Subgroups — C₂₃×#8780 #19156406

Order 5888.

310 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	2	yes	maximal
3	C₂₃×#1281	C₂	1	2	yes	maximal
4	C₂₃×#1274	C₂	1	2	yes	maximal
5	C₂₃×#431	C₂²	1	4	yes	—
6	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
8	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#376	C₂²	1	1	yes	characteristic
13	C₂₃×#335	—	2	2	no	—
14	C₂₃×#334	—	2	2	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
22	C₂₃×C₄⋊C₈	—	2	4	no	—
23	C₂₃×C₂²⋊C₈	—	2	4	no	—
24	C₂₃×#97	—	2	2	no	—
25	C₂₃×#97	—	2	2	no	—
26	C₂₃×#97	—	2	2	no	—
27	C₄×C₈×C₂₃	—	2	2	no	—
28	C₄×C₈×C₂₃	—	2	4	no	—
29	C₄×C₈×C₂₃	C₈	1	4	yes	complemented
30	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
34	C₂₃×#95	—	2	2	no	—
35	C₂²×C₄×C₂₃	—	2	1	no	—
36	C₂²×C₄×C₂₃	—	2	1	no	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	#41	1	2	yes	—
39	C₂×C₈×C₂₃	—	2	1	no	—
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	—	2	1	no	—
42	C₂×C₈×C₂₃	—	2	1	no	—
43	C₂×C₈×C₂₃	—	2	1	no	—
44	C₂×C₈×C₂₃	—	2	1	no	—
45	C₂×C₈×C₂₃	—	4	4	no	—
46	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
53	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
54	C₄²×C₂₃	—	2	1	no	—
55	C₄²×C₂₃	—	2	1	no	—
56	C₄²×C₂₃	C₂×C₈	1	2	yes	—
57	C₄²×C₂₃	M₁₆	1	2	yes	—
58	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₂₃	#41	1	2	yes	—
61	#8780	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₃	#117	1	1	yes	characteristic
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	4	2	no	—
66	C₂×C₄×C₂₃	—	4	2	no	—
67	C₂×C₄×C₂₃	—	4	2	no	—
68	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#124	1	2	yes	—
71	C₂×C₄×C₂₃	#125	1	2	yes	—
72	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
76	C₈×C₂₃	—	2	1	no	—
77	C₈×C₂₃	—	2	1	no	—
78	C₈×C₂₃	—	2	2	no	—
79	C₈×C₂₃	—	2	2	no	—
80	C₈×C₂₃	—	8	4	no	—
81	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
82	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
83	#1726	C₂×C₂₃	1	1	yes	characteristic
84	#1564	C₂×C₂₃	1	2	yes	—
85	#1281	C₂×C₂₃	1	2	yes	—
86	#1274	C₂×C₂₃	1	2	yes	—
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	#382	1	1	yes	characteristic
89	C₂²×C₂₃	#431	1	1	yes	characteristic
90	C₂²×C₂₃	#432	1	1	yes	characteristic
91	C₄×C₂₃	—	2	2	no	—
92	C₄×C₂₃	—	2	2	no	—
93	C₄×C₂₃	—	4	2	no	—
94	C₄×C₂₃	—	4	2	no	—
95	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₂₃	#423	1	1	yes	characteristic
97	C₄×C₂₃	#431	1	1	yes	characteristic
98	C₄×C₂₃	#432	1	1	yes	characteristic
99	#431	C₂²×C₂₃	1	4	yes	—
100	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
101	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
102	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
103	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
106	#376	C₂²×C₂₃	1	1	yes	characteristic
107	#335	—	2	2	no	—
108	#334	—	2	2	no	—
109	C₂×C₂₃	—	2	1	no	—
110	C₂×C₂₃	—	2	1	no	—
111	C₂×C₂₃	#1831	1	1	yes	characteristic
112	C₂×C₂₃	#1833	1	2	yes	—
113	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
118	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
119	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
120	C₄⋊C₈	—	2	4	no	—
121	C₂²⋊C₈	—	2	4	no	—
122	#97	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	4	no	—
127	C₄×C₈	C₈×C₂₃	1	4	yes	complemented
128	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
129	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
130	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
131	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
132	#95	—	2	2	no	—
133	C₂₃	#8780	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
137	C₂²×C₄	#41×C₂₃	1	2	yes	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	4	4	no	—
145	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
146	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
147	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
149	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
150	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
151	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
152	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₂×C₈×C₂₃	1	2	yes	—
156	C₄²	M₁₆×C₂₃	1	2	yes	—
157	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
158	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
159	C₄²	#41×C₂₃	1	2	yes	—
160	C₂³	C₂₃×#117	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
167	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
168	C₂×C₄	C₂₃×#124	1	2	yes	—
169	C₂×C₄	C₂₃×#125	1	2	yes	—
170	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
171	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
172	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
173	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	8	4	no	—
179	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
180	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	C₂₃×#382	1	1	yes	characteristic
183	C₂²	C₂₃×#431	1	1	yes	characteristic
184	C₂²	C₂₃×#432	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
190	C₄	C₂₃×#423	1	1	yes	characteristic
191	C₄	C₂₃×#431	1	1	yes	characteristic
192	C₄	C₂₃×#432	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₂₃×#1831	1	1	yes	characteristic
196	C₂	C₂₃×#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#382
prime radical	C₂³×C₂₃	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
29. C₄×C₈×C₂₃	C₈	semidirect
61. #8780	C₂₃	direct (N × H)
127. C₄×C₈	C₈×C₂₃	semidirect
133. C₂₃	#8780	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×#1726	2
C₂₃×#1564	2
C₂₃×#1281	2
C₂₃×#1274	2
#8780	23

Subgroups — C23×#8780 #19156406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8780 #19156406