Subgroups — C₂×#1420 #13624

Order 256.

143 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#368	C₂	1	1	yes	maximal, characteristic
4	#1420	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆	—	2	1	no	—
6	C₂×Q₃₂	—	2	1	no	—
7	C₂×Q₃₂	—	2	1	no	—
8	C₂×Q₃₂	—	2	1	no	—
9	C₂×Q₃₂	C₄	1	2	yes	—
10	C₂×Q₃₂	C₄	1	2	yes	—
11	C₂×Q₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×#108	—	2	1	no	—
14	M₆₄	—	2	1	no	—
15	M₆₄	C₂²	1	2	yes	—
16	C₂×C₃₂	—	2	1	no	—
17	#368	—	2	1	no	—
18	#368	C₂²	1	2	yes	—
19	C₂²×Q₈	—	4	1	no	—
20	C₂×Q₁₆	—	2	2	no	—
21	C₂×Q₁₆	—	2	2	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	4	1	no	—
24	C₂×Q₁₆	—	4	1	no	—
25	C₂×Q₁₆	—	4	1	no	—
26	C₂×M₁₆	—	4	1	no	—
27	C₂²×C₈	D₄	1	1	yes	characteristic
28	Q₃₂	—	2	2	no	—
29	Q₃₂	—	2	2	no	—
30	Q₃₂	—	2	2	no	—
31	Q₃₂	C₂×C₄	1	4	yes	—
32	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆	D₄	1	1	yes	characteristic
35	C₂×C₁₆	D₄	1	1	yes	characteristic
36	C₂×C₁₆	D₄	1	1	yes	characteristic
37	C₂×C₁₆	C₂³	1	1	yes	characteristic
38	#108	—	2	2	no	—
39	#108	—	4	1	no	—
40	C₃₂	—	2	2	no	—
41	C₂×Q₈	—	4	2	no	—
42	C₂×Q₈	—	4	2	no	—
43	C₂×Q₈	—	4	2	no	—
44	C₂×Q₈	—	8	1	no	—
45	C₂×Q₈	—	8	1	no	—
46	C₂×Q₈	—	8	1	no	—
47	C₂²×C₄	—	8	1	no	—
48	C₂²×C₄	D₈	1	1	yes	characteristic
49	Q₁₆	—	2	4	no	—
50	Q₁₆	—	4	2	no	—
51	Q₁₆	—	4	2	no	—
52	Q₁₆	—	4	2	no	—
53	M₁₆	—	4	2	no	—
54	M₁₆	—	8	1	no	—
55	C₂×C₈	—	8	1	no	—
56	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈	D₈	1	1	yes	characteristic
59	C₂×C₈	SD₁₆	1	1	yes	characteristic
60	C₂×C₈	SD₁₆	1	1	yes	characteristic
61	C₂×C₈	C₂×D₄	1	1	yes	characteristic
62	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
63	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
64	C₁₆	C₂²×C₄	1	1	yes	characteristic
65	C₁₆	C₂×D₄	1	1	yes	characteristic
66	C₂³	SD₃₂	1	1	yes	characteristic
67	Q₈	—	4	4	no	—
68	Q₈	—	8	2	no	—
69	Q₈	—	8	2	no	—
70	Q₈	—	8	2	no	—
71	C₂×C₄	—	8	2	no	—
72	C₂×C₄	—	8	2	no	—
73	C₂×C₄	—	8	2	no	—
74	C₂×C₄	#102	1	1	yes	characteristic
75	C₂×C₄	#102	1	1	yes	characteristic
76	C₂×C₄	D₁₆	1	1	yes	characteristic
77	C₂×C₄	D₁₆	1	1	yes	characteristic
78	C₂×C₄	SD₃₂	1	1	yes	characteristic
79	C₂×C₄	C₂×D₈	1	1	yes	characteristic
80	C₈	—	8	2	no	—
81	C₈	#102	1	1	yes	characteristic
82	C₈	#102	1	1	yes	characteristic
83	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈	C₂×SD₁₆	1	1	yes	characteristic
85	C₂²	—	2	2	no	—
86	C₂²	#357	1	1	yes	characteristic
87	C₂²	#357	1	1	yes	characteristic
88	C₂²	C₂×SD₃₂	1	1	yes	characteristic
89	C₄	—	8	4	no	—
90	C₄	#357	1	1	yes	characteristic
91	C₄	#357	1	1	yes	characteristic
92	C₄	C₂×#102	1	1	yes	characteristic
93	C₄	C₂×D₁₆	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2	1	no	—
96	C₂	#1420	1	2	yes	direct product factor
97	C₂	C₂×#357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#357
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂³	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×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
4. #1420	C₂	direct (N × H)
96. C₂	#1420	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×Q₃₂	2
C₂×M₆₄	2
C₂×#368	2
#1420	2

Subgroups — C2×#1420 #13624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1420 #13624