Subgroups — #36326385

Order 21376.

132 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	#24923045	C₂	1	1	yes	maximal, characteristic
4	#499×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	#125×C₁₆₇	—	2	1	no	—
10	#107×C₁₆₇	C₂²	1	2	yes	—
11	#106×C₁₆₇	—	2	1	no	—
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	2	no	—
17	C₁₆₇⋊C₁₆	—	2	2	no	—
18	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₆₇	—	2	2	no	—
21	C₄⋊C₄×C₁₆₇	—	4	2	no	—
22	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
23	Dic₃₃₄	—	4	2	no	—
24	Dic₃₃₄	—	4	2	no	—
25	C₂×Dic₁₆₇	—	4	2	no	—
26	C₂×C₄×C₁₆₇	—	4	2	no	—
27	C₂×C₄×C₁₆₇	D₈	1	2	yes	—
28	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₆₇	SD₁₆	1	2	yes	—
30	C₈×C₁₆₇	#41	1	2	yes	—
31	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₆₇	—	2	2	no	—
33	C₄×C₁₆₇	—	8	2	no	—
34	C₄×C₁₆₇	#124	1	1	yes	characteristic
35	C₄×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
36	Dic₁₆₇	—	8	2	no	—
37	C₂×C₁₆₇	#486	1	1	yes	characteristic
38	C₂×C₁₆₇	#510	1	2	yes	—
39	C₁₆₇	#2245	1	1	yes	characteristic, complemented, Sylow
40	#2245	—	167	1	no	maximal, Sylow
41	#499	D₁₆₇	1	1	yes	characteristic
42	#494	—	167	1	no	—
43	#358	—	167	4	no	—
44	#346	—	167	1	no	—
45	C₂×Q₁₆	—	334	1	no	—
46	C₂×Q₁₆	—	167	2	no	—
47	C₄⋊Q₈	—	334	1	no	—
48	#125	—	2	1	no	—
49	C₂×C₁₆	—	167	2	no	—
50	#107	C₂×D₁₆₇	1	2	yes	—
51	#106	—	2	1	no	—
52	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
53	C₂×Q₈	—	334	2	no	—
54	Q₁₆	—	334	4	no	—
55	Q₁₆	—	668	2	no	—
56	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	668	2	no	—
61	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
62	C₁₆	—	334	2	no	—
63	Q₈	—	668	2	no	—
64	Q₈	—	668	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	668	2	no	—
67	C₂×C₄	#14581935	1	2	yes	—
68	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
69	C₈	#14581937	1	2	yes	—
70	C₈	#14581951	1	2	yes	—
71	C₂²	C₂×#14581935	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	1336	2	no	—
75	C₄	#19128821	1	1	yes	characteristic
76	C₄	C₂×#14581937	1	1	yes	characteristic
77	C₂	#24923617	1	1	yes	characteristic
78	C₂	#24923709	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14581935
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	#499×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₈
hypercenter	#499	D₁₆₇
hypocenter	C₁₆₇	#2245
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
39. C₁₆₇	#2245	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
#24923045	2
#499×C₁₆₇	2
#2245	167