Subgroups — #11720175

Order 864.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₂₇	C₂	1	1	yes	maximal, characteristic
2	#88372	C₂	1	2	yes	maximal
3	C₄⋊Dic₂₇	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₂₇	C₂	1	2	yes	maximal, complemented
5	#63488	—	3	1	no	maximal
6	C₂³×C₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₂₇	C₄	1	4	yes	complemented
8	C₂×C₄×C₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₂₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₇	C₂²	1	4	yes	—
11	C₂²×C₄×C₉	D₃	1	1	yes	characteristic
12	#3676	—	3	2	no	—
13	C₄⋊Dic₉	—	3	2	no	—
14	C₄×Dic₉	—	3	2	no	—
15	C₂²×C₂₇	—	2	2	no	—
16	C₂²×C₂₇	C₂×C₄	1	2	yes	—
17	C₂²×C₂₇	C₂³	1	1	yes	characteristic
18	C₄×C₂₇	C₂×C₄	1	2	yes	—
19	C₄×C₂₇	C₂×C₄	1	2	yes	—
20	Dic₂₇	—	2	4	no	—
21	#926	—	9	1	no	—
22	C₂³×C₉	C₂×D₃	1	1	yes	characteristic
23	C₂×C₄×C₉	Dic₃	1	4	yes	—
24	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
25	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
26	C₂×Dic₉	—	3	4	no	—
27	C₂×C₂₇	—	2	2	no	—
28	C₂×C₂₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₂₇	#41	1	2	yes	—
30	C₂²×C₃×C₄	D₉	1	1	yes	characteristic
31	#217	—	9	2	no	—
32	C₄⋊Dic₃	—	9	2	no	—
33	C₄×Dic₃	—	9	2	no	—
34	C₂²×C₉	—	2	2	no	—
35	C₂²×C₉	C₂×Dic₃	1	2	yes	—
36	C₂²×C₉	C₂²×D₃	1	1	yes	characteristic
37	C₄×C₉	C₂×Dic₃	1	2	yes	—
38	C₄×C₉	C₂×Dic₃	1	2	yes	—
39	Dic₉	—	6	4	no	—
40	#117	—	27	1	no	Sylow
41	C₂₇	#117	1	1	yes	characteristic, complemented, Sylow
42	C₂³×C₃	C₂×D₉	1	1	yes	characteristic
43	C₂×C₃×C₄	Dic₉	1	4	yes	—
44	C₂×C₃×C₄	C₂×D₉	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₉	1	1	yes	characteristic
46	C₂×Dic₃	—	9	4	no	—
47	C₂×C₉	—	2	2	no	—
48	C₂×C₉	#235	1	2	yes	—
49	C₂×C₉	C₂²×Dic₃	1	1	yes	characteristic
50	C₂²×C₄	D₂₇	1	1	yes	characteristic
51	C₄⋊C₄	—	27	2	no	—
52	C₂²⋊C₄	—	27	2	no	—
53	C₄²	—	27	2	no	—
54	C₂²×C₃	—	2	2	no	—
55	C₂²×C₃	C₂×Dic₉	1	2	yes	—
56	C₂²×C₃	C₂²×D₉	1	1	yes	characteristic
57	C₃×C₄	C₂×Dic₉	1	2	yes	—
58	C₃×C₄	C₂×Dic₉	1	2	yes	—
59	Dic₃	—	18	4	no	—
60	C₉	#926	1	1	yes	characteristic
61	C₂³	C₂×D₂₇	1	1	yes	characteristic
62	C₂×C₄	—	27	4	no	—
63	C₂×C₄	Dic₂₇	1	4	yes	complemented
64	C₂×C₄	C₂×D₂₇	1	1	yes	characteristic
65	C₂×C₄	C₂×D₂₇	1	1	yes	characteristic
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	#3697	1	2	yes	—
68	C₂×C₃	C₂²×Dic₉	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	C₂×Dic₂₇	1	2	yes	—
71	C₂²	C₂²×D₂₇	1	1	yes	characteristic
72	C₄	—	54	4	no	—
73	C₄	C₂×Dic₂₇	1	2	yes	—
74	C₄	C₂×Dic₂₇	1	2	yes	—
75	C₃	#63488	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	#88399	1	2	yes	—
78	C₂	C₂²×Dic₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₄⋊Dic₂₇	C₂	semidirect
4. C₄×Dic₂₇	C₂	semidirect
7. C₂×C₄×C₂₇	C₄	semidirect
41. C₂₇	#117	semidirect
63. C₂×C₄	Dic₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₂₇	2
#88372	2
C₄⋊Dic₂₇	2
C₄×Dic₂₇	2
#63488	3