Subgroups — C₂²×C₂₃×#519 #19210272

Order 5888.

11230 conjugacy classes of subgroups.

The subgroup lattice has 144 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	C₂×C₂₃×#519	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×Q₈×C₂₃	C₂	1	8	yes	maximal, complemented
4	C₂²×C₂₃×#117	C₂	1	6	yes	maximal
5	C₂³×Q₈×C₂₃	C₄	1	32	yes	complemented
6	C₂³×Q₈×C₂₃	C₂²	1	4	yes	—
7	C₂³×Q₈×C₂₃	C₂²	1	24	yes	—
8	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
9	C₂₃×#519	C₂²	1	256	yes	direct product factor
10	C₂×C₄×Q₈×C₂₃	C₂²	1	192	yes	complemented
11	C₂×C₂₃×#117	C₂²	1	144	yes	—
12	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
13	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
14	C₂²×C₄²×C₂₃	C₂²	1	12	yes	—
15	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×Q₈×C₂₃	C₂×C₄	1	64	yes	—
17	C₂²×Q₈×C₂₃	C₂×C₄	1	384	yes	complemented
18	C₂²×Q₈×C₂₃	C₂³	1	48	yes	—
19	C₂²×Q₈×C₂₃	C₂³	1	64	yes	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	48	yes	—
21	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
22	C₂³×C₄×C₂₃	C₂³	1	36	yes	—
23	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
24	C₄×Q₈×C₂₃	C₂³	1	512	yes	complemented
25	C₂₃×#117	C₂³	1	384	yes	—
26	C₂×C₄⋊C₄×C₂₃	C₂³	1	144	yes	—
27	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
28	C₂×C₄²×C₂₃	C₂³	1	144	yes	—
29	C₂⁴×C₂₃	—	2	8	no	—
30	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	—
31	C₂⁴×C₂₃	C₂⁴	1	6	yes	—
32	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
33	C₂×Q₈×C₂₃	C₂²×C₄	1	384	yes	—
34	C₂×Q₈×C₂₃	C₂²×C₄	1	512	yes	complemented
35	C₂×Q₈×C₂₃	C₂⁴	1	64	yes	—
36	C₂²×C₄×C₂₃	—	2	32	no	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	48	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	288	yes	—
39	C₂²×C₄×C₂₃	C₂⁴	1	48	yes	—
40	C₂²×C₄×C₂₃	C₂⁴	1	36	yes	—
41	C₂²×C₄×C₂₃	C₂⁴	1	48	yes	—
42	C₄⋊C₄×C₂₃	C₂⁴	1	192	yes	—
43	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	—
44	C₄²×C₂₃	C₂⁴	1	192	yes	—
45	C₂²×#519	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₂₃	—	2	32	no	—
47	C₂³×C₂₃	—	2	24	no	—
48	C₂³×C₂₃	C₂³×C₄	1	24	yes	—
49	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
50	C₂³×C₂₃	2₋¹⁺⁴	1	8	yes	—
51	C₂³×C₂₃	C₂⁵	1	4	yes	—
52	C₂³×C₂₃	C₂⁵	1	3	yes	—
53	Q₈×C₂₃	C₂³×C₄	1	256	yes	—
54	C₂×C₄×C₂₃	—	2	96	no	—
55	C₂×C₄×C₂₃	C₂³×C₄	1	144	yes	—
56	C₂×C₄×C₂₃	C₂³×C₄	1	192	yes	—
57	C₂×C₄×C₂₃	C₂⁵	1	32	yes	—
58	C₂×C₄×C₂₃	C₂⁵	1	24	yes	—
59	C₂⁴×Q₈	C₂×C₂₃	1	1	yes	characteristic
60	C₂×#519	C₂×C₂₃	1	48	yes	direct product factor
61	C₂²×C₄×Q₈	C₂×C₂₃	1	8	yes	complemented
62	C₂²×#117	C₂×C₂₃	1	6	yes	—
63	C₂²×C₂₃	—	2	8	no	—
64	C₂²×C₂₃	—	2	48	no	—
65	C₂²×C₂₃	#519	1	16	yes	direct product factor
66	C₂²×C₂₃	C₂⁴×C₄	1	8	yes	—
67	C₂²×C₂₃	C₂⁴×C₄	1	6	yes	—
68	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	12	yes	—
69	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
70	C₄×C₂₃	—	2	32	no	—
71	C₄×C₂₃	C₂⁴×C₄	1	48	yes	—
72	C₂³×Q₈	C₄×C₂₃	1	32	yes	complemented
73	C₂³×Q₈	C₂²×C₂₃	1	4	yes	—
74	C₂³×Q₈	C₂²×C₂₃	1	24	yes	—
75	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
76	#519	C₂²×C₂₃	1	256	yes	direct product factor
77	C₂×C₄×Q₈	C₂²×C₂₃	1	192	yes	complemented
78	C₂×#117	C₂²×C₂₃	1	144	yes	—
79	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
80	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
