Subgroups — C₂³×#14523503 #30680683

Order 16768.

1464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523503	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂³×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₂	—	2	7	no	—
6	C₂²×Dic₂₆₂	C₂²	1	14	yes	—
7	C₂³×Dic₁₃₁	—	2	1	no	—
8	C₂×#14523503	C₂²	1	112	yes	direct product factor
9	C₂²×C₁₃₁⋊C₈	C₂²	1	14	yes	—
10	C₂²×Q₈×C₁₃₁	—	2	7	no	—
11	C₂²×Q₈×C₁₃₁	C₂²	1	14	yes	—
12	C₂³×C₄×C₁₃₁	—	2	1	no	—
13	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₆₂	—	2	42	no	—
15	C₂×Dic₂₆₂	C₂³	1	28	yes	—
16	C₂²×Dic₁₃₁	—	2	14	no	—
17	#14523503	C₂³	1	64	yes	direct product factor
18	C₂×C₁₃₁⋊C₈	C₂³	1	28	yes	—
19	C₂⁴×C₁₃₁	D₄	1	1	yes	characteristic
20	C₂×Q₈×C₁₃₁	—	2	42	no	—
21	C₂×Q₈×C₁₃₁	C₂³	1	28	yes	—
22	C₂²×C₄×C₁₃₁	—	2	14	no	—
23	C₂²×C₄×C₁₃₁	D₄	1	7	yes	—
24	C₂²×C₄×C₁₃₁	C₂³	1	7	yes	—
25	Dic₂₆₂	—	2	28	no	—
26	Dic₂₆₂	C₂⁴	1	8	yes	—
27	C₂×Dic₁₃₁	—	2	28	no	—
28	C₁₃₁⋊C₈	C₂⁴	1	8	yes	—
29	C₂³×C₁₃₁	Q₁₆	1	8	yes	complemented
30	C₂³×C₁₃₁	C₂×D₄	1	7	yes	—
31	Q₈×C₁₃₁	—	2	28	no	—
32	Q₈×C₁₃₁	C₂⁴	1	8	yes	—
33	C₂×C₄×C₁₃₁	—	2	28	no	—
34	C₂×C₄×C₁₃₁	C₂×D₄	1	21	yes	—
35	C₂×C₄×C₁₃₁	C₂⁴	1	7	yes	—
36	C₂²×C₁₃₁	C₂×Q₁₆	1	28	yes	complemented
37	C₂²×C₁₃₁	C₂²×D₄	1	7	yes	—
38	C₄×C₁₃₁	—	2	8	no	—
39	C₄×C₁₃₁	C₂²×D₄	1	7	yes	—
40	C₄×C₁₃₁	C₂⁵	1	1	yes	characteristic
41	Dic₁₃₁	—	2	8	no	—
42	C₂×C₁₃₁	C₂²×Q₁₆	1	14	yes	complemented
43	C₂×C₁₃₁	C₂³×D₄	1	1	yes	characteristic
44	C₁₃₁	C₂³×Q₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₂³×Q₁₆	—	131	1	no	maximal, Sylow
46	C₂³×Q₈	—	131	1	no	—
47	C₂³×Q₈	D₁₃₁	1	1	yes	characteristic
48	C₂²×Q₁₆	—	131	28	no	—
49	C₂³×C₈	—	131	1	no	—
50	C₂²×Q₈	—	2	7	no	—
51	C₂²×Q₈	—	262	7	no	—
52	C₂²×Q₈	—	131	14	no	—
53	C₂²×Q₈	C₂×D₁₃₁	1	14	yes	—
54	C₂³×C₄	—	2	1	no	—
55	C₂³×C₄	—	262	1	no	—
56	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
57	C₂×Q₁₆	—	131	112	no	—
58	C₂²×C₈	—	131	14	no	—
59	C₂⁴	C₂²⋊D₁₃₁	1	1	yes	characteristic
60	C₂×Q₈	—	2	42	no	—
61	C₂×Q₈	—	262	42	no	—
62	C₂×Q₈	—	131	28	no	—
63	C₂×Q₈	C₂²×D₁₃₁	1	28	yes	—
64	C₂²×C₄	—	2	14	no	—
65	C₂²×C₄	—	262	14	no	—
66	C₂²×C₄	C₂²⋊D₁₃₁	1	7	yes	—
67	C₂²×C₄	C₂²×D₁₃₁	1	7	yes	—
68	Q₁₆	—	131	64	no	—
69	C₂×C₈	—	131	28	no	—
70	C₂³	#14523503	1	8	yes	direct product factor
71	C₂³	C₂×C₂²⋊D₁₃₁	1	7	yes	—
72	Q₈	—	2	28	no	—
73	Q₈	—	262	28	no	—
74	Q₈	—	131	8	no	—
75	Q₈	C₂³×D₁₃₁	1	8	yes	—
76	C₂×C₄	—	2	28	no	—
77	C₂×C₄	—	262	28	no	—
78	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	21	yes	—
79	C₂×C₄	C₂³×D₁₃₁	1	7	yes	—
80	C₈	—	131	8	no	—
81	C₂²	C₂×#14523503	1	28	yes	direct product factor
82	C₂²	C₂²×C₂²⋊D₁₃₁	1	7	yes	—
83	C₄	—	2	8	no	—
84	C₄	—	262	8	no	—
85	C₄	C₂²×C₂²⋊D₁₃₁	1	7	yes	—
86	C₄	C₂⁴×D₁₃₁	1	1	yes	characteristic
87	C₂	C₂²×#14523503	1	14	yes	direct product factor
88	C₂	C₂³×C₂²⋊D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₁₃₁
commutator, cosocle	C₄×C₁₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₁
Fitting	C₂³×Q₈×C₁₃₁	C₂
socle, prime radical	C₂⁴×C₁₃₁	D₄
hypercenter	C₂³×Q₈	D₁₃₁
hypocenter	C₁₃₁	C₂³×Q₁₆
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#14523503	C₂	direct (N × H)
8. C₂×#14523503	C₂²	direct (N × H)
17. #14523503	C₂³	direct (N × H)
29. C₂³×C₁₃₁	Q₁₆	semidirect
36. C₂²×C₁₃₁	C₂×Q₁₆	semidirect
42. C₂×C₁₃₁	C₂²×Q₁₆	semidirect
44. C₁₃₁	C₂³×Q₁₆	semidirect
70. C₂³	#14523503	direct (N × H)
81. C₂²	C₂×#14523503	direct (N × H)
87. C₂	C₂²×#14523503	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₆₂	2
C₂²×#14523503	2
C₂³×C₁₃₁⋊C₈	2
C₂³×Q₈×C₁₃₁	2
C₂³×Q₁₆	131

Subgroups — C23×#14523503 #30680683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523503 #30680683