Subgroups — C₂₇×#124 #11720213

Order 864.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×C₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₄×C₂₇	C₂	1	4	yes	maximal, complemented
4	C₄²×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#124	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₂₇	C₂²	1	2	yes	—
7	Q₈×C₂₇	—	2	2	no	—
8	D₄×C₂₇	—	2	2	no	—
9	C₂×C₄×C₂₇	C₂²	1	2	yes	—
10	C₂×C₄×C₂₇	C₂²	1	2	yes	—
11	C₂×C₄×C₂₇	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
13	C₂×D₄×C₉	C₂×C₃	1	1	yes	characteristic
14	C₉×C₂²⋊C₄	C₂×C₃	1	4	yes	—
15	C₄²×C₉	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₂₇	—	2	2	no	—
17	C₂²×C₂₇	—	2	4	no	—
18	C₂²×C₂₇	C₂³	1	1	yes	characteristic
19	C₄×C₂₇	—	2	2	no	—
20	C₄×C₂₇	—	2	2	no	—
21	C₄×C₂₇	D₄	1	2	yes	—
22	C₃×#124	C₉	1	1	yes	characteristic
23	C₂³×C₉	C₂²×C₃	1	2	yes	—
24	Q₈×C₉	—	2	2	no	—
25	D₄×C₉	—	2	2	no	—
26	C₂×C₄×C₉	C₂²×C₃	1	2	yes	—
27	C₂×C₄×C₉	C₂²×C₃	1	2	yes	—
28	C₂×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
29	C₂×C₂₇	—	4	2	no	—
30	C₂×C₂₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₂₇	#41	1	2	yes	—
32	C₂×C₃×Q₈	C₂×C₉	1	1	yes	characteristic
33	C₂×C₃×D₄	C₂×C₉	1	1	yes	characteristic
34	C₃×C₂²⋊C₄	C₂×C₉	1	4	yes	—
35	C₃×C₄²	C₂×C₉	1	1	yes	characteristic
36	C₂²×C₉	—	2	2	no	—
37	C₂²×C₉	—	2	4	no	—
38	C₂²×C₉	C₂³×C₃	1	1	yes	characteristic
39	C₄×C₉	—	2	2	no	—
40	C₄×C₉	—	2	2	no	—
41	C₄×C₉	C₃×D₄	1	2	yes	—
42	#124	C₂₇	1	1	yes	characteristic, direct product factor, Sylow
43	C₂₇	#124	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃	C₂²×C₉	1	2	yes	—
45	C₃×Q₈	—	2	2	no	—
46	C₃×D₄	—	2	2	no	—
47	C₂×C₃×C₄	C₂²×C₉	1	2	yes	—
48	C₂×C₃×C₄	C₂²×C₉	1	2	yes	—
49	C₂×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
50	C₂×C₉	—	4	2	no	—
51	C₂×C₉	C₂×C₃×D₄	1	1	yes	characteristic
52	C₂×C₉	C₃×#41	1	2	yes	—
53	C₂×Q₈	C₂×C₂₇	1	1	yes	characteristic, complemented
54	C₂×D₄	C₂×C₂₇	1	1	yes	characteristic
55	C₂²⋊C₄	C₂×C₂₇	1	4	yes	complemented
56	C₄²	C₂×C₂₇	1	1	yes	characteristic, complemented
57	C₂²×C₃	—	2	2	no	—
58	C₂²×C₃	—	2	4	no	—
59	C₂²×C₃	C₂³×C₉	1	1	yes	characteristic
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	D₄×C₉	1	2	yes	—
63	C₉	C₃×#124	1	1	yes	characteristic
64	C₂³	C₂²×C₂₇	1	2	yes	—
65	Q₈	—	2	2	no	—
66	D₄	—	2	2	no	—
67	C₂×C₄	C₂²×C₂₇	1	2	yes	—
68	C₂×C₄	C₂²×C₂₇	1	2	yes	—
69	C₂×C₄	C₂²×C₂₇	1	1	yes	characteristic
70	C₂×C₃	—	4	2	no	—
71	C₂×C₃	C₂×D₄×C₉	1	1	yes	characteristic
72	C₂×C₃	C₉×#41	1	2	yes	—
73	C₂²	—	2	2	no	—
74	C₂²	—	2	4	no	—
75	C₂²	C₂³×C₂₇	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	D₄×C₂₇	1	2	yes	—
79	C₃	C₉×#124	1	1	yes	characteristic
80	C₂	—	4	2	no	—
81	C₂	C₂×D₄×C₂₇	1	1	yes	characteristic
82	C₂	#41×C₂₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₂₇	C₂³
commutator	C₂²	C₂³×C₂₇
Frattini, cosocle	C₂²×C₉	C₂³×C₃
socle	C₂²×C₃	C₂³×C₉
prime radical	C₂×C₃×D₄	C₂×C₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₂₇	C₂	semidirect
3. C₂²⋊C₄×C₂₇	C₂	semidirect
4. C₄²×C₂₇	C₂	semidirect
42. #124	C₂₇	direct (N × H)
43. C₂₇	#124	direct (N × H)
53. C₂×Q₈	C₂×C₂₇	semidirect
55. C₂²⋊C₄	C₂×C₂₇	semidirect
56. C₄²	C₂×C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₂₇	2
C₂×D₄×C₂₇	2
C₂²⋊C₄×C₂₇	2
C₄²×C₂₇	2
C₉×#124	3

Subgroups — C27×#124 #11720213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₇×#124 #11720213