Subgroups — #70539

Order 384.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#424	C₂	1	1	yes	maximal, characteristic
2	#5125	C₂	1	1	yes	maximal, characteristic
3	C₃⋊#338	C₂	1	1	yes	maximal, characteristic
4	#4552	C₂	1	2	yes	maximal
5	#4533	C₂	1	2	yes	maximal
6	#2103	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₃×#117	C₂²	1	1	yes	characteristic
10	C₂×#208	C₂²	1	2	yes	—
11	C₂×#208	C₂²	1	2	yes	—
12	C₃×C₄⋊C₈	—	2	2	no	—
13	#819	—	2	2	no	—
14	#430	—	3	1	no	—
15	#424	D₃	1	1	yes	characteristic
16	#344	—	3	2	no	—
17	#341	—	3	2	no	—
18	#338	—	3	1	no	—
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₃×M₁₆	—	2	2	no	—
21	C₂×C₃×C₈	—	2	1	no	—
22	C₂×C₃×C₈	—	2	2	no	—
23	C₂×C₃×C₈	D₄	1	2	yes	—
24	C₃×C₄⋊C₄	—	2	1	no	—
25	C₃×C₂²⋊C₄	Q₈	1	2	yes	—
26	C₃×C₄²	—	2	1	no	—
27	#208	—	2	2	no	—
28	#208	—	2	2	no	—
29	#208	—	2	2	no	—
30	#208	—	2	2	no	—
31	C₂×C₃⋊C₈	—	2	2	no	—
32	C₂×C₃⋊C₈	—	2	2	no	—
33	C₂×M₁₆	—	3	2	no	—
34	C₂×M₁₆	—	3	2	no	—
35	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
37	#117	C₂×D₃	1	1	yes	characteristic
38	#108	—	6	2	no	—
39	C₄⋊C₈	—	2	2	no	—
40	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₃×C₄	—	2	2	no	—
42	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃×C₄	#41	1	1	yes	characteristic
44	C₂×C₃×C₄	#41	1	2	yes	—
45	C₂×C₃×C₄	#41	1	2	yes	—
46	C₃×C₈	—	2	2	no	—
47	C₃×C₈	—	4	1	no	—
48	C₃⋊C₈	—	4	2	no	—
49	C₃⋊C₈	—	4	2	no	—
50	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	M₁₆	—	6	2	no	—
53	M₁₆	—	6	2	no	—
54	M₁₆	—	6	2	no	—
55	M₁₆	—	6	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	6	2	no	—
59	C₂×C₈	—	6	2	no	—
60	C₂×C₈	#67	1	2	yes	—
61	C₄⋊C₄	—	2	1	no	—
62	C₂²⋊C₄	Dic₆	1	2	yes	—
63	C₄²	—	2	1	no	—
64	C₂²×C₃	—	4	1	no	—
65	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂²×C₃	#125	1	2	yes	—
67	C₃×C₄	—	4	2	no	—
68	C₃×C₄	#123	1	1	yes	characteristic
69	C₃×C₄	#124	1	1	yes	characteristic
70	C₃×C₄	#126	1	2	yes	—
71	C₂³	C₂×Dic₆	1	1	yes	characteristic
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	#235	1	1	yes	characteristic
74	C₂×C₄	#237	1	2	yes	—
75	C₂×C₄	#239	1	2	yes	—
76	C₂×C₄	C₂×#67	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	4	1	no	—
79	C₈	—	12	2	no	—
80	C₈	—	12	2	no	—
81	C₂×C₃	—	2	1	no	—
82	C₂×C₃	—	2	2	no	—
83	C₂×C₃	#400	1	1	yes	characteristic
84	C₂²	—	4	1	no	—
85	C₂²	#890	1	2	yes	—
86	C₂²	#924	1	1	yes	characteristic
87	C₄	—	4	2	no	—
88	C₄	#898	1	2	yes	—
89	C₄	#935	1	1	yes	characteristic
90	C₄	#947	1	1	yes	characteristic
91	C₃	#2103	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#4986	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#947
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#424	C₂
socle	C₂×C₃	#400
prime radical	C₂³×C₃	C₂×Q₈
hypercenter	#424	D₃
hypocenter	C₃	#2103
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
91. C₃	#2103	semidirect

Maximal subgroups

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

subgroup	index
C₃×#424	2
#5125	2
C₃⋊#338	2
#4552	2
#4533	2
#2103	3