Subgroups — C₂²×C₅×#494 #11983350

Order 1280.

3262 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Q₁₆	C₂	1	4	yes	maximal
2	C₂²×C₅×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₂×C₅×#494	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₅×Q₈	C₂²	1	4	yes	—
6	C₂²×C₅×Q₁₆	—	2	32	no	—
7	C₂²×C₅×Q₁₆	C₂²	1	48	yes	—
8	C₂³×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Q₈	—	2	12	no	—
10	C₂×C₅×C₄⋊Q₈	C₂²	1	24	yes	—
11	C₂²×C₅×C₄⋊C₄	—	2	4	no	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₅×#494	C₂²	1	64	yes	direct product factor
14	C₂×C₄×C₅×C₈	C₂²	1	12	yes	—
15	C₂²×#494	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₅×Q₈	—	2	12	no	—
17	C₂²×C₅×Q₈	—	2	32	no	—
18	C₂²×C₅×Q₈	C₂³	1	24	yes	—
19	C₂³×C₄×C₅	—	2	4	no	—
20	C₂³×C₄×C₅	D₄	1	2	yes	—
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₅×Q₁₆	—	2	192	no	—
23	C₂×C₅×Q₁₆	C₂³	1	64	yes	—
24	C₂²×C₅×C₈	D₄	1	16	yes	—
25	C₂²×C₅×C₈	C₂³	1	12	yes	—
26	C₅×C₄⋊Q₈	—	2	48	no	—
27	C₅×C₄⋊Q₈	C₂³	1	32	yes	—
28	C₂×C₅×C₄⋊C₄	—	2	48	no	—
29	C₂×C₄²×C₅	D₄	1	6	yes	—
30	C₂×C₄²×C₅	C₂³	1	6	yes	—
31	C₄×C₅×C₈	C₂³	1	16	yes	—
32	C₂³×Q₁₆	C₂×C₅	1	4	yes	—
33	C₂²×C₄⋊Q₈	C₂×C₅	1	2	yes	—
34	C₂×#494	C₂×C₅	1	24	yes	direct product factor
35	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
36	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂×C₅×Q₈	—	2	24	no	—
38	C₂×C₅×Q₈	—	2	96	no	—
39	C₂×C₅×Q₈	—	4	48	no	—
40	C₂×C₅×Q₈	C₂⁴	1	16	yes	—
41	C₂²×C₄×C₅	—	2	24	no	—
42	C₂²×C₄×C₅	—	4	16	no	—
43	C₂²×C₄×C₅	Q₁₆	1	16	yes	complemented
44	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
45	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
47	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
48	C₅×Q₁₆	—	2	128	no	—
49	C₂×C₅×C₈	C₂×D₄	1	48	yes	—
50	C₂×C₅×C₈	C₂⁴	1	8	yes	—
51	C₅×C₄⋊C₄	—	2	64	no	—
52	C₄²×C₅	C₂×D₄	1	12	yes	—
53	C₄²×C₅	C₂⁴	1	4	yes	—
54	C₂³×Q₈	C₂²×C₅	1	4	yes	—
55	C₂²×Q₁₆	—	2	32	no	—
56	C₂²×Q₁₆	C₂²×C₅	1	48	yes	—
57	C₂³×C₈	C₂²×C₅	1	2	yes	—
58	C₂×C₄⋊Q₈	—	2	12	no	—
59	C₂×C₄⋊Q₈	C₂²×C₅	1	24	yes	—
60	C₂²×C₄⋊C₄	—	2	4	no	—
61	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
62	#494	C₂²×C₅	1	64	yes	direct product factor
63	C₂×C₄×C₈	C₂²×C₅	1	12	yes	—
64	C₂³×C₅	#127	1	4	yes	—
65	C₂³×C₅	C₂×Q₁₆	1	8	yes	—
66	C₂³×C₅	C₂²×D₄	1	3	yes	—
67	C₅×Q₈	—	2	32	no	—
68	C₅×Q₈	—	4	48	no	—
69	C₂×C₄×C₅	—	2	16	no	—
70	C₂×C₄×C₅	—	4	48	no	—
71	C₂×C₄×C₅	#127	1	12	yes	—
72	C₂×C₄×C₅	C₂×Q₁₆	1	48	yes	complemented
73	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
74	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
75	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
76	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
77	C₅×C₈	C₂²×D₄	1	16	yes	—
78	C₂²×Q₈	—	2	12	no	—
79	C₂²×Q₈	—	2	32	no	—
80	C₂²×Q₈	C₂³×C₅	1	24	yes	—
81	C₂³×C₄	—	2	4	no	—
82	C₂³×C₄	C₅×D₄	1	2	yes	—
83	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
84	C₂×Q₁₆	—	2	192	no	—
85	C₂×Q₁₆	C₂³×C₅	1	64	yes	—
86	C₂²×C₈	C₅×D₄	1	16	yes	—
