Subgroups — C₂×C₂₃×#2253 #19181350

Order 5888.

494 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	2	yes	maximal
2	C₂₃×#2253	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂₃×#367	C₂	1	2	yes	maximal
4	C₂×C₂₃×#366	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#125	—	2	2	no	—
7	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#499	—	2	4	no	—
10	C₂₃×#499	C₂²	1	8	yes	—
11	C₂×C₂₃×#107	C₂²	1	4	yes	—
12	C₂×C₂₃×#106	—	2	2	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#367	C₂²	1	8	yes	—
15	C₂₃×#366	C₂²	1	8	yes	—
16	C₄×C₁₆×C₂₃	C₂²	1	4	yes	—
17	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂₃×#125	—	2	8	no	—
20	C₂₃×#125	—	4	4	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
23	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₂₃	Q₈	1	4	yes	—
25	C₂×C₁₆×C₂₃	Q₈	1	4	yes	—
26	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
27	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
28	C₂₃×#107	—	2	4	no	—
29	C₂₃×#107	C₂³	1	8	yes	—
30	C₂₃×#106	—	2	8	no	—
31	C₄×C₈×C₂₃	D₄	1	2	yes	—
32	C₄×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	—	4	4	no	—
34	C₂²×C₄×C₂₃	D₈	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₂₃	Q₁₆	1	4	yes	—
37	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
40	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
42	C₄⋊C₄×C₂₃	—	2	8	no	—
43	C₄⋊C₄×C₂₃	—	4	4	no	—
44	C₄⋊C₄×C₂₃	—	4	16	no	—
45	C₄²×C₂₃	D₈	1	2	yes	—
46	C₄²×C₂₃	C₂×D₄	1	2	yes	—
47	C₁₆×C₂₃	C₂×Q₈	1	4	yes	—
48	C₁₆×C₂₃	C₂×Q₈	1	4	yes	—
49	C₂×#2253	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	—	4	8	no	—
53	C₂×C₄×C₂₃	—	8	8	no	—
54	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
55	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
56	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
58	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
59	C₈×C₂₃	C₄⋊Q₈	1	2	yes	—
60	C₈×C₂₃	C₂×Q₁₆	1	4	yes	—
61	C₈×C₂₃	C₂²×Q₈	1	1	yes	characteristic
62	C₈×C₂₃	C₂²×Q₈	1	1	yes	characteristic
63	C₂×#499	C₂×C₂₃	1	2	yes	—
64	#2253	C₂×C₂₃	1	8	yes	direct product factor
65	C₂×#367	C₂×C₂₃	1	2	yes	—
66	C₂×#366	C₂×C₂₃	1	2	yes	—
67	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
68	C₂²×C₂₃	#500	1	2	yes	—
69	C₂²×C₂₃	#508	1	4	yes	—
70	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
71	C₄×C₂₃	—	2	4	no	—
72	C₄×C₂₃	—	8	8	no	—
73	C₄×C₂₃	#500	1	2	yes	—
74	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×#125	—	2	2	no	—
77	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
78	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
79	#499	—	2	4	no	—
80	#499	C₂²×C₂₃	1	8	yes	—
81	C₂×#107	C₂²×C₂₃	1	4	yes	—
82	C₂×#106	—	2	2	no	—
83	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
84	#367	C₂²×C₂₃	1	8	yes	—
85	#366	C₂²×C₂₃	1	8	yes	—
86	C₄×C₁₆	C₂²×C₂₃	1	4	yes	—
87	C₂×C₂₃	#2253	1	4	yes	direct product factor
88	C₂×C₂₃	C₂×#500	1	1	yes	characteristic
89	C₂×C₂₃	C₂×#508	1	2	yes	—
90	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
91	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
92	#125	—	2	8	no	—
93	#125	—	4	4	no	—
94	C₂×C₄⋊C₄	—	2	4	no	—
95	C₂×C₄⋊C₄	—	4	4	no	—
96	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
97	C₂×C₁₆	Q₈×C₂₃	1	4	yes	—
98	C₂×C₁₆	Q₈×C₂₃	1	4	yes	—
99	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
100	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
101	#107	—	2	4	no	—
102	#107	C₂³×C₂₃	1	8	yes	—
103	#106	—	2	8	no	—
104	C₄×C₈	D₄×C₂₃	1	2	yes	—
105	C₄×C₈	C₂³×C₂₃	1	2	yes	—
106	C₂₃	C₂×#2253	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₄	—	4	4	no	—
108	C₂²×C₄	D₈×C₂₃	1	2	yes	—
109	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
110	C₂×C₈	Q₁₆×C₂₃	1	4	yes	—
111	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
113	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
114	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
115	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	8	no	—
117	C₄⋊C₄	—	4	4	no	—
118	C₄⋊C₄	—	4	16	no	—
119	C₄²	D₈×C₂₃	1	2	yes	—
120	C₄²	C₂×D₄×C₂₃	1	2	yes	—
121	C₁₆	C₂×Q₈×C₂₃	1	4	yes	—
122	C₁₆	C₂×Q₈×C₂₃	1	4	yes	—
123	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	8	no	—
126	C₂×C₄	—	8	8	no	—
127	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
128	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
129	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
130	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
131	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
132	C₈	C₂₃×C₄⋊Q₈	1	2	yes	—
133	C₈	C₂×Q₁₆×C₂₃	1	4	yes	—
134	C₈	C₂²×Q₈×C₂₃	1	1	yes	characteristic
135	C₈	C₂²×Q₈×C₂₃	1	1	yes	characteristic
136	C₂²	C₂₃×#500	1	2	yes	—
137	C₂²	C₂₃×#508	1	4	yes	—
138	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	8	8	no	—
141	C₄	C₂₃×#500	1	2	yes	—
142	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
143	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
144	C₂	C₂₃×#2253	1	4	yes	direct product factor
145	C₂	C₂×C₂₃×#500	1	1	yes	characteristic
146	C₂	C₂×C₂₃×#508	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2253	C₂	direct (N × H)
49. C₂×#2253	C₂₃	direct (N × H)
64. #2253	C₂×C₂₃	direct (N × H)
87. C₂×C₂₃	#2253	direct (N × H)
106. C₂₃	C₂×#2253	direct (N × H)
144. C₂	C₂₃×#2253	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#499	2
C₂₃×#2253	2
C₂×C₂₃×#367	2
C₂×C₂₃×#366	2
C₂×C₄×C₁₆×C₂₃	2
C₂×#2253	23

Subgroups — C2×C23×#2253 #19181350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2253 #19181350