Subgroups — #36324310

Order 21376.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128768	C₂	1	4	yes	maximal
2	#24923293	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581934	C₂²	1	4	yes	—
5	#19128768	—	2	8	no	—
6	C₄⋊(C₁₆₇⋊C₈)	—	2	4	no	—
7	C₂²×C₈×C₁₆₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₆₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₆₇	C₄	1	2	yes	—
11	#14581934	—	2	8	no	—
12	C₂×C₁₆₇⋊C₈	—	2	4	no	—
13	C₂²×C₄×C₁₆₇	Q₈	1	2	yes	—
14	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₁₆₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₆₇	D₄	1	4	yes	—
17	C₂×C₈×C₁₆₇	Q₈	1	4	yes	—
18	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₁₆₇	D₄	1	2	yes	—
21	C₁₆₇⋊C₈	—	4	4	no	—
22	C₂³×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₄×C₁₆₇	—	2	4	no	—
24	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₆₇	C₂×Q₈	1	2	yes	—
30	C₈×C₁₆₇	C₄⋊C₄	1	8	yes	—
31	C₂²×C₁₆₇	—	2	2	no	—
32	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₁₆₇	C₄⋊Q₈	1	2	yes	—
34	C₄×C₁₆₇	—	2	2	no	—
35	C₄×C₁₆₇	#108	1	4	yes	complemented
36	C₄×C₁₆₇	C₂×C₄⋊C₄	1	2	yes	—
37	C₄×C₁₆₇	#127	1	1	yes	characteristic
38	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₁₆₇	—	2	2	no	—
40	C₂×C₁₆₇	#384	1	1	yes	characteristic
41	C₂×C₁₆₇	C₂×#108	1	2	yes	—
42	C₁₆₇	#1848	1	1	yes	characteristic, complemented, Sylow
43	#1848	—	167	1	no	maximal, Sylow
44	C₂×#108	—	167	4	no	—
45	#423	—	167	2	no	—
46	C₂×C₄×C₈	D₁₆₇	1	1	yes	characteristic
47	C₂×M₁₆	—	167	4	no	—
48	C₂²×C₈	C₂×D₁₆₇	1	2	yes	—
49	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
50	#108	—	334	8	no	—
51	C₄⋊C₈	—	334	4	no	—
52	C₄×C₈	Dic₁₆₇	1	2	yes	—
53	C₄×C₈	Dic₁₆₇	1	2	yes	—
54	C₂²×C₄	Dic₃₃₄	1	2	yes	—
55	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
56	M₁₆	—	334	8	no	—
57	C₂×C₈	—	334	4	no	—
58	C₂×C₈	C₂×Dic₁₆₇	1	4	yes	—
59	C₂×C₈	D₆₆₈	1	4	yes	—
60	C₂×C₈	Dic₃₃₄	1	4	yes	—
61	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
62	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
63	C₄²	D₆₆₈	1	2	yes	—
64	C₂³	C₂×Dic₃₃₄	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄⋊Dic₁₆₇	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₁₆₇	1	2	yes	—
68	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
69	C₂×C₄	C₂×D₆₆₈	1	2	yes	—
70	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
71	C₂×C₄	C₂×Dic₃₃₄	1	2	yes	—
72	C₈	—	668	4	no	—
73	C₈	C₄⋊Dic₁₆₇	1	8	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂×C₄⋊Dic₁₆₇	1	1	yes	characteristic
76	C₂²	#19128832	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	#19128768	1	4	yes	complemented
79	C₄	C₂×C₄⋊Dic₁₆₇	1	2	yes	—
80	C₄	#19128829	1	1	yes	characteristic
81	C₄	#19128832	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#24923139	1	1	yes	characteristic
84	C₂	C₂×#19128768	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₈
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	C₂×C₄×C₈×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₆₇
hypocenter	C₁₆₇	#1848
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×C₁₆₇	#108	semidirect
42. C₁₆₇	#1848	semidirect
78. C₄	#19128768	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19128768	2
#24923293	2
C₂×C₄×C₈×C₁₆₇	2
#1848	167