Subgroups — #13046824

Order 1296.

140 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, complemented
2	C₄⋊(C₃⋊D₂₇)	C₂	1	1	yes	maximal, characteristic
3	C₃⋊(C₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#88548	—	3	1	no	maximal
5	#88371	—	3	3	no	maximal
6	C₂×C₃⋊D₂₇	—	2	1	no	—
7	C₃×C₄×C₂₇	—	2	1	no	—
8	C₃×C₄×C₂₇	C₂²	1	1	yes	characteristic
9	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
10	C₄⋊(C₃⋊D₉)	—	3	1	no	—
11	C₃⋊(C₉⋊C₈)	—	3	1	no	—
12	Q₈×C₂₇	D₃	1	3	yes	complemented
13	D₁₀₈	—	3	3	no	—
14	C₂₇⋊C₈	—	3	3	no	—
15	C₂×C₃×C₂₇	D₄	1	1	yes	characteristic
16	C₃⋊D₂₇	—	4	1	no	—
17	#3755	—	9	1	no	—
18	#3675	—	9	2	no	—
19	#3675	—	9	1	no	—
20	C₂×C₃⋊D₉	—	6	1	no	—
21	C₃×C₄×C₉	—	2	1	no	—
22	C₃×C₄×C₉	C₂×D₃	1	1	yes	characteristic
23	C₂×D₂₇	—	6	3	no	—
24	C₄×C₂₇	—	2	3	no	—
25	C₄×C₂₇	C₂×D₃	1	3	yes	—
26	C₃×C₂₇	SD₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₃²×Q₈	D₉	1	1	yes	characteristic
28	C₄⋊(C₃⋊D₃)	—	9	1	no	—
29	C₃⋊(C₃⋊C₈)	—	9	1	no	—
30	Q₈×C₉	D₉	1	2	yes	—
31	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
32	D₃₆	—	9	2	no	—
33	D₃₆	—	9	1	no	—
34	C₉⋊C₈	—	9	2	no	—
35	C₉⋊C₈	—	9	1	no	—
36	C₂×C₃×C₉	#67	1	1	yes	characteristic
37	C₃⋊D₉	—	12	1	no	—
38	C₂×C₂₇	#67	1	3	yes	—
39	D₂₇	—	12	3	no	—
40	#215	—	27	3	no	—
41	#215	—	27	1	no	—
42	C₂×C₃⋊D₃	—	18	1	no	—
43	C₃²×C₄	—	2	1	no	—
44	C₃²×C₄	C₂×D₉	1	1	yes	characteristic
45	C₂×D₉	—	18	2	no	—
46	C₂×D₉	—	18	1	no	—
47	C₄×C₉	—	2	2	no	—
48	C₄×C₉	—	2	1	no	—
49	C₄×C₉	C₂×D₉	1	2	yes	—
50	C₄×C₉	C₂×C₃⋊D₃	1	1	yes	characteristic
51	C₃×C₉	#215	1	1	yes	characteristic
52	C₂₇	#215	1	3	yes	complemented
53	C₃×Q₈	D₂₇	1	3	yes	complemented
54	C₃×Q₈	C₃⋊D₉	1	1	yes	characteristic
55	D₁₂	—	27	3	no	—
56	D₁₂	—	27	1	no	—
57	C₃⋊C₈	—	27	3	no	—
58	C₃⋊C₈	—	27	1	no	—
59	C₂×C₃²	#611	1	1	yes	characteristic
60	C₃⋊D₃	—	36	1	no	—
61	C₂×C₉	#611	1	2	yes	—
62	C₂×C₉	#638	1	1	yes	characteristic
63	D₉	—	36	2	no	—
64	D₉	—	36	1	no	—
65	SD₁₆	—	81	1	no	Sylow
66	C₂×D₃	—	54	3	no	—
67	C₂×D₃	—	54	1	no	—
68	C₃×C₄	—	2	3	no	—
69	C₃×C₄	—	2	1	no	—
70	C₃×C₄	C₂×D₂₇	1	3	yes	—
71	C₃×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
72	C₃²	#3675	1	1	yes	characteristic
73	C₉	#3675	1	2	yes	—
74	C₉	#3755	1	1	yes	characteristic
75	Q₈	C₃⋊D₂₇	1	1	yes	characteristic, complemented
76	D₄	—	81	1	no	—
77	C₈	—	81	1	no	—
78	C₂×C₃	#6171	1	3	yes	—
79	C₂×C₃	#6233	1	1	yes	characteristic
80	D₃	—	108	3	no	—
81	D₃	—	108	1	no	—
82	C₂²	—	162	1	no	—
83	C₄	—	2	1	no	—
84	C₄	C₂×C₃⋊D₂₇	1	1	yes	characteristic
85	C₃	#88371	1	3	yes	complemented
86	C₃	#88548	1	1	yes	characteristic
87	C₂	—	324	1	no	—
88	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	C₂×C₃²	#611
prime radical	C₄⋊(C₃⋊D₂₇)	C₂
hypercenter	Q₈	C₃⋊D₂₇
hypocenter	C₃×C₂₇	SD₁₆
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
1. C₃×Q₈×C₂₇	C₂	semidirect
3. C₃⋊(C₂₇⋊C₈)	C₂	semidirect
12. Q₈×C₂₇	D₃	semidirect
26. C₃×C₂₇	SD₁₆	semidirect
52. C₂₇	#215	semidirect
53. C₃×Q₈	D₂₇	semidirect
75. Q₈	C₃⋊D₂₇	semidirect
85. C₃	#88371	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
C₄⋊(C₃⋊D₂₇)	2
C₃⋊(C₂₇⋊C₈)	2
#88548	3
#88371	3