Subgroups — #36326367

Order 21376.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923649	C₂	1	1	yes	maximal, characteristic
2	#24923072	C₂	1	4	yes	maximal
3	C₄×C₁₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×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	C₂×C₁₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₆₇	—	2	1	no	—
10	#107×C₁₆₇	—	2	1	no	—
11	#107×C₁₆₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₃₄	—	2	2	no	—
14	Dic₆₆₈	—	2	4	no	—
15	Dic₆₆₈	—	4	2	no	—
16	C₄⋊Dic₁₆₇	—	4	1	no	—
17	C₄⋊Dic₁₆₇	—	4	1	no	—
18	C₁₆₇⋊C₁₆	—	2	2	no	—
19	C₂×Q₈×C₁₆₇	—	4	1	no	—
20	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₆₇	—	2	2	no	—
23	C₄⋊C₄×C₁₆₇	—	4	1	no	—
24	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
25	Dic₃₃₄	—	4	2	no	—
26	Dic₃₃₄	—	4	2	no	—
27	C₂×Dic₁₆₇	—	4	2	no	—
28	Q₈×C₁₆₇	—	8	2	no	—
29	C₂×C₄×C₁₆₇	—	4	2	no	—
30	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
31	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₆₇	SD₁₆	1	2	yes	—
34	C₈×C₁₆₇	#41	1	2	yes	—
35	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₆₇	—	8	2	no	—
37	C₄×C₁₆₇	SD₃₂	1	2	yes	—
38	C₄×C₁₆₇	Q₃₂	1	2	yes	—
39	C₄×C₁₆₇	#124	1	1	yes	characteristic
40	C₄×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₁₆₇	—	8	2	no	—
42	C₂×C₁₆₇	#486	1	1	yes	characteristic
43	C₂×C₁₆₇	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₁₆₇	C₂×Q₃₂	1	1	yes	characteristic
45	C₁₆₇	#2241	1	1	yes	characteristic, complemented, Sylow
46	#2241	—	167	1	no	maximal, Sylow
47	#500	D₁₆₇	1	1	yes	characteristic
48	#494	—	167	1	no	—
49	#358	—	167	4	no	—
50	C₄×C₁₆	—	167	1	no	—
51	C₂×Q₁₆	—	334	1	no	—
52	C₂×Q₁₆	—	167	2	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	334	1	no	—
55	C₂×C₁₆	—	167	2	no	—
56	#107	—	2	1	no	—
57	#107	C₂×D₁₆₇	1	2	yes	—
58	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	334	2	no	—
61	Q₁₆	—	334	4	no	—
62	Q₁₆	—	668	2	no	—
63	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	668	1	no	—
68	C₄⋊C₄	—	668	1	no	—
69	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
70	C₁₆	—	334	2	no	—
71	Q₈	—	8	2	no	—
72	Q₈	—	668	2	no	—
73	Q₈	—	668	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	668	2	no	—
76	C₂×C₄	#14581935	1	1	yes	characteristic
77	C₂×C₄	#14581935	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
79	C₈	#14581937	1	2	yes	—
80	C₈	#14581951	1	2	yes	—
81	C₂²	C₂×#14581935	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1336	2	no	—
84	C₄	#19128776	1	2	yes	—
85	C₄	#19128778	1	2	yes	—
86	C₄	#19128821	1	1	yes	characteristic
87	C₄	C₂×#14581937	1	1	yes	characteristic
88	C₂	#24923617	1	1	yes	characteristic
89	C₂	C₂×#19128776	1	1	yes	characteristic
90	C₂	C₂×#19128778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14581935
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	#500×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₈
hypercenter	#500	D₁₆₇
hypocenter	C₁₆₇	#2241
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
45. C₁₆₇	#2241	semidirect

Maximal subgroups

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

subgroup	index
#24923649	2
#24923072	2
C₄×C₁₆₇⋊C₁₆	2
#500×C₁₆₇	2
#2241	167