Subgroups — #36326010

Order 21376.

148 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	C₄×Dic₆₆₈	C₂	1	2	yes	maximal
3	#24923097	C₂	1	1	yes	maximal, characteristic
4	#24923074	C₂	1	2	yes	maximal
5	#346×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₆₈	C₂²	1	2	yes	—
7	C₄×Dic₃₃₄	—	2	2	no	—
8	Dic₁₃₃₆	C₄	1	4	yes	complemented
9	#19128766	C₂²	1	2	yes	—
10	#19128757	—	2	2	no	—
11	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₃₄	—	2	2	no	—
15	Dic₆₆₈	—	2	2	no	—
16	Dic₆₆₈	C₂×C₄	1	4	yes	—
17	C₄⋊Dic₁₆₇	—	2	2	no	—
18	Dic₁₆₇⋊C₄	—	4	2	no	—
19	C₄×Dic₁₆₇	—	4	2	no	—
20	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
22	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₆₇	—	2	1	no	—
24	C₁₆×C₁₆₇	C₂×C₄	1	2	yes	—
25	Dic₃₃₄	—	2	4	no	—
26	Dic₃₃₄	—	4	2	no	—
27	C₂×Dic₁₆₇	—	4	2	no	—
28	C₂×Dic₁₆₇	—	4	2	no	—
29	C₂×C₄×C₁₆₇	D₈	1	2	yes	—
30	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
31	C₈×C₁₆₇	—	2	1	no	—
32	C₈×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₆₇	#41	1	1	yes	characteristic
34	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₆₇	—	2	2	no	—
36	C₄×C₁₆₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₆₇	#135	1	1	yes	characteristic
38	Dic₁₆₇	—	4	4	no	—
39	Dic₁₆₇	—	8	2	no	—
40	C₂×C₁₆₇	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₆₇	#510	1	2	yes	—
42	C₁₆₇	#2176	1	1	yes	characteristic, complemented, Sylow
43	#2176	—	167	1	no	maximal, Sylow
44	C₂×Q₃₂	—	167	1	no	—
45	C₄×Q₁₆	—	167	2	no	—
46	#367	—	167	1	no	—
47	#358	—	167	2	no	—
48	#346	D₁₆₇	1	1	yes	characteristic
49	C₂×Q₁₆	—	167	2	no	—
50	C₄×Q₈	—	334	2	no	—
51	Q₃₂	—	167	4	no	—
52	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
54	#107	—	167	2	no	—
55	#103	—	334	2	no	—
56	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
57	C₂×Q₈	—	334	2	no	—
58	Q₁₆	—	334	2	no	—
59	Q₁₆	—	167	4	no	—
60	C₂×C₈	D₆₆₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
62	C₄⋊C₄	—	334	2	no	—
63	C₄⋊C₄	—	668	2	no	—
64	C₄²	—	668	2	no	—
65	C₄²	D₆₆₈	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₁₆₇	1	2	yes	—
68	Q₈	—	334	4	no	—
69	Q₈	—	668	2	no	—
70	C₂×C₄	—	668	2	no	—
71	C₂×C₄	—	668	2	no	—
72	C₂×C₄	D₁₃₃₆	1	2	yes	—
73	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₁₆₇	1	1	yes	characteristic
76	C₈	#14581950	1	1	yes	characteristic
77	C₂²	C₂×D₁₃₃₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	1336	2	no	—
80	C₄	—	668	4	no	—
81	C₄	C₄×D₆₆₈	1	1	yes	characteristic
82	C₄	#19128854	1	1	yes	characteristic
83	C₂	C₄×D₁₃₃₆	1	1	yes	characteristic
84	C₂	#24923712	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₃₆
commutator	C₈×C₁₆₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	#346×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₈
cosocle	C₂×C₈×C₁₆₇	C₂³
hypercenter	#346	D₁₆₇
hypocenter	C₁₆₇	#2176
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₁₃₃₆	C₄	semidirect
42. C₁₆₇	#2176	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
C₄×Dic₆₆₈	2
#24923097	2
#24923074	2
#346×C₁₆₇	2
#2176	167