Subgroups — #36322313

Order 21376.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923066	C₂	1	1	yes	maximal, characteristic
2	#24923064	C₂	1	1	yes	maximal, characteristic
3	#488×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	#19128751	—	2	1	no	—
5	#19128747	—	2	1	no	—
6	#125×C₁₆₇	C₄	1	1	yes	characteristic, complemented
7	#124×C₁₆₇	C₂²	1	1	yes	characteristic
8	#103×C₁₆₇	—	2	1	no	—
9	#102×C₁₆₇	—	2	1	no	—
10	C₄×C₈×C₁₆₇	C₄	1	1	yes	characteristic, complemented
11	#14581934	—	4	1	no	—
12	C₂²⋊Dic₁₆₇	—	4	1	no	—
13	C₂×Q₈×C₁₆₇	D₄	1	1	yes	characteristic
14	C₂×D₄×C₁₆₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₆₇	—	2	1	no	—
16	C₄⋊C₄×C₁₆₇	—	2	1	no	—
17	C₄⋊C₄×C₁₆₇	—	4	1	no	—
18	C₂²⋊C₄×C₁₆₇	—	4	1	no	—
19	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₆₇	—	8	1	no	—
21	C₁₆₇⋊C₈	—	8	1	no	—
22	C₂³×C₁₆₇	—	2	1	no	—
23	Q₈×C₁₆₇	—	4	1	no	—
24	D₄×C₁₆₇	—	4	1	no	—
25	C₂×C₄×C₁₆₇	—	2	1	no	—
26	C₂×C₄×C₁₆₇	—	2	1	no	—
27	C₂×C₄×C₁₆₇	—	4	1	no	—
28	C₂×C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₁₆₇	—	4	1	no	—
30	C₂²×C₁₆₇	—	4	1	no	—
31	C₂²×C₁₆₇	—	8	1	no	—
32	C₂²×C₁₆₇	#99	1	1	yes	characteristic
33	C₄×C₁₆₇	—	2	1	no	—
34	C₄×C₁₆₇	—	4	1	no	—
35	C₄×C₁₆₇	—	4	1	no	—
36	C₄×C₁₆₇	—	8	1	no	—
37	Dic₁₆₇	—	16	1	no	—
38	C₂×C₁₆₇	—	2	1	no	—
39	C₂×C₁₆₇	—	8	1	no	—
40	C₂×C₁₆₇	#354	1	1	yes	characteristic
41	C₁₆₇	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	167	1	no	maximal, Sylow
43	#488	D₁₆₇	1	1	yes	characteristic
44	#355	—	167	1	no	—
45	#354	—	167	1	no	—
46	#125	Dic₁₆₇	1	1	yes	characteristic, complemented
47	#124	C₂×D₁₆₇	1	1	yes	characteristic
48	#103	—	2	1	no	—
49	#102	—	2	1	no	—
50	#101	—	334	1	no	—
51	#99	—	334	1	no	—
52	C₄×C₈	Dic₁₆₇	1	1	yes	characteristic, complemented
53	C₂×Q₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
55	M₁₆	—	668	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	668	1	no	—
61	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	1336	1	no	—
69	C₂×C₄	C₂²⋊Dic₁₆₇	1	1	yes	characteristic
70	C₈	—	4	1	no	—
71	C₈	—	1336	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#19128747	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	8	1	no	—
79	C₄	—	2672	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#24923064	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#24923064
commutator	C₄²×C₁₆₇	C₂×C₄
Frattini	#124	C₂×D₁₆₇
Fitting	#488×C₁₆₇	C₂
socle	C₂×C₁₆₇	#354
prime radical	C₂×D₄×C₁₆₇	D₄
cosocle	#124×C₁₆₇	C₂²
hypercenter	#488	D₁₆₇
hypocenter	C₁₆₇	#1414
LCS	G ▷ C₄²×C₁₆₇ ▷ C₂×C₄×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #488

Complements

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

N	H ≅ G/N	type
6. #125×C₁₆₇	C₄	semidirect
10. C₄×C₈×C₁₆₇	C₄	semidirect
41. C₁₆₇	#1414	semidirect
46. #125	Dic₁₆₇	semidirect
52. C₄×C₈	Dic₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923066	2
#24923064	2
#488×C₁₆₇	2
#1414	167