Subgroups — C₂×#1373 #12010

Order 256.

115 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1373	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₃₂	—	2	2	no	—
5	C₂²×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#364	—	2	2	no	—
9	#364	C₂²	1	4	yes	—
10	C₄×C₁₆	C₄	1	2	yes	—
11	C₄×C₁₆	C₂²	1	2	yes	—
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈	D₄	1	1	yes	characteristic
14	C₂×C₄²	Q₈	1	1	yes	characteristic
15	M₃₂	—	2	4	no	—
16	M₃₂	—	4	2	no	—
17	C₂×C₁₆	—	2	1	no	—
18	C₂×C₁₆	—	2	1	no	—
19	C₂×C₁₆	—	4	2	no	—
20	C₂×C₁₆	C₈	1	2	yes	—
21	C₂×C₁₆	C₈	1	2	yes	—
22	C₂×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₆	C₂×C₄	1	2	yes	—
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	D₄	1	1	yes	characteristic
26	C₄×C₈	Q₈	1	1	yes	characteristic
27	C₄×C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₄	—	2	1	no	—
29	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈	M₁₆	1	1	yes	characteristic
35	C₂×C₈	M₁₆	1	1	yes	characteristic
36	C₂×C₈	D₈	1	1	yes	characteristic
37	C₂×C₈	D₈	1	1	yes	characteristic
38	C₂×C₈	Q₁₆	1	1	yes	characteristic
39	C₂×C₈	Q₁₆	1	1	yes	characteristic
40	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈	C₂×D₄	1	1	yes	characteristic
42	C₄²	—	2	1	no	—
43	C₄²	C₄⋊C₄	1	1	yes	characteristic
44	C₄²	C₂×Q₈	1	1	yes	characteristic
45	C₁₆	—	2	2	no	—
46	C₁₆	—	4	4	no	—
47	C₁₆	C₂×C₈	1	4	yes	—
48	C₂³	#108	1	1	yes	characteristic
49	C₂×C₄	—	2	4	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄	#107	1	1	yes	characteristic
55	C₂×C₄	#107	1	1	yes	characteristic
56	C₂×C₄	#108	1	1	yes	characteristic
57	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₈	C₄⋊C₈	1	1	yes	characteristic
59	C₈	C₄⋊C₈	1	1	yes	characteristic
60	C₈	#107	1	1	yes	characteristic
61	C₈	#107	1	1	yes	characteristic
62	C₈	C₂²×C₈	1	1	yes	characteristic
63	C₈	C₂×M₁₆	1	1	yes	characteristic
64	C₈	C₂×D₈	1	1	yes	characteristic
65	C₈	C₂×Q₁₆	1	1	yes	characteristic
66	C₂²	—	2	2	no	—
67	C₂²	#335	1	1	yes	characteristic
68	C₂²	#335	1	1	yes	characteristic
69	C₂²	C₂×#108	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	#335	1	1	yes	characteristic
73	C₄	#335	1	1	yes	characteristic
74	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₄	C₂×#107	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	2	1	no	—
78	C₂	#1373	1	2	yes	direct product factor
79	C₂	C₂×#335	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₈
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	#335
prime radical	C₂³	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1373	C₂	direct (N × H)
78. C₂	#1373	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×#364	2
C₂×C₄×C₁₆	2
#1373	2

Subgroups — C2×#1373 #12010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1373 #12010