Subgroups — #71589

Order 384.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	#5207	C₂	1	2	yes	maximal
3	#4530	C₂	1	4	yes	maximal, complemented
4	#1822	—	3	1	no	maximal, Sylow
5	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₃×#117	—	2	4	no	—
7	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₂²⋊C₄	C₄	1	2	yes	—
9	C₂×C₃×C₄²	C₄	1	2	yes	—
10	C₂×#208	C₂²	1	4	yes	—
11	#830	—	2	4	no	—
12	C₂×#117	D₃	1	1	yes	characteristic
13	#407	—	3	2	no	—
14	#343	—	3	4	no	—
15	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃×C₄	—	2	4	no	—
17	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
19	C₂²×C₃×C₄	D₄	1	4	yes	—
20	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
21	C₃×C₄⋊C₄	—	2	4	no	—
22	C₃×C₂²⋊C₄	—	2	4	no	—
23	C₃×C₄²	—	2	4	no	—
24	#208	—	2	8	no	—
25	C₂×C₃⋊C₈	—	2	4	no	—
26	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
27	C₂×M₁₆	—	3	4	no	—
28	#117	—	2	4	no	—
29	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
30	C₂×C₂²⋊C₄	Dic₃	1	2	yes	—
31	C₂×C₄²	Dic₃	1	2	yes	—
32	C₂²⋊C₈	—	6	4	no	—
33	C₂³×C₃	—	2	2	no	—
34	C₂³×C₃	—	4	2	no	—
35	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄	—	2	2	no	—
38	C₂×C₃×C₄	—	2	8	no	—
39	C₂×C₃×C₄	—	2	4	no	—
40	C₂×C₃×C₄	—	4	4	no	—
41	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
42	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
43	C₂×C₃×C₄	#41	1	4	yes	—
44	C₃⋊C₈	—	4	4	no	—
45	C₂⁴	C₂×Dic₃	1	1	yes	characteristic
46	C₂²×C₄	—	2	4	no	—
47	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
48	C₂²×C₄	C₂×Dic₃	1	4	yes	—
49	C₂²×C₄	#67	1	4	yes	—
50	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
51	M₁₆	—	6	8	no	—
52	C₂×C₈	—	6	4	no	—
53	C₄⋊C₄	—	2	4	no	—
54	C₂²⋊C₄	—	2	4	no	—
55	C₄²	—	2	4	no	—
56	C₂²×C₃	—	2	2	no	—
57	C₂²×C₃	—	2	2	no	—
58	C₂²×C₃	—	2	4	no	—
59	C₂²×C₃	—	4	4	no	—
60	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₃	#117	1	2	yes	—
62	C₃×C₄	—	2	2	no	—
63	C₃×C₄	—	4	4	no	—
64	C₃×C₄	#123	1	4	yes	—
65	C₂³	—	2	2	no	—
66	C₂³	—	4	2	no	—
67	C₂³	#217	1	2	yes	—
68	C₂³	C₂²×Dic₃	1	1	yes	characteristic
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	8	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	4	4	no	—
73	C₂×C₄	#217	1	2	yes	—
74	C₂×C₄	#235	1	4	yes	—
75	C₂×C₄	C₂×#67	1	2	yes	—
76	C₈	—	12	4	no	—
77	C₂×C₃	—	2	2	no	—
78	C₂×C₃	—	4	2	no	—
79	C₂×C₃	#381	1	1	yes	characteristic
80	C₂×C₃	#413	1	2	yes	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	4	4	no	—
85	C₂²	#926	1	2	yes	—
86	C₂²	C₂×#217	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	4	4	no	—
89	C₄	#929	1	4	yes	—
90	C₃	#1822	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	#5211	1	1	yes	characteristic
94	C₂	#5237	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#217
commutator	C₂³×C₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#117	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#117	D₃
hypocenter	C₃	#1822
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #4530	C₂	semidirect
90. C₃	#1822	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#117	2
#5207	2
#4530	2
#1822	3