Subgroups — #5713

Order 192.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#127	C₂	1	1	yes	maximal, characteristic
2	#936	C₂	1	4	yes	maximal, complemented
3	#935	C₂	1	4	yes	maximal, complemented
4	D₄×Dic₃	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₆	C₂	1	2	yes	maximal, complemented
6	#559	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₂²	1	4	yes	—
8	C₂×C₃×D₄	C₂²	1	2	yes	—
9	C₂²×Dic₃	C₂²	1	4	yes	—
10	C₂×Dic₆	C₂²	1	2	yes	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	#217	C₂²	1	8	yes	—
13	#217	C₂²	1	4	yes	—
14	C₄⋊Dic₃	C₂²	1	2	yes	—
15	Dic₃⋊C₄	C₂²	1	4	yes	—
16	C₄×Dic₃	C₂²	1	4	yes	—
17	#127	D₃	1	1	yes	characteristic
18	#124	—	3	4	no	—
19	#123	—	3	4	no	—
20	C₄×Q₈	—	3	2	no	—
21	C₄×D₄	—	3	4	no	—
22	C₂³×C₃	C₂³	1	4	yes	—
23	C₃×D₄	—	2	8	no	—
24	C₃×D₄	—	4	2	no	—
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂³	1	2	yes	—
27	C₂×Dic₃	—	4	4	no	—
28	C₂×Dic₃	C₂³	1	4	yes	—
29	C₂×Dic₃	C₂³	1	4	yes	—
30	Dic₆	—	2	4	no	—
31	C₂×Q₈	—	3	2	no	—
32	C₂×D₄	C₂×D₃	1	4	yes	—
33	C₂×D₄	C₂×D₃	1	2	yes	—
34	C₂²×C₄	—	3	4	no	—
35	C₄⋊C₄	—	3	2	no	—
36	C₄⋊C₄	—	3	4	no	—
37	C₂²⋊C₄	—	3	8	no	—
38	C₂²⋊C₄	—	3	4	no	—
39	C₄²	—	3	4	no	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	—	2	8	no	—
42	C₂²×C₃	—	2	4	no	—
43	C₂²×C₃	C₂⁴	1	1	yes	characteristic
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	#41	1	4	yes	—
46	Dic₃	—	2	4	no	—
47	Dic₃	—	2	4	no	—
48	C₂³	C₂²×D₃	1	4	yes	—
49	Q₈	—	6	4	no	—
50	D₄	—	2	8	no	—
51	D₄	—	4	2	no	—
52	C₂×C₄	—	3	4	no	—
53	C₂×C₄	—	3	4	no	—
54	C₂×C₄	—	12	4	no	—
55	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₃	1	2	yes	—
57	C₂×C₃	—	4	4	no	—
58	C₂×C₃	C₂×#41	1	2	yes	—
59	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂²	—	2	8	no	—
61	C₂²	—	2	4	no	—
62	C₂²	C₂³×D₃	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	6	4	no	—
65	C₄	—	6	4	no	—
66	C₄	#237	1	4	yes	—
67	C₃	#559	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	4	4	no	—
69	C₂	C₂×#237	1	2	yes	—
70	C₂	#1004	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting, prime radical	C₃×#127	C₂
hypercenter	#127	D₃
hypocenter	C₃	#559
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
2. #936	C₂	semidirect
3. #935	C₂	semidirect
4. D₄×Dic₃	C₂	semidirect
5. C₄×Dic₆	C₂	semidirect
67. C₃	#559	semidirect

Maximal subgroups

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

subgroup	index
C₃×#127	2
#936	2
#935	2
D₄×Dic₃	2
C₄×Dic₆	2
#559	3