Subgroups — C₂₃×#22685 #19170311

Order 5888.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2351	C₂	1	2	yes	maximal
4	C₂₃×#1942	C₂	1	2	yes	maximal
5	C₂₃×#1865	C₂	1	4	yes	maximal
6	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1805	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
11	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₄×Q₁₆×C₂₃	—	2	8	no	—
13	C₄×Q₁₆×C₂₃	C₄	1	8	yes	complemented
14	C₂×C₂₃×#107	C₂²	1	2	yes	—
15	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#106	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#103	C₂²	1	4	yes	—
19	C₂×C₂₃×#103	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#389	—	2	4	no	—
23	C₂₃×#389	C₂²	1	4	yes	—
24	C₂₃×#382	—	2	2	no	—
25	C₂₃×#382	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂₃×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
32	C₂×Q₁₆×C₂₃	D₄	1	4	yes	—
33	C₂×Q₁₆×C₂₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₂₃	—	2	4	no	—
38	C₄×Q₈×C₂₃	—	2	8	no	—
39	C₄×Q₈×C₂₃	—	4	4	no	—
40	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₂₃	—	2	2	no	—
48	C₂×C₄²×C₂₃	—	2	4	no	—
49	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×#107	—	2	4	no	—
52	C₂₃×#107	C₂×C₄	1	4	yes	—
53	C₂₃×#106	—	4	2	no	—
54	C₂₃×C₄⋊C₈	—	2	2	no	—
55	C₂₃×#103	—	2	8	no	—
56	C₂₃×#103	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₂₃	—	2	4	no	—
58	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
59	C₂₃×#95	—	2	2	no	—
60	C₂₃×#95	—	2	4	no	—
61	C₂₃×#95	—	2	2	no	—
62	C₂₃×#95	D₄	1	1	yes	characteristic
63	C₂₃×#95	D₄	1	1	yes	characteristic
64	C₂×Q₈×C₂₃	—	2	2	no	—
65	C₂×Q₈×C₂₃	—	2	2	no	—
66	C₂×Q₈×C₂₃	—	2	2	no	—
67	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
68	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
69	C₂×Q₈×C₂₃	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	—	2	4	no	—
73	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₂₃	C₄⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
88	C₄⋊C₄×C₂₃	—	2	8	no	—
89	C₄⋊C₄×C₂₃	—	2	4	no	—
90	C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	4	8	no	—
94	C₄⋊C₄×C₂₃	—	4	8	no	—
95	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
96	C₄²×C₂₃	—	2	8	no	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	2	2	no	—
99	C₄²×C₂₃	—	4	8	no	—
100	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₂₃	#41	1	2	yes	—
102	#22685	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₂₃	—	2	4	no	—
105	Q₈×C₂₃	—	2	4	no	—
106	Q₈×C₂₃	—	2	4	no	—
107	Q₈×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	4	4	no	—
120	C₂×C₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	—	4	4	no	—
123	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₂₃	#135	1	4	yes	—
126	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₂₃	—	2	2	no	—
133	C₈×C₂₃	—	4	2	no	—
134	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×Q₁₆	C₂×C₂₃	1	2	yes	—
136	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
137	#2351	C₂×C₂₃	1	2	yes	—
138	#1942	C₂×C₂₃	1	2	yes	—
139	#1865	C₂×C₂₃	1	4	yes	—
140	#1846	C₂×C₂₃	1	1	yes	characteristic
141	#1805	C₂×C₂₃	1	2	yes	—
142	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
143	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₂₃	D₄²	1	1	yes	characteristic
145	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
146	C₂²×C₂₃	C₂×#135	1	2	yes	—
147	C₂²×C₂₃	#578	1	2	yes	—
148	C₄×C₂₃	—	2	8	no	—
149	C₄×C₂₃	—	2	2	no	—
150	C₄×C₂₃	—	2	2	no	—
151	C₄×C₂₃	—	2	4	no	—
152	C₄×C₂₃	—	4	4	no	—
153	C₄×C₂₃	—	4	2	no	—
154	C₄×C₂₃	—	8	4	no	—
155	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₂₃	#520	1	1	yes	characteristic
157	C₄×C₂₃	#547	1	1	yes	characteristic
158	C₄×C₂₃	#554	1	1	yes	characteristic
159	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
