Subgroups — #11720121

Order 864.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₂₇	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₂₇)⋊C₄	C₂	1	2	yes	maximal
4	Dic₂₇⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#63434	—	3	1	no	maximal
7	C₂²×D₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₂₇	C₂²	1	2	yes	—
9	C₂×C₄×C₂₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₇	C₂²	1	2	yes	—
11	C₂×Dic₂₇	C₂²	1	1	yes	characteristic
12	C₄×D₂₇	C₄	1	4	yes	complemented
13	C₂×C₄×D₉	—	3	1	no	—
14	C₄²×C₉	D₃	1	1	yes	characteristic
15	(C₂×D₉)⋊C₄	—	3	2	no	—
16	Dic₉⋊C₄	—	3	2	no	—
17	C₄×Dic₉	—	3	1	no	—
18	C₂²×C₂₇	C₂³	1	1	yes	characteristic
19	C₂×D₂₇	—	2	2	no	—
20	C₂×D₂₇	C₂×C₄	1	2	yes	—
21	C₄×C₂₇	—	2	2	no	—
22	C₄×C₂₇	C₂×C₄	1	2	yes	—
23	Dic₂₇	—	2	2	no	—
24	Dic₂₇	C₂×C₄	1	2	yes	—
25	#872	—	9	1	no	—
26	C₂²×D₉	—	3	1	no	—
27	C₂×C₄×C₉	C₂×D₃	1	2	yes	—
28	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
29	C₂×Dic₉	—	3	1	no	—
30	C₂×Dic₉	—	3	2	no	—
31	C₄×D₉	—	3	4	no	—
32	C₂×C₂₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₂₇	#41	1	2	yes	—
34	D₂₇	—	2	2	no	—
35	C₂×C₄×D₃	—	9	1	no	—
36	C₃×C₄²	D₉	1	1	yes	characteristic
37	(C₂×D₃)⋊C₄	—	9	2	no	—
38	Dic₃⋊C₄	—	9	2	no	—
39	C₄×Dic₃	—	9	1	no	—
40	C₂²×C₉	C₂²×D₃	1	1	yes	characteristic
41	C₂×D₉	—	3	2	no	—
42	C₂×D₉	—	6	2	no	—
43	C₄×C₉	—	2	2	no	—
44	C₄×C₉	C₄×D₃	1	2	yes	—
45	Dic₉	—	3	2	no	—
46	Dic₉	—	6	2	no	—
47	#117	—	27	1	no	Sylow
48	C₂₇	#117	1	1	yes	characteristic, complemented, Sylow
49	C₂²×D₃	—	9	1	no	—
50	C₂×C₃×C₄	C₂×D₉	1	2	yes	—
51	C₂×C₃×C₄	C₂×D₉	1	1	yes	characteristic
52	C₂×Dic₃	—	9	1	no	—
53	C₂×Dic₃	—	9	2	no	—
54	C₄×D₃	—	9	4	no	—
55	C₂×C₉	C₂×C₄×D₃	1	1	yes	characteristic
56	C₂×C₉	#235	1	2	yes	—
57	D₉	—	6	2	no	—
58	C₂²×C₄	—	27	1	no	—
59	C₄⋊C₄	—	27	2	no	—
60	C₂²⋊C₄	—	27	2	no	—
61	C₄²	—	27	1	no	—
62	C₄²	D₂₇	1	1	yes	characteristic, complemented
63	C₂²×C₃	C₂²×D₉	1	1	yes	characteristic
64	C₂×D₃	—	9	2	no	—
65	C₂×D₃	—	18	2	no	—
66	C₃×C₄	—	2	2	no	—
67	C₃×C₄	C₄×D₉	1	2	yes	—
68	Dic₃	—	9	2	no	—
69	Dic₃	—	18	2	no	—
70	C₉	#872	1	1	yes	characteristic
71	C₂³	—	27	1	no	—
72	C₂×C₄	—	27	1	no	—
73	C₂×C₄	—	27	4	no	—
74	C₂×C₄	—	27	2	no	—
75	C₂×C₄	C₂×D₂₇	1	2	yes	—
76	C₂×C₄	C₂×D₂₇	1	1	yes	characteristic
77	C₂×C₃	C₂×C₄×D₉	1	1	yes	characteristic
78	C₂×C₃	#3697	1	2	yes	—
79	D₃	—	18	2	no	—
80	C₂²	—	27	2	no	—
81	C₂²	—	54	2	no	—
82	C₂²	C₂²×D₂₇	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	27	2	no	—
85	C₄	—	54	2	no	—
86	C₄	C₄×D₂₇	1	2	yes	—
87	C₃	#63434	1	1	yes	characteristic
88	C₂	—	54	2	no	—
89	C₂	C₂×C₄×D₂₇	1	1	yes	characteristic
90	C₂	#88399	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₇
commutator	C₂×C₂₇	C₂²×C₄
Frattini	C₂²×C₉	C₂²×D₃
Fitting	C₄²×C₂₇	C₂
socle	C₂²×C₃	C₂²×D₉
prime radical	C₂²×D₂₇	C₂²
cosocle	C₂²×C₂₇	C₂³
hypercenter	C₄²	D₂₇
hypocenter	C₂₇	#117
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×C₂₇	C₂	semidirect
4. Dic₂₇⋊C₄	C₂	semidirect
5. C₄×Dic₂₇	C₂	semidirect
12. C₄×D₂₇	C₄	semidirect
48. C₂₇	#117	semidirect
62. C₄²	D₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₂₇	2
C₄²×C₂₇	2
(C₂×D₂₇)⋊C₄	2
Dic₂₇⋊C₄	2
C₄×Dic₂₇	2
#63434	3