Subgroups — C₂₃×#20807 #19168433

Order 5888.

954 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#430	C₂	1	2	yes	maximal
2	C₂²×C₂₃×#108	C₂	1	2	yes	maximal
3	C₂₃×#2905	C₂	1	2	yes	maximal
4	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1849	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1848	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂₃×#430	—	2	8	no	—
12	C₂×C₂₃×#108	—	2	8	no	—
13	C₂×C₂₃×#108	C₂²	1	4	yes	—
14	C₂×C₂₃×#108	C₂²	1	4	yes	—
15	C₂×C₂₃×#108	C₂²	1	8	yes	—
16	C₂₃×#424	—	2	8	no	—
17	C₂₃×#423	C₂²	1	8	yes	—
18	C₂₃×#405	C₄	1	8	yes	—
19	C₂×C₂₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	—	2	4	no	—
23	C₂×M₁₆×C₂₃	—	2	8	no	—
24	C₂×M₁₆×C₂₃	—	2	8	no	—
25	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
27	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
28	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
29	C₂²×C₈×C₂₃	—	2	4	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₂₃	D₄	1	4	yes	—
32	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂₃×#117	C₂×C₄	1	4	yes	—
37	C₂₃×#117	D₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₂₃	Q₈	1	2	yes	—
39	C₂×C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
41	C₂₃×#108	—	2	8	no	—
42	C₂₃×#108	—	2	8	no	—
43	C₂₃×#108	—	2	16	no	—
44	C₂₃×C₄⋊C₈	—	2	16	no	—
45	C₂₃×#97	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
47	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₂₃	—	2	8	no	—
58	M₁₆×C₂₃	—	2	8	no	—
59	M₁₆×C₂₃	—	4	8	no	—
60	M₁₆×C₂₃	C₄⋊C₄	1	16	yes	—
61	C₂×C₈×C₂₃	—	2	8	no	—
62	C₂×C₈×C₂₃	—	4	8	no	—
63	C₂×C₈×C₂₃	C₄⋊C₄	1	16	yes	—
64	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
73	C₄⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
74	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
75	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
76	C₄²×C₂₃	C₂×D₄	1	4	yes	—
77	#20807	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	—	4	2	no	—
79	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
81	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
82	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₂₃	—	4	4	no	—
84	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
85	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₂₃	#127	1	4	yes	—
89	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
90	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
91	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
98	C₈×C₂₃	—	4	8	no	—
99	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
101	C₈×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
102	C₂×#430	C₂×C₂₃	1	2	yes	—
103	C₂²×#108	C₂×C₂₃	1	2	yes	—
104	#2905	C₂×C₂₃	1	2	yes	—
105	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
106	#1849	C₂×C₂₃	1	4	yes	complemented
107	#1848	C₂×C₂₃	1	4	yes	complemented
108	C₂²×C₂₃	—	2	2	no	—
109	C₂²×C₂₃	—	2	4	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	#384	1	4	yes	—
113	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₄×C₂₃	—	2	8	no	—
117	C₄×C₂₃	#384	1	4	yes	—
118	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₂₃	C₂×#127	1	1	yes	characteristic
121	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
123	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
124	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
126	#430	—	2	8	no	—
127	C₂×#108	—	2	8	no	—
128	C₂×#108	C₂²×C₂₃	1	4	yes	—
129	C₂×#108	C₂²×C₂₃	1	8	yes	—
130	C₂×#108	C₂²×C₂₃	1	4	yes	—
131	#424	—	2	8	no	—
132	#423	C₂²×C₂₃	1	8	yes	—
133	#405	C₄×C₂₃	1	8	yes	—
134	C₂×#97	C₂²×C₂₃	1	2	yes	—
135	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
136	C₂×C₂₃	—	2	1	no	—
137	C₂×C₂₃	—	2	1	no	—
138	C₂×C₂₃	—	2	4	no	—
139	C₂×C₂₃	C₂×#384	1	1	yes	characteristic
140	C₂×C₂₃	#2916	1	2	yes	—
141	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	8	no	—
144	C₂×M₁₆	—	2	8	no	—
145	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
146	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
147	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
148	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
151	C₂²×C₈	D₄×C₂₃	1	4	yes	—
152	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
153	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
156	#117	C₂×C₄×C₂₃	1	4	yes	—
157	#117	D₄×C₂₃	1	4	yes	—
158	C₂×C₄⋊C₄	Q₈×C₂₃	1	2	yes	—
159	C₂×C₂²⋊C₄	Q₈×C₂₃	1	2	yes	—
160	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
161	#108	—	2	8	no	—
162	#108	—	2	8	no	—
163	#108	—	2	16	no	—
164	C₄⋊C₈	—	2	16	no	—
165	#97	C₂×C₄×C₂₃	1	8	yes	—
166	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
167	C₂₃	#20807	1	1	yes	characteristic, direct product factor, Sylow
168	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
172	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
173	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
174	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
178	M₁₆	—	2	8	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	—	4	8	no	—
181	M₁₆	C₄⋊C₄×C₂₃	1	16	yes	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	4	8	no	—
184	C₂×C₈	C₄⋊C₄×C₂₃	1	16	yes	—
185	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
186	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
187	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
188	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
189	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
190	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
191	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
192	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
193	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
194	C₄⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
195	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
196	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
197	C₄²	C₂×D₄×C₂₃	1	4	yes	—
198	C₂³	—	4	2	no	—
199	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
200	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
201	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
202	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
205	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
208	C₂×C₄	C₂₃×#127	1	4	yes	—
209	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
210	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
211	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
212	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
218	C₈	—	4	8	no	—
219	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
220	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
221	C₈	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂₃×#384	1	4	yes	—
227	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
228	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
229	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	C₂₃×#384	1	4	yes	—
232	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
233	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
234	C₄	C₂×C₂₃×#127	1	1	yes	characteristic
235	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
236	C₂	—	2	4	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	1	no	—
239	C₂	C₂×C₂₃×#384	1	1	yes	characteristic
240	C₂	C₂₃×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1849	C₂	semidirect
6. C₂₃×#1848	C₂	semidirect
77. #20807	C₂₃	direct (N × H)
106. #1849	C₂×C₂₃	semidirect
107. #1848	C₂×C₂₃	semidirect
167. C₂₃	#20807	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×#430	2
C₂²×C₂₃×#108	2
C₂₃×#2905	2
C₂×C₂₃×#405	2
C₂₃×#1849	2
C₂₃×#1848	2
#20807	23

Subgroups — C23×#20807 #19168433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20807 #19168433