Subgroups — C₂³×#103 #60078

Order 256.

2751 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂²×#103	C₂	1	28	yes	maximal, direct product factor
5	C₂³×Q₈	—	2	8	no	—
6	C₂³×Q₈	—	2	7	no	—
7	C₂³×Q₈	C₄	1	16	yes	complemented
8	C₂³×Q₈	C₂²	1	14	yes	—
9	C₂⁴×C₄	—	2	1	no	—
10	C₂⁴×C₄	—	2	1	no	—
11	C₂⁴×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₈	—	2	8	no	—
13	C₂³×C₈	C₂²	1	14	yes	—
14	C₂²×C₄⋊C₄	—	2	7	no	—
15	C₂²×C₄⋊C₄	C₂²	1	14	yes	—
16	C₂×#103	C₂²	1	112	yes	direct product factor
17	C₂⁵	D₄	1	1	yes	characteristic
18	C₂²×Q₈	—	2	112	no	—
19	C₂²×Q₈	—	2	56	no	—
20	C₂²×Q₈	—	2	42	no	—
21	C₂²×Q₈	C₂×C₄	1	112	yes	complemented
22	C₂²×Q₈	C₂³	1	28	yes	—
23	C₂³×C₄	—	2	16	no	—
24	C₂³×C₄	—	2	14	no	—
25	C₂³×C₄	—	2	14	no	—
26	C₂³×C₄	—	4	8	no	—
27	C₂³×C₄	C₂×C₄	1	8	yes	—
28	C₂³×C₄	D₄	1	7	yes	—
29	C₂³×C₄	D₄	1	8	yes	—
30	C₂³×C₄	C₂³	1	7	yes	—
31	C₂²×C₈	—	2	56	no	—
32	C₂²×C₈	C₂³	1	28	yes	—
33	C₂×C₄⋊C₄	—	2	42	no	—
34	C₂×C₄⋊C₄	C₂³	1	28	yes	—
35	#103	C₂³	1	64	yes	direct product factor
36	C₂⁴	C₂²⋊C₄	1	8	yes	—
37	C₂⁴	SD₁₆	1	8	yes	—
38	C₂⁴	Q₁₆	1	8	yes	—
39	C₂⁴	C₂×D₄	1	7	yes	—
40	C₂×Q₈	—	2	336	no	—
41	C₂×Q₈	—	2	56	no	—
42	C₂×Q₈	—	2	28	no	—
43	C₂×Q₈	C₂²×C₄	1	112	yes	complemented
44	C₂×Q₈	C₂⁴	1	8	yes	—
45	C₂²×C₄	—	2	112	no	—
46	C₂²×C₄	—	2	28	no	—
47	C₂²×C₄	—	2	28	no	—
48	C₂²×C₄	—	4	56	no	—
49	C₂²×C₄	C₂²⋊C₄	1	56	yes	—
50	C₂²×C₄	C₂²×C₄	1	28	yes	—
51	C₂²×C₄	C₂×D₄	1	21	yes	—
52	C₂²×C₄	C₂×D₄	1	28	yes	—
53	C₂²×C₄	C₂⁴	1	7	yes	—
54	C₂×C₈	—	2	56	no	—
55	C₂×C₈	C₂⁴	1	8	yes	—
56	C₄⋊C₄	—	2	28	no	—
57	C₄⋊C₄	C₂⁴	1	8	yes	—
58	C₂³	#103	1	64	yes	direct product factor
59	C₂³	C₂×C₂²⋊C₄	1	28	yes	—
60	C₂³	C₂×SD₁₆	1	28	yes	—
61	C₂³	C₂×Q₁₆	1	28	yes	—
62	C₂³	C₂²×D₄	1	7	yes	—
63	Q₈	—	2	112	no	—
64	Q₈	—	2	8	no	—
65	Q₈	C₂³×C₄	1	16	yes	complemented
66	C₂×C₄	—	2	112	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	—	4	56	no	—
70	C₂×C₄	C₂×C₂²⋊C₄	1	84	yes	—
71	C₂×C₄	C₂³×C₄	1	14	yes	—
72	C₂×C₄	C₂²×D₄	1	7	yes	—
73	C₂×C₄	C₂²×D₄	1	14	yes	—
74	C₂×C₄	C₂⁵	1	1	yes	characteristic
75	C₈	—	2	8	no	—
76	C₂²	C₂×#103	1	112	yes	direct product factor
77	C₂²	C₂²×C₂²⋊C₄	1	14	yes	—
78	C₂²	C₂²×SD₁₆	1	14	yes	—
79	C₂²	C₂²×Q₁₆	1	14	yes	—
80	C₂²	C₂³×D₄	1	1	yes	characteristic
81	C₄	—	2	16	no	—
82	C₄	—	4	8	no	—
83	C₄	C₂²×C₂²⋊C₄	1	14	yes	—
84	C₄	C₂⁴×C₄	1	1	yes	characteristic
85	C₄	C₂³×D₄	1	1	yes	characteristic
86	C₂	C₂²×#103	1	28	yes	direct product factor
87	C₂	C₂³×C₂²⋊C₄	1	1	yes	characteristic
88	C₂	C₂³×SD₁₆	1	1	yes	characteristic
89	C₂	C₂³×Q₁₆	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂²×#103	C₂	direct (N × H)
7. C₂³×Q₈	C₄	semidirect
16. C₂×#103	C₂²	direct (N × H)
21. C₂²×Q₈	C₂×C₄	semidirect
35. #103	C₂³	direct (N × H)
43. C₂×Q₈	C₂²×C₄	semidirect
58. C₂³	#103	direct (N × H)
65. Q₈	C₂³×C₄	semidirect
76. C₂²	C₂×#103	direct (N × H)
86. C₂	C₂²×#103	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈	2
C₂⁴×C₈	2
C₂³×C₄⋊C₄	2
C₂²×#103	2

Subgroups — C23×#103 #60078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#103 #60078