Subgroups — C₂₃×#8129 #19155755

Order 5888.

612 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#423	C₂	1	2	yes	maximal
3	C₂₃×#1309	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₂₃	C₄	1	4	yes	—
5	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#432	—	2	2	no	—
7	C₂₃×#431	—	2	2	no	—
8	C₂₃×#423	—	2	2	no	—
9	C₂₃×#423	—	2	2	no	—
10	C₂₃×#423	C₄	1	4	yes	—
11	C₂₃×#423	C₂²	1	2	yes	—
12	C₂₃×#423	C₂²	1	4	yes	—
13	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
15	C₂₃×#407	C₄	1	2	yes	—
16	C₄×M₁₆×C₂₃	C₄	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	—	2	4	no	—
20	C₂×M₁₆×C₂₃	—	4	8	no	—
21	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₂₃	—	2	4	no	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	D₄	1	2	yes	—
27	C₂×C₄²×C₂₃	D₄	1	2	yes	—
28	C₂×C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	—	2	4	no	—
32	C₂₃×C₄⋊C₈	—	2	4	no	—
33	C₂₃×C₄⋊C₈	—	2	4	no	—
34	C₂₃×C₄⋊C₈	—	2	4	no	—
35	C₂₃×C₄⋊C₈	—	2	4	no	—
36	C₂₃×C₂²⋊C₈	—	2	4	no	—
37	C₂₃×#97	—	2	2	no	—
38	C₄×C₈×C₂₃	—	2	2	no	—
39	C₂⁴×C₂₃	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	—	2	4	no	—
42	C₂²×C₄×C₂₃	—	4	2	no	—
43	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
44	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₂₃	—	4	8	no	—
54	M₁₆×C₂₃	—	4	8	no	—
55	M₁₆×C₂₃	—	4	4	no	—
56	C₂×C₈×C₂₃	—	2	8	no	—
57	C₂×C₈×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	—	4	8	no	—
59	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₂₃	C₂×D₄	1	2	yes	—
68	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
69	#8129	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	—	2	4	no	—
71	C₂³×C₂₃	—	4	1	no	—
72	C₂³×C₂₃	#95	1	2	yes	—
73	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	4	1	no	—
84	C₂×C₄×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	#95	1	2	yes	—
86	C₂×C₄×C₂₃	#95	1	2	yes	—
87	C₂×C₄×C₂₃	#95	1	2	yes	—
88	C₂×C₄×C₂₃	#100	1	4	yes	—
89	C₂×C₄×C₂₃	#101	1	4	yes	—
90	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₈×C₂₃	—	4	8	no	—
95	C₈×C₂₃	—	4	4	no	—
96	#2920	C₂×C₂₃	1	1	yes	characteristic
97	C₂×#423	C₂×C₂₃	1	2	yes	—
98	#1309	C₂×C₂₃	1	4	yes	complemented
99	C₂²×C₂₃	—	2	2	no	—
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	4	4	no	—
104	C₂²×C₂₃	#343	1	4	yes	—
105	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	#343	1	4	yes	—
111	C₄×C₂₃	C₂×#100	1	2	yes	—
112	C₄×C₂₃	C₂×#101	1	2	yes	—
113	C₄×C₂₃	#413	1	2	yes	—
114	C₂²×M₁₆	C₄×C₂₃	1	4	yes	—
115	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
116	#432	—	2	2	no	—
117	#431	—	2	2	no	—
118	#423	—	2	2	no	—
119	#423	—	2	2	no	—
120	#423	C₄×C₂₃	1	4	yes	—
121	#423	C₂²×C₂₃	1	4	yes	—
122	#423	C₂²×C₂₃	1	2	yes	—
123	C₂×C₄⋊C₈	—	2	2	no	—
124	C₂×C₄⋊C₈	—	2	2	no	—
125	#407	C₄×C₂₃	1	2	yes	—
126	C₄×M₁₆	C₄×C₂₃	1	2	yes	—
127	C₂×C₂₃	—	2	4	no	—
128	C₂×C₂₃	—	4	1	no	—
129	C₂×C₂₃	C₂×#343	1	2	yes	—
130	C₂×C₂₃	#1742	1	1	yes	characteristic
131	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
132	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂×M₁₆	—	2	4	no	—
134	C₂×M₁₆	—	4	8	no	—
135	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
136	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
137	C₂²×C₈	—	2	4	no	—
138	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
139	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
140	C₂×C₄²	D₄×C₂₃	1	2	yes	—
141	C₂×C₄²	D₄×C₂₃	1	2	yes	—
142	C₂×C₄²	D₄×C₂₃	1	2	yes	—
143	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
144	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	—	2	4	no	—
149	C₄⋊C₈	—	2	4	no	—
150	C₂²⋊C₈	—	2	4	no	—
151	#97	—	2	2	no	—
152	C₄×C₈	—	2	2	no	—
153	C₂₃	#8129	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	C₄²×C₂₃	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
159	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
162	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
163	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
164	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
165	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
168	M₁₆	—	4	8	no	—
169	M₁₆	—	4	8	no	—
170	M₁₆	—	4	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	8	no	—
173	C₂×C₈	—	4	8	no	—
174	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
176	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
177	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
178	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
179	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
180	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
181	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₄²	C₂×D₄×C₂₃	1	2	yes	—
183	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
184	C₂³	—	2	4	no	—
185	C₂³	—	4	1	no	—
186	C₂³	C₂₃×#95	1	2	yes	—
187	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂₃×#95	1	2	yes	—
200	C₂×C₄	C₂₃×#95	1	2	yes	—
201	C₂×C₄	C₂₃×#95	1	2	yes	—
202	C₂×C₄	C₂₃×#100	1	4	yes	—
203	C₂×C₄	C₂₃×#101	1	4	yes	—
204	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
208	C₈	—	4	8	no	—
209	C₈	—	4	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	4	4	no	—
215	C₂²	C₂₃×#343	1	4	yes	—
216	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	C₂₃×#343	1	4	yes	—
222	C₄	C₂×C₂₃×#100	1	2	yes	—
223	C₄	C₂×C₂₃×#101	1	2	yes	—
224	C₄	C₂₃×#413	1	2	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	4	1	no	—
227	C₂	C₂×C₂₃×#343	1	2	yes	—
228	C₂	C₂₃×#1742	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#95
commutator	C₂³	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1309	C₂	semidirect
69. #8129	C₂₃	direct (N × H)
98. #1309	C₂×C₂₃	semidirect
153. C₂₃	#8129	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×#2920	2
C₂×C₂₃×#423	2
C₂₃×#1309	2
#8129	23

Subgroups — C23×#8129 #19155755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8129 #19155755