Subgroups — C₂₃×#21783 #19169409

Order 5888.

1230 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3499	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2889	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2884	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1912	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1909	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1886	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1884	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1827	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1796	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
12	C₂₃×#568	C₂²	1	2	yes	—
13	D₄×Q₈×C₂₃	—	2	2	no	—
14	C₂₃×#547	—	2	2	no	—
15	C₂₃×#547	—	2	4	no	—
16	C₂₃×#536	C₄	1	2	yes	complemented
17	C₂₃×#536	C₄	1	2	yes	complemented
18	C₂₃×#536	C₂²	1	1	yes	characteristic
19	C₂₃×#536	C₂²	1	1	yes	characteristic
20	C₂₃×#532	C₄	1	2	yes	complemented
21	C₂₃×#532	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×C₄⋊Q₈	C₄	1	2	yes	complemented
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂₃×#519	C₂²	1	2	yes	—
25	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#421	C₂²	1	4	yes	—
27	C₂₃×#419	C₂²	1	4	yes	—
28	C₂₃×#413	—	2	2	no	—
29	C₂₃×#413	C₂²	1	2	yes	—
30	C₂×C₂₃×#101	—	2	2	no	—
31	C₂×C₂₃×#101	C₂²	1	2	yes	—
32	C₂₃×#410	—	2	4	no	—
33	C₂₃×#410	C₂²	1	2	yes	—
34	C₂₃×#410	C₂²	1	2	yes	—
35	C₂₃×#339	C₂²	1	4	yes	—
36	C₂₃×H(3,ℤ/4)	C₂²	1	4	yes	—
37	C₂₃×2₋¹⁺⁴	—	4	4	no	—
38	C₂₃×2₋¹⁺⁴	—	4	4	no	—
39	C₂×#41×C₂₃	—	2	1	no	—
40	C₂×#41×C₂₃	—	2	2	no	—
41	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
42	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×#41×C₂₃	D₄	1	2	yes	—
44	C₂×#41×C₂₃	D₄	1	4	yes	—
45	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₂₃	—	2	1	no	—
49	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
50	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
51	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
52	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂₃×#131	—	4	4	no	—
55	C₂×M₁₆×C₂₃	—	2	4	no	—
56	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
57	C₂₃×C₄⋊Q₈	—	2	4	no	—
58	C₂₃×C₄⋊Q₈	—	2	2	no	—
59	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
61	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
62	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
63	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
64	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
65	C₂₃×#124	C₂×C₄	1	2	yes	—
66	C₂₃×#124	C₂×C₄	1	2	yes	—
67	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
68	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
69	C₂₃×#123	—	2	4	no	—
70	C₂₃×#123	—	2	2	no	—
71	C₂₃×#123	—	2	4	no	—
72	C₂₃×#123	—	2	2	no	—
73	C₂₃×C₂²⋊Q₈	—	2	4	no	—
74	C₂₃×C₂²⋊Q₈	—	2	2	no	—
75	C₂₃×C₂²⋊Q₈	—	2	4	no	—
76	C₂₃×C₂²⋊Q₈	—	2	2	no	—
77	C₂₃×#121	—	2	4	no	—
78	C₂₃×#121	—	2	2	no	—
79	C₄×Q₈×C₂₃	—	2	1	no	—
80	C₄×Q₈×C₂₃	—	4	4	no	—
81	C₄×D₄×C₂₃	—	2	2	no	—
82	C₄×D₄×C₂₃	—	2	1	no	—
83	C₄×D₄×C₂₃	—	2	4	no	—
84	C₄×D₄×C₂₃	—	2	2	no	—
85	C₂₃×#117	—	2	4	no	—
86	C₂₃×#117	C₂×C₄	1	2	yes	—
87	C₂₃×#117	C₂³	1	4	yes	—
88	C₂₃×#117	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
92	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
93	C₂₃×#104	—	4	4	no	—
94	C₂₃×#103	—	4	4	no	—
95	C₂₃×#101	—	2	2	no	—
96	C₂₃×#101	—	2	4	no	—
97	C₂₃×#100	—	2	2	no	—
98	C₂₃×#99	—	2	4	no	—
99	C₂₃×#99	—	2	4	no	—
100	#41×C₂₃	—	2	4	no	—
101	#41×C₂₃	—	2	4	no	—
102	#41×C₂₃	—	2	4	no	—
103	#41×C₂₃	—	4	8	no	—
104	#41×C₂₃	—	4	4	no	—
105	#41×C₂₃	—	4	4	no	—
106	#41×C₂₃	—	4	4	no	—
107	#41×C₂₃	—	4	8	no	—
108	#41×C₂₃	—	4	4	no	—
109	C₂×Q₈×C₂₃	—	2	2	no	—
110	C₂×Q₈×C₂₃	—	2	1	no	—
111	C₂×Q₈×C₂₃	—	2	4	no	—
