Subgroups — C₂×C₂₃×#1688 #19166770

Order 5888.

850 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×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#460	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1688	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₂₃	—	2	2	no	—
8	C₂²×D₈×C₂₃	—	2	1	no	—
9	C₂²×D₈×C₂₃	—	2	1	no	—
10	C₂×C₂₃×#127	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#493	—	2	2	no	—
13	C₂₃×#493	C₂²	1	4	yes	—
14	C₂₃×#460	—	2	4	no	—
15	C₂₃×#460	C₂²	1	8	yes	complemented
16	C₂₃×#446	—	2	2	no	—
17	C₂₃×#446	C₂²	1	4	yes	complemented
18	C₂×C₂₃×C₄⋊C₈	—	2	1	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×#97	—	2	1	no	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#334	C₂²	1	4	yes	complemented
25	C₂₃×#331	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₂₃	—	2	2	no	—
28	C₂²×D₄×C₂₃	—	2	2	no	—
29	C₂²×D₄×C₂₃	—	2	2	no	—
30	C₂×SD₁₆×C₂₃	—	2	8	no	—
31	C₂×SD₁₆×C₂₃	—	4	8	no	—
32	C₂×D₈×C₂₃	—	2	4	no	—
33	C₂×D₈×C₂₃	—	2	4	no	—
34	C₂×D₈×C₂₃	—	4	4	no	—
35	C₂×D₈×C₂₃	—	4	4	no	—
36	C₂²×C₈×C₂₃	—	2	2	no	—
37	C₂²×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
40	C₂₃×#127	—	2	2	no	—
41	C₂₃×#127	—	2	2	no	—
42	C₂₃×#127	—	2	2	no	—
43	C₂₃×#127	C₂³	1	4	yes	—
44	C₄×Q₈×C₂₃	—	2	1	no	—
45	C₄×Q₈×C₂₃	—	2	1	no	—
46	C₄×Q₈×C₂₃	D₄	1	2	yes	—
47	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₂₃	—	2	1	no	—
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×C₄⋊C₈	—	2	4	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×#97	—	2	4	no	—
59	C₄×C₈×C₂₃	D₄	1	2	yes	—
60	C₄×C₈×C₂₃	C₂³	1	2	yes	—
61	C₂⁴×C₂₃	—	4	2	no	—
62	C₂×Q₈×C₂₃	—	2	1	no	—
63	C₂×Q₈×C₂₃	—	2	4	no	—
64	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
65	C₂×D₄×C₂₃	—	2	4	no	—
66	C₂×D₄×C₂₃	—	2	4	no	—
67	C₂×D₄×C₂₃	—	2	4	no	—
68	C₂×D₄×C₂₃	—	4	2	no	—
69	C₂×D₄×C₂₃	—	4	2	no	—
70	C₂×D₄×C₂₃	—	4	2	no	—
71	C₂×D₄×C₂₃	—	4	4	no	—
72	C₂×D₄×C₂₃	—	4	4	no	—
73	C₂×D₄×C₂₃	—	4	4	no	—
74	C₂×D₄×C₂₃	—	4	4	no	—
75	C₂×D₄×C₂₃	—	8	4	no	—
76	C₂²×C₄×C₂₃	—	2	1	no	—
77	C₂²×C₄×C₂₃	—	2	1	no	—
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
81	SD₁₆×C₂₃	—	4	16	no	—
82	D₈×C₂₃	—	4	8	no	—
83	D₈×C₂₃	—	4	8	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	4	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	4	2	no	—
88	C₂×C₈×C₂₃	—	4	4	no	—
89	C₂×C₈×C₂₃	SD₁₆	1	4	yes	complemented
90	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₂₃	#41	1	1	yes	characteristic
101	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
102	C₂×#1688	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₃	—	4	4	no	—
104	C₂³×C₂₃	—	8	4	no	—
105	C₂³×C₂₃	—	8	4	no	—
106	C₂³×C₂₃	—	8	4	no	—
107	C₂³×C₂₃	#121	1	1	yes	characteristic
108	Q₈×C₂₃	—	2	4	no	—
109	Q₈×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	4	4	no	—
111	D₄×C₂₃	—	4	4	no	—
112	D₄×C₂₃	—	4	4	no	—
113	D₄×C₂₃	—	4	4	no	—
114	D₄×C₂₃	—	8	4	no	—
115	D₄×C₂₃	—	8	4	no	—
116	D₄×C₂₃	—	8	4	no	—
117	D₄×C₂₃	—	8	4	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
127	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
129	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	C₈×C₂₃	—	2	2	no	—
133	C₈×C₂₃	—	4	2	no	—
134	C₈×C₂₃	—	4	4	no	—
135	C₈×C₂₃	C₂×SD₁₆	1	4	yes	complemented
136	C₂×#493	C₂×C₂₃	1	1	yes	characteristic
137	C₂×#460	C₂×C₂₃	1	2	yes	complemented
138	C₂×#446	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#1688	C₂×C₂₃	1	8	yes	direct product factor
140	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
141	C₂×#331	C₂×C₂₃	1	2	yes	complemented
142	C₂²×C₂₃	—	8	4	no	—
143	C₂²×C₂₃	—	8	4	no	—
144	C₂²×C₂₃	—	8	4	no	—
145	C₂²×C₂₃	—	16	8	no	—
146	C₂²×C₂₃	#460	1	2	yes	—
147	C₂²×C₂₃	#469	1	2	yes	—
148	C₂²×C₂₃	#472	1	2	yes	—
149	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	4	2	no	—
153	C₄×C₂₃	—	4	2	no	—
154	C₄×C₂₃	#460	1	2	yes	—
155	C₄×C₂₃	#469	1	2	yes	—
156	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
157	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
159	C₂²×SD₁₆	—	2	2	no	—
160	C₂²×D₈	—	2	1	no	—
161	C₂²×D₈	—	2	1	no	—
162	C₂×#127	C₂²×C₂₃	1	2	yes	—
163	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
