Subgroups — C₂×C₂₃×#1689 #19166758

Order 5888.

758 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#474	C₂	1	2	yes	maximal
3	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1689	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#332	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₂₃	—	2	1	no	—
8	C₂²×Q₁₆×C₂₃	—	2	1	no	—
9	C₂²×M₁₆×C₂₃	—	2	1	no	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#494	—	2	2	no	—
13	C₂₃×#494	C₂²	1	4	yes	complemented
14	C₂₃×#474	—	2	4	no	—
15	C₂₃×#474	C₂²	1	8	yes	—
16	C₂₃×#436	—	2	2	no	—
17	C₂₃×#436	C₂²	1	4	yes	—
18	C₂×C₂₃×#107	—	2	2	no	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₂₃×#102	—	2	2	no	—
22	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#334	C₂²	1	4	yes	complemented
25	C₂₃×#332	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₂₃	—	2	2	no	—
27	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	—	2	1	no	—
29	C₂×Q₁₆×C₂₃	—	2	4	no	—
30	C₂×Q₁₆×C₂₃	—	2	4	no	—
31	C₂×Q₁₆×C₂₃	—	4	4	no	—
32	C₂×Q₁₆×C₂₃	—	4	4	no	—
33	C₂×M₁₆×C₂₃	—	2	4	no	—
34	C₂×M₁₆×C₂₃	—	4	4	no	—
35	C₂²×C₈×C₂₃	—	2	1	no	—
36	C₂²×C₈×C₂₃	—	2	2	no	—
37	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
39	C₂₃×C₄⋊Q₈	—	2	2	no	—
40	C₂₃×C₄⋊Q₈	—	2	2	no	—
41	C₂₃×C₄⋊Q₈	—	2	2	no	—
42	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
43	C₄×D₄×C₂₃	—	2	1	no	—
44	C₄×D₄×C₂₃	—	2	1	no	—
45	C₄×D₄×C₂₃	D₄	1	2	yes	—
46	C₄×D₄×C₂₃	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#107	—	2	8	no	—
53	C₂₃×C₄⋊C₈	—	2	1	no	—
54	C₂₃×C₄⋊C₈	—	2	2	no	—
55	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
56	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
57	C₂₃×#102	—	2	8	no	—
58	C₂₃×C₂²⋊C₈	—	2	4	no	—
59	C₄×C₈×C₂₃	D₄	1	2	yes	—
60	C₄×C₈×C₂₃	C₂³	1	2	yes	—
61	C₂⁴×C₂₃	—	2	1	no	—
62	C₂×Q₈×C₂₃	—	2	4	no	—
63	C₂×Q₈×C₂₃	—	4	2	no	—
64	C₂×Q₈×C₂₃	—	4	4	no	—
65	C₂×Q₈×C₂₃	—	4	4	no	—
66	C₂×D₄×C₂₃	—	2	1	no	—
67	C₂×D₄×C₂₃	—	2	4	no	—
68	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₂₃	—	2	1	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	—	4	2	no	—
73	C₂²×C₄×C₂₃	—	4	2	no	—
74	C₂²×C₄×C₂₃	—	4	2	no	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
78	Q₁₆×C₂₃	—	4	8	no	—
79	Q₁₆×C₂₃	—	4	8	no	—
80	M₁₆×C₂₃	—	4	8	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	—	4	2	no	—
85	C₂×C₈×C₂₃	—	4	4	no	—
86	C₂×C₈×C₂₃	D₈	1	4	yes	—
87	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₄⋊C₄×C₂₃	—	4	2	no	—
93	C₄⋊C₄×C₂₃	—	4	2	no	—
94	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
95	C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₂²⋊C₄×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₂₃	#41	1	1	yes	characteristic
100	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
101	C₂×#1689	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₂₃	—	2	2	no	—
103	C₂³×C₂₃	—	4	2	no	—
104	C₂³×C₂₃	—	4	2	no	—
105	C₂³×C₂₃	—	4	2	no	—
106	C₂³×C₂₃	#121	1	1	yes	characteristic
107	Q₈×C₂₃	—	4	4	no	—
108	Q₈×C₂₃	—	4	4	no	—
109	Q₈×C₂₃	—	8	4	no	—
110	D₄×C₂₃	—	2	4	no	—
111	D₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	—	4	4	no	—
119	C₂×C₄×C₂₃	—	8	4	no	—
120	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₂₃	#137	1	4	yes	—
126	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
129	C₈×C₂₃	—	2	2	no	—
130	C₈×C₂₃	—	4	2	no	—
131	C₈×C₂₃	—	4	4	no	—
132	C₈×C₂₃	C₂×D₈	1	4	yes	—
133	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
134	C₂×#474	C₂×C₂₃	1	2	yes	—
135	C₂×#436	C₂×C₂₃	1	1	yes	characteristic
136	#1689	C₂×C₂₃	1	8	yes	direct product factor
137	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
138	C₂×#332	C₂×C₂₃	1	2	yes	complemented
139	C₂²×C₂₃	—	4	2	no	—
140	C₂²×C₂₃	—	4	2	no	—
141	C₂²×C₂₃	—	4	2	no	—
142	C₂²×C₂₃	—	8	4	no	—
143	C₂²×C₂₃	#459	1	2	yes	—
144	C₂²×C₂₃	#470	1	2	yes	—
145	C₂²×C₂₃	#473	1	2	yes	—
146	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	2	1	no	—
149	C₄×C₂₃	—	4	2	no	—
150	C₄×C₂₃	—	8	4	no	—
151	C₄×C₂₃	#459	1	2	yes	—
152	C₄×C₂₃	#470	1	2	yes	—
153	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₄×C₂₃	C₂×#137	1	2	yes	—
