Subgroups — C₂×C₂₃×#3259 #19208536

Order 5888.

2010 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#540	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3259	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#478	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#469	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#466	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#459	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
9	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#540	—	2	8	no	—
13	C₂₃×#540	C₂²	1	16	yes	complemented
14	C₂×C₂₃×#127	C₂²	1	2	yes	—
15	C₂×C₂₃×#125	C₂²	1	2	yes	—
16	C₂×C₂₃×#123	—	2	4	no	—
17	C₂×C₂₃×#121	—	2	4	no	—
18	C₂×C₂₃×#121	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂₃×#478	C₂²	1	32	yes	complemented
22	C₂₃×#469	C₂²	1	16	yes	complemented
23	C₂₃×#466	C₂²	1	16	yes	complemented
24	C₂₃×#459	C₂²	1	32	yes	complemented
25	C₂×C₂₃×#107	C₂²	1	2	yes	—
26	C₂×C₂₃×#106	C₂²	1	2	yes	—
27	C₂₃×#424	C₂²	1	8	yes	complemented
28	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
29	C₂×C₂₃×#102	C₂²	1	4	yes	—
30	C₂×C₂₃×#102	C₂²	1	4	yes	—
31	C₂²×D₄×C₂₃	—	2	4	no	—
32	C₂²×D₄×C₂₃	—	2	4	no	—
33	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₂₃	—	2	4	no	—
35	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂×D₈×C₂₃	—	4	4	no	—
37	C₂×D₈×C₂₃	—	4	4	no	—
38	C₂×D₈×C₂₃	C₂³	1	8	yes	—
39	C₂×D₈×C₂₃	C₂³	1	8	yes	—
40	C₂×M₁₆×C₂₃	—	4	2	no	—
41	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
42	C₂²×C₈×C₂₃	—	4	2	no	—
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
46	C₂₃×#127	—	2	4	no	—
47	C₂₃×#127	C₂³	1	8	yes	—
48	C₂₃×#125	—	2	4	no	—
49	C₂₃×#125	C₂³	1	8	yes	—
50	C₂₃×#123	—	2	32	no	—
51	C₂₃×#121	—	2	8	no	—
52	C₂₃×#121	—	2	32	no	—
53	C₂₃×#121	C₂³	1	16	yes	—
54	C₄×D₄×C₂₃	—	2	8	no	—
55	C₄×D₄×C₂₃	C₂³	1	16	yes	—
56	C₂₃×#117	D₄	1	4	yes	—
57	C₂₃×#117	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
62	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
63	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
64	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
65	C₂₃×#107	C₂³	1	8	yes	—
66	C₂₃×#106	C₂³	1	8	yes	—
67	C₂₃×C₄⋊C₈	C₂³	1	16	yes	—
68	C₂₃×#102	C₂³	1	16	yes	—
69	C₂₃×#102	C₂³	1	16	yes	—
70	C₂⁴×C₂₃	—	2	4	no	—
71	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂×D₄×C₂₃	—	2	16	no	—
73	C₂×D₄×C₂₃	—	2	4	no	—
74	C₂×D₄×C₂₃	—	2	16	no	—
75	C₂×D₄×C₂₃	—	2	8	no	—
76	C₂×D₄×C₂₃	—	2	8	no	—
77	C₂×D₄×C₂₃	—	8	8	no	—
78	C₂×D₄×C₂₃	—	8	8	no	—
79	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₂₃	—	2	2	no	—
81	C₂²×C₄×C₂₃	—	2	2	no	—
82	C₂²×C₄×C₂₃	—	2	4	no	—
83	C₂²×C₄×C₂₃	—	2	16	no	—
84	C₂²×C₄×C₂₃	—	8	8	no	—
85	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
90	D₈×C₂₃	—	4	8	no	—
91	D₈×C₂₃	—	4	8	no	—
92	M₁₆×C₂₃	—	4	4	no	—
93	C₂×C₈×C₂₃	—	2	4	no	—
94	C₂×C₈×C₂₃	—	2	4	no	—
95	C₂×C₈×C₂₃	—	4	4	no	—
96	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
97	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
98	C₄⋊C₄×C₂₃	—	2	16	no	—
99	C₄⋊C₄×C₂₃	—	2	4	no	—
100	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
101	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
102	C₂²⋊C₄×C₂₃	—	2	16	no	—
103	C₂²⋊C₄×C₂₃	—	2	16	no	—
104	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
105	C₄²×C₂₃	C₂×D₄	1	4	yes	—
106	C₄²×C₂₃	C₂⁴	1	4	yes	—
107	C₂×#3259	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₂₃	—	2	2	no	—
109	C₂³×C₂₃	—	2	4	no	—
110	C₂³×C₂₃	—	2	8	no	—
111	C₂³×C₂₃	—	4	8	no	—
112	C₂³×C₂₃	—	4	8	no	—
113	C₂³×C₂₃	—	4	8	no	—
114	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
