Subgroups — C₂³×C₄×Q₈×C₂₃ #19210269

Order 5888.

12446 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₂₃	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₄×C₂₃	C₂	1	3	yes	maximal
4	C₂³×C₄²×C₂₃	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₂₃	C₄	1	32	yes	direct product factor
6	C₂³×Q₈×C₂₃	C₂²	1	28	yes	—
7	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
8	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	448	yes	direct product factor
11	C₂²×C₄⋊C₄×C₂₃	C₂²	1	84	yes	—
12	C₂²×C₄²×C₂₃	C₂²	1	84	yes	—
13	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
14	C₂²×Q₈×C₂₃	C₂×C₄	1	448	yes	direct product factor
15	C₂²×Q₈×C₂₃	C₂³	1	112	yes	—
16	C₂³×C₄×C₂₃	—	2	24	no	—
17	C₂³×C₄×C₂₃	C₂×C₄	1	48	yes	—
18	C₂³×C₄×C₂₃	Q₈	1	16	yes	direct product factor
19	C₂³×C₄×C₂₃	C₂³	1	42	yes	—
20	C₂³×C₄×C₂₃	C₂³	1	42	yes	—
21	C₂³×C₄×C₂₃	C₂³	1	14	yes	—
22	C₄×Q₈×C₂₃	C₂³	1	512	yes	direct product factor
23	C₂×C₄⋊C₄×C₂₃	C₂³	1	336	yes	—
24	C₂×C₄²×C₂₃	C₂³	1	336	yes	—
25	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	—
26	C₂⁴×C₂₃	C₂×Q₈	1	8	yes	—
27	C₂⁴×C₂₃	#41	1	8	yes	—
28	C₂⁴×C₂₃	C₂⁴	1	7	yes	—
29	C₂×Q₈×C₂₃	C₂²×C₄	1	896	yes	direct product factor
30	C₂×Q₈×C₂₃	C₂⁴	1	64	yes	—
31	C₂²×C₄×C₂₃	—	2	168	no	—
32	C₂²×C₄×C₂₃	C₂²×C₄	1	336	yes	—
33	C₂²×C₄×C₂₃	C₂×Q₈	1	112	yes	direct product factor
34	C₂²×C₄×C₂₃	C₂⁴	1	84	yes	—
35	C₂²×C₄×C₂₃	C₂⁴	1	84	yes	—
36	C₂²×C₄×C₂₃	C₂⁴	1	28	yes	—
37	C₄⋊C₄×C₂₃	C₂⁴	1	192	yes	—
38	C₄²×C₂₃	C₂⁴	1	192	yes	—
39	C₂³×C₄×Q₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂³×C₂₃	C₄×Q₈	1	64	yes	direct product factor
41	C₂³×C₂₃	C₂³×C₄	1	28	yes	—
42	C₂³×C₂₃	C₂²×Q₈	1	28	yes	—
43	C₂³×C₂₃	C₂×#41	1	28	yes	—
44	C₂³×C₂₃	C₂⁵	1	7	yes	—
45	Q₈×C₂₃	C₂³×C₄	1	256	yes	direct product factor
46	C₂×C₄×C₂₃	—	2	168	no	—
47	C₂×C₄×C₂₃	C₂³×C₄	1	336	yes	—
48	C₂×C₄×C₂₃	C₂²×Q₈	1	112	yes	direct product factor
49	C₂×C₄×C₂₃	C₂⁵	1	24	yes	—
50	C₂×C₄×C₂₃	C₂⁵	1	24	yes	—
51	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
52	C₂⁴×Q₈	C₂×C₂₃	1	1	yes	characteristic
53	C₂²×C₄×Q₈	C₂×C₂₃	1	56	yes	direct product factor
54	C₂³×C₄⋊C₄	C₂×C₂₃	1	3	yes	—
55	C₂³×C₄²	C₂×C₂₃	1	3	yes	—
