Subgroups — C₂₃×#10724 #19158350

Order 5888.

1294 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#407	C₂	1	2	yes	maximal
3	C₂×C₂₃×#341	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₂₃	—	2	4	no	—
6	C₂²×M₁₆×C₂₃	—	2	8	no	—
7	C₂²×M₁₆×C₂₃	C₄	1	4	yes	—
8	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
9	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂₃×#407	—	2	4	no	—
11	C₂₃×#407	C₄	1	8	yes	—
12	C₂×C₂₃×C₂²⋊C₈	—	2	4	no	—
13	C₂₃×#341	—	2	8	no	—
14	C₂⁵×C₂₃	Q₈	1	1	yes	characteristic
15	C₂³×C₄×C₂₃	—	2	4	no	—
16	C₂³×C₄×C₂₃	—	2	4	no	—
17	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₂₃	D₄	1	4	yes	—
20	C₂³×C₄×C₂₃	D₄	1	2	yes	—
21	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₂₃	—	2	32	no	—
24	C₂×M₁₆×C₂₃	—	2	8	no	—
25	C₂×M₁₆×C₂₃	—	2	4	no	—
26	C₂×M₁₆×C₂₃	—	2	16	no	—
27	C₂×M₁₆×C₂₃	—	2	8	no	—
28	C₂×M₁₆×C₂₃	—	2	8	no	—
29	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₂₃	—	2	4	no	—
31	C₂²×C₈×C₂₃	—	2	4	no	—
32	C₂²×C₈×C₂₃	—	2	4	no	—
33	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₂₃	D₄	1	8	yes	—
35	C₂₃×C₂²⋊C₈	—	2	16	no	—
36	C₂⁴×C₂₃	—	2	4	no	—
37	C₂⁴×C₂₃	—	4	4	no	—
38	C₂⁴×C₂₃	C₄⋊C₄	1	2	yes	—
39	C₂⁴×C₂₃	C₄⋊C₄	1	4	yes	—
40	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	—	2	4	no	—
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	2	16	no	—
45	C₂²×C₄×C₂₃	—	2	8	no	—
46	C₂²×C₄×C₂₃	—	2	8	no	—
47	C₂²×C₄×C₂₃	—	2	8	no	—
48	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
49	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	#41	1	4	yes	—
59	M₁₆×C₂₃	—	2	32	no	—
60	M₁₆×C₂₃	—	2	16	no	—
61	M₁₆×C₂₃	—	4	4	no	—
62	M₁₆×C₂₃	—	4	4	no	—
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	—	2	16	no	—
66	C₂×C₈×C₂₃	—	2	8	no	—
67	C₂×C₈×C₂₃	—	4	8	no	—
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	8	yes	—
69	#10724	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	—	2	4	no	—
71	C₂³×C₂₃	—	2	4	no	—
72	C₂³×C₂₃	—	2	8	no	—
73	C₂³×C₂₃	—	2	8	no	—
74	C₂³×C₂₃	—	4	8	no	—
75	C₂³×C₂₃	—	4	16	no	—
76	C₂³×C₂₃	#95	1	4	yes	—
77	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₂₃	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	8	no	—
82	C₂×C₄×C₂₃	—	2	8	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	4	no	—
85	C₂×C₄×C₂₃	—	2	16	no	—
86	C₂×C₄×C₂₃	#95	1	4	yes	—
87	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₂₃	#117	1	2	yes	—
92	C₂×C₄×C₂₃	C₄×D₄	1	8	yes	—
93	C₂×C₄×C₂₃	#120	1	4	yes	—
94	C₂×C₄×C₂₃	#121	1	4	yes	—
95	C₂×C₄×C₂₃	#123	1	4	yes	—
96	C₈×C₂₃	—	2	8	no	—
97	C₈×C₂₃	—	4	8	no	—
98	C₂³×M₁₆	C₂×C₂₃	1	1	yes	characteristic
99	C₂×#407	C₂×C₂₃	1	2	yes	—
100	C₂×#341	C₂×C₂₃	1	4	yes	complemented
101	C₂²×C₂₃	—	2	8	no	—
102	C₂²×C₂₃	—	2	4	no	—
103	C₂²×C₂₃	—	2	4	no	—
104	C₂²×C₂₃	—	2	8	no	—
105	C₂²×C₂₃	—	4	16	no	—
106	C₂²×C₂₃	—	4	8	no	—
107	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
108	C₂²×C₂₃	#380	1	2	yes	—
109	C₂²×C₂₃	#385	1	4	yes	—
110	C₂²×C₂₃	#430	1	4	yes	—
111	C₄×C₂₃	—	2	4	no	—
112	C₄×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
113	C₄×C₂₃	#379	1	1	yes	characteristic
114	C₄×C₂₃	#381	1	1	yes	characteristic
115	C₄×C₂₃	#386	1	4	yes	—
116	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
117	C₂²×M₁₆	—	2	4	no	—
118	C₂²×M₁₆	—	2	8	no	—
119	C₂²×M₁₆	C₄×C₂₃	1	4	yes	—
120	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
