Subgroups — C₂×C₂₃×#2112 #19180957

Order 5888.

1406 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2112	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	—	2	8	no	—
7	C₂³×C₈×C₂₃	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₂₃	C₄	1	4	yes	—
9	C₂³×C₈×C₂₃	C₄	1	2	yes	—
10	C₂³×C₈×C₂₃	C₂²	1	4	yes	—
11	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×M₃₂	—	2	8	no	—
14	C₂×C₂₃×M₃₂	C₂²	1	8	yes	complemented
15	C₂²×C₁₆×C₂₃	C₂²	1	4	yes	complemented
16	C₂₃×C₂²⋊C₁₆	C₂²	1	16	yes	complemented
17	C₂⁵×C₂₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	—	2	4	no	—
19	C₂³×C₄×C₂₃	—	2	4	no	—
20	C₂³×C₄×C₂₃	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	C₈	1	4	yes	—
22	C₂³×C₄×C₂₃	C₈	1	2	yes	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	—	2	4	no	—
27	C₂²×C₈×C₂₃	—	2	16	no	—
28	C₂²×C₈×C₂₃	—	2	8	no	—
29	C₂²×C₈×C₂₃	—	2	8	no	—
30	C₂²×C₈×C₂₃	—	2	16	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₂₃	D₄	1	8	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
39	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×M₃₂	—	2	16	no	—
42	C₂×C₁₆×C₂₃	—	2	8	no	—
43	C₂×C₁₆×C₂₃	C₂³	1	8	yes	complemented
44	C₂⁴×C₂₃	—	2	4	no	—
45	C₂⁴×C₂₃	—	2	8	no	—
46	C₂⁴×C₂₃	C₂×C₈	1	4	yes	—
47	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
48	C₂⁴×C₂₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₂₃	—	2	4	no	—
50	C₂²×C₄×C₂₃	—	2	16	no	—
51	C₂²×C₄×C₂₃	—	2	8	no	—
52	C₂²×C₄×C₂₃	—	2	8	no	—
53	C₂²×C₄×C₂₃	—	2	4	no	—
54	C₂²×C₄×C₂₃	—	2	8	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂×C₈	1	8	yes	—
60	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	M₁₆	1	4	yes	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₃	—	2	8	no	—
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	—	2	16	no	—
70	C₂×C₈×C₂₃	—	2	8	no	—
71	C₂×C₈×C₂₃	—	2	8	no	—
72	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
73	C₂×C₈×C₂₃	C₂²⋊C₄	1	8	yes	—
74	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
79	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
80	C₁₆×C₂₃	—	2	8	no	—
81	C₂×#2112	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	—	2	32	no	—
83	C₂³×C₂₃	—	2	8	no	—
84	C₂³×C₂₃	—	2	4	no	—
85	C₂³×C₂₃	—	2	4	no	—
86	C₂³×C₂₃	—	2	4	no	—
87	C₂³×C₂₃	—	2	16	no	—
88	C₂³×C₂₃	C₂²⋊C₈	1	4	yes	—
89	C₂³×C₂₃	M₃₂	1	8	yes	complemented
90	C₂³×C₂₃	C₂²×C₈	1	2	yes	—
91	C₂³×C₂₃	C₂²×C₈	1	4	yes	—
92	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₂₃	—	2	8	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	2	8	no	—
98	C₂×C₄×C₂₃	—	2	16	no	—
99	C₂×C₄×C₂₃	C₂²⋊C₈	1	8	yes	—
100	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
102	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
104	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
107	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₂₃	—	2	4	no	—
110	C₈×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
111	C₈×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
112	C₈×C₂₃	C₂²×D₄	1	2	yes	—
113	C₂⁴×C₈	C₂×C₂₃	1	1	yes	characteristic
114	C₂²×M₃₂	C₂×C₂₃	1	2	yes	complemented
115	#2112	C₂×C₂₃	1	8	yes	direct product factor
116	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	4	yes	complemented
117	C₂²×C₂₃	—	2	32	no	—
118	C₂²×C₂₃	—	2	4	no	—
119	C₂²×C₂₃	—	2	8	no	—
120	C₂²×C₂₃	—	2	16	no	—
121	C₂²×C₂₃	—	2	4	no	—
122	C₂²×C₂₃	—	2	4	no	—
123	C₂²×C₂₃	C₂×C₂²⋊C₈	1	4	yes	—
124	C₂²×C₂₃	C₂×C₂²⋊C₈	1	2	yes	—
125	C₂²×C₂₃	C₂×M₃₂	1	4	yes	—
126	C₂²×C₂₃	C₂×M₃₂	1	8	yes	complemented
127	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₂₃	—	2	4	no	—
129	C₄×C₂₃	C₂×C₂²⋊C₈	1	2	yes	—
130	C₄×C₂₃	C₂×C₂²⋊C₈	1	4	yes	—
131	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
133	C₂⁴×C₄	C₄×C₂₃	1	1	yes	characteristic
134	C₂³×C₈	—	2	8	no	—
135	C₂³×C₈	C₄×C₂₃	1	1	yes	characteristic
136	C₂³×C₈	C₄×C₂₃	1	2	yes	—
137	C₂³×C₈	C₄×C₂₃	1	4	yes	—
138	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
139	C₂³×C₈	C₂²×C₂₃	1	4	yes	—
140	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×M₃₂	—	2	8	no	—
142	C₂×M₃₂	C₂²×C₂₃	1	8	yes	complemented
143	C₂²×C₁₆	C₂²×C₂₃	1	4	yes	complemented
144	C₂²⋊C₁₆	C₂²×C₂₃	1	16	yes	complemented
145	C₂×C₂₃	—	2	4	no	—
146	C₂×C₂₃	—	2	8	no	—
147	C₂×C₂₃	#2112	1	4	yes	direct product factor
148	C₂×C₂₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
149	C₂×C₂₃	C₂²×M₃₂	1	2	yes	—
150	C₂⁵	C₈×C₂₃	1	1	yes	characteristic
151	C₂³×C₄	—	2	4	no	—
152	C₂³×C₄	—	2	4	no	—
153	C₂³×C₄	C₈×C₂₃	1	1	yes	characteristic
154	C₂³×C₄	C₈×C₂₃	1	4	yes	—
155	C₂³×C₄	C₈×C₂₃	1	2	yes	—
156	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
