Subgroups — #5077

Order 192.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1004	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#932	C₂	1	2	yes	maximal, complemented
4	#833	C₂	1	1	yes	maximal, characteristic, complemented
5	#805	C₂	1	2	yes	maximal, complemented
6	#453	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	—	2	2	no	—
8	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄	C₂²	1	1	yes	characteristic, complemented
10	C₂×#67	—	2	2	no	—
11	#237	—	2	2	no	—
12	D₃×D₄	—	2	2	no	—
13	#235	C₂²	1	2	yes	—
14	C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
15	#214	—	2	2	no	—
16	#213	—	2	2	no	—
17	#208	C₂²	1	2	yes	complemented
18	H(5,𝔽₂)	—	3	1	no	—
19	AGL(1,ℤ/8)	—	3	2	no	—
20	#127	D₃	1	1	yes	characteristic, complemented
21	#104	—	3	2	no	—
22	#100	—	3	1	no	—
23	C₂³×C₃	—	2	1	no	—
24	C₂³×C₃	D₄	1	2	yes	—
25	C₂²×D₃	—	2	2	no	—
26	C₃×D₄	—	2	2	no	—
27	C₃×D₄	—	2	2	no	—
28	C₃×D₄	—	4	4	no	—
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
31	#67	—	2	4	no	—
32	#67	—	2	2	no	—
33	C₂×Dic₃	—	2	2	no	—
34	D₁₂	D₄	1	2	yes	complemented
35	C₄×D₃	—	2	2	no	—
36	Dic₆	D₄	1	2	yes	complemented
37	C₃⋊C₈	—	2	2	no	—
38	#41	—	3	2	no	—
39	#41	—	6	2	no	—
40	C₂×D₄	—	2	2	no	—
41	C₂×D₄	—	6	2	no	—
42	C₂×D₄	—	6	2	no	—
43	C₂×D₄	C₂×D₃	1	1	yes	characteristic
44	C₂×D₄	C₂×D₃	1	1	yes	characteristic, complemented
45	SD₁₆	—	6	2	no	—
46	D₈	—	6	2	no	—
47	M₁₆	—	3	2	no	—
48	C₄²	C₂×D₃	1	1	yes	characteristic, complemented
49	C₂²×C₃	—	2	2	no	—
50	C₂²×C₃	—	4	2	no	—
51	C₂²×C₃	—	4	2	no	—
52	C₂²×C₃	—	4	1	no	—
53	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
54	C₂×D₃	—	2	2	no	—
55	C₂×D₃	—	8	2	no	—
56	C₃×C₄	—	2	2	no	—
57	C₃×C₄	C₂×D₄	1	2	yes	—
58	Dic₃	—	2	2	no	—
59	C₂³	—	2	1	no	—
60	C₂³	—	6	2	no	—
61	C₂³	#67	1	2	yes	—
62	Q₈	—	3	2	no	—
63	D₄	—	2	2	no	—
64	D₄	—	2	2	no	—
65	D₄	—	3	2	no	—
66	D₄	—	4	4	no	—
67	D₄	—	6	4	no	—
68	D₄	—	6	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	6	2	no	—
71	C₂×C₄	—	6	2	no	—
72	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₈	—	6	2	no	—
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	—	4	2	no	—
76	C₂×C₃	—	8	1	no	—
77	C₂×C₃	#120	1	1	yes	characteristic
78	D₃	—	4	2	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	4	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	4	1	no	—
83	C₂²	—	6	2	no	—
84	C₂²	—	24	2	no	—
85	C₂²	C₂×#67	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	6	2	no	—
88	C₄	D₃×D₄	1	2	yes	—
89	C₃	#453	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	8	1	no	—
93	C₂	—	12	2	no	—
94	C₂	#937	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#937
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#127	C₂
socle	C₂×C₃	#120
hypercenter	#127	D₃
hypocenter	C₃	#453
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #127

Complements

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

N	H ≅ G/N	type
1. #1004	C₂	semidirect
2. C₃×#127	C₂	semidirect
3. #932	C₂	semidirect
4. #833	C₂	semidirect
5. #805	C₂	semidirect
9. C₂×C₃×D₄	C₂²	semidirect
14. C₃×C₄²	C₂²	semidirect
17. #208	C₂²	semidirect
20. #127	D₃	semidirect
34. D₁₂	D₄	semidirect
36. Dic₆	D₄	semidirect
44. C₂×D₄	C₂×D₃	semidirect
48. C₄²	C₂×D₃	semidirect
89. C₃	#453	semidirect

Maximal subgroups

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

subgroup	index
#1004	2
C₃×#127	2
#932	2
#833	2
#805	2
#453	3