Subgroups — C₂×#13731 #513057

Order 512.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2252	C₂	1	2	yes	maximal
2	#13731	C₂	1	8	yes	maximal, direct product factor
3	C₂×#1423	C₂	1	4	yes	maximal
4	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	2	no	—
6	C₂²×C₃₂	C₂²	1	2	yes	—
7	#2252	—	2	4	no	—
8	#2252	C₂²	1	8	yes	—
9	C₂×#366	—	2	2	no	—
10	C₂×#366	C₂²	1	4	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1423	C₂²	1	16	yes	—
13	C₄×C₃₂	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈	—	4	2	no	—
15	C₂²×C₁₆	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂³	1	1	yes	characteristic
17	#500	—	2	8	no	—
18	#500	—	4	4	no	—
19	C₂×#107	—	2	4	no	—
20	C₂×#107	—	4	2	no	—
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₃₂	Q₈	1	8	yes	complemented
23	C₂×C₃₂	C₂³	1	4	yes	—
24	#366	—	2	8	no	—
25	#366	—	2	4	no	—
26	#366	C₂³	1	8	yes	—
27	C₄×C₁₆	D₄	1	2	yes	—
28	C₄×C₁₆	C₂³	1	2	yes	—
29	C₂²×Q₈	—	8	2	no	—
30	C₂²×C₈	D₈	1	1	yes	characteristic
31	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
32	C₄⋊Q₈	—	4	8	no	—
33	C₄⋊Q₈	—	8	4	no	—
34	C₂×C₄⋊C₄	—	4	4	no	—
35	C₂×C₄⋊C₄	—	8	2	no	—
36	C₂×C₄²	D₈	1	1	yes	characteristic
37	C₂×C₁₆	Q₁₆	1	4	yes	—
38	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₄	1	2	yes	—
40	C₂×C₁₆	C₂×Q₈	1	4	yes	—
41	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
42	#107	—	2	8	no	—
43	#107	—	4	4	no	—
44	#107	—	4	8	no	—
45	C₄×C₈	D₈	1	2	yes	—
46	C₄×C₈	C₂×D₄	1	2	yes	—
47	C₃₂	C₂×Q₈	1	8	yes	complemented
48	C₂×Q₈	—	8	8	no	—
49	C₂×Q₈	—	16	8	no	—
50	C₂²×C₄	—	8	4	no	—
51	C₂²×C₄	D₁₆	1	1	yes	characteristic
52	C₂²×C₄	D₁₆	1	1	yes	characteristic
53	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
54	C₂×C₈	Q₃₂	1	4	yes	—
55	C₂×C₈	C₄⋊Q₈	1	2	yes	—
56	C₂×C₈	C₂×D₈	1	1	yes	characteristic
57	C₂×C₈	C₂×D₈	1	2	yes	—
58	C₂×C₈	C₂×Q₁₆	1	2	yes	—
59	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
60	C₄⋊C₄	—	4	8	no	—
61	C₄⋊C₄	—	8	4	no	—
62	C₄⋊C₄	—	8	8	no	—
63	C₄²	D₁₆	1	2	yes	—
64	C₄²	C₂×D₈	1	2	yes	—
65	C₁₆	C₄⋊Q₈	1	2	yes	—
66	C₁₆	C₂×Q₁₆	1	4	yes	—
67	C₁₆	C₂²×Q₈	1	2	yes	—
68	C₂³	C₂×D₁₆	1	1	yes	characteristic
69	Q₈	—	16	16	no	—
70	C₂×C₄	—	8	8	no	—
71	C₂×C₄	—	16	8	no	—
72	C₂×C₄	D₃₂	1	4	yes	—
73	C₂×C₄	Q₆₄	1	4	yes	—
74	C₂×C₄	#500	1	2	yes	—
75	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
76	C₂×C₄	C₂×D₁₆	1	2	yes	—
77	C₂×C₄	C₂×D₁₆	1	2	yes	—
78	C₂×C₄	C₂×Q₃₂	1	2	yes	—
79	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
80	C₈	#500	1	2	yes	—
81	C₈	C₂×Q₃₂	1	4	yes	—
82	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₈	C₂²×Q₁₆	1	1	yes	characteristic
84	C₂²	#2252	1	2	yes	—
85	C₂²	C₂×D₃₂	1	2	yes	—
86	C₂²	C₂×Q₆₄	1	2	yes	—
87	C₂²	C₂²×D₁₆	1	1	yes	characteristic
88	C₄	—	16	8	no	—
89	C₄	#2252	1	2	yes	—
90	C₄	C₂×D₃₂	1	4	yes	—
91	C₄	C₂×Q₆₄	1	4	yes	—
92	C₄	C₂×#500	1	1	yes	characteristic
93	C₄	C₂²×Q₃₂	1	1	yes	characteristic
94	C₂	#13731	1	4	yes	direct product factor
95	C₂	C₂×#2252	1	1	yes	characteristic
96	C₂	C₂²×D₃₂	1	1	yes	characteristic
97	C₂	C₂²×Q₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13731	C₂	direct (N × H)
22. C₂×C₃₂	Q₈	semidirect
47. C₃₂	C₂×Q₈	semidirect
94. C₂	#13731	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×#2252	2
#13731	2
C₂×#1423	2
C₂×C₄×C₃₂	2

Subgroups — C2×#13731 #513057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13731 #513057