112	C₂×Q₈×C₂₃	—	2	4	no	—
113	C₂×Q₈×C₂₃	—	4	4	no	—
114	C₂×Q₈×C₂₃	—	4	4	no	—
115	C₂×Q₈×C₂₃	—	4	4	no	—
116	C₂×Q₈×C₂₃	—	4	4	no	—
117	C₂×Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
118	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
120	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
122	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
124	C₂×D₄×C₂₃	—	2	2	no	—
125	C₂×D₄×C₂₃	—	2	2	no	—
126	C₂×D₄×C₂₃	—	2	1	no	—
127	C₂×D₄×C₂₃	C₂²⋊C₄	1	4	yes	—
128	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
131	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
132	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	2	2	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
141	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
145	M₁₆×C₂₃	—	2	4	no	—
146	M₁₆×C₂₃	—	2	4	no	—
147	M₁₆×C₂₃	—	4	4	no	—
148	C₂×C₈×C₂₃	—	2	4	no	—
149	C₂×C₈×C₂₃	—	4	4	no	—
150	C₄⋊C₄×C₂₃	—	2	4	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	4	no	—
154	C₄⋊C₄×C₂₃	—	2	4	no	—
155	C₄⋊C₄×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	—	4	4	no	—
159	C₂²⋊C₄×C₂₃	—	2	2	no	—
160	C₂²⋊C₄×C₂₃	—	2	4	no	—
161	C₂²⋊C₄×C₂₃	—	2	2	no	—
162	C₂²⋊C₄×C₂₃	—	2	1	no	—
163	C₂²⋊C₄×C₂₃	—	2	4	no	—
164	C₂²⋊C₄×C₂₃	—	2	2	no	—
165	C₂²⋊C₄×C₂₃	—	2	4	no	—
166	C₄²×C₂₃	—	2	2	no	—
167	C₄²×C₂₃	—	2	1	no	—
168	C₄²×C₂₃	—	2	4	no	—
169	C₄²×C₂₃	—	4	4	no	—
170	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
172	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
173	#21783	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₂₃	—	2	1	no	—
175	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
176	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
177	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₂₃	—	2	4	no	—
179	Q₈×C₂₃	—	2	2	no	—
180	Q₈×C₂₃	—	2	4	no	—
181	Q₈×C₂₃	—	2	2	no	—
182	Q₈×C₂₃	—	2	4	no	—
183	Q₈×C₂₃	—	2	2	no	—
184	Q₈×C₂₃	—	4	4	no	—
185	Q₈×C₂₃	—	4	4	no	—
186	Q₈×C₂₃	—	4	2	no	—
187	D₄×C₂₃	—	2	4	no	—
188	D₄×C₂₃	—	2	4	no	—
189	D₄×C₂₃	—	2	2	no	—
190	D₄×C₂₃	—	2	2	no	—
191	D₄×C₂₃	—	2	2	no	—
192	D₄×C₂₃	—	4	4	no	—
193	D₄×C₂₃	—	4	4	no	—
194	D₄×C₂₃	—	4	4	no	—
195	D₄×C₂₃	—	4	2	no	—
196	C₂×C₄×C₂₃	—	2	1	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	2	no	—
201	C₂×C₄×C₂₃	—	2	1	no	—
202	C₂×C₄×C₂₃	—	2	1	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	4	4	no	—
205	C₂×C₄×C₂₃	—	4	4	no	—
206	C₂×C₄×C₂₃	—	4	2	no	—
207	C₂×C₄×C₂₃	—	4	4	no	—
208	C₂×C₄×C₂₃	—	4	4	no	—
209	C₂×C₄×C₂₃	—	4	4	no	—
210	C₂×C₄×C₂₃	—	4	2	no	—
211	C₂×C₄×C₂₃	—	4	2	no	—
212	C₂×C₄×C₂₃	—	4	4	no	—
213	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
215	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
219	C₈×C₂₃	—	4	4	no	—
220	#3499	C₂×C₂₃	1	1	yes	characteristic
221	#2889	C₂×C₂₃	1	1	yes	characteristic, complemented
222	#2884	C₂×C₂₃	1	1	yes	characteristic, complemented
223	#1912	C₂×C₂₃	1	2	yes	complemented
224	#1909	C₂×C₂₃	1	2	yes	complemented
225	#1886	C₂×C₂₃	1	2	yes	complemented
226	#1884	C₂×C₂₃	1	2	yes	complemented
227	#1827	C₂×C₂₃	1	2	yes	complemented
228	#1796	C₂×C₂₃	1	2	yes	complemented
229	C₂²×C₂₃	—	2	2	no	—
230	C₂²×C₂₃	—	2	2	no	—
231	C₂²×C₂₃	—	4	1	no	—
232	C₂²×C₂₃	—	4	2	no	—
233	C₂²×C₂₃	—	4	2	no	—
234	C₂²×C₂₃	—	4	2	no	—
235	C₂²×C₂₃	—	4	1	no	—
236	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₂₃	#534	1	2	yes	—
238	C₄×C₂₃	—	2	2	no	—
239	C₄×C₂₃	—	2	2	no	—
240	C₄×C₂₃	—	2	2	no	—
241	C₄×C₂₃	—	2	4	no	—
242	C₄×C₂₃	—	4	2	no	—
243	C₄×C₂₃	—	4	1	no	—
244	C₄×C₂₃	—	4	2	no	—
245	C₄×C₂₃	—	4	1	no	—
246	C₄×C₂₃	—	8	4	no	—
247	C₄×C₂₃	#518	1	2	yes	—
248	C₄×C₂₃	#534	1	2	yes	—
249	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
250	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
251	#568	C₂²×C₂₃	1	2	yes	—
