Subgroups — #5717

Order 192.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₄	C₂	1	4	yes	maximal, complemented
2	C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#938	C₂	1	4	yes	maximal, complemented
4	#937	C₂	1	4	yes	maximal, complemented
5	C₄×D₁₂	C₂	1	2	yes	maximal, complemented
6	D₄²	—	3	1	no	maximal, Sylow
7	C₂³×D₃	C₂²	1	4	yes	complemented
8	C₂×C₃×D₄	C₂²	1	4	yes	complemented
9	C₂×C₃×D₄	C₂²	1	2	yes	complemented
10	C₂×#67	C₂²	1	8	yes	complemented
11	D₃×D₄	—	2	16	no	—
12	C₂×D₁₂	C₂²	1	2	yes	—
13	C₂×C₄×D₃	C₂²	1	4	yes	complemented
14	C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
15	#217	C₂²	1	4	yes	complemented
16	(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
17	C₄⋊Dic₃	C₂²	1	2	yes	complemented
18	C₂²×D₄	—	3	4	no	—
19	#127	D₃	1	1	yes	characteristic, complemented
20	#121	—	3	4	no	—
21	#120	—	3	4	no	—
22	C₄×D₄	—	3	2	no	—
23	C₂³×C₃	C₂³	1	4	yes	—
24	C₂²×D₃	—	2	16	no	—
25	C₂²×D₃	—	4	4	no	—
26	C₂²×D₃	C₂³	1	4	yes	—
27	C₃×D₄	—	2	8	no	—
28	C₃×D₄	—	4	2	no	—
29	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂³	1	2	yes	—
31	#67	—	2	16	no	—
32	C₂×Dic₃	C₂³	1	4	yes	—
33	D₁₂	D₄	1	8	yes	complemented
34	C₄×D₃	—	2	8	no	—
35	C₂⁴	—	3	4	no	—
36	C₂×D₄	—	3	8	no	—
37	C₂×D₄	—	3	2	no	—
38	C₂×D₄	—	6	16	no	—
39	C₂×D₄	C₂×D₃	1	4	yes	complemented
40	C₂×D₄	C₂×D₃	1	2	yes	complemented
41	C₂²×C₄	—	3	4	no	—
42	C₄⋊C₄	—	3	2	no	—
43	C₂²⋊C₄	—	3	4	no	—
44	C₂²⋊C₄	—	3	4	no	—
45	C₄²	C₂×D₃	1	1	yes	characteristic, complemented
46	C₂²×C₃	—	2	8	no	—
47	C₂²×C₃	—	2	4	no	—
48	C₂²×C₃	C₂⁴	1	1	yes	characteristic
49	C₂×D₃	—	2	4	no	—
50	C₂×D₃	—	2	4	no	—
51	C₂×D₃	—	4	16	no	—
52	C₂×D₃	C₂×D₄	1	8	yes	complemented
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	C₂×D₄	1	4	yes	complemented
55	Dic₃	—	2	4	no	—
56	C₂³	—	3	4	no	—
57	C₂³	—	6	16	no	—
58	C₂³	—	12	4	no	—
59	C₂³	C₂²×D₃	1	4	yes	—
60	D₄	—	2	8	no	—
61	D₄	—	3	8	no	—
62	D₄	—	4	2	no	—
63	D₄	—	6	16	no	—
64	C₂×C₄	—	3	4	no	—
65	C₂×C₄	—	6	8	no	—
66	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₃	1	2	yes	—
68	C₂×C₃	—	4	4	no	—
69	C₂×C₃	C₂²×D₄	1	2	yes	—
70	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
71	D₃	—	2	8	no	—
72	C₂²	—	2	8	no	—
73	C₂²	—	2	4	no	—
74	C₂²	—	3	8	no	—
75	C₂²	—	6	4	no	—
76	C₂²	—	6	4	no	—
77	C₂²	—	12	16	no	—
78	C₂²	C₂³×D₃	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	6	4	no	—
81	C₄	D₃×D₄	1	4	yes	complemented
82	C₃	D₄²	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	4	4	no	—
84	C₂	—	6	8	no	—
85	C₂	C₂×D₃×D₄	1	2	yes	—
86	C₂	#1004	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#127	C₂
hypercenter	#127	D₃
hypocenter	C₃	D₄²
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. C₂×D₃×D₄	C₂	semidirect
2. C₃×#127	C₂	semidirect
3. #938	C₂	semidirect
4. #937	C₂	semidirect
5. C₄×D₁₂	C₂	semidirect
7. C₂³×D₃	C₂²	semidirect
8. C₂×C₃×D₄	C₂²	semidirect
9. C₂×C₃×D₄	C₂²	semidirect
10. C₂×#67	C₂²	semidirect
13. C₂×C₄×D₃	C₂²	semidirect
14. C₃×C₄²	C₂²	semidirect
15. #217	C₂²	semidirect
16. (C₂×D₃)⋊C₄	C₂²	semidirect
17. C₄⋊Dic₃	C₂²	semidirect
19. #127	D₃	semidirect
33. D₁₂	D₄	semidirect
39. C₂×D₄	C₂×D₃	semidirect
40. C₂×D₄	C₂×D₃	semidirect
45. C₄²	C₂×D₃	semidirect
52. C₂×D₃	C₂×D₄	semidirect
54. C₃×C₄	C₂×D₄	semidirect
81. C₄	D₃×D₄	semidirect
82. C₃	D₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₄	2
C₃×#127	2
#938	2
#937	2
C₄×D₁₂	2
D₄²	3