Subgroups — C₂×#31773737 #46838668

Order 34688.

104 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
2	#31773737	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₂₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₀₈₄	—	2	1	no	—
6	Dic₂₁₆₈	—	2	1	no	—
7	Dic₂₁₆₈	C₂²	1	2	yes	—
8	C₂₇₁⋊C₃₂	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₂₇₁	—	2	1	no	—
10	Q₃₂×C₂₇₁	—	2	1	no	—
11	Q₃₂×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×Q₈×C₂₇₁	—	4	1	no	—
17	Q₁₆×C₂₇₁	—	2	2	no	—
18	Q₁₆×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	Q₈×C₂₇₁	—	4	2	no	—
26	Q₈×C₂₇₁	—	8	1	no	—
27	C₂×C₄×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	D₁₆	1	1	yes	characteristic
32	C₄×C₂₇₁	—	8	2	no	—
33	C₄×C₂₇₁	D₁₆	1	1	yes	characteristic
34	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
35	Dic₂₇₁	—	8	2	no	—
36	C₂×C₂₇₁	Q₆₄	1	2	yes	complemented
37	C₂×C₂₇₁	C₂×D₁₆	1	1	yes	characteristic
38	C₂₇₁	C₂×Q₆₄	1	1	yes	characteristic, complemented, Sylow
39	C₂×Q₆₄	—	271	1	no	maximal, Sylow
40	C₂×Q₃₂	—	271	1	no	—
41	C₂×Q₃₂	D₂₇₁	1	1	yes	characteristic
42	Q₆₄	—	271	4	no	—
43	C₂×C₃₂	—	271	1	no	—
44	C₂×Q₁₆	—	2	1	no	—
45	C₂×Q₁₆	—	542	1	no	—
46	Q₃₂	—	2	1	no	—
47	Q₃₂	—	271	2	no	—
48	Q₃₂	—	542	1	no	—
49	Q₃₂	C₂×D₂₇₁	1	2	yes	—
50	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
51	C₃₂	—	271	2	no	—
52	C₂×Q₈	—	4	1	no	—
53	C₂×Q₈	—	1084	1	no	—
54	Q₁₆	—	2	2	no	—
55	Q₁₆	—	4	1	no	—
56	Q₁₆	—	542	2	no	—
57	Q₁₆	—	1084	1	no	—
58	C₂×C₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
59	C₁₆	C₂²⋊D₂₇₁	1	1	yes	characteristic
60	C₁₆	C₂²×D₂₇₁	1	1	yes	characteristic
61	Q₈	—	4	2	no	—
62	Q₈	—	8	1	no	—
63	Q₈	—	1084	2	no	—
64	Q₈	—	2168	1	no	—
65	C₂×C₄	—	8	1	no	—
66	C₂×C₄	—	2168	1	no	—
67	C₂×C₄	#16864639	1	1	yes	characteristic
68	C₈	#16864639	1	1	yes	characteristic
69	C₈	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
70	C₂²	#22576101	1	1	yes	characteristic
71	C₄	—	8	2	no	—
72	C₄	—	2168	2	no	—
73	C₄	#22576101	1	1	yes	characteristic
74	C₄	C₂×#16864639	1	1	yes	characteristic
75	C₂	#31773737	1	2	yes	direct product factor
76	C₂	C₂×#22576101	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22576101
commutator, cosocle	C₁₆×C₂₇₁	C₂³
Frattini	C₁₆	C₂²×D₂₇₁
Fitting	C₂×Q₃₂×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	D₁₆
hypercenter	C₂×Q₃₂	D₂₇₁
hypocenter	C₂₇₁	C₂×Q₆₄
LCS	G ▷ C₁₆×C₂₇₁ ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #31773737	C₂	direct (N × H)
36. C₂×C₂₇₁	Q₆₄	semidirect
38. C₂₇₁	C₂×Q₆₄	semidirect
75. C₂	#31773737	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
#31773737	2
C₂×C₂₇₁⋊C₃₂	2
C₂×Q₃₂×C₂₇₁	2
C₂×Q₆₄	271

Subgroups — C2×#31773737 #46838668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31773737 #46838668