Subgroups — C₂²×C₂₃×#499 #19208279

Order 5888.

1966 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#125	C₂	1	2	yes	maximal
2	C₂×C₂₃×#499	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₂₃×#107	C₂	1	2	yes	maximal
4	C₂²×C₂₃×#106	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#125	—	2	12	no	—
9	C₂×C₂₃×#125	C₂²	1	24	yes	—
10	C₂²×C₄⋊C₄×C₂₃	—	2	4	no	—
11	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
12	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#499	C₂²	1	64	yes	direct product factor
14	C₂×C₂₃×#107	C₂²	1	24	yes	—
15	C₂×C₂₃×#106	C₂²	1	24	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	12	yes	—
17	C₂³×C₄×C₂₃	—	2	4	no	—
18	C₂³×C₄×C₂₃	D₄	1	2	yes	—
19	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	Q₈	1	8	yes	—
21	C₂²×C₈×C₂₃	Q₈	1	8	yes	—
22	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
23	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
24	C₂₃×#125	—	2	48	no	—
25	C₂₃×#125	C₂³	1	32	yes	—
26	C₂×C₄⋊C₄×C₂₃	—	2	48	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
29	C₂×C₄²×C₂₃	D₄	1	6	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
31	C₂₃×#107	C₂³	1	32	yes	—
32	C₂₃×#106	C₂³	1	32	yes	—
33	C₄×C₈×C₂₃	C₂³	1	16	yes	—
34	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	—	2	8	no	—
36	C₂²×C₄×C₂₃	—	2	24	no	—
37	C₂²×C₄×C₂₃	—	4	16	no	—
38	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
43	C₂×C₈×C₂₃	C₂×Q₈	1	24	yes	—
44	C₂×C₈×C₂₃	C₂×Q₈	1	24	yes	—
45	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
46	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
47	C₄⋊C₄×C₂₃	—	2	64	no	—
48	C₄⋊C₄×C₂₃	—	2	24	no	—
49	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
50	C₄²×C₂₃	C₂×D₄	1	12	yes	—
51	C₄²×C₂₃	C₂⁴	1	4	yes	—
52	C₂²×#499	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₂₃	C₄⋊Q₈	1	4	yes	—
54	C₂³×C₂₃	#135	1	8	yes	—
55	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
56	C₂×C₄×C₂₃	—	2	24	no	—
57	C₂×C₄×C₂₃	—	2	16	no	—
58	C₂×C₄×C₂₃	—	4	48	no	—
59	C₂×C₄×C₂₃	C₄⋊Q₈	1	12	yes	—
60	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
61	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
62	C₂×C₄×C₂₃	C₂²×Q₈	1	6	yes	—
63	C₂×C₄×C₂₃	C₂²×Q₈	1	6	yes	—
64	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
65	C₈×C₂₃	C₂²×Q₈	1	8	yes	—
66	C₈×C₂₃	C₂²×Q₈	1	8	yes	—
67	C₂²×#125	C₂×C₂₃	1	2	yes	—
68	C₂×#499	C₂×C₂₃	1	24	yes	direct product factor
69	C₂²×#107	C₂×C₂₃	1	2	yes	—
70	C₂²×#106	C₂×C₂₃	1	2	yes	—
71	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
72	C₂²×C₂₃	#499	1	16	yes	direct product factor
73	C₂²×C₂₃	C₂×C₄⋊Q₈	1	6	yes	—
74	C₂²×C₂₃	C₂×#135	1	12	yes	—
75	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
76	C₄×C₂₃	—	2	8	no	—
77	C₄×C₂₃	—	4	16	no	—
78	C₄×C₂₃	C₂×C₄⋊Q₈	1	6	yes	—
79	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
80	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
81	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
82	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
83	C₂×#125	—	2	12	no	—
84	C₂×#125	C₂²×C₂₃	1	24	yes	—
85	C₂²×C₄⋊C₄	—	2	4	no	—
86	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
87	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
88	#499	C₂²×C₂₃	1	64	yes	direct product factor
89	C₂×#107	C₂²×C₂₃	1	24	yes	—
