Subgroups — #36326389

Order 21376.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19128778	C₂	1	2	yes	maximal
3	#24923647	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₆₈	C₂²	1	2	yes	—
7	C₂×#14581940	—	2	2	no	—
8	#19128830	—	2	2	no	—
9	Dic₁₃₃₆	—	2	2	no	—
10	#19128778	—	2	4	no	—
11	C₂×C₁₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₆₇	C₂²	1	2	yes	—
14	Q₃₂×C₁₆₇	—	2	2	no	—
15	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₃₄	—	2	2	no	—
17	Dic₆₆₈	—	2	2	no	—
18	Dic₆₆₈	—	2	2	no	—
19	#14581940	—	4	4	no	—
20	C₂×C₁₆₇⋊C₈	D₄	1	1	yes	characteristic
21	Dic₁₆₇⋊C₄	—	4	4	no	—
22	C₄×Dic₁₆₇	D₄	1	1	yes	characteristic
23	C₁₆₇⋊C₁₆	D₄	1	2	yes	—
24	C₂×Q₈×C₁₆₇	—	2	2	no	—
25	Q₁₆×C₁₆₇	—	2	2	no	—
26	Q₁₆×C₁₆₇	—	2	2	no	—
27	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
28	C₁₆×C₁₆₇	D₄	1	2	yes	—
29	Dic₃₃₄	—	4	2	no	—
30	Dic₃₃₄	—	4	2	no	—
31	C₂×Dic₁₆₇	—	4	2	no	—
32	C₂×Dic₁₆₇	D₈	1	2	yes	—
33	C₁₆₇⋊C₈	D₈	1	2	yes	—
34	Q₈×C₁₆₇	—	4	2	no	—
35	Q₈×C₁₆₇	—	4	2	no	—
36	C₂×C₄×C₁₆₇	—	4	2	no	—
37	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₆₇	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
41	C₄×C₁₆₇	—	8	2	no	—
42	C₄×C₁₆₇	#127	1	1	yes	characteristic
43	C₄×C₁₆₇	C₂×D₈	1	1	yes	characteristic
44	Dic₁₆₇	—	8	2	no	—
45	Dic₁₆₇	Q₃₂	1	4	yes	complemented
46	C₂×C₁₆₇	#493	1	1	yes	characteristic
47	C₂×C₁₆₇	C₂×Q₃₂	1	2	yes	—
48	C₁₆₇	#2247	1	1	yes	characteristic, complemented, Sylow
49	#2247	—	167	1	no	maximal, Sylow
50	C₂×Q₃₂	—	167	2	no	—
51	C₂×Q₃₂	—	167	1	no	—
52	C₂×Q₃₂	D₁₆₇	1	1	yes	characteristic
53	#494	—	167	2	no	—
54	C₄×C₁₆	—	167	1	no	—
55	C₂×Q₁₆	—	334	2	no	—
56	C₂×Q₁₆	—	167	2	no	—
57	C₂×Q₁₆	C₂×D₁₆₇	1	2	yes	—
58	C₄⋊Q₈	—	334	2	no	—
59	Q₃₂	—	2	2	no	—
60	Q₃₂	—	334	4	no	—
61	Q₃₂	—	334	2	no	—
62	C₂×C₁₆	—	167	1	no	—
63	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
64	C₄×C₈	—	167	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	334	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	334	2	no	—
70	Q₁₆	—	334	2	no	—
71	Q₁₆	—	668	4	no	—
72	C₂×C₈	—	167	1	no	—
73	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
74	C₄⋊C₄	—	668	4	no	—
75	C₄²	—	167	1	no	—
76	C₁₆	—	167	2	no	—
77	C₁₆	C₂²⋊D₁₆₇	1	2	yes	—
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	—	668	2	no	—
81	Q₈	—	668	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	668	2	no	—
84	C₂×C₄	—	167	2	no	—
85	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
86	C₈	—	167	2	no	—
87	C₈	D₄×D₁₆₇	1	1	yes	characteristic
88	C₈	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
89	C₂²	D₈×D₁₆₇	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	1336	2	no	—
92	C₄	—	167	4	no	—
93	C₄	#19128828	1	1	yes	characteristic
94	C₄	D₈×D₁₆₇	1	1	yes	characteristic
95	C₂	#24923644	1	1	yes	characteristic
96	C₂	Q₃₂×D₁₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₆₇
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	C₂×Q₃₂×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₈
hypercenter	C₂×Q₃₂	D₁₆₇
hypocenter	C₁₆₇	#2247
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
45. Dic₁₆₇	Q₃₂	semidirect
48. C₁₆₇	#2247	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
C₂×#19128778	2
#24923647	2
C₁₆×Dic₁₆₇	2
C₂×Q₃₂×C₁₆₇	2
#2247	167