252	D₄×Q₈	—	2	2	no	—
253	#547	—	2	2	no	—
254	#547	—	2	4	no	—
255	#536	C₄×C₂₃	1	2	yes	complemented
256	#536	C₄×C₂₃	1	2	yes	complemented
257	#536	C₂²×C₂₃	1	1	yes	characteristic
258	#536	C₂²×C₂₃	1	1	yes	characteristic
259	#532	C₄×C₂₃	1	2	yes	complemented
260	#532	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄⋊Q₈	C₄×C₂₃	1	2	yes	complemented
262	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
263	#519	C₂²×C₂₃	1	2	yes	—
264	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
265	#421	C₂²×C₂₃	1	4	yes	—
266	#419	C₂²×C₂₃	1	4	yes	—
267	#413	—	2	2	no	—
268	#413	C₂²×C₂₃	1	2	yes	—
269	C₂×#101	—	2	2	no	—
270	C₂×#101	C₂²×C₂₃	1	2	yes	—
271	#410	—	2	4	no	—
272	#410	C₂²×C₂₃	1	2	yes	—
273	#410	C₂²×C₂₃	1	2	yes	—
274	#339	C₂²×C₂₃	1	4	yes	—
275	H(3,ℤ/4)	C₂²×C₂₃	1	4	yes	—
276	C₂×C₂₃	—	2	1	no	—
277	C₂×C₂₃	—	2	2	no	—
278	C₂×C₂₃	—	4	2	no	—
279	C₂×C₂₃	—	4	2	no	—
280	C₂×C₂₃	—	8	1	no	—
281	C₂×C₂₃	#2308	1	1	yes	characteristic
282	2₋¹⁺⁴	—	4	4	no	—
283	2₋¹⁺⁴	—	4	4	no	—
284	C₂×#41	—	2	1	no	—
285	C₂×#41	—	2	2	no	—
286	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
287	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
288	C₂×#41	D₄×C₂₃	1	2	yes	—
289	C₂×#41	D₄×C₂₃	1	4	yes	—
290	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
291	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
292	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
293	C₂²×Q₈	—	2	1	no	—
294	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
295	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
296	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
297	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
299	#131	—	4	4	no	—
300	C₂×M₁₆	—	2	4	no	—
301	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
302	C₄⋊Q₈	—	2	4	no	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
305	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
306	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
307	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
308	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
309	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
310	#124	C₂×C₄×C₂₃	1	2	yes	—
311	#124	C₂×C₄×C₂₃	1	2	yes	—
312	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
313	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
314	#123	—	2	4	no	—
315	#123	—	2	2	no	—
316	#123	—	2	4	no	—
317	#123	—	2	2	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	C₂²⋊Q₈	—	2	2	no	—
320	C₂²⋊Q₈	—	2	4	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	#121	—	2	4	no	—
323	#121	—	2	2	no	—
324	C₄×Q₈	—	2	1	no	—
325	C₄×Q₈	—	4	4	no	—
326	C₄×D₄	—	2	2	no	—
327	C₄×D₄	—	2	1	no	—
328	C₄×D₄	—	2	4	no	—
329	C₄×D₄	—	2	2	no	—
330	#117	—	2	4	no	—
331	#117	C₂×C₄×C₂₃	1	2	yes	—
332	#117	C₂³×C₂₃	1	4	yes	—
333	#117	C₂³×C₂₃	1	1	yes	characteristic
334	C₂×C₄⋊C₄	—	2	4	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
337	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
338	#104	—	4	4	no	—
339	#103	—	4	4	no	—
340	#101	—	2	2	no	—
341	#101	—	2	4	no	—
342	#100	—	2	2	no	—
343	#99	—	2	4	no	—
344	#99	—	2	4	no	—
345	C₂₃	#21783	1	1	yes	characteristic, direct product factor, Sylow
346	#41	—	2	4	no	—
347	#41	—	2	4	no	—
348	#41	—	2	4	no	—
349	#41	—	4	8	no	—
350	#41	—	4	4	no	—
351	#41	—	4	4	no	—
352	#41	—	4	4	no	—
353	#41	—	4	8	no	—
354	#41	—	4	4	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	4	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	4	4	no	—
360	C₂×Q₈	—	4	4	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	—	4	4	no	—
363	C₂×Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
364	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
365	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
