Subgroups — #36325898

Order 21376.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#24923224	C₂	1	1	yes	maximal, characteristic
3	Dic₁₆₇⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581934	C₄	1	1	yes	characteristic
6	#19128791	C₄	1	1	yes	characteristic
7	C₂×C₁₆₇⋊C₁₆	—	2	2	no	—
8	C₂×C₁₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#19128742	C₄	1	2	yes	—
10	C₈×Dic₁₆₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
12	M₃₂×C₁₆₇	—	2	2	no	—
13	C₂×C₁₆×C₁₆₇	C₂²	1	2	yes	—
14	#14581934	—	2	2	no	—
15	C₂×C₁₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₆₇	C₈	1	2	yes	—
17	C₂²⋊Dic₁₆₇	C₈	1	2	yes	—
18	C₄×Dic₁₆₇	C₂×C₄	1	2	yes	—
19	C₁₆₇⋊C₁₆	—	2	2	no	—
20	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
23	C₂×C₈×C₁₆₇	Q₈	1	2	yes	—
24	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
25	C₁₆×C₁₆₇	—	2	2	no	—
26	C₂×Dic₁₆₇	C₂×C₈	1	4	yes	—
27	C₁₆₇⋊C₈	—	2	2	no	—
28	C₂³×C₁₆₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₆₇	M₁₆	1	2	yes	—
32	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₆₇	C₄⋊C₄	1	2	yes	—
34	C₈×C₁₆₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₆₇	—	2	2	no	—
37	C₂²×C₁₆₇	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₆₇	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₆₇	C₄⋊C₈	1	2	yes	—
40	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₆₇	C₂×M₁₆	1	1	yes	characteristic
42	Dic₁₆₇	—	2	4	no	—
43	C₂×C₁₆₇	—	2	2	no	—
44	C₂×C₁₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₆₇	#504	1	2	yes	—
46	C₁₆₇	#2151	1	1	yes	characteristic, complemented, Sylow
47	#2151	—	167	1	no	maximal, Sylow
48	C₂×M₃₂	D₁₆₇	1	1	yes	characteristic
49	C₂²×C₁₆	—	167	1	no	—
50	#405	—	167	1	no	—
51	C₄⋊C₁₆	—	167	4	no	—
52	C₂×M₁₆	—	167	1	no	—
53	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
54	#117	—	167	1	no	—
55	M₃₂	—	2	2	no	—
56	C₂×C₁₆	—	334	2	no	—
57	C₂×C₁₆	—	167	2	no	—
58	C₂×C₁₆	C₂×D₁₆₇	1	2	yes	—
59	#97	—	167	2	no	—
60	C₄×C₈	—	167	2	no	—
61	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
62	M₁₆	—	334	2	no	—
63	C₂×C₈	—	167	2	no	—
64	C₂×C₈	C₄×D₁₆₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₆₇	1	2	yes	—
66	C₂×C₈	Dic₃₃₄	1	2	yes	—
67	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
68	C₄⋊C₄	—	167	2	no	—
69	C₂²⋊C₄	—	167	2	no	—
70	C₄²	—	167	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	334	2	no	—
73	C₂³	C₈×D₁₆₇	1	1	yes	characteristic
74	C₂×C₄	—	167	4	no	—
75	C₂×C₄	Dic₁₆₇⋊C₄	1	2	yes	—
76	C₂×C₄	C₈×D₁₆₇	1	1	yes	characteristic
77	C₂×C₄	#14581933	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
79	C₈	—	334	2	no	—
80	C₈	Dic₁₆₇⋊C₄	1	2	yes	—
81	C₈	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
82	C₈	C₂×Dic₃₃₄	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	Dic₁₆₇⋊C₈	1	2	yes	—
85	C₂²	C₂×C₈×D₁₆₇	1	1	yes	characteristic
86	C₄	—	334	4	no	—
87	C₄	Dic₁₆₇⋊C₈	1	2	yes	—
88	C₄	C₂×Dic₁₆₇⋊C₄	1	1	yes	characteristic
89	C₄	C₂×#14581933	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×Dic₁₆₇⋊C₈	1	1	yes	characteristic
92	C₂	#24923687	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₆₇
commutator, socle	C₂²×C₁₆₇	C₂²×C₈
Fitting	C₂×M₃₂×C₁₆₇	C₂
prime radical	C₂³×C₁₆₇	C₂×C₈
cosocle	C₂×C₈×C₁₆₇	C₂³
hypercenter	C₂×M₃₂	D₁₆₇
hypocenter	C₁₆₇	#2151
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. Dic₁₆₇⋊C₁₆	C₂	semidirect
46. C₁₆₇	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆₇⋊C₁₆	2
#24923224	2
Dic₁₆₇⋊C₁₆	2
C₂×M₃₂×C₁₆₇	2
#2151	167