Subgroups — #5700

Order 192.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#896	C₂	1	4	yes	maximal, complemented
3	#895	C₂	1	4	yes	maximal, complemented
4	#894	C₂	1	4	yes	maximal, complemented
5	#875	C₂	1	2	yes	maximal, complemented
6	#562	—	3	1	no	maximal, Sylow
7	C₂×D₁₂	C₂²	1	2	yes	—
8	C₂×D₁₂	C₂²	1	4	yes	—
9	C₂×C₄×D₃	C₂²	1	4	yes	—
10	C₂×Dic₆	C₂²	1	2	yes	—
11	C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₄²	C₂²	1	1	yes	characteristic
14	(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
15	(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
16	(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
17	Dic₃⋊C₄	C₂²	1	4	yes	—
18	#125	D₃	1	1	yes	characteristic, complemented
19	#124	—	3	2	no	—
20	#123	—	3	4	no	—
21	C₂²⋊Q₈	—	3	4	no	—
22	#121	—	3	4	no	—
23	C₂²×D₃	C₂³	1	4	yes	—
24	C₂×C₃×C₄	C₂³	1	4	yes	—
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂³	1	2	yes	—
27	C₂×Dic₃	C₂³	1	4	yes	—
28	D₁₂	—	4	2	no	—
29	D₁₂	—	4	4	no	—
30	C₄×D₃	—	4	4	no	—
31	Dic₆	—	4	2	no	—
32	C₂×Q₈	—	3	2	no	—
33	C₂×D₄	—	3	4	no	—
34	C₂×D₄	—	3	2	no	—
35	C₂²×C₄	—	3	4	no	—
36	C₄⋊C₄	—	3	4	no	—
37	C₄⋊C₄	C₂×D₃	1	4	yes	—
38	C₄⋊C₄	C₂×D₃	1	2	yes	—
39	C₂²⋊C₄	—	3	4	no	—
40	C₂²⋊C₄	—	3	4	no	—
41	C₂²⋊C₄	—	3	4	no	—
42	C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃	C₂⁴	1	1	yes	characteristic
44	C₂×D₃	—	2	4	no	—
45	C₂×D₃	—	2	4	no	—
46	C₂×D₃	—	2	4	no	—
47	C₃×C₄	—	2	2	no	—
48	C₃×C₄	—	2	4	no	—
49	C₃×C₄	—	2	1	no	—
50	Dic₃	—	2	4	no	—
51	C₂³	—	3	4	no	—
52	Q₈	—	12	2	no	—
53	D₄	—	12	4	no	—
54	D₄	—	12	2	no	—
55	C₂×C₄	—	3	4	no	—
56	C₂×C₄	—	12	4	no	—
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₂×C₄	C₂²×D₃	1	2	yes	—
59	C₂×C₄	C₂²×D₃	1	4	yes	—
60	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
61	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
62	D₃	—	4	4	no	—
63	C₂²	—	6	4	no	—
64	C₂²	—	6	4	no	—
65	C₂²	—	6	4	no	—
66	C₂²	C₂³×D₃	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	6	4	no	—
71	C₃	#562	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	12	4	no	—
73	C₂	#1007	1	1	yes	characteristic
74	C₂	#1009	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#125	C₂
hypercenter	#125	D₃
hypocenter	C₃	#562
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃×#125	C₂	semidirect
2. #896	C₂	semidirect
3. #895	C₂	semidirect
4. #894	C₂	semidirect
5. #875	C₂	semidirect
18. #125	D₃	semidirect
71. C₃	#562	semidirect

Maximal subgroups

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

subgroup	index
C₃×#125	2
#896	2
#895	2
#894	2
#875	2
#562	3