Subgroups — #36324276

Order 21376.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923293	C₂	1	2	yes	maximal
2	#24923034	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581934	C₂²	1	4	yes	—
5	C₄⋊(C₁₆₇⋊C₈)	—	2	4	no	—
6	C₂²×C₈×C₁₆₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₆₇	C₄	1	2	yes	—
9	C₄×C₈×C₁₆₇	C₄	1	2	yes	—
10	#14581934	—	2	8	no	—
11	C₂×C₁₆₇⋊C₈	—	2	4	no	—
12	C₂²×C₄×C₁₆₇	Q₈	1	2	yes	—
13	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
14	C₂×C₈×C₁₆₇	—	2	4	no	—
15	C₂×C₈×C₁₆₇	C₂×C₄	1	4	yes	—
16	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
17	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₁₆₇	D₄	1	2	yes	—
19	C₁₆₇⋊C₈	—	4	4	no	—
20	C₂³×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₄×C₁₆₇	—	2	4	no	—
22	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₆₇	#41	1	4	yes	—
27	C₈×C₁₆₇	—	2	4	no	—
28	C₂²×C₁₆₇	—	2	2	no	—
29	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₁₆₇	#125	1	2	yes	—
31	C₄×C₁₆₇	—	2	2	no	—
32	C₄×C₁₆₇	#108	1	4	yes	—
33	C₄×C₁₆₇	#117	1	2	yes	—
34	C₄×C₁₆₇	#124	1	2	yes	—
35	C₂×C₁₆₇	—	2	2	no	—
36	C₂×C₁₆₇	#382	1	1	yes	characteristic
37	C₂×C₁₆₇	C₂×#108	1	2	yes	—
38	C₁₆₇	#1837	1	1	yes	characteristic, complemented, Sylow
39	#1837	—	167	1	no	maximal, Sylow
40	#423	—	167	2	no	—
41	C₂×C₄×C₈	D₁₆₇	1	1	yes	characteristic
42	#341	—	167	4	no	—
43	C₂×M₁₆	—	167	4	no	—
44	C₂²×C₈	C₂×D₁₆₇	1	2	yes	—
45	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
46	C₄⋊C₈	—	334	4	no	—
47	C₄×C₈	Dic₁₆₇	1	2	yes	—
48	C₄×C₈	Dic₁₆₇	1	2	yes	—
49	C₂²×C₄	Dic₃₃₄	1	2	yes	—
50	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
51	M₁₆	—	334	8	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	334	4	no	—
54	C₂×C₈	C₂×Dic₁₆₇	1	4	yes	—
55	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
56	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
57	C₄²	D₆₆₈	1	2	yes	—
58	C₂³	C₂×Dic₃₃₄	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	C₄⋊Dic₁₆₇	1	2	yes	—
61	C₂×C₄	C₄⋊Dic₁₆₇	1	2	yes	—
62	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
63	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
64	C₂×C₄	#14581950	1	4	yes	—
65	C₈	—	2	4	no	—
66	C₈	—	668	4	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₄⋊Dic₁₆₇	1	1	yes	characteristic
69	C₂²	#19128824	1	2	yes	—
70	C₄	—	2	2	no	—
71	C₄	#19128768	1	4	yes	—
72	C₄	#19128791	1	2	yes	—
73	C₄	#19128820	1	2	yes	—
74	C₂	—	2	2	no	—
75	C₂	#24923135	1	1	yes	characteristic
76	C₂	C₂×#19128768	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₈
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	C₂×C₄×C₈×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₆₇
hypocenter	C₁₆₇	#1837
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
38. C₁₆₇	#1837	semidirect

Maximal subgroups

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

subgroup	index
#24923293	2
#24923034	2
C₂×C₄×C₈×C₁₆₇	2
#1837	167