56	C₂²×C₂₃	C₂×C₄×Q₈	1	112	yes	direct product factor
57	C₂²×C₂₃	C₂⁴×C₄	1	14	yes	—
58	C₂²×C₂₃	C₂³×Q₈	1	14	yes	—
59	C₂²×C₂₃	C₂²×#41	1	14	yes	—
60	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
61	C₄×C₂₃	—	2	24	no	—
62	C₄×C₂₃	C₂⁴×C₄	1	48	yes	—
63	C₄×C₂₃	C₂³×Q₈	1	16	yes	direct product factor
64	C₂³×Q₈	C₄×C₂₃	1	32	yes	direct product factor
65	C₂³×Q₈	C₂²×C₂₃	1	28	yes	—
66	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
67	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
68	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
69	C₂×C₄×Q₈	C₂²×C₂₃	1	448	yes	direct product factor
70	C₂²×C₄⋊C₄	C₂²×C₂₃	1	84	yes	—
71	C₂²×C₄²	C₂²×C₂₃	1	84	yes	—
72	C₂×C₂₃	C₂²×C₄×Q₈	1	28	yes	direct product factor
73	C₂×C₂₃	C₂⁵×C₄	1	1	yes	characteristic
74	C₂×C₂₃	C₂⁴×Q₈	1	1	yes	characteristic
75	C₂×C₂₃	C₂³×#41	1	1	yes	characteristic
76	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
77	C₂²×Q₈	C₂×C₄×C₂₃	1	448	yes	direct product factor
78	C₂²×Q₈	C₂³×C₂₃	1	112	yes	—
79	C₂³×C₄	—	2	24	no	—
80	C₂³×C₄	C₂×C₄×C₂₃	1	48	yes	—
81	C₂³×C₄	Q₈×C₂₃	1	16	yes	direct product factor
82	C₂³×C₄	C₂³×C₂₃	1	42	yes	—
83	C₂³×C₄	C₂³×C₂₃	1	42	yes	—
84	C₂³×C₄	C₂³×C₂₃	1	14	yes	—
85	C₄×Q₈	C₂³×C₂₃	1	512	yes	direct product factor
86	C₂×C₄⋊C₄	C₂³×C₂₃	1	336	yes	—
87	C₂×C₄²	C₂³×C₂₃	1	336	yes	—
88	C₂₃	C₂³×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
89	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	—
90	C₂⁴	C₂×Q₈×C₂₃	1	8	yes	—
91	C₂⁴	#41×C₂₃	1	8	yes	—
92	C₂⁴	C₂⁴×C₂₃	1	7	yes	—
93	C₂×Q₈	C₂²×C₄×C₂₃	1	896	yes	direct product factor
94	C₂×Q₈	C₂⁴×C₂₃	1	64	yes	—
95	C₂²×C₄	—	2	168	no	—
96	C₂²×C₄	C₂²×C₄×C₂₃	1	336	yes	—
97	C₂²×C₄	C₂×Q₈×C₂₃	1	112	yes	direct product factor
98	C₂²×C₄	C₂⁴×C₂₃	1	84	yes	—
99	C₂²×C₄	C₂⁴×C₂₃	1	84	yes	—
100	C₂²×C₄	C₂⁴×C₂₃	1	28	yes	—
101	C₄⋊C₄	C₂⁴×C₂₃	1	192	yes	—
102	C₄²	C₂⁴×C₂₃	1	192	yes	—
103	C₂³	C₄×Q₈×C₂₃	1	64	yes	direct product factor
104	C₂³	C₂³×C₄×C₂₃	1	28	yes	—
105	C₂³	C₂²×Q₈×C₂₃	1	28	yes	—
106	C₂³	C₂×#41×C₂₃	1	28	yes	—
107	C₂³	C₂⁵×C₂₃	1	7	yes	—
108	Q₈	C₂³×C₄×C₂₃	1	256	yes	direct product factor
109	C₂×C₄	—	2	168	no	—
