Subgroups — C₂×C₂₃×#1383 #19160188

Order 5888.

614 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1383	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₁₆×C₂₃	—	2	1	no	—
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
8	C₂×C₂₃×M₃₂	—	2	2	no	—
9	C₂×C₂₃×M₃₂	—	2	2	no	—
10	C₂×C₂₃×M₃₂	C₄	1	2	yes	complemented
11	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₂₃	C₄	1	2	yes	complemented
13	C₂×C₂₃×#108	—	2	1	no	—
14	C₂×C₂₃×#108	—	2	2	no	—
15	C₂×C₂₃×#108	C₄	1	4	yes	complemented
16	C₂×C₂₃×#108	C₂²	1	2	yes	—
17	C₂×C₂₃×#106	—	2	1	no	—
18	C₂×C₂₃×#106	—	2	2	no	—
19	C₂×C₂₃×#106	C₄	1	4	yes	—
20	C₂×C₂₃×#106	C₂²	1	2	yes	—
21	C₂³×C₄×C₂₃	—	4	1	no	—
22	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×M₁₆×C₂₃	—	2	4	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	4	2	no	—
26	C₂×M₁₆×C₂₃	—	4	1	no	—
27	C₂²×C₈×C₂₃	—	4	1	no	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	2	yes	—
34	C₂²×C₈×C₂₃	D₄	1	2	yes	—
35	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
42	C₂₃×M₃₂	—	2	4	no	—
43	C₂₃×M₃₂	—	2	2	no	—
44	C₂₃×M₃₂	—	2	2	no	—
45	C₂×C₁₆×C₂₃	—	2	4	no	—
46	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
47	C₂₃×#108	—	2	4	no	—
48	C₂₃×#108	—	2	2	no	—
49	C₂₃×#108	C₂×C₄	1	4	yes	complemented
50	C₂₃×#106	—	2	4	no	—
51	C₂₃×#106	—	2	2	no	—
52	C₂₃×#106	C₂×C₄	1	4	yes	—
53	C₂⁴×C₂₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	4	4	no	—
55	C₂²×C₄×C₂₃	—	4	2	no	—
56	C₂²×C₄×C₂₃	—	8	4	no	—
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	D₈	1	2	yes	—
62	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	M₁₆×C₂₃	—	2	4	no	—
68	M₁₆×C₂₃	—	4	4	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	4	4	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	C₄²	1	2	yes	—
73	C₂×C₈×C₂₃	C₄²	1	2	yes	—
74	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	—	4	4	no	—
92	C₄⋊C₄×C₂₃	—	4	2	no	—
93	C₁₆×C₂₃	—	2	4	no	—
94	C₂×#1383	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	—	2	4	no	—
96	C₂³×C₂₃	#102	1	2	yes	—
97	C₂³×C₂₃	#103	1	2	yes	—
98	C₂³×C₂₃	#106	1	1	yes	characteristic
99	C₂³×C₂₃	#106	1	1	yes	characteristic
100	C₂³×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	8	8	no	—
103	C₂×C₄×C₂₃	—	8	4	no	—
104	C₂×C₄×C₂₃	#95	1	2	yes	—
105	C₂×C₄×C₂₃	#95	1	2	yes	—
106	C₂×C₄×C₂₃	#102	1	2	yes	—
107	C₂×C₄×C₂₃	#102	1	2	yes	—
108	C₂×C₄×C₂₃	#102	1	2	yes	—
109	C₂×C₄×C₂₃	#103	1	2	yes	—
110	C₂×C₄×C₂₃	#103	1	2	yes	—
111	C₂×C₄×C₂₃	#103	1	2	yes	—
112	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#107	1	2	yes	—
115	C₂×C₄×C₂₃	#107	1	2	yes	—
116	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
122	C₈×C₂₃	—	4	4	no	—
123	C₈×C₂₃	#95	1	2	yes	—
124	C₈×C₂₃	#95	1	2	yes	—
125	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
126	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic
130	C₂²×#108	C₂×C₂₃	1	1	yes	characteristic
131	C₂²×#106	C₂×C₂₃	1	1	yes	characteristic
132	#1383	C₂×C₂₃	1	4	yes	direct product factor
133	C₂²×C₂₃	—	2	4	no	—
134	C₂²×C₂₃	—	2	4	no	—
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	2	2	no	—
137	C₂²×C₂₃	#340	1	2	yes	—
138	C₂²×C₂₃	#340	1	2	yes	—
139	C₂²×C₂₃	#365	1	4	yes	—
140	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
141	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
142	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
143	C₄×C₂₃	—	8	4	no	—
144	C₄×C₂₃	#340	1	2	yes	—
145	C₄×C₂₃	#340	1	2	yes	—
146	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
147	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
148	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
149	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
150	C₂²×M₁₆	—	2	1	no	—
151	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₂²×C₄⋊C₄	—	2	1	no	—
153	C₂×M₃₂	—	2	2	no	—
154	C₂×M₃₂	—	2	2	no	—
155	C₂×M₃₂	C₄×C₂₃	1	2	yes	complemented
156	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
157	C₂²×C₁₆	C₄×C₂₃	1	2	yes	complemented
158	C₂×#108	—	2	1	no	—
159	C₂×#108	—	2	2	no	—
160	C₂×#108	C₄×C₂₃	1	4	yes	complemented
161	C₂×#108	C₂²×C₂₃	1	2	yes	—
162	C₂×#106	—	2	1	no	—
163	C₂×#106	—	2	2	no	—
164	C₂×#106	C₄×C₂₃	1	4	yes	—
165	C₂×#106	C₂²×C₂₃	1	2	yes	—
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	2	1	no	—
169	C₂×C₂₃	#1383	1	4	yes	direct product factor
170	C₂×C₂₃	C₂×#340	1	1	yes	characteristic
171	C₂×C₂₃	C₂×#365	1	2	yes	—
172	C₂³×C₄	—	4	1	no	—
173	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	4	2	no	—
177	C₂×M₁₆	—	4	1	no	—
178	C₂²×C₈	—	4	1	no	—
179	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
