Subgroups — #1722×C₁₆₇ #36320082

Order 21376.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×C₁₆₇	C₂	1	2	yes	maximal, complemented
2	#494×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#492×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	#486×C₁₆₇	C₂	1	2	yes	maximal
5	#322×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆×C₁₆₇	—	2	2	no	—
7	C₂×SD₁₆×C₁₆₇	—	2	2	no	—
8	C₄⋊Q₈×C₁₆₇	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
10	#127×C₁₆₇	C₂²	1	1	yes	characteristic
11	#107×C₁₆₇	—	2	4	no	—
12	#102×C₁₆₇	—	2	4	no	—
13	C₄×C₈×C₁₆₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₁₆₇	—	2	1	no	—
16	C₂×Q₈×C₁₆₇	—	2	2	no	—
17	C₂×D₄×C₁₆₇	—	2	2	no	—
18	C₂×D₄×C₁₆₇	—	2	1	no	—
19	Q₁₆×C₁₆₇	—	4	4	no	—
20	SD₁₆×C₁₆₇	—	4	4	no	—
21	C₂×C₈×C₁₆₇	D₄	1	4	yes	—
22	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₆₇	—	2	2	no	—
24	C₄⋊C₄×C₁₆₇	—	2	2	no	—
25	C₄⋊C₄×C₁₆₇	—	2	2	no	—
26	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
27	C₂³×C₁₆₇	—	4	1	no	—
28	Q₈×C₁₆₇	—	4	2	no	—
29	Q₈×C₁₆₇	—	4	4	no	—
30	D₄×C₁₆₇	—	4	2	no	—
31	D₄×C₁₆₇	—	4	4	no	—
32	C₂×C₄×C₁₆₇	—	4	2	no	—
33	C₂×C₄×C₁₆₇	—	4	1	no	—
34	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
36	C₈×C₁₆₇	—	2	4	no	—
37	C₈×C₁₆₇	D₈	1	4	yes	—
38	C₂²×C₁₆₇	—	8	1	no	—
39	C₂²×C₁₆₇	—	8	2	no	—
40	C₂²×C₁₆₇	#127	1	1	yes	characteristic
41	C₄×C₁₆₇	—	8	1	no	—
42	C₄×C₁₆₇	—	8	2	no	—
43	C₄×C₁₆₇	C₂×D₈	1	2	yes	—
44	C₄×C₁₆₇	#137	1	4	yes	—
45	C₂×C₁₆₇	—	16	1	no	—
46	C₂×C₁₆₇	#493	1	1	yes	characteristic
47	C₂×C₁₆₇	#497	1	2	yes	—
48	C₁₆₇	#1722	1	1	yes	characteristic, direct product factor, Sylow
49	#1722	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	#500	C₂×C₁₆₇	1	2	yes	complemented
51	#494	C₂×C₁₆₇	1	1	yes	characteristic, complemented
52	#492	C₂×C₁₆₇	1	1	yes	characteristic
53	#486	C₂×C₁₆₇	1	2	yes	—
54	#322	C₂×C₁₆₇	1	1	yes	characteristic, complemented
55	C₂×Q₁₆	—	2	2	no	—
56	C₂×SD₁₆	—	2	2	no	—
57	C₄⋊Q₈	C₂²×C₁₆₇	1	2	yes	—
58	C₄⋊Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
59	#127	C₂²×C₁₆₇	1	1	yes	characteristic
60	#107	—	2	4	no	—
61	#102	—	2	4	no	—
62	C₄×C₈	C₂²×C₁₆₇	1	2	yes	—
63	C₄×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	1	no	—
68	Q₁₆	—	4	4	no	—
69	SD₁₆	—	4	4	no	—
70	C₂×C₈	D₄×C₁₆₇	1	4	yes	—
71	C₂×C₈	D₄×C₁₆₇	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄²	C₂³×C₁₆₇	1	1	yes	characteristic
76	C₂³	—	4	1	no	—
77	Q₈	—	4	2	no	—
78	Q₈	—	4	4	no	—
79	D₄	—	4	2	no	—
80	D₄	—	4	4	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
84	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₈×C₁₆₇	1	4	yes	—
87	C₂²	—	8	1	no	—
88	C₂²	—	8	2	no	—
89	C₂²	#127×C₁₆₇	1	1	yes	characteristic
90	C₄	—	8	1	no	—
91	C₄	—	8	2	no	—
92	C₄	C₂×D₈×C₁₆₇	1	2	yes	—
93	C₄	#137×C₁₆₇	1	4	yes	—
94	C₂	—	16	1	no	—
95	C₂	#493×C₁₆₇	1	1	yes	characteristic
96	C₂	#497×C₁₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₆₇	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₁₆₇
prime radical	#127×C₁₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆₇ ◁ C₄²×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #500×C₁₆₇	C₂	semidirect
2. #494×C₁₆₇	C₂	semidirect
5. #322×C₁₆₇	C₂	semidirect
48. C₁₆₇	#1722	direct (N × H)
49. #1722	C₁₆₇	direct (N × H)
50. #500	C₂×C₁₆₇	semidirect
51. #494	C₂×C₁₆₇	semidirect
54. #322	C₂×C₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#500×C₁₆₇	2
#494×C₁₆₇	2
#492×C₁₆₇	2
#486×C₁₆₇	2
#322×C₁₆₇	2
#1722	167

Subgroups — #1722×C167 #36320082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1722×C₁₆₇ #36320082