Subgroups — #4468

Order 192.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#830	C₂	1	1	yes	maximal, characteristic, complemented
4	#323	—	3	1	no	maximal, Sylow
5	C₂³×D₃	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈	—	2	1	no	—
9	(C₂×D₃)⋊C₄	—	2	1	no	—
10	(C₂×D₃)⋊C₄	—	2	1	no	—
11	C₂×C₃⋊C₈	—	2	1	no	—
12	C₂×C₂²⋊C₄	—	3	1	no	—
13	C₂²⋊C₈	—	3	1	no	—
14	C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
15	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×D₃	—	2	1	no	—
17	C₂²×D₃	—	2	1	no	—
18	C₂²×D₃	—	4	1	no	—
19	C₂²×D₃	—	4	1	no	—
20	C₂²×D₃	C₈	1	2	yes	complemented
21	C₂×C₃×C₄	—	4	1	no	—
22	C₂×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×Dic₃	—	2	1	no	—
25	C₂×Dic₃	—	4	1	no	—
26	C₃×C₈	—	4	1	no	—
27	C₃⋊C₈	—	4	1	no	—
28	C₂⁴	—	3	1	no	—
29	C₂²×C₄	—	3	1	no	—
30	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	6	1	no	—
33	C₂²⋊C₄	—	6	1	no	—
34	C₂²⋊C₄	—	6	1	no	—
35	C₂²×C₃	—	2	1	no	—
36	C₂²×C₃	—	2	1	no	—
37	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₃	M₁₆	1	1	yes	characteristic
40	C₂×D₃	—	2	2	no	—
41	C₂×D₃	—	4	2	no	—
42	C₂×D₃	—	4	1	no	—
43	C₂×D₃	—	4	1	no	—
44	C₂×D₃	—	4	2	no	—
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	—	2	1	no	—
47	Dic₃	—	4	1	no	—
48	C₂³	—	3	2	no	—
49	C₂³	—	6	1	no	—
50	C₂³	—	6	1	no	—
51	C₂³	—	12	1	no	—
52	C₂³	—	12	1	no	—
53	C₂³	C₄×D₃	1	1	yes	characteristic
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	6	1	no	—
56	C₂×C₄	—	12	1	no	—
57	C₂×C₄	D₁₂	1	1	yes	characteristic
58	C₂×C₄	#67	1	1	yes	characteristic
59	C₈	—	4	1	no	—
60	C₈	—	12	1	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₃	#99	1	1	yes	characteristic
65	C₂×C₃	#100	1	1	yes	characteristic
66	D₃	—	4	2	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	6	2	no	—
70	C₂²	—	12	2	no	—
71	C₂²	—	12	2	no	—
72	C₂²	—	12	1	no	—
73	C₂²	—	12	1	no	—
74	C₂²	D₃×C₈	1	1	yes	characteristic
75	C₂²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
76	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	12	1	no	—
80	C₃	#323	1	1	yes	characteristic, complemented, Sylow
81	C₂	—	2	1	no	—
82	C₂	—	2	1	no	—
83	C₂	—	12	2	no	—
84	C₂	#806	1	1	yes	characteristic
85	C₂	(C₂×D₃)⋊C₈	1	1	yes	characteristic
86	C₂	#823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₃×C₂²⋊C₈	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	C₂³×D₃	C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂²⋊C₈	D₃
hypocenter	C₃	#323
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₂²⋊C₈	C₂	semidirect
3. #830	C₂	semidirect
14. C₂²⋊C₈	D₃	semidirect
20. C₂²×D₃	C₈	semidirect
80. C₃	#323	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₃)⋊C₄	2
C₃×C₂²⋊C₈	2
#830	2
#323	3