Subgroups — C₂²×C₂₃×#498 #19208278

Order 5888.

2398 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×C₂₃×#498	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₂₃×#106	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₂₃	—	2	2	no	—
6	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₂₃×C₄⋊Q₈	—	2	12	no	—
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	24	yes	—
9	C₂²×C₄⋊C₄×C₂₃	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
11	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#498	C₂²	1	64	yes	direct product factor
13	C₂×C₂₃×#106	C₂²	1	48	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	12	yes	—
15	C₂²×Q₈×C₂₃	—	2	24	no	—
16	C₂²×Q₈×C₂₃	—	4	16	no	—
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	16	yes	complemented
21	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
22	C₂₃×C₄⋊Q₈	—	2	48	no	—
23	C₂₃×C₄⋊Q₈	C₂³	1	32	yes	—
24	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
26	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
27	C₂×C₄²×C₂₃	D₄	1	6	yes	—
28	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
29	C₂₃×#106	C₂³	1	64	yes	—
30	C₄×C₈×C₂₃	C₂³	1	16	yes	—
31	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×Q₈×C₂₃	—	2	32	no	—
33	C₂×Q₈×C₂₃	—	4	96	no	—
34	C₂²×C₄×C₂₃	—	2	24	no	—
35	C₂²×C₄×C₂₃	—	4	16	no	—
36	C₂²×C₄×C₂₃	SD₁₆	1	16	yes	—
37	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
39	C₂²×C₄×C₂₃	C₂×Q₈	1	8	yes	—
40	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
41	C₂×C₈×C₂₃	C₂×Q₈	1	48	yes	complemented
42	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
43	C₄⋊C₄×C₂₃	—	2	32	no	—
44	C₄⋊C₄×C₂₃	—	2	24	no	—
45	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
46	C₄²×C₂₃	C₂×D₄	1	12	yes	—
47	C₄²×C₂₃	C₂⁴	1	4	yes	—
48	C₂²×#498	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₂₃	C₄⋊Q₈	1	4	yes	—
50	C₂³×C₂₃	C₂×SD₁₆	1	8	yes	—
51	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
52	Q₈×C₂₃	—	4	64	no	—
53	C₂×C₄×C₂₃	—	2	16	no	—
54	C₂×C₄×C₂₃	—	4	48	no	—
55	C₂×C₄×C₂₃	C₄⋊Q₈	1	12	yes	—
56	C₂×C₄×C₂₃	C₂×SD₁₆	1	48	yes	—
57	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
58	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
59	C₂×C₄×C₂₃	C₂²×Q₈	1	12	yes	—
60	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
61	C₈×C₂₃	C₂²×Q₈	1	16	yes	complemented
62	C₂²×C₄⋊Q₈	C₂×C₂₃	1	2	yes	—
63	C₂×#498	C₂×C₂₃	1	24	yes	direct product factor
64	C₂²×#106	C₂×C₂₃	1	4	yes	—
65	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
66	C₂²×C₂₃	#498	1	16	yes	direct product factor
67	C₂²×C₂₃	C₂×C₄⋊Q₈	1	6	yes	—
68	C₂²×C₂₃	C₂²×SD₁₆	1	12	yes	—
69	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
70	C₄×C₂₃	—	4	16	no	—
71	C₄×C₂₃	C₂×C₄⋊Q₈	1	6	yes	—
72	C₄×C₂₃	C₂²×SD₁₆	1	16	yes	—
73	C₄×C₂₃	C₂³×Q₈	1	2	yes	—
74	C₂³×Q₈	—	2	2	no	—
75	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
76	C₂×C₄⋊Q₈	—	2	12	no	—
77	C₂×C₄⋊Q₈	C₂²×C₂₃	1	24	yes	—
78	C₂²×C₄⋊C₄	—	2	2	no	—
79	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
80	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
81	#498	C₂²×C₂₃	1	64	yes	direct product factor
82	C₂×#106	C₂²×C₂₃	1	48	yes	—
83	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	—
