Subgroups — #13146091

Order 1728.

1176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#127	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	2	yes	maximal, complemented
3	#10589410	—	3	13	no	maximal
4	C₂×C₃³×D₄	—	2	2	no	—
5	C₂×C₃³×D₄	C₄	1	2	yes	—
6	C₃³×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
8	C₃²×#127	D₃	1	13	yes	—
9	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	26	no	—
10	C₂³×C₃³	—	2	2	no	—
11	C₃³×D₄	—	4	2	no	—
12	C₃³×D₄	—	4	2	no	—
13	C₃³×D₄	—	4	2	no	—
14	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃³×C₄	D₄	1	2	yes	—
16	C₃⋊(C₃⋊(C₃⋊C₈))	—	4	2	no	—
17	#4544	—	9	13	no	—
18	C₂×C₃²×D₄	—	2	26	no	—
19	C₂×C₃²×D₄	Dic₃	1	26	yes	—
20	C₃²×C₄²	C₂×D₃	1	13	yes	—
21	C₂×C₃⋊(C₃⋊C₈)	—	6	26	no	—
22	C₂²×C₃³	—	4	2	no	—
23	C₂²×C₃³	—	4	4	no	—
24	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
25	C₃³×C₄	—	2	1	no	—
26	C₃³×C₄	D₈	1	2	yes	—
27	C₃³×C₄	SD₁₆	1	2	yes	—
28	C₃×#127	C₃⋊D₃	1	13	yes	—
29	C₄⋊(C₃⋊C₈)	—	9	26	no	—
30	C₂³×C₃²	—	2	26	no	—
31	C₃²×D₄	—	4	26	no	—
32	C₃²×D₄	—	4	26	no	—
33	C₃²×D₄	—	4	26	no	—
34	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
35	C₂×C₃²×C₄	#67	1	26	yes	—
36	C₃⋊(C₃⋊C₈)	—	12	26	no	—
37	#331	—	27	1	no	Sylow
38	C₂×C₃³	—	8	2	no	—
39	C₂×C₃³	#100	1	1	yes	characteristic
40	C₂×C₃³	#102	1	2	yes	—
41	C₂×C₃×D₄	—	2	26	no	—
42	C₂×C₃×D₄	C₃⋊Dic₃	1	26	yes	—
43	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
44	C₂×C₃⋊C₈	—	18	26	no	—
45	C₂²×C₃²	—	4	52	no	—
46	C₂²×C₃²	—	4	26	no	—
47	C₂²×C₃²	#217	1	13	yes	—
48	C₃²×C₄	—	2	13	no	—
49	C₃²×C₄	#213	1	26	yes	—
50	C₃²×C₄	#214	1	26	yes	—
51	#127	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
52	C₄⋊C₈	—	27	2	no	—
53	C₃³	#331	1	1	yes	characteristic, complemented, Sylow
54	C₂³×C₃	—	2	26	no	—
55	C₃×D₄	—	4	26	no	—
56	C₃×D₄	—	4	26	no	—
57	C₃×D₄	—	4	26	no	—
58	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
59	C₂×C₃×C₄	#638	1	26	yes	—
60	C₃⋊C₈	—	36	26	no	—
61	C₂×C₃²	—	8	26	no	—
62	C₂×C₃²	#832	1	26	yes	—
63	C₂×C₃²	#833	1	13	yes	—
64	C₂×D₄	—	2	2	no	—
65	C₂×D₄	C₃⋊(C₃⋊Dic₃)	1	2	yes	—
66	C₂×C₈	—	54	2	no	—
67	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
68	C₂²×C₃	—	4	52	no	—
69	C₂²×C₃	—	4	26	no	—
70	C₂²×C₃	#3757	1	13	yes	—
71	C₃×C₄	—	2	13	no	—
72	C₃×C₄	#3753	1	26	yes	—
73	C₃×C₄	#3754	1	26	yes	—
74	C₃²	#4544	1	13	yes	complemented
75	C₂³	—	2	2	no	—
76	D₄	—	4	2	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	2	no	—
79	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
80	C₂×C₄	#6312	1	2	yes	—
81	C₈	—	108	2	no	—
82	C₂×C₃	—	8	26	no	—
83	C₂×C₃	#63659	1	26	yes	—
84	C₂×C₃	#63660	1	13	yes	—
85	C₂²	—	4	4	no	—
86	C₂²	—	4	2	no	—
87	C₂²	#88864	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	#88860	1	2	yes	—
90	C₄	#88861	1	2	yes	—
91	C₃	#10589410	1	13	yes	complemented
92	C₂	—	8	2	no	—
93	C₂	#11722202	1	2	yes	—
94	C₂	#11722203	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#88864
commutator	C₂×C₃³×C₄	C₂×C₄
Frattini	C₄²	C₂×C₃⋊(C₃⋊D₃)
Fitting, prime radical	C₃³×#127	C₂
socle	C₂²×C₃³	C₂²⋊C₄
cosocle	C₃³×C₄²	C₂²
hypercenter	#127	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#331
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	semidirect
53. C₃³	#331	semidirect
74. C₃²	#4544	semidirect
91. C₃	#10589410	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#127	2
C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	2
#10589410	3