Subgroups — C₂×#366×C₁₆₇ #36320516

Order 21376.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#107×C₁₆₇	C₂	1	2	yes	maximal
3	#366×C₁₆₇	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₁₆₇	—	2	2	no	—
6	C₂×C₁₆×C₁₆₇	C₄	1	4	yes	complemented
7	C₂×C₁₆×C₁₆₇	C₂²	1	2	yes	—
8	#107×C₁₆₇	—	2	2	no	—
9	#107×C₁₆₇	C₂²	1	4	yes	—
10	C₂²×C₄×C₁₆₇	—	4	2	no	—
11	C₂²×C₄×C₁₆₇	D₄	1	1	yes	characteristic
12	C₂×C₈×C₁₆₇	C₂×C₄	1	2	yes	—
13	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆₇	Q₈	1	2	yes	—
15	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
16	C₄⋊C₄×C₁₆₇	—	2	4	no	—
17	C₄⋊C₄×C₁₆₇	—	4	2	no	—
18	C₁₆×C₁₆₇	C₂×C₄	1	4	yes	complemented
19	C₂³×C₁₆₇	D₈	1	1	yes	characteristic
20	C₂×C₄×C₁₆₇	—	4	4	no	—
21	C₂×C₄×C₁₆₇	—	8	4	no	—
22	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₆₇	Q₁₆	1	2	yes	—
25	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
26	C₈×C₁₆₇	C₄⋊C₄	1	2	yes	—
27	C₈×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₁₆₇	#107	1	2	yes	—
30	C₂²×C₁₆₇	D₁₆	1	2	yes	—
31	C₂²×C₁₆₇	Q₃₂	1	2	yes	—
32	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₆₇	—	8	4	no	—
34	C₄×C₁₆₇	#107	1	2	yes	—
35	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₁₆₇	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₆₇	#366	1	4	yes	direct product factor
38	C₂×C₁₆₇	C₂×#107	1	1	yes	characteristic
39	C₂×C₁₆₇	C₂×D₁₆	1	1	yes	characteristic
40	C₂×C₁₆₇	C₂×Q₃₂	1	1	yes	characteristic
41	C₁₆₇	C₂×#366	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×#366	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₁₆	C₂×C₁₆₇	1	1	yes	characteristic
44	C₂×#107	C₂×C₁₆₇	1	2	yes	—
45	#366	C₂×C₁₆₇	1	4	yes	direct product factor
46	C₂²×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₆	C₄×C₁₆₇	1	4	yes	complemented
49	C₂×C₁₆	C₂²×C₁₆₇	1	2	yes	—
50	#107	—	2	2	no	—
51	#107	C₂²×C₁₆₇	1	4	yes	—
52	C₂²×C₄	—	4	2	no	—
53	C₂²×C₄	D₄×C₁₆₇	1	1	yes	characteristic
54	C₂×C₈	C₂×C₄×C₁₆₇	1	2	yes	—
55	C₂×C₈	D₄×C₁₆₇	1	1	yes	characteristic
56	C₂×C₈	Q₈×C₁₆₇	1	2	yes	—
57	C₂×C₈	C₂³×C₁₆₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	4	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₁₆	C₂×C₄×C₁₆₇	1	4	yes	complemented
61	C₂³	D₈×C₁₆₇	1	1	yes	characteristic
62	C₂×C₄	—	4	4	no	—
63	C₂×C₄	—	8	4	no	—
64	C₂×C₄	C₄⋊C₄×C₁₆₇	1	2	yes	—
65	C₂×C₄	D₈×C₁₆₇	1	1	yes	characteristic
66	C₂×C₄	Q₁₆×C₁₆₇	1	2	yes	—
67	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
68	C₈	C₄⋊C₄×C₁₆₇	1	2	yes	—
69	C₈	C₂²×C₄×C₁₆₇	1	1	yes	characteristic
70	C₈	C₂×Q₈×C₁₆₇	1	1	yes	characteristic
71	C₂²	#107×C₁₆₇	1	2	yes	—
72	C₂²	D₁₆×C₁₆₇	1	2	yes	—
73	C₂²	Q₃₂×C₁₆₇	1	2	yes	—
74	C₂²	C₂×D₈×C₁₆₇	1	1	yes	characteristic
75	C₄	—	8	4	no	—
76	C₄	#107×C₁₆₇	1	2	yes	—
77	C₄	C₂×C₄⋊C₄×C₁₆₇	1	1	yes	characteristic
78	C₄	C₂×Q₁₆×C₁₆₇	1	1	yes	characteristic
79	C₂	#366×C₁₆₇	1	4	yes	direct product factor
80	C₂	C₂×#107×C₁₆₇	1	1	yes	characteristic
81	C₂	C₂×D₁₆×C₁₆₇	1	1	yes	characteristic
82	C₂	C₂×Q₃₂×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₆₇	D₈
commutator	C₈	C₂²×C₄×C₁₆₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₆₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₆₇ ◁ C₂²×C₄×C₁₆₇ ◁ C₂²×C₈×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #366×C₁₆₇	C₂	direct (N × H)
6. C₂×C₁₆×C₁₆₇	C₄	semidirect
18. C₁₆×C₁₆₇	C₂×C₄	semidirect
37. C₂×C₁₆₇	#366	direct (N × H)
41. C₁₆₇	C₂×#366	direct (N × H)
42. C₂×#366	C₁₆₇	direct (N × H)
45. #366	C₂×C₁₆₇	direct (N × H)
48. C₂×C₁₆	C₄×C₁₆₇	semidirect
60. C₁₆	C₂×C₄×C₁₆₇	semidirect
79. C₂	#366×C₁₆₇	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×C₁₆×C₁₆₇	2
C₂×#107×C₁₆₇	2
#366×C₁₆₇	2
C₂×#366	167

Subgroups — C2×#366×C167 #36320516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#366×C₁₆₇ #36320516