Subgroups — C₂×#24923107 #36326439

Order 21376.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#24923107	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₆₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₆₈	—	2	1	no	—
6	Dic₁₃₃₆	—	2	1	no	—
7	Dic₁₃₃₆	C₂²	1	2	yes	complemented
8	C₁₆₇⋊C₃₂	C₂²	1	2	yes	complemented
9	C₂×D₈×C₁₆₇	—	2	1	no	—
10	D₁₆×C₁₆₇	—	2	1	no	—
11	D₁₆×C₁₆₇	C₂²	1	2	yes	—
12	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₃₄	—	4	1	no	—
14	Dic₆₆₈	—	2	2	no	—
15	Dic₆₆₈	—	4	1	no	—
16	C₂×D₄×C₁₆₇	—	4	1	no	—
17	D₈×C₁₆₇	—	2	2	no	—
18	D₈×C₁₆₇	—	4	1	no	—
19	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
20	C₁₆×C₁₆₇	D₄	1	1	yes	characteristic
21	C₁₆×C₁₆₇	C₂³	1	1	yes	characteristic
22	Dic₃₃₄	—	4	2	no	—
23	Dic₃₃₄	—	8	1	no	—
24	C₂×Dic₁₆₇	—	8	1	no	—
25	C₂³×C₁₆₇	—	8	1	no	—
26	D₄×C₁₆₇	—	4	2	no	—
27	D₄×C₁₆₇	—	8	1	no	—
28	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
29	C₈×C₁₆₇	D₈	1	1	yes	characteristic
30	C₈×C₁₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₆₇	—	8	2	no	—
32	C₂²×C₁₆₇	—	16	2	no	—
33	C₂²×C₁₆₇	D₁₆	1	1	yes	characteristic
34	C₄×C₁₆₇	D₁₆	1	1	yes	characteristic
35	C₄×C₁₆₇	C₂×D₈	1	1	yes	characteristic
36	Dic₁₆₇	—	8	2	no	—
37	C₂×C₁₆₇	—	16	2	no	—
38	C₂×C₁₆₇	SD₆₄	1	2	yes	complemented
39	C₂×C₁₆₇	C₂×D₁₆	1	1	yes	characteristic
40	C₁₆₇	C₂×SD₆₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×SD₆₄	—	167	1	no	maximal, Sylow
42	C₂×Q₃₂	—	167	1	no	—
43	C₂×D₁₆	D₁₆₇	1	1	yes	characteristic
44	SD₆₄	—	167	4	no	—
45	C₂×C₃₂	—	167	1	no	—
46	C₂×Q₁₆	—	334	1	no	—
47	C₂×D₈	—	2	1	no	—
48	Q₃₂	—	334	1	no	—
49	Q₃₂	—	167	2	no	—
50	D₁₆	—	2	1	no	—
51	D₁₆	C₂×D₁₆₇	1	2	yes	—
52	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
53	C₃₂	—	167	2	no	—
54	C₂×Q₈	—	668	1	no	—
55	C₂×D₄	—	4	1	no	—
56	Q₁₆	—	334	2	no	—
57	Q₁₆	—	668	1	no	—
58	D₈	—	2	2	no	—
59	D₈	—	4	1	no	—
60	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
61	C₁₆	C₂²⋊D₁₆₇	1	1	yes	characteristic
62	C₁₆	C₂²×D₁₆₇	1	1	yes	characteristic
63	C₂³	—	8	1	no	—
64	Q₈	—	1336	1	no	—
65	Q₈	—	668	2	no	—
66	D₄	—	4	2	no	—
67	D₄	—	8	1	no	—
68	C₂×C₄	—	1336	1	no	—
69	C₂×C₄	#14581935	1	1	yes	characteristic
70	C₈	#14581935	1	1	yes	characteristic
71	C₈	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
72	C₂²	—	8	2	no	—
73	C₂²	—	16	2	no	—
74	C₂²	#19128773	1	1	yes	characteristic
75	C₄	—	1336	2	no	—
76	C₄	#19128773	1	1	yes	characteristic
77	C₄	C₂×#14581935	1	1	yes	characteristic
78	C₂	—	16	2	no	—
79	C₂	#24923107	1	2	yes	direct product factor
80	C₂	C₂×#19128773	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19128773
commutator, cosocle	C₁₆×C₁₆₇	C₂³
Frattini	C₁₆	C₂²×D₁₆₇
Fitting, prime radical	C₂×D₁₆×C₁₆₇	C₂
socle	C₂²×C₁₆₇	D₁₆
hypercenter	C₂×D₁₆	D₁₆₇
hypocenter	C₁₆₇	C₂×SD₆₄
LCS	G ▷ C₁₆×C₁₆₇ ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
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×Dic₁₃₃₆	C₂	semidirect
2. #24923107	C₂	direct (N × H)
3. C₂×C₁₆₇⋊C₃₂	C₂	semidirect
7. Dic₁₃₃₆	C₂²	semidirect
8. C₁₆₇⋊C₃₂	C₂²	semidirect
38. C₂×C₁₆₇	SD₆₄	semidirect
40. C₁₆₇	C₂×SD₆₄	semidirect
79. C₂	#24923107	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×Dic₁₃₃₆	2
#24923107	2
C₂×C₁₆₇⋊C₃₂	2
C₂×D₁₆×C₁₆₇	2
C₂×SD₆₄	167

Subgroups — C2×#24923107 #36326439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24923107 #36326439