164	#493	—	2	2	no	—
165	#493	C₂²×C₂₃	1	4	yes	—
166	#460	—	2	4	no	—
167	#460	C₂²×C₂₃	1	8	yes	complemented
168	#446	—	2	2	no	—
169	#446	C₂²×C₂₃	1	4	yes	complemented
170	C₂×C₄⋊C₈	—	2	1	no	—
171	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
173	C₂×#102	—	2	2	no	—
174	C₂×#97	—	2	1	no	—
175	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
176	#334	C₂²×C₂₃	1	4	yes	complemented
177	#331	C₂²×C₂₃	1	8	yes	complemented
178	C₂×C₂₃	—	16	4	no	—
179	C₂×C₂₃	#1688	1	4	yes	direct product factor
180	C₂×C₂₃	C₂×#460	1	1	yes	characteristic
181	C₂×C₂₃	C₂×#469	1	1	yes	characteristic
182	C₂×C₂₃	C₂×#472	1	1	yes	characteristic
183	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	—	2	2	no	—
187	C₂×SD₁₆	—	2	8	no	—
188	C₂×SD₁₆	—	4	8	no	—
189	C₂×D₈	—	2	4	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂×D₈	—	4	4	no	—
192	C₂×D₈	—	4	4	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
196	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
197	#127	—	2	2	no	—
198	#127	—	2	2	no	—
199	#127	—	2	2	no	—
200	#127	C₂³×C₂₃	1	4	yes	—
201	C₄×Q₈	—	2	1	no	—
202	C₄×Q₈	—	2	1	no	—
203	C₄×Q₈	D₄×C₂₃	1	2	yes	—
204	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
213	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
214	#102	—	2	8	no	—
215	#97	—	2	4	no	—
216	C₄×C₈	D₄×C₂₃	1	2	yes	—
217	C₄×C₈	C₂³×C₂₃	1	2	yes	—
218	C₂₃	C₂×#1688	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	—	4	2	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
239	SD₁₆	—	4	16	no	—
240	D₈	—	4	8	no	—
241	D₈	—	4	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	SD₁₆×C₂₃	1	4	yes	complemented
248	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
258	C₄²	#41×C₂₃	1	1	yes	characteristic
259	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
260	C₂³	—	4	4	no	—
261	C₂³	—	8	4	no	—
262	C₂³	—	8	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	C₂₃×#121	1	1	yes	characteristic
265	Q₈	—	2	4	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
281	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
282	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
283	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
284	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
285	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	4	4	no	—
291	C₈	—	4	2	no	—
292	C₈	C₂×SD₁₆×C₂₃	1	4	yes	complemented
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	16	8	no	—
297	C₂²	C₂₃×#460	1	2	yes	—
298	C₂²	C₂₃×#469	1	2	yes	—
299	C₂²	C₂₃×#472	1	2	yes	—
300	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	C₂₃×#460	1	2	yes	—
306	C₄	C₂₃×#469	1	2	yes	—
307	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
308	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
310	C₂	—	16	4	no	—
311	C₂	C₂₃×#1688	1	4	yes	direct product factor
312	C₂	C₂×C₂₃×#460	1	1	yes	characteristic
313	C₂	C₂×C₂₃×#469	1	1	yes	characteristic
314	C₂	C₂×C₂₃×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#493	C₂
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
2. C₂×C₂₃×#460	C₂	semidirect
3. C₂×C₂₃×#446	C₂	semidirect
4. C₂₃×#1688	C₂	direct (N × H)
5. C₂×C₂₃×#334	C₂	semidirect
6. C₂×C₂₃×#331	C₂	semidirect
15. C₂₃×#460	C₂²	semidirect
17. C₂₃×#446	C₂²	semidirect
24. C₂₃×#334	C₂²	semidirect
25. C₂₃×#331	C₂²	semidirect
89. C₂×C₈×C₂₃	SD₁₆	semidirect
102. C₂×#1688	C₂₃	direct (N × H)
135. C₈×C₂₃	C₂×SD₁₆	semidirect
137. C₂×#460	C₂×C₂₃	semidirect
138. C₂×#446	C₂×C₂₃	semidirect
139. #1688	C₂×C₂₃	direct (N × H)
140. C₂×#334	C₂×C₂₃	semidirect
141. C₂×#331	C₂×C₂₃	semidirect
167. #460	C₂²×C₂₃	semidirect
169. #446	C₂²×C₂₃	semidirect
176. #334	C₂²×C₂₃	semidirect
177. #331	C₂²×C₂₃	semidirect
179. C₂×C₂₃	#1688	direct (N × H)
218. C₂₃	C₂×#1688	direct (N × H)
247. C₂×C₈	SD₁₆×C₂₃	semidirect
292. C₈	C₂×SD₁₆×C₂₃	semidirect
311. C₂	C₂₃×#1688	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×#493	2
C₂×C₂₃×#460	2
C₂×C₂₃×#446	2
C₂₃×#1688	2
C₂×C₂₃×#334	2
C₂×C₂₃×#331	2
C₂×#1688	23

Subgroups — C2×C23×#1688 #19166770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1688 #19166770