Subgroups — #36322109

Order 21376.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₆₈	C₂	1	1	yes	maximal, characteristic
2	#24923020	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₆₈	C₄	1	1	yes	characteristic
5	C₄×Dic₃₃₄	—	2	1	no	—
6	#19128766	C₄	1	1	yes	characteristic
7	C₄⋊(C₁₆₇⋊C₈)	—	2	1	no	—
8	#19128757	—	2	1	no	—
9	C₂×C₁₆×C₁₆₇	—	2	1	no	—
10	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₃₄	—	2	1	no	—
12	Dic₆₆₈	—	2	1	no	—
13	Dic₆₆₈	C₈	1	2	yes	complemented
14	C₂×C₁₆₇⋊C₈	—	4	1	no	—
15	C₄⋊Dic₁₆₇	—	2	1	no	—
16	Dic₁₆₇⋊C₄	—	4	1	no	—
17	C₄×Dic₁₆₇	—	4	1	no	—
18	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
20	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
21	C₁₆×C₁₆₇	—	2	2	no	—
22	Dic₃₃₄	—	2	2	no	—
23	Dic₃₃₄	—	4	1	no	—
24	C₂×Dic₁₆₇	—	4	1	no	—
25	C₂×Dic₁₆₇	—	4	1	no	—
26	C₁₆₇⋊C₈	—	8	1	no	—
27	C₂×C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₆₇	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₆₇	—	2	1	no	—
31	C₈×C₁₆₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₆₇	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₆₇	#102	1	1	yes	characteristic
34	C₄×C₁₆₇	—	2	1	no	—
35	C₄×C₁₆₇	—	2	1	no	—
36	C₄×C₁₆₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₁₆₇	#104	1	1	yes	characteristic
38	Dic₁₆₇	—	4	2	no	—
39	Dic₁₆₇	—	8	1	no	—
40	C₂×C₁₆₇	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₁₆₇	#360	1	1	yes	characteristic
42	C₂×C₁₆₇	#362	1	1	yes	characteristic
43	C₁₆₇	#1334	1	1	yes	characteristic, complemented, Sylow
44	#1334	—	167	1	no	maximal, Sylow
45	C₄×Q₁₆	—	167	1	no	—
46	#346	D₁₆₇	1	1	yes	characteristic
47	#334	—	167	1	no	—
48	C₂×Q₁₆	—	167	1	no	—
49	C₄×Q₈	—	334	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	#107	—	167	1	no	—
52	C₄⋊C₈	—	334	1	no	—
53	#103	—	334	1	no	—
54	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
55	C₂×Q₈	—	334	1	no	—
56	Q₁₆	—	334	1	no	—
57	Q₁₆	—	167	2	no	—
58	C₂×C₈	—	668	1	no	—
59	C₂×C₈	C₄×D₁₆₇	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
61	C₄⋊C₄	—	334	1	no	—
62	C₄⋊C₄	—	668	1	no	—
63	C₄²	—	668	1	no	—
64	C₄²	D₆₆₈	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	Q₈	—	334	2	no	—
67	Q₈	—	668	1	no	—
68	C₂×C₄	—	668	1	no	—
69	C₂×C₄	—	668	1	no	—
70	C₂×C₄	(C₂×D₁₆₇)⋊C₄	1	1	yes	characteristic
71	C₂×C₄	D₁₃₃₆	1	1	yes	characteristic
72	C₂×C₄	#14581939	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	1336	1	no	—
75	C₈	C₈×D₁₆₇	1	1	yes	characteristic
76	C₈	#14581933	1	1	yes	characteristic
77	C₂²	#19128754	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	1336	1	no	—
81	C₄	—	668	2	no	—
82	C₄	(C₂×D₁₆₇)⋊C₈	1	1	yes	characteristic
83	C₄	#19128760	1	1	yes	characteristic
84	C₂	#24922998	1	1	yes	characteristic
85	C₂	#24923080	1	1	yes	characteristic
86	C₂	#24923086	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19128754
commutator	C₈×C₁₆₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₇
Fitting	#346×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	#102
cosocle	C₄×C₈×C₁₆₇	C₂²
hypercenter	#346	D₁₆₇
hypocenter	C₁₆₇	#1334
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. Dic₆₆₈	C₈	semidirect
43. C₁₆₇	#1334	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₆₈	2
#24923020	2
#346×C₁₆₇	2
#1334	167