87	C₂²×C₈	C₂³×C₅	1	12	yes	—
88	C₄⋊Q₈	—	2	48	no	—
89	C₄⋊Q₈	C₂³×C₅	1	32	yes	—
90	C₂×C₄⋊C₄	—	2	48	no	—
91	C₂×C₄²	C₅×D₄	1	6	yes	—
92	C₂×C₄²	C₂³×C₅	1	6	yes	—
93	C₄×C₈	C₂³×C₅	1	16	yes	—
94	C₂²×C₅	#494	1	16	yes	direct product factor
95	C₂²×C₅	C₂×#127	1	6	yes	—
96	C₂²×C₅	C₂²×Q₁₆	1	12	yes	—
97	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
98	C₄×C₅	—	4	16	no	—
99	C₄×C₅	C₂×#127	1	6	yes	—
100	C₄×C₅	C₂²×Q₁₆	1	16	yes	complemented
101	C₄×C₅	C₂³×D₄	1	2	yes	—
102	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
103	C₂×Q₈	—	2	24	no	—
104	C₂×Q₈	—	2	96	no	—
105	C₂×Q₈	—	4	48	no	—
106	C₂×Q₈	C₂⁴×C₅	1	16	yes	—
107	C₂²×C₄	—	2	24	no	—
108	C₂²×C₄	—	4	16	no	—
109	C₂²×C₄	C₅×Q₁₆	1	16	yes	complemented
110	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
111	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
112	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
113	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
114	Q₁₆	—	2	128	no	—
115	C₂×C₈	C₂×C₅×D₄	1	48	yes	—
116	C₂×C₈	C₂⁴×C₅	1	8	yes	—
117	C₄⋊C₄	—	2	64	no	—
118	C₄²	C₂×C₅×D₄	1	12	yes	—
119	C₄²	C₂⁴×C₅	1	4	yes	—
120	C₂×C₅	C₂×#494	1	12	yes	direct product factor
121	C₂×C₅	C₂²×#127	1	1	yes	characteristic
122	C₂×C₅	C₂³×Q₁₆	1	2	yes	—
123	C₂³	C₅×#127	1	4	yes	—
124	C₂³	C₂×C₅×Q₁₆	1	8	yes	—
125	C₂³	C₂²×C₅×D₄	1	3	yes	—
126	Q₈	—	2	32	no	—
127	Q₈	—	4	48	no	—
128	C₂×C₄	—	2	16	no	—
129	C₂×C₄	—	4	48	no	—
130	C₂×C₄	C₅×#127	1	12	yes	—
131	C₂×C₄	C₂×C₅×Q₁₆	1	48	yes	complemented
132	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
133	C₂×C₄	C₂²×C₅×D₄	1	12	yes	—
134	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
135	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
136	C₈	C₂²×C₅×D₄	1	16	yes	—
137	C₅	C₂²×#494	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²	C₅×#494	1	16	yes	direct product factor
139	C₂²	C₂×C₅×#127	1	6	yes	—
140	C₂²	C₂²×C₅×Q₁₆	1	12	yes	—
141	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
142	C₄	—	4	16	no	—
143	C₄	C₂×C₅×#127	1	6	yes	—
144	C₄	C₂²×C₅×Q₁₆	1	16	yes	complemented
145	C₄	C₂³×C₅×D₄	1	2	yes	—
146	C₂	C₂×C₅×#494	1	12	yes	direct product factor
147	C₂	C₂²×C₅×#127	1	1	yes	characteristic
148	C₂	C₂³×C₅×Q₁₆	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#494	C₂	direct (N × H)
13. C₅×#494	C₂²	direct (N × H)
15. C₂²×#494	C₅	direct (N × H)
34. C₂×#494	C₂×C₅	direct (N × H)
43. C₂²×C₄×C₅	Q₁₆	semidirect
62. #494	C₂²×C₅	direct (N × H)
72. C₂×C₄×C₅	C₂×Q₁₆	semidirect
94. C₂²×C₅	#494	direct (N × H)
100. C₄×C₅	C₂²×Q₁₆	semidirect
109. C₂²×C₄	C₅×Q₁₆	semidirect
120. C₂×C₅	C₂×#494	direct (N × H)
131. C₂×C₄	C₂×C₅×Q₁₆	semidirect
137. C₅	C₂²×#494	direct (N × H)
138. C₂²	C₅×#494	direct (N × H)
144. C₄	C₂²×C₅×Q₁₆	semidirect
146. C₂	C₂×C₅×#494	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×Q₁₆	2
C₂²×C₅×C₄⋊Q₈	2
C₂×C₅×#494	2
C₂²×C₄×C₅×C₈	2
C₂²×#494	5

Subgroups — C22×C5×#494 #11983350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#494 #11983350