Subgroups — #36322106

Order 21376.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₃₃₆	C₂	1	1	yes	maximal, characteristic
2	#24923020	C₂	1	1	yes	maximal, characteristic, complemented
3	#346×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₃₆	C₄	1	1	yes	characteristic
5	C₄×D₆₆₈	—	2	1	no	—
6	#19128766	C₄	1	1	yes	characteristic
7	C₄⋊(C₁₆₇⋊C₈)	—	2	1	no	—
8	#19128754	—	2	1	no	—
9	C₂×C₁₆×C₁₆₇	—	2	1	no	—
10	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₂×D₆₆₈	—	2	1	no	—
12	C₂×C₄×D₁₆₇	—	4	1	no	—
13	D₁₃₃₆	—	2	1	no	—
14	D₁₃₃₆	C₈	1	2	yes	complemented
15	C₂×C₁₆₇⋊C₈	—	4	1	no	—
16	C₄⋊Dic₁₆₇	—	2	1	no	—
17	(C₂×D₁₆₇)⋊C₄	—	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	C₂²×D₁₆₇	—	4	1	no	—
23	D₆₆₈	—	2	2	no	—
24	D₆₆₈	—	4	1	no	—
25	C₂×Dic₁₆₇	—	4	1	no	—
26	C₄×D₁₆₇	—	8	2	no	—
27	C₁₆₇⋊C₈	—	8	1	no	—
28	C₂×C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₆₇	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₆₇	—	2	1	no	—
32	C₈×C₁₆₇	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₆₇	M₁₆	1	1	yes	characteristic
34	C₂²×C₁₆₇	#102	1	1	yes	characteristic
35	C₂×D₁₆₇	—	4	2	no	—
36	C₂×D₁₆₇	—	8	1	no	—
37	C₂×D₁₆₇	—	8	1	no	—
38	C₄×C₁₆₇	—	2	1	no	—
39	C₄×C₁₆₇	—	2	1	no	—
40	C₄×C₁₆₇	C₂²⋊C₈	1	1	yes	characteristic
41	C₄×C₁₆₇	#104	1	1	yes	characteristic
42	Dic₁₆₇	—	8	1	no	—
43	C₂×C₁₆₇	D₄⋊C₈	1	1	yes	characteristic
44	C₂×C₁₆₇	#360	1	1	yes	characteristic
45	C₂×C₁₆₇	#361	1	1	yes	characteristic
46	D₁₆₇	—	8	2	no	—
47	C₁₆₇	#1333	1	1	yes	characteristic, complemented, Sylow
48	#1333	—	167	1	no	maximal, Sylow
49	C₄×D₈	—	167	1	no	—
50	#346	D₁₆₇	1	1	yes	characteristic, complemented
51	#334	—	167	1	no	—
52	C₂×D₈	—	167	1	no	—
53	C₄×D₄	—	334	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	#107	—	167	1	no	—
56	C₄⋊C₈	—	334	1	no	—
57	#102	—	334	1	no	—
58	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
59	C₂×D₄	—	334	1	no	—
60	C₂²×C₄	—	668	1	no	—
61	D₈	—	334	1	no	—
62	D₈	—	167	2	no	—
63	C₂×C₈	—	668	1	no	—
64	C₂×C₈	C₄×D₁₆₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
66	C₄⋊C₄	—	334	1	no	—
67	C₂²⋊C₄	—	668	1	no	—
68	C₄²	D₆₆₈	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₂³	—	668	1	no	—
71	D₄	—	334	2	no	—
72	D₄	—	668	1	no	—
73	C₂×C₄	—	1336	2	no	—
74	C₂×C₄	—	668	1	no	—
75	C₂×C₄	(C₂×D₁₆₇)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	D₁₃₃₆	1	1	yes	characteristic
77	C₂×C₄	#14581939	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	1336	1	no	—
80	C₈	C₈×D₁₆₇	1	1	yes	characteristic
81	C₈	#14581933	1	1	yes	characteristic
82	C₂²	—	1336	1	no	—
83	C₂²	—	1336	1	no	—
84	C₂²	—	668	2	no	—
85	C₂²	#19128754	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	1336	1	no	—
89	C₄	(C₂×D₁₆₇)⋊C₈	1	1	yes	characteristic
90	C₄	#19128760	1	1	yes	characteristic
91	C₂	—	1336	2	no	—
92	C₂	#24922998	1	1	yes	characteristic
93	C₂	#24923080	1	1	yes	characteristic
94	C₂	#24923083	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	C₂²×C₁₆₇	#102
prime radical	C₂×D₁₃₃₆	C₄
cosocle	C₄×C₈×C₁₆₇	C₂²
hypercenter	#346	D₁₆₇
hypocenter	C₁₆₇	#1333
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
2. #24923020	C₂	semidirect
3. #346×C₁₆₇	C₂	semidirect
14. D₁₃₃₆	C₈	semidirect
47. C₁₆₇	#1333	semidirect
50. #346	D₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₃₃₆	2
#24923020	2
#346×C₁₆₇	2
#1333	167