115	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
116	D₄×C₂₃	—	2	8	no	—
117	D₄×C₂₃	—	2	8	no	—
118	D₄×C₂₃	—	4	8	no	—
119	D₄×C₂₃	—	8	16	no	—
120	D₄×C₂₃	—	8	16	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
122	C₂×C₄×C₂₃	—	2	8	no	—
123	C₂×C₄×C₂₃	—	2	4	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	2	no	—
126	C₂×C₄×C₂₃	—	2	8	no	—
127	C₂×C₄×C₂₃	—	4	8	no	—
128	C₂×C₄×C₂₃	—	8	16	no	—
129	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
130	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
133	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
134	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
135	C₈×C₂₃	—	2	4	no	—
136	C₈×C₂₃	—	2	4	no	—
137	C₂×#540	C₂×C₂₃	1	2	yes	complemented
138	#3259	C₂×C₂₃	1	16	yes	direct product factor
139	C₂×#478	C₂×C₂₃	1	4	yes	complemented
140	C₂×#469	C₂×C₂₃	1	2	yes	complemented
141	C₂×#466	C₂×C₂₃	1	2	yes	complemented
142	C₂×#459	C₂×C₂₃	1	4	yes	complemented
143	C₂×#424	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂²×C₂₃	—	2	2	no	—
145	C₂²×C₂₃	—	2	4	no	—
146	C₂²×C₂₃	—	4	4	no	—
147	C₂²×C₂₃	—	4	8	no	—
148	C₂²×C₂₃	—	4	8	no	—
149	C₂²×C₂₃	—	4	8	no	—
150	C₂²×C₂₃	—	8	16	no	—
151	C₂²×C₂₃	#537	1	2	yes	—
152	C₂²×C₂₃	#576	1	4	yes	—
153	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
154	C₄×C₂₃	—	2	2	no	—
155	C₄×C₂₃	—	2	8	no	—
156	C₄×C₂₃	—	4	8	no	—
157	C₄×C₂₃	#537	1	2	yes	—
158	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
160	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
161	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
162	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
163	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
164	#540	—	2	8	no	—
165	#540	C₂²×C₂₃	1	16	yes	complemented
166	C₂×#127	C₂²×C₂₃	1	2	yes	—
167	C₂×#125	C₂²×C₂₃	1	2	yes	—
168	C₂×#123	—	2	4	no	—
169	C₂×#121	—	2	4	no	—
170	C₂×#121	C₂²×C₂₃	1	4	yes	—
171	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
172	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
173	#478	C₂²×C₂₃	1	32	yes	complemented
174	#469	C₂²×C₂₃	1	16	yes	complemented
175	#466	C₂²×C₂₃	1	16	yes	complemented
176	#459	C₂²×C₂₃	1	32	yes	complemented
177	C₂×#107	C₂²×C₂₃	1	2	yes	—
178	C₂×#106	C₂²×C₂₃	1	2	yes	—
179	#424	C₂²×C₂₃	1	8	yes	complemented
180	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
181	C₂×#102	C₂²×C₂₃	1	4	yes	—
182	C₂×#102	C₂²×C₂₃	1	4	yes	—
183	C₂×C₂₃	—	4	2	no	—
184	C₂×C₂₃	—	8	8	no	—
185	C₂×C₂₃	#3259	1	4	yes	direct product factor
186	C₂×C₂₃	C₂×#537	1	1	yes	characteristic
187	C₂×C₂₃	C₂×#576	1	2	yes	—
188	C₂²×D₄	—	2	4	no	—
189	C₂²×D₄	—	2	4	no	—
190	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
191	C₂³×C₄	—	2	4	no	—
192	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
193	C₂×D₈	—	4	4	no	—
194	C₂×D₈	—	4	4	no	—
195	C₂×D₈	C₂³×C₂₃	1	8	yes	—
196	C₂×D₈	C₂³×C₂₃	1	8	yes	—
197	C₂×M₁₆	—	4	2	no	—
198	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
199	C₂²×C₈	—	4	2	no	—
200	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
201	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
202	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
203	#127	—	2	4	no	—
204	#127	C₂³×C₂₃	1	8	yes	—
205	#125	—	2	4	no	—
206	#125	C₂³×C₂₃	1	8	yes	—
207	#123	—	2	32	no	—
208	#121	—	2	8	no	—
209	#121	—	2	32	no	—
210	#121	C₂³×C₂₃	1	16	yes	—
211	C₄×D₄	—	2	8	no	—
212	C₄×D₄	C₂³×C₂₃	1	16	yes	—
213	#117	D₄×C₂₃	1	4	yes	—
214	#117	C₂³×C₂₃	1	4	yes	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
218	C₂×C₂²⋊C₄	—	2	4	no	—
219	C₂×C₂²⋊C₄	—	2	4	no	—
220	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
221	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
