Subgroups — #5827

Order 192.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃	C₂	1	2	yes	maximal, complemented
3	D₄×Dic₃	C₂	1	6	yes	maximal, complemented
4	#926	C₂	1	3	yes	maximal, complemented
5	C₂×C₄⋊Dic₃	C₂	1	3	yes	maximal, complemented
6	#520	—	3	1	no	maximal, Sylow
7	C₃×#41	C₄	1	8	yes	complemented
8	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄	C₂²	1	3	yes	—
10	C₂²×C₃×C₄	C₂²	1	3	yes	—
11	C₂²×Dic₃	C₂²	1	6	yes	—
12	#217	C₂²	1	6	yes	—
13	C₄⋊Dic₃	C₂²	1	3	yes	—
14	C₄⋊Dic₃	C₂²	1	6	yes	—
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₄×Dic₃	C₂²	1	6	yes	—
17	C₂×#41	D₃	1	1	yes	characteristic
18	C₄×Q₈	—	3	2	no	—
19	C₄×D₄	—	3	6	no	—
20	#117	—	3	3	no	—
21	C₂×C₄⋊C₄	—	3	3	no	—
22	C₂³×C₃	C₂³	1	3	yes	—
23	C₃×Q₈	C₂×C₄	1	4	yes	complemented
24	C₃×D₄	C₂×C₄	1	12	yes	complemented
25	C₂×C₃×C₄	C₂×C₄	1	12	yes	complemented
26	C₂×C₃×C₄	C₂³	1	3	yes	—
27	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×Dic₃	—	4	6	no	—
29	C₂×Dic₃	C₂³	1	2	yes	—
30	C₂×Dic₃	C₂³	1	6	yes	—
31	#41	Dic₃	1	8	yes	complemented
32	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
33	C₂×D₄	C₂×D₃	1	3	yes	—
34	C₂²×C₄	—	3	6	no	—
35	C₂²×C₄	C₂×D₃	1	3	yes	—
36	C₄⋊C₄	—	3	3	no	—
37	C₄⋊C₄	—	3	6	no	—
38	C₄⋊C₄	—	3	1	no	—
39	C₂²⋊C₄	—	3	6	no	—
40	C₄²	—	3	6	no	—
41	C₂²×C₃	—	2	3	no	—
42	C₂²×C₃	—	2	3	no	—
43	C₂²×C₃	C₂²×C₄	1	6	yes	—
44	C₂²×C₃	C₂⁴	1	1	yes	characteristic
45	C₃×C₄	C₂²×C₄	1	2	yes	—
46	C₃×C₄	C₂²×C₄	1	6	yes	—
47	Dic₃	—	2	6	no	—
48	Dic₃	—	2	2	no	—
49	C₂³	C₂²×D₃	1	3	yes	—
50	Q₈	C₂×Dic₃	1	4	yes	complemented
51	D₄	C₂×Dic₃	1	12	yes	complemented
52	C₂×C₄	—	3	2	no	—
53	C₂×C₄	—	3	6	no	—
54	C₂×C₄	—	12	6	no	—
55	C₂×C₄	C₂×Dic₃	1	12	yes	complemented
56	C₂×C₄	C₂²×D₃	1	3	yes	—
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₂×C₃	—	2	6	no	—
59	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
60	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂²	—	2	3	no	—
63	C₂²	—	2	3	no	—
64	C₂²	C₂²×Dic₃	1	6	yes	—
65	C₂²	C₂³×D₃	1	1	yes	characteristic
66	C₄	—	6	6	no	—
67	C₄	—	6	2	no	—
68	C₄	C₂²×Dic₃	1	2	yes	—
69	C₄	C₂²×Dic₃	1	6	yes	—
70	C₃	#520	1	1	yes	characteristic, complemented, Sylow
71	C₂	—	2	6	no	—
72	C₂	#1009	1	1	yes	characteristic
73	C₂	#1010	1	1	yes	characteristic
74	C₂	C₂³×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄
Fitting, prime radical	C₂×C₃×#41	C₂
socle, cosocle	C₂²×C₃	C₂⁴
hypercenter	C₂×#41	D₃
hypocenter	C₃	#520
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₃	C₂	semidirect
3. D₄×Dic₃	C₂	semidirect
4. #926	C₂	semidirect
5. C₂×C₄⋊Dic₃	C₂	semidirect
7. C₃×#41	C₄	semidirect
23. C₃×Q₈	C₂×C₄	semidirect
24. C₃×D₄	C₂×C₄	semidirect
25. C₂×C₃×C₄	C₂×C₄	semidirect
31. #41	Dic₃	semidirect
50. Q₈	C₂×Dic₃	semidirect
51. D₄	C₂×Dic₃	semidirect
55. C₂×C₄	C₂×Dic₃	semidirect
70. C₃	#520	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#41	2
Q₈×Dic₃	2
D₄×Dic₃	2
#926	2
C₂×C₄⋊Dic₃	2
#520	3