Subgroups — C₂₃×#2486 #15670105

Order 2944.

624 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×#123	C₂	1	2	yes	maximal
2	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#398	C₂	1	2	yes	maximal, complemented
5	C₂₃×#397	C₂	1	2	yes	maximal, complemented
6	C₂₃×#389	C₂	1	2	yes	maximal, complemented
7	C₂₃×#381	C₂	1	2	yes	maximal, complemented
8	C₂₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#380	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#123	—	2	8	no	—
15	C₄×D₄×C₂₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#95	C₂²	1	4	yes	—
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	4	yes	—
30	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
31	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
32	C₂×D₄×C₂₃	—	2	2	no	—
33	C₂×D₄×C₂₃	—	2	2	no	—
34	C₂×D₄×C₂₃	D₄	1	4	yes	—
35	C₂²×C₄×C₂₃	—	2	4	no	—
36	C₂²×C₄×C₂₃	—	2	4	no	—
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	D₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₂₃	—	2	4	no	—
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	8	no	—
53	C₄⋊C₄×C₂₃	—	2	4	no	—
54	C₂²⋊C₄×C₂₃	—	2	2	no	—
55	C₂²⋊C₄×C₂₃	—	2	4	no	—
56	C₂²⋊C₄×C₂₃	—	2	8	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₄²×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	—	2	1	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	2	1	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	C₂×D₄	1	2	yes	—
66	C₂³×C₂₃	C₂×D₄	1	2	yes	—
67	C₂³×C₂₃	C₂×D₄	1	2	yes	—
68	C₂³×C₂₃	#41	1	2	yes	—
69	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
70	D₄×C₂₃	—	2	8	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	8	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	4	no	—
85	C₂×C₄×C₂₃	—	4	8	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₂₃	#41	1	4	yes	—
91	#2486	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	—	2	4	no	—
98	C₂²×C₂₃	—	4	1	no	—
99	C₂²×C₂₃	—	4	1	no	—
100	C₂²×C₂₃	—	4	2	no	—
101	C₂²×C₂₃	#121	1	4	yes	—
102	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
105	C₂²×C₂₃	C₂×#41	1	2	yes	—
106	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	4	4	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	—	4	4	no	—
113	C₄×C₂₃	#123	1	4	yes	complemented
114	C₂×#123	C₂×C₂₃	1	2	yes	—
115	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
116	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#398	C₂×C₂₃	1	2	yes	complemented
118	#397	C₂×C₂₃	1	2	yes	complemented
119	#389	C₂×C₂₃	1	2	yes	complemented
120	#381	C₂×C₂₃	1	2	yes	complemented
121	#380	C₂×C₂₃	1	2	yes	complemented
122	#380	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₂×C₂₃	—	2	4	no	—
124	C₂×C₂₃	—	4	2	no	—
125	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
126	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
127	C₂×C₂₃	#536	1	1	yes	characteristic
128	C₂×C₂₃	#547	1	2	yes	—
129	C₂×C₂₃	#552	1	2	yes	—
130	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
131	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
132	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
133	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
134	#123	—	2	8	no	—
135	C₄×D₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
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	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
141	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
146	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
147	#95	C₂²×C₂₃	1	4	yes	—
148	#95	C₂²×C₂₃	1	2	yes	—
149	#95	C₂²×C₂₃	1	4	yes	—
150	C₂₃	#2486	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
152	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	D₄×C₂₃	1	4	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	D₄×C₂₃	1	4	yes	—
163	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
166	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
167	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	8	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	4	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	1	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	1	no	—
185	C₂³	—	2	2	no	—
186	C₂³	C₂×D₄×C₂₃	1	2	yes	—
187	C₂³	C₂×D₄×C₂₃	1	2	yes	—
188	C₂³	C₂×D₄×C₂₃	1	2	yes	—
189	C₂³	#41×C₂₃	1	2	yes	—
190	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
191	D₄	—	2	8	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
210	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
211	C₂×C₄	#41×C₂₃	1	4	yes	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	C₂₃×#121	1	4	yes	—
222	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
223	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
224	C₂²	C₂×#41×C₂₃	1	2	yes	—
225	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
226	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	4	no	—
230	C₄	—	4	4	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	C₂₃×#123	1	4	yes	complemented
234	C₂	—	2	4	no	—
235	C₂	—	4	2	no	—
236	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
237	C₂	C₂×C₂₃×#123	1	1	yes	characteristic
238	C₂	C₂₃×#536	1	1	yes	characteristic
239	C₂	C₂₃×#547	1	2	yes	—
240	C₂	C₂₃×#552	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
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
3. C₂²×C₄⋊C₄×C₂₃	C₂	semidirect
4. C₂₃×#398	C₂	semidirect
5. C₂₃×#397	C₂	semidirect
6. C₂₃×#389	C₂	semidirect
7. C₂₃×#381	C₂	semidirect
8. C₂₃×#380	C₂	semidirect
9. C₂₃×#380	C₂	semidirect
91. #2486	C₂₃	direct (N × H)
113. C₄×C₂₃	#123	semidirect
116. C₂²×C₄⋊C₄	C₂×C₂₃	semidirect
117. #398	C₂×C₂₃	semidirect
118. #397	C₂×C₂₃	semidirect
119. #389	C₂×C₂₃	semidirect
120. #381	C₂×C₂₃	semidirect
121. #380	C₂×C₂₃	semidirect
122. #380	C₂×C₂₃	semidirect
150. C₂₃	#2486	direct (N × H)
233. C₄	C₂₃×#123	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#123	2
C₂×C₄×D₄×C₂₃	2
C₂²×C₄⋊C₄×C₂₃	2
C₂₃×#398	2
C₂₃×#397	2
C₂₃×#389	2
C₂₃×#381	2
C₂₃×#380	2
#2486	23

Subgroups — C23×#2486 #15670105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2486 #15670105