81	C₂²×C₄²	C₂²×C₂₃	1	12	yes	—
82	C₂×C₂₃	—	2	8	no	—
83	C₂×C₂₃	C₂×#519	1	12	yes	direct product factor
84	C₂×C₂₃	C₂⁵×C₄	1	1	yes	characteristic
85	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	2	yes	—
86	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
87	C₂²×Q₈	C₂×C₄×C₂₃	1	64	yes	—
88	C₂²×Q₈	C₂×C₄×C₂₃	1	384	yes	complemented
89	C₂²×Q₈	C₂³×C₂₃	1	48	yes	—
90	C₂²×Q₈	C₂³×C₂₃	1	64	yes	—
91	C₂³×C₄	C₂×C₄×C₂₃	1	48	yes	—
92	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
93	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
94	C₂³×C₄	C₂³×C₂₃	1	36	yes	—
95	C₄×Q₈	C₂³×C₂₃	1	512	yes	complemented
96	#117	C₂³×C₂₃	1	384	yes	—
97	C₂×C₄⋊C₄	C₂³×C₂₃	1	144	yes	—
98	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
99	C₂×C₄²	C₂³×C₂₃	1	144	yes	—
100	C₂₃	C₂²×#519	1	1	yes	characteristic, direct product factor, Sylow
101	C₂⁴	—	2	8	no	—
102	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	—
103	C₂⁴	C₂⁴×C₂₃	1	6	yes	—
104	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
105	C₂×Q₈	C₂²×C₄×C₂₃	1	384	yes	—
106	C₂×Q₈	C₂²×C₄×C₂₃	1	512	yes	complemented
107	C₂×Q₈	C₂⁴×C₂₃	1	64	yes	—
108	C₂²×C₄	—	2	32	no	—
109	C₂²×C₄	C₂²×C₄×C₂₃	1	48	yes	—
110	C₂²×C₄	C₂²×C₄×C₂₃	1	288	yes	—
111	C₂²×C₄	C₂⁴×C₂₃	1	48	yes	—
112	C₂²×C₄	C₂⁴×C₂₃	1	36	yes	—
113	C₂²×C₄	C₂⁴×C₂₃	1	48	yes	—
114	C₄⋊C₄	C₂⁴×C₂₃	1	192	yes	—
115	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	—
116	C₄²	C₂⁴×C₂₃	1	192	yes	—
117	C₂³	—	2	24	no	—
118	C₂³	—	2	32	no	—
119	C₂³	C₂³×C₄×C₂₃	1	24	yes	—
120	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
121	C₂³	C₂₃×2₋¹⁺⁴	1	8	yes	—
122	C₂³	C₂⁵×C₂₃	1	4	yes	—
123	C₂³	C₂⁵×C₂₃	1	3	yes	—
124	Q₈	C₂³×C₄×C₂₃	1	256	yes	—
125	C₂×C₄	—	2	96	no	—
126	C₂×C₄	C₂³×C₄×C₂₃	1	144	yes	—
127	C₂×C₄	C₂³×C₄×C₂₃	1	192	yes	—
128	C₂×C₄	C₂⁵×C₂₃	1	32	yes	—
129	C₂×C₄	C₂⁵×C₂₃	1	24	yes	—
130	C₂²	—	2	48	no	—
131	C₂²	—	2	8	no	—
132	C₂²	C₂₃×#519	1	16	yes	direct product factor
133	C₂²	C₂⁴×C₄×C₂₃	1	8	yes	—
134	C₂²	C₂⁴×C₄×C₂₃	1	6	yes	—
135	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	12	yes	—
136	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
137	C₄	—	2	32	no	—
138	C₄	C₂⁴×C₄×C₂₃	1	48	yes	—
139	C₂	—	2	8	no	—
140	C₂	C₂×C₂₃×#519	1	12	yes	direct product factor
141	C₂	C₂⁵×C₄×C₂₃	1	1	yes	characteristic
142	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#519	C₂	direct (N × H)
3. C₂²×C₄×Q₈×C₂₃	C₂	semidirect
5. C₂³×Q₈×C₂₃	C₄	semidirect
9. C₂₃×#519	C₂²	direct (N × H)
10. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
17. C₂²×Q₈×C₂₃	C₂×C₄	semidirect
24. C₄×Q₈×C₂₃	C₂³	semidirect
34. C₂×Q₈×C₂₃	C₂²×C₄	semidirect
45. C₂²×#519	C₂₃	direct (N × H)
60. C₂×#519	C₂×C₂₃	direct (N × H)
61. C₂²×C₄×Q₈	C₂×C₂₃	semidirect
65. C₂²×C₂₃	#519	direct (N × H)
72. C₂³×Q₈	C₄×C₂₃	semidirect
76. #519	C₂²×C₂₃	direct (N × H)
77. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
83. C₂×C₂₃	C₂×#519	direct (N × H)
88. C₂²×Q₈	C₂×C₄×C₂₃	semidirect
95. C₄×Q₈	C₂³×C₂₃	semidirect
100. C₂₃	C₂²×#519	direct (N × H)
106. C₂×Q₈	C₂²×C₄×C₂₃	semidirect
132. C₂²	C₂₃×#519	direct (N × H)
140. C₂	C₂×C₂₃×#519	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
C₂×C₂₃×#519	2
C₂²×C₄×Q₈×C₂₃	2
C₂²×C₂₃×#117	2
C₂²×#519	23

Subgroups — C22×C23×#519 #19210272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#519 #19210272