Subgroups — C₂₃×#12890 #19160516

Order 5888.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1380	C₂	1	4	yes	maximal
5	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
6	C₂×C₂₃×#107	—	2	1	no	—
7	C₂×C₂₃×#107	C₂²	1	2	yes	—
8	C₂×C₂₃×#107	C₂²	1	2	yes	—
9	C₂×C₂₃×#106	—	2	1	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#384	—	2	1	no	—
12	C₂₃×#382	—	2	1	no	—
13	C₂₃×#346	C₄	1	4	yes	complemented
14	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
19	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₂₃	—	2	2	no	—
21	C₂×C₁₆×C₂₃	—	2	2	no	—
22	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
23	C₂₃×#107	—	2	4	no	—
24	C₂₃×#107	—	2	4	no	—
25	C₂₃×#107	—	4	2	no	—
26	C₂₃×#106	—	4	2	no	—
27	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₂₃	Q₈	1	2	yes	—
29	C₂₃×#95	—	4	2	no	—
30	C₂²×C₄×C₂₃	—	4	2	no	—
31	C₂²×C₄×C₂₃	—	4	2	no	—
32	C₂²×C₄×C₂₃	D₈	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
36	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₂₃	#41	1	2	yes	—
40	C₂×C₈×C₂₃	#41	1	2	yes	—
41	C₄⋊C₄×C₂₃	—	4	2	no	—
42	C₄⋊C₄×C₂₃	—	4	2	no	—
43	C₄⋊C₄×C₂₃	—	4	2	no	—
44	C₄⋊C₄×C₂₃	—	4	2	no	—
45	C₄⋊C₄×C₂₃	—	8	4	no	—
46	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₂₃	Q₁₆	1	2	yes	—
48	C₁₆×C₂₃	—	2	4	no	—
49	#12890	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	8	2	no	—
54	C₂×C₄×C₂₃	—	8	2	no	—
55	C₂×C₄×C₂₃	—	8	4	no	—
56	C₂×C₄×C₂₃	—	8	2	no	—
57	C₂×C₄×C₂₃	—	8	2	no	—
58	C₂×C₄×C₂₃	#107	1	4	yes	—
59	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₂₃	#106	1	4	yes	—
66	C₈×C₂₃	#117	1	2	yes	—
67	C₈×C₂₃	#125	1	2	yes	—
68	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
69	#1846	C₂×C₂₃	1	1	yes	characteristic
70	#1845	C₂×C₂₃	1	1	yes	characteristic
71	#1380	C₂×C₂₃	1	4	yes	—
72	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
73	C₂²×C₂₃	#486	1	1	yes	characteristic
74	C₂²×C₂₃	#486	1	1	yes	characteristic
75	C₂²×C₂₃	#509	1	2	yes	—
76	C₂²×C₂₃	#510	1	2	yes	—
77	C₄×C₂₃	—	2	4	no	—
78	C₄×C₂₃	—	16	2	no	—
79	C₄×C₂₃	—	16	2	no	—
80	C₄×C₂₃	#382	1	1	yes	characteristic
81	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
82	C₄×C₂₃	#487	1	1	yes	characteristic
83	C₄×C₂₃	#487	1	1	yes	characteristic
84	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
85	C₂×#107	—	2	1	no	—
86	C₂×#107	C₂²×C₂₃	1	2	yes	—
87	C₂×#107	C₂²×C₂₃	1	2	yes	—
88	C₂×#106	—	2	1	no	—
89	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
90	#384	—	2	1	no	—
91	#382	—	2	1	no	—
92	#346	C₄×C₂₃	1	4	yes	complemented
93	C₂×C₂₃	#1834	1	1	yes	characteristic
94	C₂×C₂₃	#2159	1	2	yes	—
95	C₂×C₂₃	#2243	1	2	yes	—
96	C₂×C₂₃	#2244	1	1	yes	characteristic
97	C₂×C₂₃	#2245	1	1	yes	characteristic
98	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	4	2	no	—
103	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
107	#107	—	2	4	no	—
108	#107	—	2	4	no	—
109	#107	—	4	2	no	—
110	#106	—	4	2	no	—
111	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
112	C₄×C₈	Q₈×C₂₃	1	2	yes	—
113	#95	—	4	2	no	—
114	C₂₃	#12890	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	D₈×C₂₃	1	2	yes	—
118	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
120	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
121	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
122	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
124	C₂×C₈	#41×C₂₃	1	2	yes	—
125	C₂×C₈	#41×C₂₃	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
132	C₄²	Q₁₆×C₂₃	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	C₂₃×#107	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
145	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
146	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
148	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
149	C₈	C₂₃×#106	1	4	yes	—
150	C₈	C₂₃×#117	1	2	yes	—
151	C₈	C₂₃×#125	1	2	yes	—
152	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
153	C₂²	C₂₃×#486	1	1	yes	characteristic
154	C₂²	C₂₃×#486	1	1	yes	characteristic
155	C₂²	C₂₃×#509	1	2	yes	—
156	C₂²	C₂₃×#510	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	16	2	no	—
160	C₄	C₂₃×#382	1	1	yes	characteristic
161	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
162	C₄	C₂₃×#487	1	1	yes	characteristic
163	C₄	C₂₃×#487	1	1	yes	characteristic
164	C₂	C₂₃×#1834	1	1	yes	characteristic
165	C₂	C₂₃×#2159	1	2	yes	—
166	C₂	C₂₃×#2243	1	2	yes	—
167	C₂	C₂₃×#2244	1	1	yes	characteristic
168	C₂	C₂₃×#2245	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×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
13. C₂₃×#346	C₄	semidirect
49. #12890	C₂₃	direct (N × H)
92. #346	C₄×C₂₃	semidirect
114. C₂₃	#12890	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×#346	2
C₂₃×#1846	2
C₂₃×#1845	2
C₂₃×#1380	2
#12890	23

Subgroups — C23×#12890 #19160516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12890 #19160516