121	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
122	#407	—	2	4	no	—
123	#407	C₄×C₂₃	1	8	yes	—
124	C₂×C₂²⋊C₈	—	2	4	no	—
125	#341	—	2	8	no	—
126	C₂×C₂₃	—	2	4	no	—
127	C₂×C₂₃	—	4	4	no	—
128	C₂×C₂₃	C₂²⋊#95	1	1	yes	characteristic
129	C₂×C₂₃	#1738	1	2	yes	—
130	C₂×C₂₃	#1811	1	4	yes	—
131	C₂⁵	Q₈×C₂₃	1	1	yes	characteristic
132	C₂³×C₄	—	2	4	no	—
133	C₂³×C₄	—	2	4	no	—
134	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
135	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
136	C₂³×C₄	D₄×C₂₃	1	4	yes	—
137	C₂³×C₄	D₄×C₂₃	1	2	yes	—
138	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
139	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
140	C₂×M₁₆	—	2	32	no	—
141	C₂×M₁₆	—	2	8	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	16	no	—
144	C₂×M₁₆	—	2	8	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
151	C₂²×C₈	D₄×C₂₃	1	8	yes	—
152	C₂²⋊C₈	—	2	16	no	—
153	C₂₃	#10724	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	—	2	4	no	—
155	C₂⁴	—	4	4	no	—
156	C₂⁴	C₄⋊C₄×C₂₃	1	2	yes	—
157	C₂⁴	C₄⋊C₄×C₂₃	1	4	yes	—
158	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	16	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
167	C₂²×C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
168	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
169	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
170	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
171	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
176	C₂²×C₄	#41×C₂₃	1	4	yes	—
177	M₁₆	—	2	32	no	—
178	M₁₆	—	2	16	no	—
179	M₁₆	—	4	4	no	—
180	M₁₆	—	4	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	16	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	4	8	no	—
186	C₂×C₈	C₂²⋊C₄×C₂₃	1	8	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	2	8	no	—
190	C₂³	—	2	8	no	—
191	C₂³	—	4	8	no	—
192	C₂³	—	4	16	no	—
193	C₂³	C₂₃×#95	1	4	yes	—
194	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
195	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
196	C₂³	C₂₃×C₂²⋊Q₈	1	4	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	16	no	—
203	C₂×C₄	C₂₃×#95	1	4	yes	—
204	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
208	C₂×C₄	C₂₃×#117	1	2	yes	—
209	C₂×C₄	C₄×D₄×C₂₃	1	8	yes	—
210	C₂×C₄	C₂₃×#120	1	4	yes	—
211	C₂×C₄	C₂₃×#121	1	4	yes	—
212	C₂×C₄	C₂₃×#123	1	4	yes	—
213	C₈	—	2	8	no	—
214	C₈	—	4	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	4	16	no	—
220	C₂²	—	4	8	no	—
221	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
222	C₂²	C₂₃×#380	1	2	yes	—
223	C₂²	C₂₃×#385	1	4	yes	—
224	C₂²	C₂₃×#430	1	4	yes	—
225	C₄	—	2	4	no	—
226	C₄	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
227	C₄	C₂₃×#379	1	1	yes	characteristic
228	C₄	C₂₃×#381	1	1	yes	characteristic
229	C₄	C₂₃×#386	1	4	yes	—
230	C₂	—	2	4	no	—
231	C₂	—	4	4	no	—
232	C₂	C₂₃×C₂²⋊#95	1	1	yes	characteristic
233	C₂	C₂₃×#1738	1	2	yes	—
234	C₂	C₂₃×#1811	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁵×C₂₃	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#341	C₂	semidirect
69. #10724	C₂₃	direct (N × H)
100. C₂×#341	C₂×C₂₃	semidirect
153. C₂₃	#10724	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×M₁₆×C₂₃	2
C₂×C₂₃×#407	2
C₂×C₂₃×#341	2
#10724	23

Subgroups — C23×#10724 #19158350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10724 #19158350