Subgroups — #13046823

Order 1296.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×C₂₇	C₂	1	1	yes	maximal, characteristic
2	#10619657	C₂	1	1	yes	maximal, characteristic
3	C₃⋊(C₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#88547	—	3	1	no	maximal
5	#88370	—	3	3	no	maximal
6	C₃×C₄×C₂₇	—	2	1	no	—
7	C₃×C₄×C₂₇	C₂²	1	1	yes	characteristic
8	C₃⋊Dic₂₇	—	2	1	no	—
9	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
10	#6226	—	3	1	no	—
11	C₃⋊(C₉⋊C₈)	—	3	1	no	—
12	Q₈×C₂₇	D₃	1	3	yes	—
13	Dic₅₄	—	3	3	no	—
14	C₂₇⋊C₈	—	3	3	no	—
15	C₂×C₃×C₂₇	D₄	1	1	yes	characteristic
16	#3756	—	9	1	no	—
17	#3674	—	9	2	no	—
18	#3674	—	9	1	no	—
19	C₃×C₄×C₉	—	2	1	no	—
20	C₃×C₄×C₉	C₂×D₃	1	1	yes	characteristic
21	C₃⋊Dic₉	—	6	1	no	—
22	C₄×C₂₇	—	2	3	no	—
23	C₄×C₂₇	C₂×D₃	1	3	yes	—
24	Dic₂₇	—	6	3	no	—
25	C₃×C₂₇	Q₁₆	1	1	yes	characteristic, complemented, Sylow
26	C₃²×Q₈	D₉	1	1	yes	characteristic
27	#634	—	9	1	no	—
28	C₃⋊(C₃⋊C₈)	—	9	1	no	—
29	Q₈×C₉	D₉	1	2	yes	—
30	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
31	Dic₁₈	—	9	2	no	—
32	Dic₁₈	—	9	1	no	—
33	C₉⋊C₈	—	9	2	no	—
34	C₉⋊C₈	—	9	1	no	—
35	C₂×C₃×C₉	#67	1	1	yes	characteristic
36	C₂×C₂₇	#67	1	3	yes	—
37	#216	—	27	3	no	—
38	#216	—	27	1	no	—
39	C₃²×C₄	—	2	1	no	—
40	C₃²×C₄	C₂×D₉	1	1	yes	characteristic
41	C₃⋊Dic₃	—	18	1	no	—
42	C₄×C₉	—	2	2	no	—
43	C₄×C₉	—	2	1	no	—
44	C₄×C₉	C₂×D₉	1	2	yes	—
45	C₄×C₉	C₂×C₃⋊D₃	1	1	yes	characteristic
46	Dic₉	—	18	2	no	—
47	Dic₉	—	18	1	no	—
48	C₃×C₉	#216	1	1	yes	characteristic
49	C₂₇	#216	1	3	yes	complemented
50	C₃×Q₈	D₂₇	1	3	yes	—
51	C₃×Q₈	C₃⋊D₉	1	1	yes	characteristic
52	Dic₆	—	27	3	no	—
53	Dic₆	—	27	1	no	—
54	C₃⋊C₈	—	27	3	no	—
55	C₃⋊C₈	—	27	1	no	—
56	C₂×C₃²	#611	1	1	yes	characteristic
57	C₂×C₉	#611	1	2	yes	—
58	C₂×C₉	#638	1	1	yes	characteristic
59	Q₁₆	—	81	1	no	Sylow
60	C₃×C₄	—	2	3	no	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	C₂×D₂₇	1	3	yes	—
63	C₃×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
64	Dic₃	—	54	3	no	—
65	Dic₃	—	54	1	no	—
66	C₃²	#3674	1	1	yes	characteristic
67	C₉	#3674	1	2	yes	—
68	C₉	#3756	1	1	yes	characteristic
69	Q₈	—	81	1	no	—
70	Q₈	C₃⋊D₂₇	1	1	yes	characteristic
71	C₈	—	81	1	no	—
72	C₂×C₃	#6171	1	3	yes	—
73	C₂×C₃	#6233	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	162	1	no	—
76	C₄	C₂×C₃⋊D₂₇	1	1	yes	characteristic
77	C₃	#88370	1	3	yes	complemented
78	C₃	#88547	1	1	yes	characteristic
79	C₂	#10619677	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10619677
commutator, cosocle	C₃×C₄×C₂₇	C₂²
Frattini	C₄×C₉	C₂×C₃⋊D₃
Fitting	C₃×Q₈×C₂₇	C₂
socle, prime radical	C₂×C₃²	#611
hypercenter	Q₈	C₃⋊D₂₇
hypocenter	C₃×C₂₇	Q₁₆
LCS	G ▷ C₃×C₄×C₂₇ ▷ C₂×C₃×C₂₇ ▷ C₃×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
25. C₃×C₂₇	Q₁₆	semidirect
49. C₂₇	#216	semidirect
77. C₃	#88370	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×C₂₇	2
#10619657	2
C₃⋊(C₂₇⋊C₈)	2
#88547	3
#88370	3