Subgroups — C₂×#11720123 #13136213

Order 1728.

996 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₀₈	C₂	1	6	yes	maximal, complemented
2	C₂×C₄²×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720123	C₂	1	8	yes	maximal, direct product factor
4	C₂×#63436	—	3	1	no	maximal
5	C₂³×D₂₇	C₂²	1	4	yes	—
6	C₂²×C₄×C₂₇	C₂²	1	3	yes	—
7	C₂×D₁₀₈	—	2	24	no	—
8	C₂×D₁₀₈	C₂²	1	24	yes	complemented
9	C₄²×C₂₇	C₂²	1	4	yes	complemented
10	C₂²×D₃₆	—	3	6	no	—
11	C₂×C₄²×C₉	D₃	1	1	yes	characteristic
12	#63436	—	3	8	no	—
13	C₂³×C₂₇	C₂³	1	1	yes	characteristic
14	C₂²×D₂₇	—	2	24	no	—
15	C₂²×D₂₇	C₂³	1	8	yes	—
16	C₂×C₄×C₂₇	D₄	1	12	yes	complemented
17	C₂×C₄×C₂₇	C₂³	1	6	yes	—
18	D₁₀₈	—	2	48	no	—
19	C₂×#874	—	9	1	no	—
20	C₂³×D₉	—	3	4	no	—
21	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
22	C₂×D₃₆	—	3	24	no	—
23	C₂×D₃₆	—	6	24	no	—
24	C₄²×C₉	C₂×D₃	1	4	yes	—
25	C₂²×C₂₇	C₂×D₄	1	6	yes	—
26	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
27	C₂×D₂₇	—	2	24	no	—
28	C₂×D₂₇	—	4	16	no	—
29	C₄×C₂₇	C₂×D₄	1	12	yes	complemented
30	C₂²×D₁₂	—	9	6	no	—
31	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
32	#874	—	9	8	no	—
33	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
34	C₂²×D₉	—	3	8	no	—
35	C₂²×D₉	—	6	24	no	—
36	C₂×C₄×C₉	D₁₂	1	12	yes	—
37	C₂×C₄×C₉	C₂²×D₃	1	6	yes	—
38	D₃₆	—	6	48	no	—
39	C₂×#127	—	27	1	no	Sylow
40	C₂×C₂₇	#127	1	4	yes	complemented
41	C₂×C₂₇	C₂²×D₄	1	3	yes	—
42	D₂₇	—	4	8	no	—
43	C₂³×D₃	—	9	4	no	—
44	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
45	C₂×D₁₂	—	9	24	no	—
46	C₂×D₁₂	—	18	24	no	—
47	C₃×C₄²	C₂×D₉	1	4	yes	—
48	C₂²×C₉	C₂×D₁₂	1	6	yes	—
49	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
50	C₂×D₉	—	6	24	no	—
51	C₂×D₉	—	12	16	no	—
52	C₄×C₉	C₂×D₁₂	1	12	yes	—
53	C₂²×D₄	—	27	6	no	—
54	#127	—	27	8	no	—
55	C₂×C₄²	D₂₇	1	1	yes	characteristic, complemented
56	C₂₇	C₂×#127	1	1	yes	characteristic, complemented, Sylow
57	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
58	C₂²×D₃	—	9	8	no	—
59	C₂²×D₃	—	18	24	no	—
60	C₂×C₃×C₄	D₃₆	1	12	yes	—
61	C₂×C₃×C₄	C₂²×D₉	1	6	yes	—
62	D₁₂	—	18	48	no	—
63	C₂×C₉	#874	1	4	yes	—
64	C₂×C₉	C₂²×D₁₂	1	3	yes	—
65	D₉	—	12	8	no	—
66	C₂⁴	—	27	4	no	—
67	C₂×D₄	—	27	24	no	—
68	C₂×D₄	—	54	24	no	—
69	C₂²×C₄	C₂×D₂₇	1	3	yes	—
70	C₄²	C₂×D₂₇	1	4	yes	complemented
71	C₂²×C₃	C₂×D₃₆	1	6	yes	—
72	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
73	C₂×D₃	—	18	24	no	—
74	C₂×D₃	—	36	16	no	—
75	C₃×C₄	C₂×D₃₆	1	12	yes	—
76	C₉	C₂×#874	1	1	yes	characteristic
77	C₂³	—	27	8	no	—
78	C₂³	—	54	24	no	—
79	C₂³	C₂²×D₂₇	1	1	yes	characteristic
80	D₄	—	54	48	no	—
81	C₂×C₄	D₁₀₈	1	12	yes	complemented
82	C₂×C₄	C₂²×D₂₇	1	6	yes	—
83	C₂×C₃	#63436	1	4	yes	—
84	C₂×C₃	C₂²×D₃₆	1	3	yes	—
85	D₃	—	36	8	no	—
86	C₂²	—	54	24	no	—
87	C₂²	—	108	16	no	—
88	C₂²	C₂×D₁₀₈	1	6	yes	—
89	C₂²	C₂³×D₂₇	1	1	yes	characteristic
90	C₄	C₂×D₁₀₈	1	12	yes	complemented
91	C₃	C₂×#63436	1	1	yes	characteristic
92	C₂	—	108	8	no	—
93	C₂	#11720123	1	4	yes	direct product factor
94	C₂	C₂²×D₁₀₈	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₇
commutator, cosocle	C₂²×C₂₇	C₂⁴
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₂×C₄²×C₂₇	C₂
socle	C₂³×C₃	C₂²×D₉
hypercenter	C₂×C₄²	D₂₇
hypocenter	C₂₇	C₂×#127
LCS	G ▷ C₂²×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₀₈	C₂	semidirect
2. C₂×C₄²×C₂₇	C₂	semidirect
3. #11720123	C₂	direct (N × H)
8. C₂×D₁₀₈	C₂²	semidirect
9. C₄²×C₂₇	C₂²	semidirect
16. C₂×C₄×C₂₇	D₄	semidirect
29. C₄×C₂₇	C₂×D₄	semidirect
40. C₂×C₂₇	#127	semidirect
55. C₂×C₄²	D₂₇	semidirect
56. C₂₇	C₂×#127	semidirect
70. C₄²	C₂×D₂₇	semidirect
81. C₂×C₄	D₁₀₈	semidirect
90. C₄	C₂×D₁₀₈	semidirect
93. C₂	#11720123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₀₈	2
C₂×C₄²×C₂₇	2
#11720123	2
C₂×#63436	3

Subgroups — C2×#11720123 #13136213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720123 #13136213