Subgroups — #36326390

Order 21376.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128778	C₂	1	4	yes	maximal
2	#24923649	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#494×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₆₈	—	2	1	no	—
6	C₂×Dic₆₆₈	C₂²	1	2	yes	—
7	#19128832	—	2	1	no	—
8	#19128778	—	2	8	no	—
9	C₂×C₁₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₁₆₇	—	2	1	no	—
11	C₂×Q₁₆×C₁₆₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₆₇	—	2	1	no	—
13	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₃₄	—	2	2	no	—
15	Dic₆₆₈	—	2	4	no	—
16	Dic₆₆₈	—	4	2	no	—
17	C₄⋊Dic₁₆₇	—	4	2	no	—
18	C₁₆₇⋊C₁₆	D₄	1	4	yes	—
19	C₂×Q₈×C₁₆₇	—	2	2	no	—
20	Q₁₆×C₁₆₇	—	2	4	no	—
21	Q₁₆×C₁₆₇	—	4	2	no	—
22	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₆₇	—	4	2	no	—
25	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
26	Dic₃₃₄	—	4	2	no	—
27	Dic₃₃₄	—	4	2	no	—
28	C₂×Dic₁₆₇	—	4	2	no	—
29	Q₈×C₁₆₇	—	4	2	no	—
30	Q₈×C₁₆₇	—	4	2	no	—
31	C₂×C₄×C₁₆₇	—	4	2	no	—
32	C₂×C₄×C₁₆₇	D₈	1	2	yes	—
33	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
34	C₈×C₁₆₇	D₈	1	2	yes	—
35	C₈×C₁₆₇	C₂×D₄	1	2	yes	—
36	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
37	C₄×C₁₆₇	—	8	2	no	—
38	C₄×C₁₆₇	Q₃₂	1	4	yes	complemented
39	C₄×C₁₆₇	#127	1	1	yes	characteristic
40	C₄×C₁₆₇	C₂×D₈	1	1	yes	characteristic
41	Dic₁₆₇	—	8	2	no	—
42	C₂×C₁₆₇	#493	1	1	yes	characteristic
43	C₂×C₁₆₇	C₂×Q₃₂	1	2	yes	—
44	C₁₆₇	#2247	1	1	yes	characteristic, complemented, Sylow
45	#2247	—	167	1	no	maximal, Sylow
46	C₂×Q₃₂	—	167	4	no	—
47	#494	—	167	1	no	—
48	#494	D₁₆₇	1	1	yes	characteristic
49	C₄×C₁₆	—	167	1	no	—
50	C₂×Q₁₆	—	2	1	no	—
51	C₂×Q₁₆	—	334	1	no	—
52	C₂×Q₁₆	—	167	2	no	—
53	C₂×Q₁₆	C₂×D₁₆₇	1	2	yes	—
54	C₄⋊Q₈	—	2	1	no	—
55	C₄⋊Q₈	—	334	1	no	—
56	Q₃₂	—	334	8	no	—
57	C₂×C₁₆	—	167	2	no	—
58	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	334	2	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	2	no	—
63	Q₁₆	—	334	4	no	—
64	Q₁₆	—	668	2	no	—
65	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	—	668	2	no	—
69	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
70	C₁₆	—	167	4	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	668	2	no	—
74	Q₈	—	668	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	668	2	no	—
77	C₂×C₄	#14581935	1	2	yes	—
78	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
79	C₈	#14581935	1	2	yes	—
80	C₈	D₄×D₁₆₇	1	2	yes	—
81	C₂²	C₂×#14581935	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1336	2	no	—
84	C₄	#19128778	1	4	yes	complemented
85	C₄	#19128828	1	1	yes	characteristic
86	C₄	C₂×#14581935	1	1	yes	characteristic
87	C₂	#24923645	1	1	yes	characteristic
88	C₂	C₂×#19128778	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14581935
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	#494×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₈
hypercenter	#494	D₁₆₇
hypocenter	C₁₆₇	#2247
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
38. C₄×C₁₆₇	Q₃₂	semidirect
44. C₁₆₇	#2247	semidirect
84. C₄	#19128778	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128778	2
#24923649	2
C₄×C₁₆₇⋊C₁₆	2
#494×C₁₆₇	2
#2247	167