Subgroups — C₂₃×#21995 #19169621

Order 5888.

658 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×#2921	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#433	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2905	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1950	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1855	C₂	1	4	yes	maximal
6	C₂₃×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
12	C₂₃×#433	—	2	4	no	—
13	C₂₃×#433	C₄	1	8	yes	—
14	C₂×C₂₃×#108	C₂²	1	4	yes	—
15	C₂×C₂₃×#108	C₂²	1	4	yes	—
16	C₂₃×#424	—	2	4	no	—
17	C₂₃×#423	C₂²	1	1	yes	characteristic
18	C₂₃×#423	C₂²	1	1	yes	characteristic
19	C₂₃×#423	C₂²	1	2	yes	—
20	C₂₃×#423	C₂²	1	1	yes	characteristic
21	C₂₃×#423	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂₃×#407	C₂²	1	2	yes	—
24	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#341	C₂²	1	4	yes	—
29	C₂₃×#341	C₂²	1	4	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	—	2	2	no	—
32	C₂×M₁₆×C₂₃	—	4	4	no	—
33	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
34	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
35	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
36	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	—	2	2	no	—
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
40	C₂₃×#117	C₂×C₄	1	4	yes	—
41	C₂₃×#117	D₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₂₃	Q₈	1	2	yes	—
43	C₂×C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×#108	—	4	4	no	—
47	C₂₃×#108	—	4	4	no	—
48	C₂₃×C₄⋊C₈	—	2	2	no	—
49	C₂₃×C₄⋊C₈	—	2	2	no	—
50	C₂₃×C₄⋊C₈	—	2	4	no	—
51	C₂₃×C₄⋊C₈	—	2	2	no	—
52	C₂₃×C₄⋊C₈	—	2	2	no	—
53	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₂₃×C₂²⋊C₈	—	2	4	no	—
55	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₂₃×#97	—	2	2	no	—
57	C₂₃×#97	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₂₃	—	2	2	no	—
59	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
60	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	—	2	4	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	2	4	no	—
73	M₁₆×C₂₃	—	4	4	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	4	4	no	—
83	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
85	C₄⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
86	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
87	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₂₃	C₂×D₄	1	2	yes	—
90	C₄²×C₂₃	C₂×D₄	1	2	yes	—
91	#21995	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	—	4	2	no	—
94	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
108	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
109	C₈×C₂₃	—	2	4	no	—
110	C₈×C₂₃	—	2	4	no	—
111	C₈×C₂₃	—	4	2	no	—
112	C₈×C₂₃	—	4	2	no	—
113	C₈×C₂₃	—	4	2	no	—
114	C₈×C₂₃	—	4	2	no	—
115	#2921	C₂×C₂₃	1	1	yes	characteristic
116	C₂×#433	C₂×C₂₃	1	1	yes	characteristic
117	#2905	C₂×C₂₃	1	1	yes	characteristic
118	#1950	C₂×C₂₃	1	4	yes	complemented
119	#1855	C₂×C₂₃	1	4	yes	—
120	#1849	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#1838	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#1837	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	2	2	no	—
126	C₂²×C₂₃	—	2	4	no	—
127	C₂²×C₂₃	—	8	2	no	—
128	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₂₃	#544	1	2	yes	—
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	#520	1	2	yes	—
134	C₄×C₂₃	#535	1	1	yes	characteristic
135	C₄×C₂₃	#536	1	1	yes	characteristic
136	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
137	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
138	#433	—	2	4	no	—
139	#433	C₄×C₂₃	1	8	yes	—
140	C₂×#108	C₂²×C₂₃	1	4	yes	—
141	C₂×#108	C₂²×C₂₃	1	4	yes	—
142	#424	—	2	4	no	—
143	#423	C₂²×C₂₃	1	1	yes	characteristic
144	#423	C₂²×C₂₃	1	1	yes	characteristic
145	#423	C₂²×C₂₃	1	1	yes	characteristic
146	#423	C₂²×C₂₃	1	1	yes	characteristic
147	#423	C₂²×C₂₃	1	2	yes	—
148	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
149	#407	C₂²×C₂₃	1	2	yes	—
150	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
151	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
152	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	#341	C₂²×C₂₃	1	4	yes	—
155	#341	C₂²×C₂₃	1	4	yes	—
156	C₂×C₂₃	—	2	2	no	—
157	C₂×C₂₃	—	4	2	no	—
158	C₂×C₂₃	#2317	1	1	yes	characteristic
159	C₂×C₂₃	#2916	1	2	yes	—
160	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	4	4	no	—
163	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
164	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
165	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
166	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
169	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
170	#117	C₂×C₄×C₂₃	1	4	yes	—
171	#117	D₄×C₂₃	1	4	yes	—
172	C₂×C₄⋊C₄	Q₈×C₂₃	1	2	yes	—
173	C₂×C₂²⋊C₄	Q₈×C₂₃	1	2	yes	—
174	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
176	#108	—	4	4	no	—
177	#108	—	4	4	no	—
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
184	C₂²⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
186	#97	—	2	2	no	—
187	#97	C₂×C₄×C₂₃	1	4	yes	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
190	C₂₃	#21995	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
196	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
198	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	4	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
215	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
216	C₄⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
217	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
218	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
219	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
220	C₄²	C₂×D₄×C₂₃	1	2	yes	—
221	C₄²	C₂×D₄×C₂₃	1	2	yes	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
225	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
233	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
235	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
238	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
239	C₈	—	2	4	no	—
240	C₈	—	2	4	no	—
241	C₈	—	4	2	no	—
242	C₈	—	4	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	4	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
250	C₂²	C₂₃×#544	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	C₂₃×#520	1	2	yes	—
255	C₄	C₂₃×#535	1	1	yes	characteristic
256	C₄	C₂₃×#536	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	4	2	no	—
259	C₂	C₂₃×#2317	1	1	yes	characteristic
260	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
4. C₂₃×#1950	C₂	semidirect
6. C₂₃×#1849	C₂	semidirect
7. C₂₃×#1848	C₂	semidirect
8. C₂₃×#1838	C₂	semidirect
9. C₂₃×#1837	C₂	semidirect
91. #21995	C₂₃	direct (N × H)
118. #1950	C₂×C₂₃	semidirect
120. #1849	C₂×C₂₃	semidirect
121. #1848	C₂×C₂₃	semidirect
122. #1838	C₂×C₂₃	semidirect
123. #1837	C₂×C₂₃	semidirect
190. C₂₃	#21995	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×#2921	2
C₂×C₂₃×#433	2
C₂₃×#2905	2
C₂₃×#1950	2
C₂₃×#1855	2
C₂₃×#1849	2
C₂₃×#1848	2
C₂₃×#1838	2
C₂₃×#1837	2
#21995	23

Subgroups — C23×#21995 #19169621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21995 #19169621