Subgroups — C₂₃×#8007 #19155633

Order 5888.

542 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	C₂₃×#2920	C₂	1	2	yes	maximal
2	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1293	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×#432	—	2	2	no	—
6	C₂₃×#432	C₄	1	4	yes	—
7	C₂₃×#431	—	2	2	no	—
8	C₂₃×#431	—	2	2	no	—
9	C₂₃×#431	—	2	2	no	—
10	C₂₃×#431	C₄	1	4	yes	—
11	C₂₃×#423	—	2	2	no	—
12	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
13	C₂₃×#407	C₂²	1	4	yes	—
14	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₄×M₁₆×C₂₃	—	2	2	no	—
16	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
17	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
19	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	—	2	4	no	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	D₄	1	2	yes	—
27	C₂×C₄²×C₂₃	D₄	1	2	yes	—
28	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
30	C₂₃×C₄⋊C₈	—	2	4	no	—
31	C₂₃×C₄⋊C₈	—	2	4	no	—
32	C₂₃×C₄⋊C₈	—	2	4	no	—
33	C₂₃×C₂²⋊C₈	—	2	4	no	—
34	C₂₃×C₂²⋊C₈	—	2	4	no	—
35	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
36	C₂₃×#97	—	2	4	no	—
37	C₄×C₈×C₂₃	—	2	2	no	—
38	C₄×C₈×C₂₃	—	2	2	no	—
39	C₄×C₈×C₂₃	—	2	4	no	—
40	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	—	2	4	no	—
42	C₂²×C₄×C₂₃	—	4	2	no	—
43	C₂²×C₄×C₂₃	—	4	2	no	—
44	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₂₃	—	4	8	no	—
57	C₂×C₈×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	—	2	8	no	—
59	C₂×C₈×C₂₃	—	4	4	no	—
60	C₄²×C₂₃	C₄²	1	2	yes	—
61	C₄²×C₂₃	C₄²	1	1	yes	characteristic
62	C₄²×C₂₃	C₄²	1	1	yes	characteristic
63	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	#8007	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₂₃	—	2	2	no	—
72	C₂³×C₂₃	—	4	1	no	—
73	C₂³×C₂₃	—	4	1	no	—
74	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	4	1	no	—
84	C₂×C₄×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	—	4	2	no	—
86	C₂×C₄×C₂₃	—	4	1	no	—
87	C₂×C₄×C₂₃	#95	1	2	yes	—
88	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
89	C₂×C₄×C₂₃	#95	1	2	yes	—
90	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
91	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#108	1	4	yes	—
94	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₈×C₂₃	—	4	8	no	—
98	C₈×C₂₃	—	4	4	no	—
99	#2920	C₂×C₂₃	1	2	yes	—
100	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
101	#1293	C₂×C₂₃	1	4	yes	—
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	—	2	2	no	—
107	C₂²×C₂₃	—	4	4	no	—
108	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
109	C₂²×C₂₃	C₂×#108	1	2	yes	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	2	2	no	—
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	#338	1	2	yes	—
116	C₄×C₂₃	#341	1	2	yes	—
117	C₄×C₂₃	#341	1	2	yes	—
118	C₄×C₂₃	#410	1	2	yes	—
119	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
120	#432	—	2	2	no	—
121	#432	C₄×C₂₃	1	4	yes	—
122	#431	—	2	2	no	—
123	#431	—	2	2	no	—
124	#431	—	2	2	no	—
125	#431	C₄×C₂₃	1	4	yes	—
126	#423	—	2	2	no	—
127	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
128	#407	C₂²×C₂₃	1	4	yes	—
129	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
130	C₄×M₁₆	—	2	2	no	—
131	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
132	C₂×C₂₃	—	2	2	no	—
133	C₂×C₂₃	—	4	2	no	—
134	C₂×C₂₃	C₂×#338	1	1	yes	characteristic
135	C₂×C₂₃	C₂×#341	1	1	yes	characteristic
136	C₂×C₂₃	#1740	1	1	yes	characteristic
137	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
138	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
139	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
140	C₂×M₁₆	—	2	4	no	—
141	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
142	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
143	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
146	C₂×C₄²	D₄×C₂₃	1	2	yes	—
147	C₂×C₄²	D₄×C₂₃	1	2	yes	—
148	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
149	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	2	4	no	—
153	C₂²⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
156	#97	—	2	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	4	no	—
160	C₂₃	#8007	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
167	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
170	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
171	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
175	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
177	M₁₆	—	4	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	4	4	no	—
181	C₄²	C₄²×C₂₃	1	2	yes	—
182	C₄²	C₄²×C₂₃	1	1	yes	characteristic
183	C₄²	C₄²×C₂₃	1	1	yes	characteristic
184	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
185	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
186	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
187	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
188	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
189	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
195	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
208	C₂×C₄	C₂₃×#95	1	2	yes	—
209	C₂×C₄	C₂₃×#95	1	2	yes	—
210	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
211	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
212	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
213	C₂×C₄	C₂₃×#108	1	4	yes	—
214	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
217	C₈	—	4	8	no	—
218	C₈	—	4	4	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
226	C₂²	C₂×C₂₃×#108	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	C₂₃×#338	1	2	yes	—
233	C₄	C₂₃×#341	1	2	yes	—
234	C₄	C₂₃×#341	1	2	yes	—
235	C₄	C₂₃×#410	1	2	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	4	2	no	—
238	C₂	C₂×C₂₃×#338	1	1	yes	characteristic
239	C₂	C₂×C₂₃×#341	1	1	yes	characteristic
240	C₂	C₂₃×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#95
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#95
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
70. #8007	C₂₃	direct (N × H)
160. C₂₃	#8007	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×#431	2
C₂₃×#1293	2
#8007	23

Subgroups — C23×#8007 #19155633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8007 #19155633