Subgroups — C₂₃×#21970 #19169596

Order 5888.

702 conjugacy classes of subgroups.

The subgroup lattice has 262 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	2	yes	maximal
2	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1950	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1811	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#432	—	2	2	no	—
8	C₂₃×#432	—	2	2	no	—
9	C₂₃×#431	—	2	2	no	—
10	C₂₃×#431	—	2	2	no	—
11	C₂×C₂₃×#108	C₂²	1	8	yes	—
12	C₂₃×#423	C₄	1	2	yes	—
13	C₂₃×#423	C₄	1	4	yes	—
14	C₂₃×#423	C₄	1	2	yes	—
15	C₂₃×#423	C₂²	1	4	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
17	C₂₃×#407	C₂²	1	4	yes	—
18	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
19	C₂₃×#341	C₂²	1	8	yes	—
20	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	—	4	4	no	—
23	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	D₄	1	2	yes	—
32	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
36	C₂₃×#108	—	4	8	no	—
37	C₂₃×C₄⋊C₈	—	2	4	no	—
38	C₂₃×C₄⋊C₈	—	2	4	no	—
39	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
40	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
41	C₂₃×C₂²⋊C₈	—	2	4	no	—
42	C₂₃×C₂²⋊C₈	—	2	4	no	—
43	C₂₃×#97	—	2	4	no	—
44	C₄×C₈×C₂₃	—	2	4	no	—
45	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	2	no	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	4	4	no	—
49	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₂₃	—	2	8	no	—
62	M₁₆×C₂₃	—	2	8	no	—
63	M₁₆×C₂₃	—	4	4	no	—
64	M₁₆×C₂₃	—	4	4	no	—
65	C₂×C₈×C₂₃	—	2	8	no	—
66	C₂×C₈×C₂₃	—	2	8	no	—
67	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
68	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
71	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
72	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
74	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₂₃	C₂×D₄	1	2	yes	—
77	C₄²×C₂₃	C₂×D₄	1	2	yes	—
78	#21970	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₃	—	2	2	no	—
80	C₂³×C₂₃	—	4	2	no	—
81	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
106	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
107	C₈×C₂₃	—	2	8	no	—
108	C₈×C₂₃	—	4	8	no	—
109	#2920	C₂×C₂₃	1	2	yes	—
110	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
111	#1950	C₂×C₂₃	1	8	yes	complemented
112	#1811	C₂×C₂₃	1	4	yes	—
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	—	4	4	no	—
119	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₂₃	#544	1	2	yes	—
121	C₄×C₂₃	—	2	1	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	#430	1	4	yes	—
127	C₄×C₂₃	#518	1	1	yes	characteristic
128	C₄×C₂₃	#519	1	1	yes	characteristic
129	C₄×C₂₃	#537	1	2	yes	—
130	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
131	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
132	#432	—	2	2	no	—
133	#432	—	2	2	no	—
134	#431	—	2	2	no	—
135	#431	—	2	2	no	—
136	C₂×#108	C₂²×C₂₃	1	8	yes	—
137	#423	C₄×C₂₃	1	2	yes	—
138	#423	C₄×C₂₃	1	4	yes	—
139	#423	C₄×C₂₃	1	2	yes	—
140	#423	C₂²×C₂₃	1	4	yes	—
141	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
142	#407	C₂²×C₂₃	1	4	yes	—
143	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
144	#341	C₂²×C₂₃	1	8	yes	—
145	C₂×C₂₃	—	2	2	no	—
146	C₂×C₂₃	—	4	1	no	—
147	C₂×C₂₃	—	4	1	no	—
148	C₂×C₂₃	#2316	1	1	yes	characteristic
149	C₂×C₂₃	C₂×#430	1	2	yes	—
150	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
151	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
152	C₂×M₁₆	—	4	4	no	—
153	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
154	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
155	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
156	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
157	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
158	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
161	C₂×C₄²	D₄×C₂₃	1	2	yes	—
162	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
164	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
166	#108	—	4	8	no	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
170	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	#97	—	2	4	no	—
174	C₄×C₈	—	2	4	no	—
175	C₂₃	#21970	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
181	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
182	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
183	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
190	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
191	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
192	M₁₆	—	2	8	no	—
193	M₁₆	—	2	8	no	—
194	M₁₆	—	4	4	no	—
195	M₁₆	—	4	4	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
199	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
200	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
201	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
202	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
203	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
204	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
205	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
206	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
207	C₄²	C₂×D₄×C₂₃	1	2	yes	—
208	C₄²	C₂×D₄×C₂₃	1	2	yes	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
233	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
236	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
237	C₈	—	2	8	no	—
238	C₈	—	4	8	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	4	4	no	—
245	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
246	C₂²	C₂₃×#544	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	C₂₃×#430	1	4	yes	—
253	C₄	C₂₃×#518	1	1	yes	characteristic
254	C₄	C₂₃×#519	1	1	yes	characteristic
255	C₄	C₂₃×#537	1	2	yes	—
256	C₂	—	2	2	no	—
257	C₂	—	4	1	no	—
258	C₂	—	4	1	no	—
259	C₂	C₂₃×#2316	1	1	yes	characteristic
260	C₂	C₂×C₂₃×#430	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 ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1950	C₂	semidirect
78. #21970	C₂₃	direct (N × H)
111. #1950	C₂×C₂₃	semidirect
175. C₂₃	#21970	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×#1950	2
C₂₃×#1811	2
#21970	23

Subgroups — C23×#21970 #19169596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21970 #19169596