Subgroups — C₂₃×#10815 #19158441

Order 5888.

626 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	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1742	C₂	1	2	yes	maximal
3	C₂×C₂₃×#341	C₂	1	2	yes	maximal
4	C₂×C₂₃×#338	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
6	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
9	C₂₃×#405	—	2	4	no	—
10	C₂×C₂₃×#97	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
12	C₂₃×#343	—	2	4	no	—
13	C₂₃×#341	—	2	4	no	—
14	C₂₃×#338	C₄	1	8	yes	complemented
15	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	—	2	2	no	—
17	C₂×M₁₆×C₂₃	—	2	2	no	—
18	C₂×M₁₆×C₂₃	—	2	4	no	—
19	C₂×M₁₆×C₂₃	—	2	4	no	—
20	C₂×M₁₆×C₂₃	—	2	4	no	—
21	C₂×M₁₆×C₂₃	—	2	4	no	—
22	C₂×M₁₆×C₂₃	—	2	4	no	—
23	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
24	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₂²×C₈×C₂₃	—	2	4	no	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	D₄	1	4	yes	—
32	C₂₃×#117	—	2	2	no	—
33	C₂₃×#117	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₂₃×#97	—	2	4	no	—
38	C₄×C₈×C₂₃	—	2	4	no	—
39	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	#41	1	2	yes	—
46	C₂²×C₄×C₂₃	#41	1	2	yes	—
47	C₂²×C₄×C₂₃	#41	1	2	yes	—
48	C₂²×C₄×C₂₃	#41	1	2	yes	—
49	M₁₆×C₂₃	—	2	8	no	—
50	M₁₆×C₂₃	—	2	8	no	—
51	M₁₆×C₂₃	—	2	8	no	—
52	M₁₆×C₂₃	—	4	4	no	—
53	M₁₆×C₂₃	—	4	4	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	1	no	—
56	C₂×C₈×C₂₃	—	2	1	no	—
57	C₂×C₈×C₂₃	—	2	1	no	—
58	C₂×C₈×C₂₃	—	2	1	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	2	8	no	—
61	C₂×C₈×C₂₃	—	2	8	no	—
62	C₂×C₈×C₂₃	—	4	4	no	—
63	C₂×C₈×C₂₃	—	4	4	no	—
64	C₂×C₈×C₂₃	C₄²	1	4	yes	—
65	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₂₃	—	2	4	no	—
68	C₂²⋊C₄×C₂₃	—	2	4	no	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	—	2	2	no	—
71	#10815	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	—	4	2	no	—
73	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
75	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
76	C₂³×C₂₃	#125	1	2	yes	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	4	4	no	—
79	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#117	1	4	yes	—
83	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
87	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₂₃	#121	1	2	yes	—
89	C₂×C₄×C₂₃	#123	1	4	yes	—
90	C₂×C₄×C₂₃	#124	1	2	yes	—
91	C₂×C₄×C₂₃	#126	1	4	yes	—
92	C₈×C₂₃	—	2	4	no	—
93	C₈×C₂₃	—	2	2	no	—
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	4	8	no	—
96	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
97	#1742	C₂×C₂₃	1	2	yes	—
98	C₂×#341	C₂×C₂₃	1	2	yes	—
99	C₂×#338	C₂×C₂₃	1	2	yes	—
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	4	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₂₃	#380	1	1	yes	characteristic
106	C₂²×C₂₃	#382	1	1	yes	characteristic
107	C₂²×C₂₃	#387	1	4	yes	—
108	C₄×C₂₃	—	4	4	no	—
109	C₄×C₂₃	#376	1	1	yes	characteristic
110	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₂₃	#381	1	1	yes	characteristic
112	C₄×C₂₃	#383	1	1	yes	characteristic
113	C₄×C₂₃	#388	1	4	yes	—
114	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
115	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
116	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
117	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
118	#405	—	2	4	no	—
119	C₂×#97	C₄×C₂₃	1	2	yes	—
120	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
121	#343	—	2	4	no	—
122	#341	—	2	4	no	—
123	#338	C₄×C₂₃	1	8	yes	complemented
124	C₂×C₂₃	—	2	1	no	—
125	C₂×C₂₃	—	2	1	no	—
126	C₂×C₂₃	—	2	4	no	—
127	C₂×C₂₃	#1434	1	1	yes	characteristic
128	C₂×C₂₃	#1813	1	2	yes	—
129	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	—	2	4	no	—
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	—	2	4	no	—
136	C₂×M₁₆	—	2	4	no	—
137	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
138	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
143	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
145	C₂²×C₈	D₄×C₂₃	1	4	yes	—
146	#117	—	2	2	no	—
147	#117	—	2	2	no	—
148	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
149	C₂×C₂²⋊C₄	Q₈×C₂₃	1	2	yes	—
150	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
151	#97	—	2	4	no	—
152	C₄×C₈	—	2	4	no	—
153	C₂₃	#10815	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
159	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	#41×C₂₃	1	2	yes	—
161	C₂²×C₄	#41×C₂₃	1	2	yes	—
162	C₂²×C₄	#41×C₂₃	1	2	yes	—
163	C₂²×C₄	#41×C₂₃	1	2	yes	—
164	M₁₆	—	2	8	no	—
165	M₁₆	—	2	8	no	—
166	M₁₆	—	2	8	no	—
167	M₁₆	—	4	4	no	—
168	M₁₆	—	4	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	C₄²×C₂₃	1	4	yes	—
180	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
181	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₂³	—	4	2	no	—
187	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
189	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
190	C₂³	C₂₃×#125	1	2	yes	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
196	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
197	C₂×C₄	C₂₃×#117	1	4	yes	—
198	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
199	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
200	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
201	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
202	C₂×C₄	C₂₃×#121	1	2	yes	—
203	C₂×C₄	C₂₃×#123	1	4	yes	—
204	C₂×C₄	C₂₃×#124	1	2	yes	—
205	C₂×C₄	C₂₃×#126	1	4	yes	—
206	C₈	—	2	4	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	4	8	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂²	C₂₃×#380	1	1	yes	characteristic
216	C₂²	C₂₃×#382	1	1	yes	characteristic
217	C₂²	C₂₃×#387	1	4	yes	—
218	C₄	—	4	4	no	—
219	C₄	C₂₃×#376	1	1	yes	characteristic
220	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₄	C₂₃×#381	1	1	yes	characteristic
222	C₄	C₂₃×#383	1	1	yes	characteristic
223	C₄	C₂₃×#388	1	4	yes	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	—	2	4	no	—
227	C₂	C₂₃×#1434	1	1	yes	characteristic
228	C₂	C₂₃×#1813	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#380
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₂₃×#338	C₄	semidirect
71. #10815	C₂₃	direct (N × H)
123. #338	C₄×C₂₃	semidirect
153. C₂₃	#10815	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×#405	2
C₂₃×#1742	2
C₂×C₂₃×#341	2
C₂×C₂₃×#338	2
#10815	23

Subgroups — C23×#10815 #19158441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10815 #19158441