182	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
183	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₂₃	1	2	yes	—
185	C₂²×C₈	D₄×C₂₃	1	2	yes	—
186	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
187	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	4	2	no	—
192	C₂×C₄⋊C₄	—	4	1	no	—
193	M₃₂	—	2	4	no	—
194	M₃₂	—	2	2	no	—
195	M₃₂	—	2	2	no	—
196	C₂×C₁₆	—	2	4	no	—
197	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
198	#108	—	2	4	no	—
199	#108	—	2	2	no	—
200	#108	C₂×C₄×C₂₃	1	4	yes	complemented
201	#106	—	2	4	no	—
202	#106	—	2	2	no	—
203	#106	C₂×C₄×C₂₃	1	4	yes	—
204	C₂₃	C₂×#1383	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	SD₁₆×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	8	4	no	—
209	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
211	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	D₈×C₂₃	1	2	yes	—
214	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	4	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	C₄²×C₂₃	1	2	yes	—
225	C₂×C₈	C₄²×C₂₃	1	2	yes	—
226	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
227	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
228	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
229	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
230	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
231	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
232	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
233	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
234	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
235	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
236	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₁₆	—	2	4	no	—
246	C₂³	—	2	4	no	—
247	C₂³	C₂₃×#102	1	2	yes	—
248	C₂³	C₂₃×#103	1	2	yes	—
249	C₂³	C₂₃×#106	1	1	yes	characteristic
250	C₂³	C₂₃×#106	1	1	yes	characteristic
251	C₂³	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	8	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	C₂₃×#95	1	2	yes	—
256	C₂×C₄	C₂₃×#95	1	2	yes	—
257	C₂×C₄	C₂₃×#102	1	2	yes	—
258	C₂×C₄	C₂₃×#102	1	2	yes	—
259	C₂×C₄	C₂₃×#102	1	2	yes	—
260	C₂×C₄	C₂₃×#103	1	2	yes	—
261	C₂×C₄	C₂₃×#103	1	2	yes	—
262	C₂×C₄	C₂₃×#103	1	2	yes	—
263	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
264	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
265	C₂×C₄	C₂₃×#107	1	2	yes	—
266	C₂×C₄	C₂₃×#107	1	2	yes	—
267	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
270	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
272	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
273	C₈	—	4	4	no	—
274	C₈	C₂₃×#95	1	2	yes	—
275	C₈	C₂₃×#95	1	2	yes	—
276	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
277	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
278	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
279	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	C₂₃×#340	1	2	yes	—
285	C₂²	C₂₃×#340	1	2	yes	—
286	C₂²	C₂₃×#365	1	4	yes	—
287	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
288	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
289	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
290	C₄	—	8	4	no	—
291	C₄	C₂₃×#340	1	2	yes	—
292	C₄	C₂₃×#340	1	2	yes	—
293	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
294	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
295	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
296	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	1	no	—
300	C₂	C₂₃×#1383	1	4	yes	direct product factor
301	C₂	C₂×C₂₃×#340	1	1	yes	characteristic
302	C₂	C₂×C₂₃×#365	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	#106
commutator	C₈	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1383	C₂	direct (N × H)
10. C₂×C₂₃×M₃₂	C₄	semidirect
12. C₂²×C₁₆×C₂₃	C₄	semidirect
15. C₂×C₂₃×#108	C₄	semidirect
46. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
49. C₂₃×#108	C₂×C₄	semidirect
94. C₂×#1383	C₂₃	direct (N × H)
132. #1383	C₂×C₂₃	direct (N × H)
155. C₂×M₃₂	C₄×C₂₃	semidirect
157. C₂²×C₁₆	C₄×C₂₃	semidirect
160. C₂×#108	C₄×C₂₃	semidirect
169. C₂×C₂₃	#1383	direct (N × H)
197. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
200. #108	C₂×C₄×C₂₃	semidirect
204. C₂₃	C₂×#1383	direct (N × H)
300. C₂	C₂₃×#1383	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×M₃₂	2
C₂²×C₂₃×#108	2
C₂²×C₂₃×#106	2
C₂₃×#1383	2
C₂×#1383	23

Subgroups — C2×C23×#1383 #19160188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1383 #19160188