110	C₂×C₄	C₂³×C₄×C₂₃	1	336	yes	—
111	C₂×C₄	C₂²×Q₈×C₂₃	1	112	yes	direct product factor
112	C₂×C₄	C₂⁵×C₂₃	1	24	yes	—
113	C₂×C₄	C₂⁵×C₂₃	1	24	yes	—
114	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
115	C₂²	C₂×C₄×Q₈×C₂₃	1	112	yes	direct product factor
116	C₂²	C₂⁴×C₄×C₂₃	1	14	yes	—
117	C₂²	C₂³×Q₈×C₂₃	1	14	yes	—
118	C₂²	C₂²×#41×C₂₃	1	14	yes	—
119	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
120	C₄	—	2	24	no	—
121	C₄	C₂⁴×C₄×C₂₃	1	48	yes	—
122	C₄	C₂³×Q₈×C₂₃	1	16	yes	direct product factor
123	C₂	C₂²×C₄×Q₈×C₂₃	1	28	yes	direct product factor
124	C₂	C₂⁵×C₄×C₂₃	1	1	yes	characteristic
125	C₂	C₂⁴×Q₈×C₂₃	1	1	yes	characteristic
126	C₂	C₂³×#41×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄×C₂₃	C₂²
commutator	C₂	C₂⁵×C₄×C₂₃
Frattini, cosocle	C₂²	C₂⁶×C₂₃
socle, prime radical	C₂⁵×C₂₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₂₃	C₂	direct (N × H)
5. C₂³×Q₈×C₂₃	C₄	direct (N × H)
10. C₂×C₄×Q₈×C₂₃	C₂²	direct (N × H)
14. C₂²×Q₈×C₂₃	C₂×C₄	direct (N × H)
18. C₂³×C₄×C₂₃	Q₈	direct (N × H)
22. C₄×Q₈×C₂₃	C₂³	direct (N × H)
29. C₂×Q₈×C₂₃	C₂²×C₄	direct (N × H)
33. C₂²×C₄×C₂₃	C₂×Q₈	direct (N × H)
39. C₂³×C₄×Q₈	C₂₃	direct (N × H)
40. C₂³×C₂₃	C₄×Q₈	direct (N × H)
45. Q₈×C₂₃	C₂³×C₄	direct (N × H)
48. C₂×C₄×C₂₃	C₂²×Q₈	direct (N × H)
53. C₂²×C₄×Q₈	C₂×C₂₃	direct (N × H)
56. C₂²×C₂₃	C₂×C₄×Q₈	direct (N × H)
63. C₄×C₂₃	C₂³×Q₈	direct (N × H)
64. C₂³×Q₈	C₄×C₂₃	direct (N × H)
69. C₂×C₄×Q₈	C₂²×C₂₃	direct (N × H)
72. C₂×C₂₃	C₂²×C₄×Q₈	direct (N × H)
77. C₂²×Q₈	C₂×C₄×C₂₃	direct (N × H)
81. C₂³×C₄	Q₈×C₂₃	direct (N × H)
85. C₄×Q₈	C₂³×C₂₃	direct (N × H)
88. C₂₃	C₂³×C₄×Q₈	direct (N × H)
93. C₂×Q₈	C₂²×C₄×C₂₃	direct (N × H)
97. C₂²×C₄	C₂×Q₈×C₂₃	direct (N × H)
103. C₂³	C₄×Q₈×C₂₃	direct (N × H)
108. Q₈	C₂³×C₄×C₂₃	direct (N × H)
111. C₂×C₄	C₂²×Q₈×C₂₃	direct (N × H)
115. C₂²	C₂×C₄×Q₈×C₂₃	direct (N × H)
122. C₄	C₂³×Q₈×C₂₃	direct (N × H)
123. C₂	C₂²×C₄×Q₈×C₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₂₃	2
C₂²×C₄×Q₈×C₂₃	2
C₂³×C₄⋊C₄×C₂₃	2
C₂³×C₄²×C₂₃	2
C₂³×C₄×Q₈	23

Subgroups — C23×C4×Q8×C23 #19210269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×Q₈×C₂₃ #19210269