Subgroups — #88401

Order 432.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#6171	C₂	1	2	yes	maximal, complemented
3	C₂×Dic₂₇	C₂	1	2	yes	maximal, complemented
4	C₄×D₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₅₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#3699	—	3	1	no	maximal
7	C₂²×C₂₇	C₂²	1	2	yes	—
8	C₂×D₂₇	C₂²	1	1	yes	characteristic
9	C₄×C₂₇	C₂²	1	1	yes	characteristic
10	Dic₂₇	C₂²	1	2	yes	—
11	Dic₂₇	C₂²	1	1	yes	characteristic
12	D₄×C₉	D₃	1	1	yes	characteristic
13	#611	—	3	2	no	—
14	C₂×Dic₉	—	3	2	no	—
15	C₄×D₉	—	3	1	no	—
16	Dic₁₈	—	3	1	no	—
17	C₂×C₂₇	—	2	2	no	—
18	C₂×C₂₇	C₂³	1	1	yes	characteristic
19	D₂₇	—	2	1	no	—
20	#237	—	9	1	no	—
21	C₂²×C₉	C₂×D₃	1	2	yes	—
22	C₂×D₉	—	3	1	no	—
23	C₄×C₉	C₂×D₃	1	1	yes	characteristic
24	Dic₉	—	3	2	no	—
25	Dic₉	—	3	1	no	—
26	C₂₇	#41	1	1	yes	characteristic, complemented, Sylow
27	C₃×D₄	D₉	1	1	yes	characteristic
28	#67	—	9	2	no	—
29	C₂×Dic₃	—	9	2	no	—
30	C₄×D₃	—	9	1	no	—
31	Dic₆	—	9	1	no	—
32	C₂×C₉	—	2	2	no	—
33	C₂×C₉	C₂²×D₃	1	1	yes	characteristic
34	D₉	—	6	1	no	—
35	#41	—	27	1	no	Sylow
36	C₂²×C₃	C₂×D₉	1	2	yes	—
37	C₂×D₃	—	9	1	no	—
38	C₃×C₄	C₂×D₉	1	1	yes	characteristic
39	Dic₃	—	9	2	no	—
40	Dic₃	—	9	1	no	—
41	C₉	#237	1	1	yes	characteristic
42	Q₈	—	27	1	no	—
43	D₄	—	27	2	no	—
44	D₄	D₂₇	1	1	yes	characteristic, complemented
45	C₂×C₄	—	27	2	no	—
46	C₂×C₄	—	27	1	no	—
47	C₂×C₃	—	2	2	no	—
48	C₂×C₃	C₂²×D₉	1	1	yes	characteristic
49	D₃	—	18	1	no	—
50	C₂²	—	27	1	no	—
51	C₂²	C₂×D₂₇	1	2	yes	—
52	C₄	—	27	1	no	—
53	C₄	—	27	2	no	—
54	C₄	C₂×D₂₇	1	1	yes	characteristic
55	C₃	#3699	1	1	yes	characteristic
56	C₂	—	2	2	no	—
57	C₂	—	54	1	no	—
58	C₂	C₂²×D₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₇
commutator, cosocle	C₂×C₂₇	C₂³
Frattini	C₂×C₉	C₂²×D₃
Fitting	D₄×C₂₇	C₂
socle	C₂×C₃	C₂²×D₉
hypercenter	D₄	D₂₇
hypocenter	C₂₇	#41
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. D₄×C₂₇	C₂	semidirect
2. #6171	C₂	semidirect
3. C₂×Dic₂₇	C₂	semidirect
4. C₄×D₂₇	C₂	semidirect
5. Dic₅₄	C₂	semidirect
26. C₂₇	#41	semidirect
44. D₄	D₂₇	semidirect

Maximal subgroups

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

subgroup	index
D₄×C₂₇	2
#6171	2
C₂×Dic₂₇	2
C₄×D₂₇	2
Dic₅₄	2
#3699	3