Subgroups — C₂×#31773735 #46838665

Order 34688.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#31773735	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₂₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₀₈₄	—	2	1	no	—
6	Dic₂₁₆₈	—	2	1	no	—
7	Dic₂₁₆₈	C₂²	1	2	yes	complemented
8	C₂₇₁⋊C₃₂	C₂²	1	2	yes	complemented
9	C₂×D₈×C₂₇₁	—	2	1	no	—
10	D₁₆×C₂₇₁	—	2	1	no	—
11	D₁₆×C₂₇₁	C₂²	1	2	yes	—
12	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₅₄₂	—	4	1	no	—
14	Dic₁₀₈₄	—	2	2	no	—
15	Dic₁₀₈₄	—	4	1	no	—
16	C₂×D₄×C₂₇₁	—	4	1	no	—
17	D₈×C₂₇₁	—	2	2	no	—
18	D₈×C₂₇₁	—	4	1	no	—
19	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
20	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
21	C₁₆×C₂₇₁	C₂³	1	1	yes	characteristic
22	Dic₅₄₂	—	4	2	no	—
23	Dic₅₄₂	—	8	1	no	—
24	C₂×Dic₂₇₁	—	8	1	no	—
25	C₂³×C₂₇₁	—	8	1	no	—
26	D₄×C₂₇₁	—	4	2	no	—
27	D₄×C₂₇₁	—	8	1	no	—
28	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
29	C₈×C₂₇₁	D₈	1	1	yes	characteristic
30	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₂₇₁	—	8	2	no	—
32	C₂²×C₂₇₁	—	16	2	no	—
33	C₂²×C₂₇₁	D₁₆	1	1	yes	characteristic
34	C₄×C₂₇₁	D₁₆	1	1	yes	characteristic
35	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
36	Dic₂₇₁	—	8	2	no	—
37	C₂×C₂₇₁	—	16	2	no	—
38	C₂×C₂₇₁	SD₆₄	1	2	yes	complemented
39	C₂×C₂₇₁	C₂×D₁₆	1	1	yes	characteristic
40	C₂₇₁	C₂×SD₆₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×SD₆₄	—	271	1	no	maximal, Sylow
42	C₂×Q₃₂	—	271	1	no	—
43	C₂×D₁₆	D₂₇₁	1	1	yes	characteristic
44	SD₆₄	—	271	4	no	—
45	C₂×C₃₂	—	271	1	no	—
46	C₂×Q₁₆	—	542	1	no	—
47	C₂×D₈	—	2	1	no	—
48	Q₃₂	—	271	2	no	—
49	Q₃₂	—	542	1	no	—
50	D₁₆	—	2	1	no	—
51	D₁₆	C₂×D₂₇₁	1	2	yes	—
52	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
53	C₃₂	—	271	2	no	—
54	C₂×Q₈	—	1084	1	no	—
55	C₂×D₄	—	4	1	no	—
56	Q₁₆	—	542	2	no	—
57	Q₁₆	—	1084	1	no	—
58	D₈	—	2	2	no	—
59	D₈	—	4	1	no	—
60	C₂×C₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
61	C₁₆	C₂²⋊D₂₇₁	1	1	yes	characteristic
62	C₁₆	C₂²×D₂₇₁	1	1	yes	characteristic
63	C₂³	—	8	1	no	—
64	Q₈	—	1084	2	no	—
65	Q₈	—	2168	1	no	—
66	D₄	—	4	2	no	—
67	D₄	—	8	1	no	—
68	C₂×C₄	—	2168	1	no	—
69	C₂×C₄	#16864639	1	1	yes	characteristic
70	C₈	#16864639	1	1	yes	characteristic
71	C₈	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
72	C₂²	—	8	2	no	—
73	C₂²	—	16	2	no	—
74	C₂²	#22576101	1	1	yes	characteristic
75	C₄	—	2168	2	no	—
76	C₄	#22576101	1	1	yes	characteristic
77	C₄	C₂×#16864639	1	1	yes	characteristic
78	C₂	—	16	2	no	—
79	C₂	#31773735	1	2	yes	direct product factor
80	C₂	C₂×#22576101	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22576101
commutator, cosocle	C₁₆×C₂₇₁	C₂³
Frattini	C₁₆	C₂²×D₂₇₁
Fitting, prime radical	C₂×D₁₆×C₂₇₁	C₂
socle	C₂²×C₂₇₁	D₁₆
hypercenter	C₂×D₁₆	D₂₇₁
hypocenter	C₂₇₁	C₂×SD₆₄
LCS	G ▷ C₁₆×C₂₇₁ ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₁₆₈	C₂	semidirect
2. #31773735	C₂	direct (N × H)
3. C₂×C₂₇₁⋊C₃₂	C₂	semidirect
7. Dic₂₁₆₈	C₂²	semidirect
8. C₂₇₁⋊C₃₂	C₂²	semidirect
38. C₂×C₂₇₁	SD₆₄	semidirect
40. C₂₇₁	C₂×SD₆₄	semidirect
79. C₂	#31773735	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×Dic₂₁₆₈	2
#31773735	2
C₂×C₂₇₁⋊C₃₂	2
C₂×D₁₆×C₂₇₁	2
C₂×SD₆₄	271

Subgroups — C2×#31773735 #46838665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31773735 #46838665