Subgroups — C₂²×C₂₃×#500 #19208280

Order 5888.

2398 conjugacy classes of subgroups.

The subgroup lattice has 164 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×#500	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₂₃×#107	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×#500	C₂²	1	64	yes	direct product factor
13	C₂×C₂₃×#107	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	1	yes	characteristic
19	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	Q₈	1	16	yes	complemented
22	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
23	C₂₃×C₄⋊Q₈	—	2	48	no	—
24	C₂₃×C₄⋊Q₈	C₂³	1	32	yes	—
25	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
27	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
28	C₂×C₄²×C₂₃	D₄	1	6	yes	—
29	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
30	C₂₃×#107	C₂³	1	64	yes	—
31	C₄×C₈×C₂₃	C₂³	1	16	yes	—
32	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Q₈×C₂₃	—	2	32	no	—
34	C₂×Q₈×C₂₃	—	4	96	no	—
35	C₂²×C₄×C₂₃	—	2	24	no	—
36	C₂²×C₄×C₂₃	—	4	16	no	—
37	C₂²×C₄×C₂₃	D₈	1	8	yes	—
38	C₂²×C₄×C₂₃	Q₁₆	1	8	yes	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₂₃	C₂×Q₈	1	8	yes	—
43	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
44	C₂×C₈×C₂₃	C₂×Q₈	1	48	yes	complemented
45	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
46	C₄⋊C₄×C₂₃	—	2	32	no	—
47	C₄⋊C₄×C₂₃	—	2	24	no	—
48	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
49	C₄²×C₂₃	C₂×D₄	1	12	yes	—
50	C₄²×C₂₃	C₂⁴	1	4	yes	—
51	C₂²×#500	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	C₄⋊Q₈	1	4	yes	—
53	C₂³×C₂₃	C₂×D₈	1	4	yes	—
54	C₂³×C₂₃	C₂×Q₁₆	1	4	yes	—
55	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
56	Q₈×C₂₃	—	4	64	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	C₂×D₈	1	24	yes	—
61	C₂×C₄×C₂₃	C₂×Q₁₆	1	24	yes	—
62	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
63	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
64	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
65	C₂×C₄×C₂₃	C₂²×Q₈	1	12	yes	—
66	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
67	C₈×C₂₃	C₂²×Q₈	1	16	yes	complemented
68	C₂²×C₄⋊Q₈	C₂×C₂₃	1	2	yes	—
69	C₂×#500	C₂×C₂₃	1	24	yes	direct product factor
70	C₂²×#107	C₂×C₂₃	1	4	yes	—
71	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
72	C₂²×C₂₃	#500	1	16	yes	direct product factor
73	C₂²×C₂₃	C₂×C₄⋊Q₈	1	6	yes	—
74	C₂²×C₂₃	C₂²×D₈	1	6	yes	—
75	C₂²×C₂₃	C₂²×Q₁₆	1	6	yes	—
76	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
77	C₄×C₂₃	—	4	16	no	—
78	C₄×C₂₃	C₂×C₄⋊Q₈	1	6	yes	—
79	C₄×C₂₃	C₂²×D₈	1	8	yes	—
80	C₄×C₂₃	C₂²×Q₁₆	1	8	yes	—
81	C₄×C₂₃	C₂³×Q₈	1	2	yes	—
82	C₂³×Q₈	—	2	2	no	—
83	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
84	C₂×C₄⋊Q₈	—	2	12	no	—
85	C₂×C₄⋊Q₈	C₂²×C₂₃	1	24	yes	—
86	C₂²×C₄⋊C₄	—	2	2	no	—
87	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
88	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
89	#500	C₂²×C₂₃	1	64	yes	direct product factor
90	C₂×#107	C₂²×C₂₃	1	48	yes	—
91	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	—
92	C₂×C₂₃	C₂×#500	1	12	yes	direct product factor
93	C₂×C₂₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
95	C₂×C₂₃	C₂³×Q₁₆	1	1	yes	characteristic
96	C₂²×Q₈	—	2	24	no	—
97	C₂²×Q₈	—	4	16	no	—
98	C₂³×C₄	—	2	4	no	—
99	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
