Subgroups — #13046839

Order 1296.

221 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#733	C₂	1	1	yes	maximal, characteristic
2	C₂×#733⋊C₄	C₂	1	2	yes	maximal, complemented
3	#3676⋊C₃	—	3	3	no	maximal
4	C₃×#3676	—	3	1	no	maximal
5	C₂²×#733	—	2	2	no	—
6	C₂²×#733	C₄	1	2	yes	complemented
7	C₂²×#733	C₂²	1	1	yes	characteristic
8	#733⋊C₄	—	2	2	no	—
9	C₂³×3₋¹⁺²	D₃	1	3	yes	—
10	C₂³×C₃×C₉	D₃	1	1	yes	characteristic
11	C₂×3₋¹⁺²⋊C₄	—	3	6	no	—
12	C₂×C₃×Dic₉	—	3	2	no	—
13	C₂×#733	—	2	2	no	—
14	C₂×#733	C₂×C₄	1	1	yes	characteristic
15	C₂×#733	D₄	1	2	yes	—
16	C₃×#217	—	9	1	no	—
17	#3676	—	3	1	no	—
18	#3676	—	3	3	no	—
19	C₂²×3₋¹⁺²	—	2	6	no	—
20	C₂²×3₋¹⁺²	Dic₃	1	6	yes	—
21	C₂²×3₋¹⁺²	C₂×D₃	1	3	yes	—
22	C₂²×C₃×C₉	—	2	2	no	—
23	C₂²×C₃×C₉	Dic₃	1	2	yes	—
24	C₂²×C₃×C₉	C₂×D₃	1	1	yes	characteristic
25	3₋¹⁺²⋊C₄	—	6	6	no	—
26	C₃×Dic₉	—	6	2	no	—
27	#733	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
28	C₂³×C₃²	C₃⋊D₃	1	1	yes	characteristic
29	C₂×C₃×Dic₃	—	9	2	no	—
30	C₂³×C₉	—	3	1	no	—
31	C₂³×C₉	—	3	3	no	—
32	C₂×Dic₉	—	3	2	no	—
33	C₂×Dic₉	—	3	6	no	—
34	C₂×3₋¹⁺²	—	2	6	no	—
35	C₂×3₋¹⁺²	C₂×Dic₃	1	3	yes	—
36	C₂×3₋¹⁺²	#67	1	6	yes	—
37	C₂×C₃×C₉	—	2	2	no	—
38	C₂×C₃×C₉	C₂×Dic₃	1	1	yes	characteristic
39	C₂×C₃×C₉	#67	1	2	yes	—
40	C₃×C₂²⋊C₄	—	27	1	no	—
41	#217	—	9	1	no	—
42	C₂²×C₃²	—	2	2	no	—
43	C₂²×C₃²	C₃⋊Dic₃	1	2	yes	—
44	C₂²×C₃²	C₂×C₃⋊D₃	1	1	yes	characteristic
45	C₃×Dic₃	—	18	2	no	—
46	C₂²×C₉	—	3	2	no	—
47	C₂²×C₉	—	3	6	no	—
48	C₂²×C₉	—	3	1	no	—
49	C₂²×C₉	—	3	3	no	—
50	C₂²×C₉	—	6	2	no	—
51	C₂²×C₉	—	6	6	no	—
52	Dic₉	—	6	2	no	—
53	Dic₉	—	6	6	no	—
54	3₋¹⁺²	#217	1	3	yes	—
55	C₃×C₉	#217	1	1	yes	characteristic
56	C₂³×C₃	—	3	1	no	—
57	C₂³×C₃	#272	1	1	yes	characteristic
58	C₂×C₃×C₄	—	27	2	no	—
59	C₂×Dic₃	—	9	2	no	—
60	C₂×C₃²	—	2	2	no	—
61	C₂×C₃²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
62	C₂×C₃²	#638	1	2	yes	—
63	C₂×C₉	—	3	1	no	—
64	C₂×C₉	—	3	3	no	—
65	C₂×C₉	—	3	2	no	—
66	C₂×C₉	—	3	6	no	—
67	C₂×C₉	—	6	2	no	—
68	C₂×C₉	—	6	6	no	—
69	C₂²⋊C₄	—	27	1	no	Sylow
70	C₂²×C₃	—	2	2	no	—
71	C₂²×C₃	—	3	2	no	—
72	C₂²×C₃	—	3	1	no	—
73	C₂²×C₃	—	6	2	no	—
74	C₂²×C₃	#1084	1	2	yes	—
75	C₂²×C₃	C₂×#272	1	1	yes	characteristic
76	C₃×C₄	—	54	2	no	—
77	Dic₃	—	18	2	no	—
78	C₃²	#3757	1	1	yes	characteristic
79	C₉	—	3	1	no	—
80	C₉	—	3	3	no	—
81	C₂³	#733⋊C₂	1	1	yes	characteristic
82	C₂×C₄	—	27	2	no	—
83	C₂×C₃	—	2	2	no	—
84	C₂×C₃	—	3	1	no	—
85	C₂×C₃	—	3	2	no	—
86	C₂×C₃	—	6	2	no	—
87	C₂×C₃	C₂×#1084	1	1	yes	characteristic
88	C₂×C₃	#6235	1	2	yes	—
89	C₂²	—	2	2	no	—
90	C₂²	#733⋊C₄	1	2	yes	complemented
91	C₂²	C₂×#733⋊C₂	1	1	yes	characteristic
92	C₄	—	54	2	no	—
93	C₃	—	3	1	no	—
94	C₃	#88552	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	C₂×#733⋊C₄	1	1	yes	characteristic
97	C₂	#10619685	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#733⋊C₂
commutator	C₂×#733	C₂×C₄
Frattini	C₂²×C₃²	C₂×C₃⋊D₃
Fitting	C₂³×#733	C₂
socle	C₂²×C₃	C₂×#272
prime radical	C₂³×C₃²	C₃⋊D₃
cosocle	C₂²×#733	C₂²
hypercenter	C₂³	#733⋊C₂
hypocenter	#733	C₂²⋊C₄
LCS	G ▷ C₂×#733 ▷ #733
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×#733⋊C₄	C₂	semidirect
6. C₂²×#733	C₄	semidirect
27. #733	C₂²⋊C₄	semidirect
90. C₂²	#733⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#733	2
C₂×#733⋊C₄	2
#3676⋊C₃	3
C₃×#3676	3