Subgroups — #69841

Order 384.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic
2	#4924	C₂	1	2	yes	maximal
3	#4552	C₂	1	4	yes	maximal
4	#1838	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₂²	1	2	yes	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#208	C₂²	1	4	yes	—
8	C₃×#97	C₄	1	4	yes	—
9	C₄⋊(C₃⋊C₈)	—	2	4	no	—
10	#423	—	3	2	no	—
11	C₂×#97	D₃	1	1	yes	characteristic
12	#341	—	3	4	no	—
13	C₂²×C₃×C₄	Q₈	1	2	yes	—
14	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
15	C₂×C₃×C₈	—	2	4	no	—
16	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
17	C₃×C₄²	C₂×C₄	1	2	yes	—
18	C₃×C₄²	D₄	1	2	yes	—
19	#208	—	2	8	no	—
20	C₂×C₃⋊C₈	—	2	4	no	—
21	C₂×M₁₆	—	3	4	no	—
22	C₂²×C₈	C₂×D₃	1	2	yes	—
23	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
24	C₄⋊C₈	—	6	4	no	—
25	#97	Dic₃	1	4	yes	—
26	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₃×C₄	—	4	2	no	—
28	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
29	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
31	C₂×C₃×C₄	#41	1	4	yes	—
32	C₃×C₈	—	2	4	no	—
33	C₃⋊C₈	—	4	4	no	—
34	C₂²×C₄	Dic₆	1	2	yes	—
35	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
36	M₁₆	—	6	8	no	—
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	—	6	4	no	—
39	C₂×C₈	C₂×Dic₃	1	4	yes	—
40	C₄²	D₁₂	1	2	yes	—
41	C₄²	C₂×Dic₃	1	2	yes	—
42	C₂²×C₃	—	2	2	no	—
43	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₃	#125	1	2	yes	—
45	C₃×C₄	—	2	4	no	—
46	C₃×C₄	#117	1	2	yes	—
47	C₃×C₄	#124	1	2	yes	—
48	C₂³	C₂×Dic₆	1	1	yes	characteristic
49	C₂×C₄	—	4	2	no	—
50	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
51	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
52	C₂×C₄	#235	1	4	yes	—
53	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
54	C₈	—	2	4	no	—
55	C₈	—	12	4	no	—
56	C₂×C₃	—	2	2	no	—
57	C₂×C₃	#382	1	1	yes	characteristic
58	C₂×C₃	#430	1	2	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	#870	1	2	yes	—
61	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
62	C₄	—	2	4	no	—
63	C₄	#875	1	2	yes	—
64	C₄	#926	1	2	yes	—
65	C₃	#1838	1	1	yes	characteristic, complemented, Sylow
66	C₂	—	2	2	no	—
67	C₂	#4936	1	1	yes	characteristic
68	C₂	#5125	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#97	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#97	D₃
hypocenter	C₃	#1838
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
65. C₃	#1838	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#97	2
#4924	2
#4552	2
#1838	3