Subgroups — C₂₃×#2570 #15670189

Order 2944.

380 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#400	C₂	1	2	yes	maximal
2	C₂₃×#398	C₂	1	2	yes	maximal
3	C₂₃×#389	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#389	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#387	C₂	1	2	yes	maximal
6	C₂₃×#387	C₂	1	2	yes	maximal
7	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#95	C₂²	1	1	yes	characteristic
27	C₂₃×#95	C₂²	1	1	yes	characteristic
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂₃×#95	C₂²	1	2	yes	—
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₃	—	2	4	no	—
49	C₄⋊C₄×C₂₃	—	2	4	no	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₄⋊C₄×C₂₃	—	4	2	no	—
56	C₄²×C₂₃	—	2	2	no	—
57	C₄²×C₂₃	—	2	2	no	—
58	C₄²×C₂₃	—	4	2	no	—
59	C₄²×C₂₃	Q₈	1	4	yes	—
60	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₂₃	—	2	1	no	—
62	C₂×C₄×C₂₃	—	2	1	no	—
63	C₂×C₄×C₂₃	—	2	1	no	—
64	C₂×C₄×C₂₃	—	2	1	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	2	1	no	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	1	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₂₃	#41	1	2	yes	—
91	C₂×C₄×C₂₃	#41	1	2	yes	—
92	C₂×C₄×C₂₃	#41	1	2	yes	—
93	C₂×C₄×C₂₃	#41	1	2	yes	—
94	#2570	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
97	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
98	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
99	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
102	C₄×C₂₃	—	2	2	no	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	4	1	no	—
107	C₄×C₂₃	—	4	1	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	#125	1	4	yes	—
113	#400	C₂×C₂₃	1	2	yes	—
114	#398	C₂×C₂₃	1	2	yes	—
115	#389	C₂×C₂₃	1	1	yes	characteristic
116	#389	C₂×C₂₃	1	1	yes	characteristic
117	#387	C₂×C₂₃	1	2	yes	—
118	#387	C₂×C₂₃	1	2	yes	—
119	#384	C₂×C₂₃	1	1	yes	characteristic
120	#382	C₂×C₂₃	1	1	yes	characteristic
121	#382	C₂×C₂₃	1	1	yes	characteristic
122	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
123	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
124	C₂×C₂₃	C₂×#125	1	1	yes	characteristic
125	C₂×C₂₃	#533	1	1	yes	characteristic
126	C₂×C₂₃	#544	1	1	yes	characteristic
127	C₂×C₂₃	#552	1	1	yes	characteristic
128	C₂×C₂₃	#553	1	1	yes	characteristic
129	C₂×C₂₃	#556	1	1	yes	characteristic
130	C₂×C₂₃	#559	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
141	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
144	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
145	#95	C₂²×C₂₃	1	1	yes	characteristic
146	#95	C₂²×C₂₃	1	1	yes	characteristic
147	#95	C₂²×C₂₃	1	2	yes	—
148	#95	C₂²×C₂₃	1	2	yes	—
149	#95	C₂²×C₂₃	1	2	yes	—
150	#95	C₂²×C₂₃	1	2	yes	—
151	#95	C₂²×C₂₃	1	2	yes	—
152	#95	C₂²×C₂₃	1	2	yes	—
153	#95	C₂²×C₂₃	1	2	yes	—
154	C₂₃	#2570	1	1	yes	characteristic, direct product factor, Sylow
155	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
157	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	Q₈×C₂₃	1	4	yes	—
180	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
208	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
209	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
210	C₂×C₄	#41×C₂₃	1	2	yes	—
211	C₂×C₄	#41×C₂₃	1	2	yes	—
212	C₂×C₄	#41×C₂₃	1	2	yes	—
213	C₂×C₄	#41×C₂₃	1	2	yes	—
214	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
215	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
216	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
217	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
218	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
219	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
220	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	C₂₃×#125	1	4	yes	—
232	C₂	C₂×C₂₃×#125	1	1	yes	characteristic
233	C₂	C₂₃×#533	1	1	yes	characteristic
234	C₂	C₂₃×#544	1	1	yes	characteristic
235	C₂	C₂₃×#552	1	1	yes	characteristic
236	C₂	C₂₃×#553	1	1	yes	characteristic
237	C₂	C₂₃×#556	1	1	yes	characteristic
238	C₂	C₂₃×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
94. #2570	C₂₃	direct (N × H)
154. C₂₃	#2570	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×#400	2
C₂₃×#398	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#384	2
C₂₃×#382	2
C₄×C₄⋊C₄×C₂₃	2
#2570	23

Subgroups — C23×#2570 #15670189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2570 #15670189