366	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
367	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
368	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
369	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	C₂²⋊C₄×C₂₃	1	4	yes	—
374	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
375	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
376	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
377	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
378	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
386	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
387	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
391	M₁₆	—	2	4	no	—
392	M₁₆	—	2	4	no	—
393	M₁₆	—	4	4	no	—
394	C₂×C₈	—	2	4	no	—
395	C₂×C₈	—	4	4	no	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	4	4	no	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₂²⋊C₄	—	2	4	no	—
407	C₂²⋊C₄	—	2	2	no	—
408	C₂²⋊C₄	—	2	1	no	—
409	C₂²⋊C₄	—	2	4	no	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	1	no	—
414	C₄²	—	2	4	no	—
415	C₄²	—	4	4	no	—
416	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
417	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
418	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
419	C₂³	—	2	1	no	—
420	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
421	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
422	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
423	Q₈	—	2	4	no	—
424	Q₈	—	2	2	no	—
425	Q₈	—	2	4	no	—
426	Q₈	—	2	2	no	—
427	Q₈	—	2	4	no	—
428	Q₈	—	2	2	no	—
429	Q₈	—	4	4	no	—
430	Q₈	—	4	4	no	—
431	Q₈	—	4	2	no	—
432	D₄	—	2	4	no	—
433	D₄	—	2	4	no	—
434	D₄	—	2	2	no	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	4	4	no	—
438	D₄	—	4	4	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
459	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
460	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
461	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
463	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
464	C₈	—	4	4	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	1	no	—
472	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
473	C₂²	C₂₃×#534	1	2	yes	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	4	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	1	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	1	no	—
482	C₄	—	8	4	no	—
483	C₄	C₂₃×#518	1	2	yes	—
484	C₄	C₂₃×#534	1	2	yes	—
485	C₂	—	2	1	no	—
486	C₂	—	2	2	no	—
487	C₂	—	4	2	no	—
488	C₂	—	4	2	no	—
489	C₂	—	8	1	no	—
490	C₂	C₂₃×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2308
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2889	C₂	semidirect
3. C₂₃×#2884	C₂	semidirect
4. C₂₃×#1912	C₂	semidirect
5. C₂₃×#1909	C₂	semidirect
6. C₂₃×#1886	C₂	semidirect
7. C₂₃×#1884	C₂	semidirect
8. C₂₃×#1827	C₂	semidirect
9. C₂₃×#1796	C₂	semidirect
16. C₂₃×#536	C₄	semidirect
17. C₂₃×#536	C₄	semidirect
20. C₂₃×#532	C₄	semidirect
22. C₂×C₂₃×C₄⋊Q₈	C₄	semidirect
173. #21783	C₂₃	direct (N × H)
221. #2889	C₂×C₂₃	semidirect
222. #2884	C₂×C₂₃	semidirect
223. #1912	C₂×C₂₃	semidirect
224. #1909	C₂×C₂₃	semidirect
225. #1886	C₂×C₂₃	semidirect
226. #1884	C₂×C₂₃	semidirect
227. #1827	C₂×C₂₃	semidirect
228. #1796	C₂×C₂₃	semidirect
255. #536	C₄×C₂₃	semidirect
256. #536	C₄×C₂₃	semidirect
259. #532	C₄×C₂₃	semidirect
261. C₂×C₄⋊Q₈	C₄×C₂₃	semidirect
345. C₂₃	#21783	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3499	2
C₂₃×#2889	2
C₂₃×#2884	2
C₂₃×#1912	2
C₂₃×#1909	2
C₂₃×#1886	2
C₂₃×#1884	2
C₂₃×#1827	2
C₂₃×#1796	2
#21783	23

Subgroups — C23×#21783 #19169409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21783 #19169409