Subgroups — C₂₃×#12556 #19160182

Order 5888.

606 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#427	C₂	1	2	yes	maximal
3	C₂₃×#1380	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
5	C₂×C₂₃×#117	—	2	2	no	—
6	C₂²×C₁₆×C₂₃	—	2	4	no	—
7	C₂²×C₁₆×C₂₃	C₄	1	4	yes	complemented
8	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂₃×#427	—	2	4	no	—
10	C₂₃×#427	C₄	1	8	yes	complemented
11	C₂×C₂₃×#107	C₂²	1	4	yes	—
12	C₂×C₂₃×#106	—	2	2	no	—
13	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	D₄	1	4	yes	—
18	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
19	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
20	C₂₃×#117	—	2	8	no	—
21	C₂₃×#117	—	4	4	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
23	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
24	C₂×C₄²×C₂₃	—	4	2	no	—
25	C₂×C₁₆×C₂₃	—	2	8	no	—
26	C₂×C₁₆×C₂₃	—	2	4	no	—
27	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
28	C₂₃×#107	—	2	4	no	—
29	C₂₃×#107	C₂×C₄	1	8	yes	—
30	C₂₃×#106	—	2	8	no	—
31	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	—	4	4	no	—
33	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
34	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
37	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₄²	1	4	yes	—
40	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
41	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
44	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊C₄×C₂₃	—	2	8	no	—
49	C₄⋊C₄×C₂₃	—	4	4	no	—
50	C₂²⋊C₄×C₂₃	—	4	8	no	—
51	C₄²×C₂₃	—	4	8	no	—
52	C₁₆×C₂₃	—	2	4	no	—
53	#12556	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₂₃	—	2	4	no	—
55	C₂³×C₂₃	#102	1	4	yes	—
56	C₂³×C₂₃	#107	1	2	yes	—
57	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₂₃	—	4	8	no	—
59	C₂×C₄×C₂₃	—	8	8	no	—
60	C₂×C₄×C₂₃	#95	1	4	yes	—
61	C₂×C₄×C₂₃	#102	1	4	yes	—
62	C₂×C₄×C₂₃	#103	1	8	yes	—
63	C₂×C₄×C₂₃	#106	1	4	yes	—
64	C₂×C₄×C₂₃	#107	1	2	yes	—
65	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
68	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₂₃	#95	1	4	yes	—
70	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₁₆	C₂×C₂₃	1	1	yes	characteristic
74	C₂×#427	C₂×C₂₃	1	2	yes	—
75	#1380	C₂×C₂₃	1	4	yes	complemented
76	C₂²×C₂₃	—	2	8	no	—
77	C₂²×C₂₃	—	2	4	no	—
78	C₂²×C₂₃	#340	1	4	yes	—
79	C₂²×C₂₃	C₂×#102	1	2	yes	—
80	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
81	C₂²×C₂₃	#508	1	4	yes	—
82	C₄×C₂₃	—	8	8	no	—
83	C₄×C₂₃	#340	1	4	yes	—
84	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
85	C₄×C₂₃	C₂×#103	1	2	yes	—
86	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
87	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
88	C₂×#117	—	2	2	no	—
89	C₂²×C₁₆	—	2	4	no	—
90	C₂²×C₁₆	C₄×C₂₃	1	4	yes	complemented
91	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
92	#427	—	2	4	no	—
93	#427	C₄×C₂₃	1	8	yes	complemented
94	C₂×#107	C₂²×C₂₃	1	4	yes	—
95	C₂×#106	—	2	2	no	—
96	C₂×C₂₃	—	2	4	no	—
97	C₂×C₂₃	C₂×#340	1	1	yes	characteristic
98	C₂×C₂₃	#2138	1	4	yes	—
99	C₂×C₂₃	#2158	1	2	yes	—
100	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
102	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
103	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
104	C₂²×C₈	D₄×C₂₃	1	4	yes	—
105	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
106	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
107	#117	—	2	8	no	—
108	#117	—	4	4	no	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₂²⋊C₄	—	4	2	no	—
111	C₂×C₄²	—	4	2	no	—
112	C₂×C₁₆	—	2	8	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
115	#107	—	2	4	no	—
116	#107	C₂×C₄×C₂₃	1	8	yes	—
117	#106	—	2	8	no	—
118	C₂₃	#12556	1	1	yes	characteristic, direct product factor, Sylow
119	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
122	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
123	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
124	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
125	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
126	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
127	C₂×C₈	C₄²×C₂₃	1	4	yes	—
128	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
129	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
130	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
131	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
132	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
134	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
135	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₂²⋊C₄	—	4	8	no	—
139	C₄²	—	4	8	no	—
140	C₁₆	—	2	4	no	—
141	C₂³	—	2	4	no	—
142	C₂³	C₂₃×#102	1	4	yes	—
143	C₂³	C₂₃×#107	1	2	yes	—
144	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	8	8	no	—
147	C₂×C₄	C₂₃×#95	1	4	yes	—
148	C₂×C₄	C₂₃×#102	1	4	yes	—
149	C₂×C₄	C₂₃×#103	1	8	yes	—
150	C₂×C₄	C₂₃×#106	1	4	yes	—
151	C₂×C₄	C₂₃×#107	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
155	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
156	C₈	C₂₃×#95	1	4	yes	—
157	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
158	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
159	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
160	C₂²	—	2	4	no	—
161	C₂²	—	2	8	no	—
162	C₂²	C₂₃×#340	1	4	yes	—
163	C₂²	C₂×C₂₃×#102	1	2	yes	—
164	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
165	C₂²	C₂₃×#508	1	4	yes	—
166	C₄	—	8	8	no	—
167	C₄	C₂₃×#340	1	4	yes	—
168	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
169	C₄	C₂×C₂₃×#103	1	2	yes	—
170	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
171	C₂	—	2	4	no	—
172	C₂	C₂×C₂₃×#340	1	1	yes	characteristic
173	C₂	C₂₃×#2138	1	4	yes	—
174	C₂	C₂₃×#2158	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	D₈
commutator	C₈	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×D₈
prime radical	C₂⁴×C₂₃	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1380	C₂	semidirect
7. C₂²×C₁₆×C₂₃	C₄	semidirect
10. C₂₃×#427	C₄	semidirect
53. #12556	C₂₃	direct (N × H)
75. #1380	C₂×C₂₃	semidirect
90. C₂²×C₁₆	C₄×C₂₃	semidirect
93. #427	C₄×C₂₃	semidirect
118. C₂₃	#12556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₂₃	2
C₂×C₂₃×#427	2
C₂₃×#1380	2
#12556	23

Subgroups — C23×#12556 #19160182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12556 #19160182