Subgroups — C₂²×Q₈×Dic₂₇₁ #46849540

Order 34688.

1664 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₇₁	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊Dic₂₇₁	C₂	1	3	yes	maximal
3	C₂²×C₄×Dic₂₇₁	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₇₁	C₂²	1	3	yes	—
6	C₂³×Dic₂₇₁	C₂²	1	1	yes	characteristic
7	Q₈×Dic₂₇₁	C₂²	1	64	yes	direct product factor
8	C₂×C₄⋊Dic₂₇₁	C₂²	1	36	yes	—
9	C₂×C₄×Dic₂₇₁	C₂²	1	36	yes	—
10	C₂²×Q₈×C₂₇₁	C₄	1	16	yes	complemented
11	C₂²×Q₈×C₂₇₁	C₂²	1	12	yes	—
12	C₂³×C₄×C₂₇₁	C₂²	1	3	yes	—
13	C₂²×Dic₂₇₁	—	2	12	no	—
14	C₂²×Dic₂₇₁	Q₈	1	8	yes	direct product factor
15	C₂²×Dic₂₇₁	C₂³	1	18	yes	—
16	C₂²×Dic₂₇₁	C₂³	1	6	yes	—
17	C₄⋊Dic₂₇₁	C₂³	1	48	yes	—
18	C₄×Dic₂₇₁	C₂³	1	48	yes	—
19	C₂⁴×C₂₇₁	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₂₇₁	C₂×C₄	1	96	yes	complemented
21	C₂×Q₈×C₂₇₁	C₂³	1	16	yes	—
22	C₂²×C₄×C₂₇₁	C₂×C₄	1	24	yes	—
23	C₂²×C₄×C₂₇₁	C₂³	1	18	yes	—
24	C₂×Dic₂₇₁	—	2	36	no	—
25	C₂×Dic₂₇₁	C₂×Q₈	1	24	yes	direct product factor
26	C₂×Dic₂₇₁	C₂⁴	1	12	yes	—
27	C₂×Dic₂₇₁	C₂⁴	1	4	yes	—
28	C₂³×C₂₇₁	C₂²×C₄	1	4	yes	—
29	C₂³×C₂₇₁	C₂×Q₈	1	4	yes	—
30	C₂³×C₂₇₁	#41	1	4	yes	—
31	C₂³×C₂₇₁	C₂⁴	1	3	yes	—
32	Q₈×C₂₇₁	C₂²×C₄	1	64	yes	complemented
33	C₂×C₄×C₂₇₁	C₂²×C₄	1	72	yes	—
34	C₂×C₄×C₂₇₁	C₂⁴	1	12	yes	—
35	C₂²×C₂₇₁	C₄×Q₈	1	16	yes	complemented
36	C₂²×C₂₇₁	C₂³×C₄	1	6	yes	—
37	C₂²×C₂₇₁	C₂²×Q₈	1	6	yes	—
38	C₂²×C₂₇₁	C₂×#41	1	6	yes	—
39	C₂²×C₂₇₁	C₂⁵	1	1	yes	characteristic
40	C₄×C₂₇₁	C₂³×C₄	1	24	yes	—
41	Dic₂₇₁	—	2	12	no	—
42	Dic₂₇₁	C₂²×Q₈	1	8	yes	direct product factor
43	C₂×C₂₇₁	C₂×C₄×Q₈	1	12	yes	complemented
44	C₂×C₂₇₁	C₂⁴×C₄	1	1	yes	characteristic
45	C₂×C₂₇₁	C₂³×Q₈	1	1	yes	characteristic
46	C₂×C₂₇₁	C₂²×#41	1	1	yes	characteristic
47	C₂₇₁	C₂²×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₄×Q₈	—	271	1	no	maximal, Sylow
49	C₂³×Q₈	D₂₇₁	1	1	yes	characteristic
50	C₂×C₄×Q₈	—	271	24	no	—
51	C₂²×C₄⋊C₄	—	271	3	no	—
52	C₂²×C₄²	—	271	3	no	—
53	C₂²×Q₈	Dic₂₇₁	1	16	yes	direct product factor
54	C₂²×Q₈	C₂×D₂₇₁	1	12	yes	—
55	C₂³×C₄	—	271	1	no	—
56	C₂³×C₄	—	271	3	no	—
57	C₂³×C₄	C₂×D₂₇₁	1	3	yes	—
58	C₄×Q₈	—	271	64	no	—
59	C₂×C₄⋊C₄	—	271	36	no	—
