Subgroups — #5720

Order 192.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#939	C₂	1	6	yes	maximal, complemented
3	#937	C₂	1	3	yes	maximal, complemented
4	#935	C₂	1	3	yes	maximal, complemented
5	#876	C₂	1	2	yes	maximal, complemented
6	#560	—	3	1	no	maximal, Sylow
7	C₂³×D₃	C₂²	1	1	yes	characteristic
8	C₂×C₃×D₄	C₂²	1	3	yes	—
9	C₂×C₃×D₄	C₂²	1	3	yes	complemented
10	C₂×#67	C₂²	1	6	yes	—
11	C₂²×Dic₃	C₂²	1	3	yes	complemented
12	C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
13	#217	C₂²	1	3	yes	—
14	#217	C₂²	1	3	yes	complemented
15	(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
16	Dic₃⋊C₄	C₂²	1	6	yes	complemented
17	#127	D₃	1	1	yes	characteristic, complemented
18	#126	—	3	2	no	—
19	#123	—	3	3	no	—
20	#121	—	3	6	no	—
21	#120	—	3	3	no	—
22	C₂³×C₃	C₂³	1	1	yes	characteristic
23	C₂³×C₃	C₂³	1	3	yes	—
24	C₂²×D₃	—	4	3	no	—
25	C₂²×D₃	C₂³	1	2	yes	—
26	C₃×D₄	—	4	3	no	—
27	C₃×D₄	—	4	3	no	—
28	C₂×C₃×C₄	C₂³	1	3	yes	—
29	#67	—	4	6	no	—
30	C₂×Dic₃	—	4	3	no	—
31	C₂×Dic₃	C₂³	1	6	yes	—
32	C₂⁴	—	3	1	no	—
33	C₂×D₄	—	3	6	no	—
34	C₂×D₄	C₂×D₃	1	3	yes	—
35	C₂×D₄	C₂×D₃	1	3	yes	complemented
36	C₂²×C₄	—	3	3	no	—
37	C₄⋊C₄	—	3	6	no	—
38	C₂²⋊C₄	—	3	3	no	—
39	C₂²⋊C₄	—	3	3	no	—
40	C₂²⋊C₄	—	3	6	no	—
41	C₄²	C₂×D₃	1	1	yes	characteristic, complemented
42	C₂²×C₃	—	2	3	no	—
43	C₂²×C₃	—	2	6	no	—
44	C₂²×C₃	—	2	3	no	—
45	C₂²×C₃	C₂⁴	1	1	yes	characteristic
46	C₂×D₃	—	2	6	no	—
47	C₂×D₃	—	4	4	no	—
48	C₃×C₄	—	2	3	no	—
49	Dic₃	—	2	6	no	—
50	C₂³	—	3	2	no	—
51	C₂³	—	12	3	no	—
52	C₂³	C₂²×D₃	1	1	yes	characteristic
53	C₂³	C₂²×D₃	1	3	yes	—
54	D₄	—	4	3	no	—
55	D₄	—	4	3	no	—
56	D₄	—	12	6	no	—
57	C₂×C₄	—	3	6	no	—
58	C₂×C₄	—	12	3	no	—
59	C₂×C₄	C₂²×D₃	1	3	yes	—
60	C₂×C₃	—	4	1	no	—
61	C₂×C₃	—	4	3	no	—
62	C₂×C₃	H(5,𝔽₂)	1	3	yes	—
63	D₃	—	4	2	no	—
64	C₂²	—	2	3	no	—
65	C₂²	—	2	6	no	—
66	C₂²	—	2	3	no	—
67	C₂²	—	6	6	no	—
68	C₂²	—	12	4	no	—
69	C₂²	C₂³×D₃	1	1	yes	characteristic
70	C₄	—	2	3	no	—
71	C₄	—	6	6	no	—
72	C₃	#560	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	12	2	no	—
76	C₂	#1004	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#127	C₂
hypercenter	#127	D₃
hypocenter	C₃	#560
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₃×#127	C₂	semidirect
2. #939	C₂	semidirect
3. #937	C₂	semidirect
4. #935	C₂	semidirect
5. #876	C₂	semidirect
9. C₂×C₃×D₄	C₂²	semidirect
11. C₂²×Dic₃	C₂²	semidirect
12. C₃×C₄²	C₂²	semidirect
14. #217	C₂²	semidirect
16. Dic₃⋊C₄	C₂²	semidirect
17. #127	D₃	semidirect
35. C₂×D₄	C₂×D₃	semidirect
41. C₄²	C₂×D₃	semidirect
72. C₃	#560	semidirect

Maximal subgroups

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

subgroup	index
C₃×#127	2
#939	2
#937	2
#935	2
#876	2
#560	3