92	C₂×C₄×C₅	—	4	8	no	—
93	C₂×C₄×C₅	#127	1	16	yes	—
94	C₂×C₄×C₅	C₂²×D₄	1	24	yes	—
95	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
96	C₂⁵	C₂³×C₅	1	6	yes	—
97	C₂²×Q₈	—	2	12	no	—
98	C₂²×Q₈	C₂³×C₅	1	4	yes	—
99	C₂²×D₄	—	2	12	no	—
100	C₂²×D₄	—	2	48	no	—
101	C₂²×D₄	—	2	24	no	—
102	C₂²×D₄	—	4	48	no	—
103	C₂²×D₄	C₂³×C₅	1	12	yes	—
104	C₂²×D₄	C₂³×C₅	1	48	yes	—
105	C₂³×C₄	C₂³×C₅	1	2	yes	—
106	C₂³×C₄	C₂³×C₅	1	6	yes	—
107	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
108	#127	—	2	96	no	—
109	#124	—	2	96	no	—
110	C₂×C₂²⋊C₄	—	2	48	no	—
111	C₂×C₂²⋊C₄	C₂³×C₅	1	48	yes	—
112	C₂×C₄²	C₅×D₄	1	24	yes	complemented
113	C₂×C₄²	C₂³×C₅	1	12	yes	—
114	#95	C₂³×C₅	1	16	yes	—
115	C₂²×C₅	—	4	12	no	—
116	C₂²×C₅	—	4	48	no	—
117	C₂²×C₅	—	4	24	no	—
118	C₂²×C₅	—	8	48	no	—
119	C₂²×C₅	C₂×#127	1	4	yes	—
120	C₂²×C₅	#535	1	24	yes	—
121	C₂²×C₅	C₂³×D₄	1	3	yes	—
122	C₂²×C₅	C₂×H(5,𝔽₂)	1	4	yes	—
123	C₄×C₅	—	2	24	no	—
124	C₄×C₅	—	4	4	no	—
125	C₄×C₅	C₂×#127	1	8	yes	—
126	C₂⁴	—	2	12	no	—
127	C₂⁴	—	2	24	no	—
128	C₂⁴	—	2	48	no	—
129	C₂⁴	C₂⁴×C₅	1	12	yes	—
130	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
131	C₂×Q₈	—	2	24	no	—
132	C₂×Q₈	—	4	16	no	—
133	C₂×D₄	—	2	24	no	—
134	C₂×D₄	—	2	96	no	—
135	C₂×D₄	—	2	48	no	—
136	C₂×D₄	—	4	96	no	—
137	C₂×D₄	—	4	192	no	—
138	C₂×D₄	—	4	48	no	—
139	C₂²×C₄	—	2	24	no	—
140	C₂²×C₄	—	2	12	no	—
141	C₂²×C₄	C₂×C₅×D₄	1	24	yes	—
142	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
143	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
144	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
145	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
146	C₂²⋊C₄	—	2	96	no	—
147	C₄²	C₂×C₅×D₄	1	48	yes	complemented
148	C₂×C₅	—	8	12	no	—
149	C₂×C₅	#2657	1	8	yes	direct product factor
150	C₂×C₅	C₂²×#127	1	1	yes	characteristic
151	C₂×C₅	C₂×#535	1	6	yes	—
152	C₂³	—	2	12	no	—
153	C₂³	—	2	24	no	—
154	C₂³	—	2	48	no	—
155	C₂³	—	4	24	no	—
156	C₂³	—	4	96	no	—
157	C₂³	—	4	48	no	—
158	C₂³	C₂²×C₅×D₄	1	6	yes	—
159	C₂³	C₅×H(5,𝔽₂)	1	8	yes	—
160	C₂³	C₂⁵×C₅	1	1	yes	characteristic
161	Q₈	—	4	16	no	—
162	D₄	—	4	192	no	—
163	D₄	—	4	48	no	—
164	C₂×C₄	—	2	48	no	—
165	C₂×C₄	—	2	24	no	—
166	C₂×C₄	—	2	12	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	C₅×#127	1	16	yes	—
169	C₂×C₄	C₂²×C₅×D₄	1	24	yes	—
170	C₂×C₄	C₂²×C₅×D₄	1	12	yes	—
171	C₅	C₂×#2657	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²	—	4	12	no	—
173	C₂²	—	4	48	no	—
174	C₂²	—	4	24	no	—
175	C₂²	—	8	48	no	—
176	C₂²	C₂×C₅×#127	1	4	yes	—
177	C₂²	C₅×#535	1	24	yes	—
178	C₂²	C₂³×C₅×D₄	1	3	yes	—
179	C₂²	C₂×C₅×H(5,𝔽₂)	1	4	yes	—
180	C₄	—	2	24	no	—
181	C₄	—	4	4	no	—
182	C₄	C₂×C₅×#127	1	8	yes	—
183	C₂	—	8	12	no	—
184	C₂	C₅×#2657	1	8	yes	direct product factor
185	C₂	C₂²×C₅×#127	1	1	yes	characteristic
186	C₂	C₂×C₅×#535	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#127	C₂	semidirect
2. C₂²×C₅×#124	C₂	semidirect
3. C₅×#2657	C₂	direct (N × H)
4. C₂×C₅×#392	C₂	semidirect
5. C₂×C₅×#376	C₂	semidirect
10. C₂×C₅×#127	C₂²	semidirect
12. C₂×C₅×#124	C₂²	semidirect
15. C₅×#392	C₂²	semidirect
16. C₅×#376	C₂²	semidirect
18. C₂×#2657	C₅	direct (N × H)
35. C₂×C₄²×C₅	D₄	semidirect
38. C₂²×#127	C₂×C₅	semidirect
39. C₂²×#124	C₂×C₅	semidirect
40. #2657	C₂×C₅	direct (N × H)
41. C₂×#392	C₂×C₅	semidirect
42. C₂×#376	C₂×C₅	semidirect
64. C₄²×C₅	C₂×D₄	semidirect
69. C₂×#127	C₂²×C₅	semidirect
71. C₂×#124	C₂²×C₅	semidirect
74. #392	C₂²×C₅	semidirect
75. #376	C₂²×C₅	semidirect
112. C₂×C₄²	C₅×D₄	semidirect
147. C₄²	C₂×C₅×D₄	semidirect
149. C₂×C₅	#2657	direct (N × H)
171. C₅	C₂×#2657	direct (N × H)
184. C₂	C₅×#2657	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#127	2
C₂²×C₅×#124	2
C₅×#2657	2
C₂×C₅×#392	2
C₂×C₅×#376	2
C₂×#2657	5

Subgroups — C2×C5×#2657 #11957342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2657 #11957342