222	#107	C₂³×C₂₃	1	8	yes	—
223	#106	C₂³×C₂₃	1	8	yes	—
224	C₄⋊C₈	C₂³×C₂₃	1	16	yes	—
225	#102	C₂³×C₂₃	1	16	yes	—
226	#102	C₂³×C₂₃	1	16	yes	—
227	C₂₃	C₂×#3259	1	1	yes	characteristic, direct product factor, Sylow
228	C₂⁴	—	2	4	no	—
229	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂×D₄	—	2	16	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	16	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	8	8	no	—
236	C₂×D₄	—	8	8	no	—
237	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	16	no	—
242	C₂²×C₄	—	8	8	no	—
243	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
244	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
248	D₈	—	4	8	no	—
249	D₈	—	4	8	no	—
250	M₁₆	—	4	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	4	no	—
254	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
255	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
256	C₄⋊C₄	—	2	16	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
259	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
260	C₂²⋊C₄	—	2	16	no	—
261	C₂²⋊C₄	—	2	16	no	—
262	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
263	C₄²	C₂×D₄×C₂₃	1	4	yes	—
264	C₄²	C₂⁴×C₂₃	1	4	yes	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	8	no	—
268	C₂³	—	4	8	no	—
269	C₂³	—	4	8	no	—
270	C₂³	—	4	8	no	—
271	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
272	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
273	D₄	—	2	8	no	—
274	D₄	—	2	8	no	—
275	D₄	—	4	8	no	—
276	D₄	—	8	16	no	—
277	D₄	—	8	16	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	8	16	no	—
286	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
287	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
288	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
289	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
290	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
291	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	8	no	—
300	C₂²	—	8	16	no	—
301	C₂²	C₂₃×#537	1	2	yes	—
302	C₂²	C₂₃×#576	1	4	yes	—
303	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	8	no	—
306	C₄	—	4	8	no	—
307	C₄	C₂₃×#537	1	2	yes	—
308	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
309	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	—	8	8	no	—
312	C₂	C₂₃×#3259	1	4	yes	direct product factor
313	C₂	C₂×C₂₃×#537	1	1	yes	characteristic
314	C₂	C₂×C₂₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#540	C₂	semidirect
2. C₂₃×#3259	C₂	direct (N × H)
3. C₂×C₂₃×#478	C₂	semidirect
4. C₂×C₂₃×#469	C₂	semidirect
5. C₂×C₂₃×#466	C₂	semidirect
6. C₂×C₂₃×#459	C₂	semidirect
7. C₂×C₂₃×#424	C₂	semidirect
13. C₂₃×#540	C₂²	semidirect
21. C₂₃×#478	C₂²	semidirect
22. C₂₃×#469	C₂²	semidirect
23. C₂₃×#466	C₂²	semidirect
24. C₂₃×#459	C₂²	semidirect
27. C₂₃×#424	C₂²	semidirect
107. C₂×#3259	C₂₃	direct (N × H)
137. C₂×#540	C₂×C₂₃	semidirect
138. #3259	C₂×C₂₃	direct (N × H)
139. C₂×#478	C₂×C₂₃	semidirect
140. C₂×#469	C₂×C₂₃	semidirect
141. C₂×#466	C₂×C₂₃	semidirect
142. C₂×#459	C₂×C₂₃	semidirect
143. C₂×#424	C₂×C₂₃	semidirect
165. #540	C₂²×C₂₃	semidirect
173. #478	C₂²×C₂₃	semidirect
174. #469	C₂²×C₂₃	semidirect
175. #466	C₂²×C₂₃	semidirect
176. #459	C₂²×C₂₃	semidirect
179. #424	C₂²×C₂₃	semidirect
185. C₂×C₂₃	#3259	direct (N × H)
227. C₂₃	C₂×#3259	direct (N × H)
312. C₂	C₂₃×#3259	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#540	2
C₂₃×#3259	2
C₂×C₂₃×#478	2
C₂×C₂₃×#469	2
C₂×C₂₃×#466	2
C₂×C₂₃×#459	2
C₂×C₂₃×#424	2
C₂×#3259	23

Subgroups — C2×C23×#3259 #19208536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3259 #19208536