60	C₂×C₄²	—	271	36	no	—
61	C₂⁴	C₂²×D₂₇₁	1	1	yes	characteristic
62	C₂×Q₈	C₂×Dic₂₇₁	1	96	yes	direct product factor
63	C₂×Q₈	C₂²×D₂₇₁	1	16	yes	—
64	C₂²×C₄	—	271	8	no	—
65	C₂²×C₄	—	271	6	no	—
66	C₂²×C₄	—	271	18	no	—
67	C₂²×C₄	—	542	12	no	—
68	C₂²×C₄	C₂×Dic₂₇₁	1	24	yes	—
69	C₂²×C₄	C₂²×D₂₇₁	1	18	yes	—
70	C₄⋊C₄	—	271	48	no	—
71	C₄²	—	271	48	no	—
72	C₂³	C₂²×Dic₂₇₁	1	4	yes	—
73	C₂³	Q₈×D₂₇₁	1	4	yes	—
74	C₂³	#16864655	1	4	yes	—
75	C₂³	C₂³×D₂₇₁	1	3	yes	—
76	Q₈	C₂²×Dic₂₇₁	1	64	yes	direct product factor
77	C₂×C₄	—	271	24	no	—
78	C₂×C₄	—	271	4	no	—
79	C₂×C₄	—	271	12	no	—
80	C₂×C₄	—	542	36	no	—
81	C₂×C₄	C₂²×Dic₂₇₁	1	72	yes	—
82	C₂×C₄	C₂³×D₂₇₁	1	12	yes	—
83	C₂²	Q₈×Dic₂₇₁	1	16	yes	direct product factor
84	C₂²	C₂³×Dic₂₇₁	1	6	yes	—
85	C₂²	C₂×Q₈×D₂₇₁	1	6	yes	—
86	C₂²	C₂×#16864655	1	6	yes	—
87	C₂²	C₂⁴×D₂₇₁	1	1	yes	characteristic
88	C₄	—	271	8	no	—
89	C₄	—	542	12	no	—
90	C₄	C₂³×Dic₂₇₁	1	24	yes	—
91	C₂	C₂×Q₈×Dic₂₇₁	1	12	yes	direct product factor
92	C₂	C₂⁴×Dic₂₇₁	1	1	yes	characteristic
93	C₂	C₂²×Q₈×D₂₇₁	1	1	yes	characteristic
94	C₂	C₂²×#16864655	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₂₇₁
commutator	C₂×C₂₇₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₂₇₁
Fitting	C₂³×Q₈×C₂₇₁	C₂
socle, prime radical	C₂⁴×C₂₇₁	C₂³
cosocle	C₂²×C₂₇₁	C₂⁵
hypercenter	C₂³×Q₈	D₂₇₁
hypocenter	C₂₇₁	C₂²×C₄×Q₈
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₂₇₁	C₂	direct (N × H)
7. Q₈×Dic₂₇₁	C₂²	direct (N × H)
10. C₂²×Q₈×C₂₇₁	C₄	semidirect
14. C₂²×Dic₂₇₁	Q₈	direct (N × H)
20. C₂×Q₈×C₂₇₁	C₂×C₄	semidirect
25. C₂×Dic₂₇₁	C₂×Q₈	direct (N × H)
32. Q₈×C₂₇₁	C₂²×C₄	semidirect
35. C₂²×C₂₇₁	C₄×Q₈	semidirect
42. Dic₂₇₁	C₂²×Q₈	direct (N × H)
43. C₂×C₂₇₁	C₂×C₄×Q₈	semidirect
47. C₂₇₁	C₂²×C₄×Q₈	semidirect
53. C₂²×Q₈	Dic₂₇₁	direct (N × H)
62. C₂×Q₈	C₂×Dic₂₇₁	direct (N × H)
76. Q₈	C₂²×Dic₂₇₁	direct (N × H)
83. C₂²	Q₈×Dic₂₇₁	direct (N × H)
91. C₂	C₂×Q₈×Dic₂₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₂₇₁	2
C₂²×C₄⋊Dic₂₇₁	2
C₂²×C₄×Dic₂₇₁	2
C₂³×Q₈×C₂₇₁	2
C₂²×C₄×Q₈	271

Subgroups — C22×Q8×Dic271 #46849540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×Dic₂₇₁ #46849540