Subgroups — C₂³×#92481 #16860429

Order 3968.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#92481	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₃₁	—	2	1	no	—
6	C₂²×Q₈×C₃₁	—	2	7	no	—
7	C₂²×Q₈×C₃₁	C₂²	1	14	yes	complemented
8	C₂³×C₄×C₃₁	—	2	1	no	—
9	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×D₁₂₄	—	2	7	no	—
11	C₂²×D₁₂₄	C₂²	1	14	yes	—
12	C₂×#92481	C₂²	1	112	yes	direct product factor
13	C₂²×C₃₁⋊C₈	C₂²	1	14	yes	complemented
14	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
15	C₂³×D₃₁	—	2	14	no	—
16	C₂³×D₃₁	—	4	8	no	—
17	C₂×Q₈×C₃₁	—	2	42	no	—
18	C₂×Q₈×C₃₁	C₂³	1	28	yes	complemented
19	C₂²×C₄×C₃₁	—	2	14	no	—
20	C₂²×C₄×C₃₁	D₄	1	7	yes	—
21	C₂²×C₄×C₃₁	C₂³	1	7	yes	—
22	C₂×D₁₂₄	—	2	42	no	—
23	C₂×D₁₂₄	C₂³	1	28	yes	—
24	#92481	C₂³	1	64	yes	direct product factor
25	C₂×C₃₁⋊C₈	C₂³	1	28	yes	complemented
26	C₂³×C₃₁	SD₁₆	1	8	yes	complemented
27	C₂³×C₃₁	C₂×D₄	1	7	yes	—
28	C₂²×D₃₁	—	2	28	no	—
29	C₂²×D₃₁	—	4	56	no	—
30	Q₈×C₃₁	—	2	28	no	—
31	Q₈×C₃₁	C₂⁴	1	8	yes	complemented
32	C₂×C₄×C₃₁	—	2	28	no	—
33	C₂×C₄×C₃₁	C₂×D₄	1	21	yes	—
34	C₂×C₄×C₃₁	C₂⁴	1	7	yes	—
35	D₁₂₄	—	2	28	no	—
36	D₁₂₄	C₂⁴	1	8	yes	—
37	C₃₁⋊C₈	C₂⁴	1	8	yes	complemented
38	C₂³×SD₁₆	—	31	1	no	maximal, Sylow
39	C₂²×C₃₁	C₂×SD₁₆	1	28	yes	complemented
40	C₂²×C₃₁	C₂²×D₄	1	7	yes	—
41	C₂×D₃₁	—	2	8	no	—
42	C₂×D₃₁	—	4	56	no	—
43	C₄×C₃₁	—	2	8	no	—
44	C₄×C₃₁	C₂²×D₄	1	7	yes	—
45	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
46	C₂³×Q₈	D₃₁	1	1	yes	characteristic, complemented
47	C₂³×D₄	—	31	1	no	—
48	C₂²×SD₁₆	—	31	28	no	—
49	C₂³×C₈	—	31	1	no	—
50	C₂×C₃₁	C₂²×SD₁₆	1	14	yes	complemented
51	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
52	D₃₁	—	4	8	no	—
53	C₂⁵	—	62	1	no	—
54	C₂²×Q₈	—	2	7	no	—
55	C₂²×Q₈	C₂×D₃₁	1	14	yes	complemented
56	C₂²×D₄	—	31	14	no	—
57	C₂²×D₄	—	62	7	no	—
58	C₂³×C₄	—	2	1	no	—
59	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
60	C₂×SD₁₆	—	31	112	no	—
61	C₂²×C₈	—	31	14	no	—
62	C₃₁	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Sylow
63	C₂⁴	—	62	14	no	—
64	C₂⁴	—	124	8	no	—
65	C₂⁴	C₂²⋊D₃₁	1	1	yes	characteristic
66	C₂×Q₈	—	2	42	no	—
67	C₂×Q₈	C₂²×D₃₁	1	28	yes	complemented
68	C₂×D₄	—	31	28	no	—
69	C₂×D₄	—	62	42	no	—
70	C₂²×C₄	—	2	14	no	—
71	C₂²×C₄	C₂²⋊D₃₁	1	7	yes	—
72	C₂²×C₄	C₂²×D₃₁	1	7	yes	—
73	SD₁₆	—	31	64	no	—
74	C₂×C₈	—	31	28	no	—
75	C₂³	—	62	28	no	—
76	C₂³	—	124	56	no	—
77	C₂³	#92481	1	8	yes	direct product factor
78	C₂³	C₂×C₂²⋊D₃₁	1	7	yes	—
79	Q₈	—	2	28	no	—
80	Q₈	C₂³×D₃₁	1	8	yes	complemented
81	D₄	—	31	8	no	—
82	D₄	—	62	28	no	—
83	C₂×C₄	—	2	28	no	—
84	C₂×C₄	C₂×C₂²⋊D₃₁	1	21	yes	—
85	C₂×C₄	C₂³×D₃₁	1	7	yes	—
86	C₈	—	31	8	no	—
87	C₂²	—	62	8	no	—
88	C₂²	—	124	56	no	—
89	C₂²	C₂×#92481	1	28	yes	direct product factor
90	C₂²	C₂²×C₂²⋊D₃₁	1	7	yes	—
91	C₄	—	2	8	no	—
92	C₄	C₂²×C₂²⋊D₃₁	1	7	yes	—
93	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
94	C₂	—	124	8	no	—
95	C₂	C₂²×#92481	1	14	yes	direct product factor
96	C₂	C₂³×C₂²⋊D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₃₁
commutator, cosocle	C₄×C₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₃₁
Fitting	C₂³×Q₈×C₃₁	C₂
socle	C₂⁴×C₃₁	D₄
prime radical	C₂³×D₁₂₄	C₂
hypercenter	C₂³×Q₈	D₃₁
hypocenter	C₃₁	C₂³×SD₁₆
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
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×#92481	C₂	direct (N × H)
3. C₂³×C₃₁⋊C₈	C₂	semidirect
4. C₂³×Q₈×C₃₁	C₂	semidirect
7. C₂²×Q₈×C₃₁	C₂²	semidirect
12. C₂×#92481	C₂²	direct (N × H)
13. C₂²×C₃₁⋊C₈	C₂²	semidirect
18. C₂×Q₈×C₃₁	C₂³	semidirect
24. #92481	C₂³	direct (N × H)
25. C₂×C₃₁⋊C₈	C₂³	semidirect
26. C₂³×C₃₁	SD₁₆	semidirect
31. Q₈×C₃₁	C₂⁴	semidirect
37. C₃₁⋊C₈	C₂⁴	semidirect
39. C₂²×C₃₁	C₂×SD₁₆	semidirect
46. C₂³×Q₈	D₃₁	semidirect
50. C₂×C₃₁	C₂²×SD₁₆	semidirect
55. C₂²×Q₈	C₂×D₃₁	semidirect
62. C₃₁	C₂³×SD₁₆	semidirect
67. C₂×Q₈	C₂²×D₃₁	semidirect
77. C₂³	#92481	direct (N × H)
80. Q₈	C₂³×D₃₁	semidirect
89. C₂²	C₂×#92481	direct (N × H)
95. C₂	C₂²×#92481	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×D₁₂₄	2
C₂²×#92481	2
C₂³×C₃₁⋊C₈	2
C₂³×Q₈×C₃₁	2
C₂³×SD₁₆	31

Subgroups — C23×#92481 #16860429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#92481 #16860429