Subgroups — C₂₃×#13281 #19160907

Order 5888.

366 conjugacy classes of subgroups.

The subgroup lattice has 172 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	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1380	C₂	1	4	yes	maximal
5	C₂×C₂₃×#125	—	2	1	no	—
6	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂₃×#499	—	2	2	no	—
8	C₂₃×#499	C₄	1	4	yes	complemented
9	C₂×C₂₃×#107	C₂²	1	2	yes	—
10	C₂×C₂₃×#107	C₂²	1	2	yes	—
11	C₂×C₂₃×#106	—	2	1	no	—
12	C₂×C₂₃×#106	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#382	—	2	1	no	—
15	C₄×C₁₆×C₂₃	—	2	2	no	—
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
18	C₂₃×#125	—	2	4	no	—
19	C₂₃×#125	—	4	2	no	—
20	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
23	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₂₃	—	2	2	no	—
25	C₂×C₁₆×C₂₃	—	2	2	no	—
26	C₂×C₁₆×C₂₃	Q₈	1	4	yes	—
27	C₂₃×#107	—	2	2	no	—
28	C₂₃×#107	—	2	4	no	—
29	C₂₃×#107	C₂×C₄	1	4	yes	—
30	C₂₃×#106	—	2	4	no	—
31	C₂₃×#106	—	4	2	no	—
32	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₂₃	D₄	1	2	yes	—
34	C₂₃×#95	—	4	2	no	—
35	C₂²×C₄×C₂₃	—	4	2	no	—
36	C₂²×C₄×C₂₃	—	4	2	no	—
37	C₂²×C₄×C₂₃	D₈	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₂₃	#41	1	2	yes	—
46	C₄⋊C₄×C₂₃	—	2	4	no	—
47	C₄⋊C₄×C₂₃	—	4	8	no	—
48	C₄⋊C₄×C₂₃	—	4	2	no	—
49	C₄⋊C₄×C₂₃	—	4	2	no	—
50	C₄⋊C₄×C₂₃	—	4	2	no	—
51	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₂₃	SD₁₆	1	2	yes	—
53	C₁₆×C₂₃	—	2	4	no	—
54	#13281	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	—	4	4	no	—
58	C₂×C₄×C₂₃	—	8	4	no	—
59	C₂×C₄×C₂₃	—	8	2	no	—
60	C₂×C₄×C₂₃	—	8	4	no	—
61	C₂×C₄×C₂₃	#102	1	4	yes	—
62	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₂₃	#103	1	4	yes	—
69	C₈×C₂₃	C₄×Q₈	1	2	yes	—
70	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
72	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
73	#1846	C₂×C₂₃	1	1	yes	characteristic
74	#1380	C₂×C₂₃	1	4	yes	—
75	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
76	C₂²×C₂₃	#486	1	1	yes	characteristic
77	C₂²×C₂₃	#500	1	1	yes	characteristic
78	C₂²×C₂₃	#508	1	4	yes	—
79	C₄×C₂₃	—	2	4	no	—
80	C₄×C₂₃	—	8	4	no	—
81	C₄×C₂₃	—	16	2	no	—
82	C₄×C₂₃	#391	1	1	yes	characteristic
83	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
84	C₄×C₂₃	#487	1	1	yes	characteristic
85	C₄×C₂₃	#498	1	1	yes	characteristic
86	C₂×#125	—	2	1	no	—
87	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
88	#499	—	2	2	no	—
89	#499	C₄×C₂₃	1	4	yes	complemented
90	C₂×#107	C₂²×C₂₃	1	2	yes	—
91	C₂×#107	C₂²×C₂₃	1	2	yes	—
92	C₂×#106	—	2	1	no	—
93	C₂×#106	—	2	1	no	—
94	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
95	#382	—	2	1	no	—
96	C₄×C₁₆	—	2	2	no	—
97	C₂×C₂₃	#1990	1	1	yes	characteristic
98	C₂×C₂₃	#2138	1	2	yes	—
99	C₂×C₂₃	#2242	1	2	yes	—
100	C₂×C₂₃	#2253	1	2	yes	—
101	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
103	#125	—	2	4	no	—
104	#125	—	4	2	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	Q₈×C₂₃	1	4	yes	—
112	#107	—	2	2	no	—
113	#107	—	2	4	no	—
114	#107	C₂×C₄×C₂₃	1	4	yes	—
115	#106	—	2	4	no	—
116	#106	—	4	2	no	—
117	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
118	C₄×C₈	D₄×C₂₃	1	2	yes	—
119	#95	—	4	2	no	—
120	C₂₃	#13281	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	D₈×C₂₃	1	2	yes	—
124	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
126	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
127	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
128	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
129	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
131	C₂×C₈	#41×C₂₃	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	8	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
138	C₄²	SD₁₆×C₂₃	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	C₂₃×#102	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
149	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
153	C₈	C₂₃×#103	1	4	yes	—
154	C₈	C₄×Q₈×C₂₃	1	2	yes	—
155	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
156	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
157	C₂²	C₂₃×#486	1	1	yes	characteristic
158	C₂²	C₂₃×#500	1	1	yes	characteristic
159	C₂²	C₂₃×#508	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	8	4	no	—
162	C₄	—	16	2	no	—
163	C₄	C₂₃×#391	1	1	yes	characteristic
164	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
165	C₄	C₂₃×#487	1	1	yes	characteristic
166	C₄	C₂₃×#498	1	1	yes	characteristic
167	C₂	C₂₃×#1990	1	1	yes	characteristic
168	C₂	C₂₃×#2138	1	2	yes	—
169	C₂	C₂₃×#2242	1	2	yes	—
170	C₂	C₂₃×#2253	1	2	yes	—

	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
8. C₂₃×#499	C₄	semidirect
54. #13281	C₂₃	direct (N × H)
89. #499	C₄×C₂₃	semidirect
120. C₂₃	#13281	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
C₂×C₄×C₁₆×C₂₃	2
C₂₃×#1846	2
C₂₃×#1380	2
#13281	23

Subgroups — C23×#13281 #19160907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13281 #19160907