157	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
158	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₈	—	2	4	no	—
160	C₂²×C₈	—	2	16	no	—
161	C₂²×C₈	—	2	8	no	—
162	C₂²×C₈	—	2	8	no	—
163	C₂²×C₈	—	2	16	no	—
164	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
166	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
167	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
168	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
169	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
170	C₂²×C₈	D₄×C₂₃	1	8	yes	—
171	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
172	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
173	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
174	M₃₂	—	2	16	no	—
175	C₂×C₁₆	—	2	8	no	—
176	C₂×C₁₆	C₂³×C₂₃	1	8	yes	complemented
177	C₂₃	C₂×#2112	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴	—	2	8	no	—
179	C₂⁴	—	2	4	no	—
180	C₂⁴	C₂×C₈×C₂₃	1	4	yes	—
181	C₂⁴	C₂×C₈×C₂₃	1	2	yes	—
182	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	16	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
191	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
192	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
193	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
194	C₂²×C₄	C₂×C₈×C₂₃	1	8	yes	—
195	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
196	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	M₁₆×C₂₃	1	4	yes	—
198	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	2	16	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	C₂²⋊C₄×C₂₃	1	8	yes	—
207	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
208	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
209	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
212	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
213	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
214	C₁₆	—	2	8	no	—
215	C₂³	—	2	32	no	—
216	C₂³	—	2	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	16	no	—
221	C₂³	C₂₃×C₂²⋊C₈	1	4	yes	—
222	C₂³	C₂₃×M₃₂	1	8	yes	complemented
223	C₂³	C₂²×C₈×C₂₃	1	2	yes	—
224	C₂³	C₂²×C₈×C₂₃	1	4	yes	—
225	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	C₂₃×C₂²⋊C₈	1	8	yes	—
233	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
236	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
237	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
239	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
240	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
241	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
244	C₈	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
245	C₈	C₂²×D₄×C₂₃	1	2	yes	—
246	C₂²	—	2	32	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	16	no	—
249	C₂²	—	2	8	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₂×C₂₃×C₂²⋊C₈	1	4	yes	—
253	C₂²	C₂×C₂₃×C₂²⋊C₈	1	2	yes	—
254	C₂²	C₂×C₂₃×M₃₂	1	4	yes	—
255	C₂²	C₂×C₂₃×M₃₂	1	8	yes	complemented
256	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	C₂×C₂₃×C₂²⋊C₈	1	4	yes	—
259	C₄	C₂×C₂₃×C₂²⋊C₈	1	2	yes	—
260	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
261	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
262	C₂	—	2	4	no	—
263	C₂	—	2	8	no	—
264	C₂	C₂₃×#2112	1	4	yes	direct product factor
265	C₂	C₂²×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
266	C₂	C₂²×C₂₃×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₃	C₂³
commutator	C₂²	C₂³×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×C₈
prime radical	C₂⁵×C₂₃	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×M₃₂	C₂	semidirect
3. C₂₃×#2112	C₂	direct (N × H)
4. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
14. C₂×C₂₃×M₃₂	C₂²	semidirect
15. C₂²×C₁₆×C₂₃	C₂²	semidirect
16. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
43. C₂×C₁₆×C₂₃	C₂³	semidirect
81. C₂×#2112	C₂₃	direct (N × H)
89. C₂³×C₂₃	M₃₂	semidirect
114. C₂²×M₃₂	C₂×C₂₃	semidirect
115. #2112	C₂×C₂₃	direct (N × H)
116. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
126. C₂²×C₂₃	C₂×M₃₂	semidirect
142. C₂×M₃₂	C₂²×C₂₃	semidirect
143. C₂²×C₁₆	C₂²×C₂₃	semidirect
144. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
147. C₂×C₂₃	#2112	direct (N × H)
176. C₂×C₁₆	C₂³×C₂₃	semidirect
177. C₂₃	C₂×#2112	direct (N × H)
222. C₂³	C₂₃×M₃₂	semidirect
255. C₂²	C₂×C₂₃×M₃₂	semidirect
264. C₂	C₂₃×#2112	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₂₃	2
C₂²×C₂₃×M₃₂	2
C₂₃×#2112	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂×#2112	23

Subgroups — C2×C23×#2112 #19180957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2112 #19180957