Subgroups — #36326391

Order 21376.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₃₆	C₂	1	4	yes	maximal
2	#24923649	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₆₈	—	2	2	no	—
5	C₂×Dic₆₆₈	C₂²	1	4	yes	—
6	#19128832	—	2	2	no	—
7	Dic₁₃₃₆	—	2	8	no	—
8	C₂×C₁₆×C₁₆₇	C₂²	1	2	yes	—
9	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₃₄	—	2	4	no	—
11	Dic₆₆₈	—	2	8	no	—
12	Dic₆₆₈	—	4	4	no	—
13	C₄⋊Dic₁₆₇	—	4	4	no	—
14	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
16	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₆₇	D₄	1	4	yes	—
18	Dic₃₃₄	—	4	4	no	—
19	Dic₃₃₄	—	4	4	no	—
20	C₂×Dic₁₆₇	—	4	4	no	—
21	C₂×C₄×C₁₆₇	D₈	1	2	yes	—
22	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
23	C₈×C₁₆₇	D₈	1	2	yes	—
24	C₈×C₁₆₇	C₂×D₄	1	2	yes	—
25	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
26	C₄×C₁₆₇	Q₃₂	1	4	yes	complemented
27	C₄×C₁₆₇	#127	1	1	yes	characteristic
28	C₄×C₁₆₇	C₂×D₈	1	1	yes	characteristic
29	Dic₁₆₇	—	8	4	no	—
30	C₂×C₁₆₇	#493	1	1	yes	characteristic
31	C₂×C₁₆₇	C₂×Q₃₂	1	2	yes	—
32	C₁₆₇	#2247	1	1	yes	characteristic, complemented, Sylow
33	#2247	—	167	1	no	maximal, Sylow
34	C₂×Q₃₂	—	167	4	no	—
35	#494	—	167	2	no	—
36	C₄×C₁₆	D₁₆₇	1	1	yes	characteristic
37	C₂×Q₁₆	—	334	2	no	—
38	C₂×Q₁₆	—	167	4	no	—
39	C₄⋊Q₈	—	334	2	no	—
40	Q₃₂	—	334	8	no	—
41	C₂×C₁₆	C₂×D₁₆₇	1	2	yes	—
42	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
43	C₂×Q₈	—	334	4	no	—
44	Q₁₆	—	334	8	no	—
45	Q₁₆	—	668	4	no	—
46	C₂×C₈	D₆₆₈	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
48	C₄⋊C₄	—	668	4	no	—
49	C₄²	D₆₆₈	1	1	yes	characteristic
50	C₁₆	D₆₆₈	1	4	yes	—
51	Q₈	—	668	4	no	—
52	Q₈	—	668	4	no	—
53	C₂×C₄	—	668	4	no	—
54	C₂×C₄	D₁₃₃₆	1	2	yes	—
55	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
56	C₈	D₁₃₃₆	1	2	yes	—
57	C₈	C₂×D₆₆₈	1	2	yes	—
58	C₂²	C₂×D₁₃₃₆	1	1	yes	characteristic
59	C₄	—	1336	4	no	—
60	C₄	Dic₁₃₃₆	1	4	yes	complemented
61	C₄	#19128829	1	1	yes	characteristic
62	C₄	C₂×D₁₃₃₆	1	1	yes	characteristic
63	C₂	#24923646	1	1	yes	characteristic
64	C₂	C₂×Dic₁₃₃₆	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₃₆
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	C₄×C₁₆×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₆₇
hypocenter	C₁₆₇	#2247
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₁₆₇	Q₃₂	semidirect
32. C₁₆₇	#2247	semidirect
60. C₄	Dic₁₃₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₃₆	2
#24923649	2
C₄×C₁₆×C₁₆₇	2
#2247	167