Subgroups — #5594

Order 192.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	#953	C₂	1	2	yes	maximal, complemented
3	#937	C₂	1	3	yes	maximal, complemented
4	#936	C₂	1	3	yes	maximal, complemented
5	#885	C₂	1	6	yes	maximal, complemented
6	H(3,𝔽₄)	—	3	1	no	maximal, Sylow
7	C₂⁴×C₃	C₂²	1	1	yes	characteristic, complemented
8	C₂³×D₃	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₃×D₄	C₂²	1	3	yes	—
10	C₂×#67	C₂²	1	6	yes	—
11	C₂×Dic₆	C₂²	1	3	yes	complemented
12	C₃×C₂²⋊C₄	C₂²	1	3	yes	complemented
13	#217	C₂²	1	6	yes	complemented
14	#217	C₂²	1	3	yes	complemented
15	(C₂×D₃)⋊C₄	C₂²	1	6	yes	complemented
16	C₄×Dic₃	C₂²	1	3	yes	complemented
17	#124	—	3	3	no	—
18	#124	—	3	6	no	—
19	#120	—	3	2	no	—
20	#120	—	3	3	no	—
21	#120	D₃	1	1	yes	characteristic, complemented
22	C₂³×C₃	—	4	3	no	—
23	C₂³×C₃	C₂³	1	1	yes	characteristic
24	C₂³×C₃	C₂³	1	3	yes	—
25	C₂²×D₃	—	4	3	no	—
26	C₂²×D₃	C₂³	1	2	yes	—
27	C₃×D₄	—	4	3	no	—
28	C₂×C₃×C₄	C₂³	1	3	yes	—
29	#67	—	4	6	no	—
30	C₂×Dic₃	C₂³	1	6	yes	—
31	Dic₆	—	4	3	no	—
32	C₂⁴	—	3	1	no	—
33	C₂⁴	C₂×D₃	1	1	yes	characteristic, complemented
34	C₂×Q₈	—	3	3	no	—
35	C₂×D₄	—	3	6	no	—
36	C₂×D₄	C₂×D₃	1	3	yes	—
37	C₂²⋊C₄	—	3	3	no	—
38	C₂²⋊C₄	—	3	6	no	—
39	C₂²⋊C₄	—	3	6	no	—
40	C₂²⋊C₄	C₂×D₃	1	3	yes	complemented
41	C₄²	—	3	3	no	—
42	C₂²×C₃	—	2	6	no	—
43	C₂²×C₃	—	2	3	no	—
44	C₂²×C₃	—	2	3	no	—
45	C₂²×C₃	—	4	4	no	—
46	C₂²×C₃	C₂⁴	1	1	yes	characteristic
47	C₂×D₃	—	2	6	no	—
48	C₂×D₃	—	4	4	no	—
49	C₃×C₄	—	2	3	no	—
50	Dic₃	—	2	6	no	—
51	C₂³	—	3	2	no	—
52	C₂³	—	4	3	no	—
53	C₂³	—	12	3	no	—
54	C₂³	C₂²×D₃	1	1	yes	characteristic
55	C₂³	C₂²×D₃	1	3	yes	—
56	Q₈	—	12	3	no	—
57	D₄	—	4	3	no	—
58	D₄	—	12	6	no	—
59	C₂×C₄	—	3	6	no	—
60	C₂×C₄	C₂²×D₃	1	3	yes	—
61	C₂×C₃	—	4	1	no	—
62	C₂×C₃	—	4	3	no	—
63	C₂×C₃	H(5,𝔽₂)	1	3	yes	—
64	D₃	—	4	2	no	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	6	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	6	6	no	—
70	C₂²	—	12	4	no	—
71	C₂²	C₂³×D₃	1	1	yes	characteristic
72	C₄	—	2	3	no	—
73	C₄	—	6	6	no	—
74	C₃	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	12	2	no	—
78	C₂	#1004	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#120	C₂
hypercenter	#120	D₃
hypocenter	C₃	H(3,𝔽₄)
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. C₃×#120	C₂	semidirect
2. #953	C₂	semidirect
3. #937	C₂	semidirect
4. #936	C₂	semidirect
5. #885	C₂	semidirect
7. C₂⁴×C₃	C₂²	semidirect
8. C₂³×D₃	C₂²	semidirect
11. C₂×Dic₆	C₂²	semidirect
12. C₃×C₂²⋊C₄	C₂²	semidirect
13. #217	C₂²	semidirect
14. #217	C₂²	semidirect
15. (C₂×D₃)⋊C₄	C₂²	semidirect
16. C₄×Dic₃	C₂²	semidirect
21. #120	D₃	semidirect
33. C₂⁴	C₂×D₃	semidirect
40. C₂²⋊C₄	C₂×D₃	semidirect
74. C₃	H(3,𝔽₄)	semidirect

Maximal subgroups

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

subgroup	index
C₃×#120	2
#953	2
#937	2
#936	2
#885	2
H(3,𝔽₄)	3