161	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
162	C₄×Q₁₆	—	2	8	no	—
163	C₄×Q₁₆	C₄×C₂₃	1	8	yes	complemented
164	C₂×#107	C₂²×C₂₃	1	2	yes	—
165	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×#106	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×#103	C₂²×C₂₃	1	4	yes	—
169	C₂×#103	C₂²×C₂₃	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
172	#389	—	2	4	no	—
173	#389	C₂²×C₂₃	1	4	yes	—
174	#382	—	2	2	no	—
175	#382	C₂²×C₂₃	1	2	yes	—
176	C₄×C₄⋊C₄	—	2	2	no	—
177	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
178	#340	C₂²×C₂₃	1	4	yes	—
179	#336	C₂²×C₂₃	1	2	yes	—
180	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₂₃	C₄×#135	1	1	yes	characteristic
182	C₂×C₂₃	#2950	1	1	yes	characteristic
183	C₂×C₂₃	#3285	1	2	yes	—
184	C₂×C₂₃	#3383	1	2	yes	—
185	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
186	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
187	C₂×Q₁₆	D₄×C₂₃	1	4	yes	—
188	C₂×Q₁₆	Q₈×C₂₃	1	4	yes	—
189	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
192	C₄×Q₈	—	2	4	no	—
193	C₄×Q₈	—	2	8	no	—
194	C₄×Q₈	—	4	4	no	—
195	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
201	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
205	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	C₂×C₄×C₂₃	1	4	yes	—
208	#106	—	4	2	no	—
209	C₄⋊C₈	—	2	2	no	—
210	#103	—	2	8	no	—
211	#103	C₂×C₄×C₂₃	1	8	yes	—
212	C₄×C₈	—	2	4	no	—
213	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
214	#95	—	2	2	no	—
215	#95	—	2	4	no	—
216	#95	—	2	2	no	—
217	#95	D₄×C₂₃	1	1	yes	characteristic
218	#95	D₄×C₂₃	1	1	yes	characteristic
219	C₂₃	#22685	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
224	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
225	C₂×Q₈	C₂×Q₈×C₂₃	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
234	Q₁₆	C₄⋊C₄×C₂₃	1	16	yes	complemented
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
242	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
243	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	—	4	8	no	—
251	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
252	C₄²	—	2	8	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	8	no	—
256	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
257	C₄²	#41×C₂₃	1	2	yes	—
258	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
259	Q₈	—	2	4	no	—
260	Q₈	—	2	4	no	—
261	Q₈	—	2	4	no	—
262	Q₈	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
280	C₂×C₄	C₂₃×#135	1	4	yes	—
281	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
285	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
290	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
291	C₂²	D₄²×C₂₃	1	1	yes	characteristic
292	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
293	C₂²	C₂×C₂₃×#135	1	2	yes	—
294	C₂²	C₂₃×#578	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	2	8	no	—
299	C₄	—	4	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
303	C₄	C₂₃×#520	1	1	yes	characteristic
304	C₄	C₂₃×#547	1	1	yes	characteristic
305	C₄	C₂₃×#554	1	1	yes	characteristic
306	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
307	C₂	C₄×C₂₃×#135	1	1	yes	characteristic
308	C₂	C₂₃×#2950	1	1	yes	characteristic
309	C₂	C₂₃×#3285	1	2	yes	—
310	C₂	C₂₃×#3383	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×Q₁₆×C₂₃	C₄	semidirect
78. Q₁₆×C₂₃	C₄⋊C₄	semidirect
102. #22685	C₂₃	direct (N × H)
163. C₄×Q₁₆	C₄×C₂₃	semidirect
219. C₂₃	#22685	direct (N × H)
234. Q₁₆	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₂₃	2
C₂₃×#2351	2
C₂₃×#1942	2
C₂₃×#1865	2
C₂₃×#1846	2
C₂₃×#1805	2
C₈×C₄⋊C₄×C₂₃	2
#22685	23

Subgroups — C23×#22685 #19170311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22685 #19170311