84	C₂×C₂₃	C₂×#498	1	12	yes	direct product factor
85	C₂×C₂₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₂₃	C₂³×SD₁₆	1	2	yes	—
87	C₂²×Q₈	—	2	24	no	—
88	C₂²×Q₈	—	4	16	no	—
89	C₂³×C₄	—	2	4	no	—
90	C₂³×C₄	D₄×C₂₃	1	2	yes	—
91	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
92	C₂²×C₈	Q₈×C₂₃	1	16	yes	complemented
93	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
94	C₄⋊Q₈	—	2	48	no	—
95	C₄⋊Q₈	C₂³×C₂₃	1	32	yes	—
96	C₂×C₄⋊C₄	—	2	24	no	—
97	C₂×C₄⋊C₄	—	2	12	no	—
98	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
99	C₂×C₄²	D₄×C₂₃	1	6	yes	—
100	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
101	#106	C₂³×C₂₃	1	64	yes	—
102	C₄×C₈	C₂³×C₂₃	1	16	yes	—
103	C₂₃	C₂²×#498	1	1	yes	characteristic, direct product factor, Sylow
104	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
105	C₂×Q₈	—	2	32	no	—
106	C₂×Q₈	—	4	96	no	—
107	C₂²×C₄	—	2	24	no	—
108	C₂²×C₄	—	4	16	no	—
109	C₂²×C₄	SD₁₆×C₂₃	1	16	yes	—
110	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
111	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
112	C₂²×C₄	C₂×Q₈×C₂₃	1	8	yes	—
113	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
114	C₂×C₈	C₂×Q₈×C₂₃	1	48	yes	complemented
115	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
116	C₄⋊C₄	—	2	32	no	—
117	C₄⋊C₄	—	2	24	no	—
118	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
119	C₄²	C₂×D₄×C₂₃	1	12	yes	—
120	C₄²	C₂⁴×C₂₃	1	4	yes	—
121	C₂³	C₂₃×C₄⋊Q₈	1	4	yes	—
122	C₂³	C₂×SD₁₆×C₂₃	1	8	yes	—
123	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
124	Q₈	—	4	64	no	—
125	C₂×C₄	—	2	16	no	—
126	C₂×C₄	—	4	48	no	—
127	C₂×C₄	C₂₃×C₄⋊Q₈	1	12	yes	—
128	C₂×C₄	C₂×SD₁₆×C₂₃	1	48	yes	—
129	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
130	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
131	C₂×C₄	C₂²×Q₈×C₂₃	1	12	yes	—
132	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
133	C₈	C₂²×Q₈×C₂₃	1	16	yes	complemented
134	C₂²	C₂₃×#498	1	16	yes	direct product factor
135	C₂²	C₂×C₂₃×C₄⋊Q₈	1	6	yes	—
136	C₂²	C₂²×SD₁₆×C₂₃	1	12	yes	—
137	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
138	C₄	—	4	16	no	—
139	C₄	C₂×C₂₃×C₄⋊Q₈	1	6	yes	—
140	C₄	C₂²×SD₁₆×C₂₃	1	16	yes	—
141	C₄	C₂³×Q₈×C₂₃	1	2	yes	—
142	C₂	C₂×C₂₃×#498	1	12	yes	direct product factor
143	C₂	C₂²×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
144	C₂	C₂³×SD₁₆×C₂₃	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×#498	C₂	direct (N × H)
12. C₂₃×#498	C₂²	direct (N × H)
20. C₂²×C₈×C₂₃	Q₈	semidirect
41. C₂×C₈×C₂₃	C₂×Q₈	semidirect
48. C₂²×#498	C₂₃	direct (N × H)
61. C₈×C₂₃	C₂²×Q₈	semidirect
63. C₂×#498	C₂×C₂₃	direct (N × H)
66. C₂²×C₂₃	#498	direct (N × H)
81. #498	C₂²×C₂₃	direct (N × H)
84. C₂×C₂₃	C₂×#498	direct (N × H)
92. C₂²×C₈	Q₈×C₂₃	semidirect
103. C₂₃	C₂²×#498	direct (N × H)
114. C₂×C₈	C₂×Q₈×C₂₃	semidirect
133. C₈	C₂²×Q₈×C₂₃	semidirect
134. C₂²	C₂₃×#498	direct (N × H)
142. C₂	C₂×C₂₃×#498	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×C₄⋊Q₈	2
C₂×C₂₃×#498	2
C₂²×C₂₃×#106	2
C₂²×C₄×C₈×C₂₃	2
C₂²×#498	23

Subgroups — C22×C23×#498 #19208278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#498 #19208278