100	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
101	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
102	C₂²×C₈	Q₈×C₂₃	1	16	yes	complemented
103	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
104	C₄⋊Q₈	—	2	48	no	—
105	C₄⋊Q₈	C₂³×C₂₃	1	32	yes	—
106	C₂×C₄⋊C₄	—	2	24	no	—
107	C₂×C₄⋊C₄	—	2	12	no	—
108	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
109	C₂×C₄²	D₄×C₂₃	1	6	yes	—
110	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
111	#107	C₂³×C₂₃	1	64	yes	—
112	C₄×C₈	C₂³×C₂₃	1	16	yes	—
113	C₂₃	C₂²×#500	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
115	C₂×Q₈	—	2	32	no	—
116	C₂×Q₈	—	4	96	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	—	4	16	no	—
119	C₂²×C₄	D₈×C₂₃	1	8	yes	—
120	C₂²×C₄	Q₁₆×C₂₃	1	8	yes	—
121	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
122	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
123	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
124	C₂²×C₄	C₂×Q₈×C₂₃	1	8	yes	—
125	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
126	C₂×C₈	C₂×Q₈×C₂₃	1	48	yes	complemented
127	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
128	C₄⋊C₄	—	2	32	no	—
129	C₄⋊C₄	—	2	24	no	—
130	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
131	C₄²	C₂×D₄×C₂₃	1	12	yes	—
132	C₄²	C₂⁴×C₂₃	1	4	yes	—
133	C₂³	C₂₃×C₄⋊Q₈	1	4	yes	—
134	C₂³	C₂×D₈×C₂₃	1	4	yes	—
135	C₂³	C₂×Q₁₆×C₂₃	1	4	yes	—
136	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
137	Q₈	—	4	64	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	—	4	48	no	—
140	C₂×C₄	C₂₃×C₄⋊Q₈	1	12	yes	—
141	C₂×C₄	C₂×D₈×C₂₃	1	24	yes	—
142	C₂×C₄	C₂×Q₁₆×C₂₃	1	24	yes	—
143	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
144	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
145	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
146	C₂×C₄	C₂²×Q₈×C₂₃	1	12	yes	—
147	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
148	C₈	C₂²×Q₈×C₂₃	1	16	yes	complemented
149	C₂²	C₂₃×#500	1	16	yes	direct product factor
150	C₂²	C₂×C₂₃×C₄⋊Q₈	1	6	yes	—
151	C₂²	C₂²×D₈×C₂₃	1	6	yes	—
152	C₂²	C₂²×Q₁₆×C₂₃	1	6	yes	—
153	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
154	C₄	—	4	16	no	—
155	C₄	C₂×C₂₃×C₄⋊Q₈	1	6	yes	—
156	C₄	C₂²×D₈×C₂₃	1	8	yes	—
157	C₄	C₂²×Q₁₆×C₂₃	1	8	yes	—
158	C₄	C₂³×Q₈×C₂₃	1	2	yes	—
159	C₂	C₂×C₂₃×#500	1	12	yes	direct product factor
160	C₂	C₂²×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
161	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic
162	C₂	C₂³×Q₁₆×C₂₃	1	1	yes	characteristic

	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×#500	C₂	direct (N × H)
12. C₂₃×#500	C₂²	direct (N × H)
21. C₂²×C₈×C₂₃	Q₈	semidirect
44. C₂×C₈×C₂₃	C₂×Q₈	semidirect
51. C₂²×#500	C₂₃	direct (N × H)
67. C₈×C₂₃	C₂²×Q₈	semidirect
69. C₂×#500	C₂×C₂₃	direct (N × H)
72. C₂²×C₂₃	#500	direct (N × H)
89. #500	C₂²×C₂₃	direct (N × H)
92. C₂×C₂₃	C₂×#500	direct (N × H)
102. C₂²×C₈	Q₈×C₂₃	semidirect
113. C₂₃	C₂²×#500	direct (N × H)
126. C₂×C₈	C₂×Q₈×C₂₃	semidirect
148. C₈	C₂²×Q₈×C₂₃	semidirect
149. C₂²	C₂₃×#500	direct (N × H)
159. C₂	C₂×C₂₃×#500	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×#500	2
C₂²×C₂₃×#107	2
C₂²×C₄×C₈×C₂₃	2
C₂²×#500	23

Subgroups — C22×C23×#500 #19208280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#500 #19208280