Subgroups — C₂₃×#11672 #19159298

Order 5888.

334 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×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1332	C₂	1	4	yes	maximal, complemented
5	C₂₃×#533	—	2	1	no	—
6	C₂²×C₁₆×C₂₃	—	2	1	no	—
7	C₂₃×#500	C₄	1	1	yes	characteristic
8	C₂₃×#494	C₄	1	1	yes	characteristic
9	C₂₃×#487	—	2	1	no	—
10	C₂₃×#467	—	2	1	no	—
11	C₂₃×#467	C₄	1	2	yes	—
12	C₄×Q₁₆×C₂₃	—	2	1	no	—
13	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₂₃	—	2	1	no	—
17	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#335	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₂₃	—	2	1	no	—
21	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₂₃	—	4	1	no	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂₃×C₄⋊Q₈	—	2	1	no	—
26	C₂₃×C₄⋊Q₈	—	2	1	no	—
27	C₂₃×#125	—	4	1	no	—
28	C₂₃×C₂²⋊Q₈	—	2	2	no	—
29	C₂₃×C₂²⋊Q₈	—	4	1	no	—
30	C₄×Q₈×C₂₃	—	2	2	no	—
31	C₄×Q₈×C₂₃	—	4	1	no	—
32	C₂₃×#117	—	4	1	no	—
33	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₂₃	—	2	4	no	—
35	C₂×C₁₆×C₂₃	—	2	2	no	—
36	C₂₃×#107	—	2	1	no	—
37	C₂₃×#107	C₂×C₄	1	2	yes	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₄⋊C₈	—	4	1	no	—
40	C₂₃×#103	—	2	2	no	—
41	C₂₃×#103	—	2	2	no	—
42	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	4	1	no	—
48	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₂₃	—	2	4	no	—
52	Q₁₆×C₂₃	—	4	2	no	—
53	M₁₆×C₂₃	—	8	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	4	2	no	—
57	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	4	1	no	—
63	C₄⋊C₄×C₂₃	—	4	1	no	—
64	C₄⋊C₄×C₂₃	—	4	2	no	—
65	C₄⋊C₄×C₂₃	—	4	1	no	—
66	C₄⋊C₄×C₂₃	—	4	2	no	—
67	C₂²⋊C₄×C₂₃	—	4	2	no	—
68	C₄²×C₂₃	—	4	2	no	—
69	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₂₃	D₈	1	1	yes	characteristic
71	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
72	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₂₃	—	2	4	no	—
74	#11672	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₃	#102	1	1	yes	characteristic
76	Q₈×C₂₃	—	4	2	no	—
77	Q₈×C₂₃	—	4	2	no	—
78	Q₈×C₂₃	—	8	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
88	C₂×C₄×C₂₃	Q₃₂	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₂₃	—	2	1	no	—
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	8	2	no	—
95	C₈×C₂₃	#104	1	2	yes	—
96	C₈×C₂₃	#131	1	2	yes	—
97	#3170	C₂×C₂₃	1	1	yes	characteristic
98	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
99	#1567	C₂×C₂₃	1	1	yes	characteristic
100	#1332	C₂×C₂₃	1	4	yes	complemented
101	C₂²×C₂₃	—	2	1	no	—
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	#358	1	2	yes	—
104	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	8	2	no	—
108	C₄×C₂₃	—	8	2	no	—
109	C₄×C₂₃	#358	1	2	yes	—
110	C₄×C₂₃	#359	1	2	yes	—
111	C₄×C₂₃	#408	1	1	yes	characteristic
112	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
113	C₄×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
114	C₄×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
115	#533	—	2	1	no	—
116	C₂²×C₁₆	—	2	1	no	—
117	#500	C₄×C₂₃	1	1	yes	characteristic
118	#494	C₄×C₂₃	1	1	yes	characteristic
119	#487	—	2	1	no	—
120	#467	—	2	1	no	—
121	#467	C₄×C₂₃	1	2	yes	—
122	C₄×Q₁₆	—	2	1	no	—
123	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
124	#423	—	2	1	no	—
125	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
126	C₄×C₁₆	—	2	1	no	—
127	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
128	#335	—	2	1	no	—
129	#335	C₂²×C₂₃	1	2	yes	—
130	C₂×C₂₃	—	2	2	no	—
131	C₂×C₂₃	#1492	1	1	yes	characteristic
132	C₂×C₂₃	C₂×#358	1	1	yes	characteristic
133	C₂×C₂₃	C₂×#359	1	1	yes	characteristic
134	C₂×Q₁₆	—	2	1	no	—
135	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
136	C₂×M₁₆	—	4	1	no	—
137	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	—	2	1	no	—
141	#125	—	4	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	4	1	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	1	no	—
146	#117	—	4	1	no	—
147	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
148	C₂×C₁₆	—	2	4	no	—
149	C₂×C₁₆	—	2	2	no	—
150	#107	—	2	1	no	—
151	#107	C₂×C₄×C₂₃	1	2	yes	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	#103	—	2	2	no	—
155	#103	—	2	2	no	—
156	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
157	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
158	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
159	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
160	C₂₃	#11672	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	2	no	—
168	M₁₆	—	8	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
185	C₄²	D₈×C₂₃	1	1	yes	characteristic
186	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₁₆	—	2	4	no	—
189	C₂³	C₂₃×#102	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
199	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
200	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
201	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
202	C₂×C₄	C₂₃×Q₃₂	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	8	2	no	—
209	C₈	C₂₃×#104	1	2	yes	—
210	C₈	C₂₃×#131	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂₃×#358	1	2	yes	—
214	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	C₂₃×#358	1	2	yes	—
220	C₄	C₂₃×#359	1	2	yes	—
221	C₄	C₂₃×#408	1	1	yes	characteristic
222	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
223	C₄	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
224	C₄	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	C₂₃×#1492	1	1	yes	characteristic
227	C₂	C₂×C₂₃×#358	1	1	yes	characteristic
228	C₂	C₂×C₂₃×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#102
prime radical	C₂³×C₂₃	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1332	C₂	semidirect
74. #11672	C₂₃	direct (N × H)
100. #1332	C₂×C₂₃	semidirect
160. C₂₃	#11672	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×#3170	2
C₂×C₄×C₁₆×C₂₃	2
C₂₃×#1567	2
C₂₃×#1332	2
#11672	23

Subgroups — C23×#11672 #19159298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11672 #19159298