156	C₂²×Q₁₆	—	2	1	no	—
157	C₂²×Q₁₆	—	2	1	no	—
158	C₂²×M₁₆	—	2	1	no	—
159	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
160	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
161	#494	—	2	2	no	—
162	#494	C₂²×C₂₃	1	4	yes	complemented
163	#474	—	2	4	no	—
164	#474	C₂²×C₂₃	1	8	yes	—
165	#436	—	2	2	no	—
166	#436	C₂²×C₂₃	1	4	yes	—
167	C₂×#107	—	2	2	no	—
168	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
170	C₂×#102	—	2	2	no	—
171	C₂×C₂²⋊C₈	—	2	1	no	—
172	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
173	#334	C₂²×C₂₃	1	4	yes	complemented
174	#332	C₂²×C₂₃	1	8	yes	complemented
175	C₂×C₂₃	—	8	2	no	—
176	C₂×C₂₃	#1689	1	4	yes	direct product factor
177	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
178	C₂×C₂₃	C₂×#470	1	1	yes	characteristic
179	C₂×C₂₃	C₂×#473	1	1	yes	characteristic
180	C₂²×Q₈	—	2	2	no	—
181	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
182	C₂³×C₄	—	2	1	no	—
183	C₂×Q₁₆	—	2	4	no	—
184	C₂×Q₁₆	—	2	4	no	—
185	C₂×Q₁₆	—	4	4	no	—
186	C₂×Q₁₆	—	4	4	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	—	4	4	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
192	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	D₄×C₂₃	1	2	yes	—
200	C₄×D₄	C₂³×C₂₃	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
206	#107	—	2	8	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
210	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
211	#102	—	2	8	no	—
212	C₂²⋊C₈	—	2	4	no	—
213	C₄×C₈	D₄×C₂₃	1	2	yes	—
214	C₄×C₈	C₂³×C₂₃	1	2	yes	—
215	C₂₃	C₂×#1689	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	—	2	1	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
233	Q₁₆	—	4	8	no	—
234	Q₁₆	—	4	8	no	—
235	M₁₆	—	4	8	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	D₈×C₂₃	1	4	yes	—
242	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
254	C₄²	#41×C₂₃	1	1	yes	characteristic
255	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
256	C₂³	—	2	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	C₂₃×#121	1	1	yes	characteristic
261	Q₈	—	4	4	no	—
262	Q₈	—	4	4	no	—
263	Q₈	—	8	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	4	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
275	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
276	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
277	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
278	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
279	C₂×C₄	C₂₃×#137	1	4	yes	—
280	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	4	no	—
286	C₈	C₂×D₈×C₂₃	1	4	yes	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	4	no	—
291	C₂²	C₂₃×#459	1	2	yes	—
292	C₂²	C₂₃×#470	1	2	yes	—
293	C₂²	C₂₃×#473	1	2	yes	—
294	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	8	4	no	—
299	C₄	C₂₃×#459	1	2	yes	—
300	C₄	C₂₃×#470	1	2	yes	—
301	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
302	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₄	C₂×C₂₃×#137	1	2	yes	—
304	C₂	—	8	2	no	—
305	C₂	C₂₃×#1689	1	4	yes	direct product factor
306	C₂	C₂×C₂₃×#459	1	1	yes	characteristic
307	C₂	C₂×C₂₃×#470	1	1	yes	characteristic
308	C₂	C₂×C₂₃×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#494	C₂	semidirect
4. C₂₃×#1689	C₂	direct (N × H)
5. C₂×C₂₃×#334	C₂	semidirect
6. C₂×C₂₃×#332	C₂	semidirect
13. C₂₃×#494	C₂²	semidirect
24. C₂₃×#334	C₂²	semidirect
25. C₂₃×#332	C₂²	semidirect
101. C₂×#1689	C₂₃	direct (N × H)
133. C₂×#494	C₂×C₂₃	semidirect
136. #1689	C₂×C₂₃	direct (N × H)
137. C₂×#334	C₂×C₂₃	semidirect
138. C₂×#332	C₂×C₂₃	semidirect
162. #494	C₂²×C₂₃	semidirect
173. #334	C₂²×C₂₃	semidirect
174. #332	C₂²×C₂₃	semidirect
176. C₂×C₂₃	#1689	direct (N × H)
215. C₂₃	C₂×#1689	direct (N × H)
305. C₂	C₂₃×#1689	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×#494	2
C₂×C₂₃×#474	2
C₂×C₂₃×#436	2
C₂₃×#1689	2
C₂×C₂₃×#334	2
C₂×C₂₃×#332	2
C₂×#1689	23

Subgroups — C2×C23×#1689 #19166758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1689 #19166758