90	C₂×#106	C₂²×C₂₃	1	24	yes	—
91	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	—
92	C₂×C₂₃	C₂×#499	1	12	yes	direct product factor
93	C₂×C₂₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₂₃	C₂²×#135	1	2	yes	—
95	C₂³×C₄	—	2	4	no	—
96	C₂³×C₄	D₄×C₂₃	1	2	yes	—
97	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
98	C₂²×C₈	Q₈×C₂₃	1	8	yes	—
99	C₂²×C₈	Q₈×C₂₃	1	8	yes	—
100	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
101	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
102	#125	—	2	48	no	—
103	#125	C₂³×C₂₃	1	32	yes	—
104	C₂×C₄⋊C₄	—	2	48	no	—
105	C₂×C₄⋊C₄	—	2	12	no	—
106	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
107	C₂×C₄²	D₄×C₂₃	1	6	yes	—
108	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
109	#107	C₂³×C₂₃	1	32	yes	—
110	#106	C₂³×C₂₃	1	32	yes	—
111	C₄×C₈	C₂³×C₂₃	1	16	yes	—
112	C₂₃	C₂²×#499	1	1	yes	characteristic, direct product factor, Sylow
113	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	24	no	—
116	C₂²×C₄	—	4	16	no	—
117	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
118	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
119	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
120	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
121	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
122	C₂×C₈	C₂×Q₈×C₂₃	1	24	yes	—
123	C₂×C₈	C₂×Q₈×C₂₃	1	24	yes	—
124	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
125	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
126	C₄⋊C₄	—	2	64	no	—
127	C₄⋊C₄	—	2	24	no	—
128	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
129	C₄²	C₂×D₄×C₂₃	1	12	yes	—
130	C₄²	C₂⁴×C₂₃	1	4	yes	—
131	C₂³	C₂₃×C₄⋊Q₈	1	4	yes	—
132	C₂³	C₂₃×#135	1	8	yes	—
133	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
134	C₂×C₄	—	2	24	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	4	48	no	—
137	C₂×C₄	C₂₃×C₄⋊Q₈	1	12	yes	—
138	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
139	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
140	C₂×C₄	C₂²×Q₈×C₂₃	1	6	yes	—
141	C₂×C₄	C₂²×Q₈×C₂₃	1	6	yes	—
142	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
143	C₈	C₂²×Q₈×C₂₃	1	8	yes	—
144	C₈	C₂²×Q₈×C₂₃	1	8	yes	—
145	C₂²	C₂₃×#499	1	16	yes	direct product factor
146	C₂²	C₂×C₂₃×C₄⋊Q₈	1	6	yes	—
147	C₂²	C₂×C₂₃×#135	1	12	yes	—
148	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
149	C₄	—	2	8	no	—
150	C₄	—	4	16	no	—
151	C₄	C₂×C₂₃×C₄⋊Q₈	1	6	yes	—
152	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
153	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
154	C₂	C₂×C₂₃×#499	1	12	yes	direct product factor
155	C₂	C₂²×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
156	C₂	C₂²×C₂₃×#135	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#499	C₂	direct (N × H)
13. C₂₃×#499	C₂²	direct (N × H)
52. C₂²×#499	C₂₃	direct (N × H)
68. C₂×#499	C₂×C₂₃	direct (N × H)
72. C₂²×C₂₃	#499	direct (N × H)
88. #499	C₂²×C₂₃	direct (N × H)
92. C₂×C₂₃	C₂×#499	direct (N × H)
112. C₂₃	C₂²×#499	direct (N × H)
145. C₂²	C₂₃×#499	direct (N × H)
154. C₂	C₂×C₂₃×#499	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#125	2
C₂×C₂₃×#499	2
C₂²×C₂₃×#107	2
C₂²×C₂₃×#106	2
C₂²×C₄×C₈×C₂₃	2
C₂²×#499	23

Subgroups — C22×C23×#499 #19208279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#499 #19208279