Subgroups — C₂₃×#26354 #19173980

Order 5888.

904 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2439	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2361	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2092	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2086	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2065	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1919	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1895	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1875	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1853	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1851	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	—	2	1	no	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#476	—	2	4	no	—
24	C₂₃×#428	—	2	4	no	—
25	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×#396	—	2	1	no	—
39	C₂₃×#393	—	2	1	no	—
40	C₂₃×#391	—	2	1	no	—
41	C₂₃×#390	C₂²	1	1	yes	characteristic
42	C₂₃×#389	C₂²	1	1	yes	characteristic
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#389	C₂²	1	1	yes	characteristic
45	C₂₃×#388	—	2	1	no	—
46	C₂₃×#387	—	2	1	no	—
47	C₂₃×#386	—	2	1	no	—
48	C₂₃×#385	—	2	1	no	—
49	C₂₃×#382	C₂²	1	1	yes	characteristic
50	C₂₃×#380	C₂²	1	1	yes	characteristic
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	—	2	1	no	—
62	C₂²×D₄×C₂₃	—	2	1	no	—
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₂₃	—	2	2	no	—
68	C₂×M₁₆×C₂₃	—	2	2	no	—
69	C₂×M₁₆×C₂₃	—	2	2	no	—
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×C₄⋊Q₈	—	2	2	no	—
75	C₂₃×C₄⋊Q₈	—	4	1	no	—
76	C₂₃×#121	—	2	2	no	—
77	C₂₃×#121	Q₈	1	4	yes	complemented
78	C₄×D₄×C₂₃	—	2	2	no	—
79	C₄×D₄×C₂₃	—	4	1	no	—
80	C₄×D₄×C₂₃	—	4	4	no	—
81	C₂₃×#117	—	2	2	no	—
82	C₂₃×#117	—	4	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂₃×#107	—	2	2	no	—
110	C₂₃×#107	—	2	2	no	—
111	C₂₃×#106	—	2	2	no	—
112	C₂₃×#106	—	2	2	no	—
113	C₂₃×#106	—	4	1	no	—
114	C₂₃×#106	D₄	1	4	yes	complemented
115	C₂₃×C₄⋊C₈	—	2	2	no	—
116	C₂₃×C₄⋊C₈	—	2	2	no	—
117	C₂₃×#102	—	2	2	no	—
118	C₂₃×#102	—	2	2	no	—
119	C₂₃×#102	—	2	2	no	—
120	C₂₃×#102	—	4	1	no	—
121	C₂₃×C₂²⋊C₈	—	2	2	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	D₄	1	1	yes	characteristic
130	C₂₃×#95	D₄	1	1	yes	characteristic
131	C₂⁴×C₂₃	—	2	1	no	—
132	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Q₈×C₂₃	—	4	2	no	—
134	C₂×Q₈×C₂₃	—	4	2	no	—
135	C₂×Q₈×C₂₃	—	8	1	no	—
136	C₂×D₄×C₂₃	—	2	4	no	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	2	1	no	—
139	C₂×D₄×C₂₃	—	2	1	no	—
140	C₂×D₄×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	1	no	—
142	C₂×D₄×C₂₃	—	4	2	no	—
143	C₂×D₄×C₂₃	—	4	2	no	—
144	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	4	2	no	—
157	C₂²×C₄×C₂₃	—	4	2	no	—
158	C₂²×C₄×C₂₃	—	4	2	no	—
159	C₂²×C₄×C₂₃	—	4	2	no	—
160	C₂²×C₄×C₂₃	—	8	1	no	—
161	C₂²×C₄×C₂₃	—	8	1	no	—
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
168	C₂²×C₄×C₂₃	#41	1	2	yes	—
169	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
170	M₁₆×C₂₃	—	2	8	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₂₃	#41	1	2	yes	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	4	2	no	—
195	C₄⋊C₄×C₂₃	—	4	2	no	—
196	C₄⋊C₄×C₂₃	—	4	2	no	—
197	C₄⋊C₄×C₂₃	—	4	2	no	—
198	C₄⋊C₄×C₂₃	—	8	1	no	—
199	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
200	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
201	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
202	C₂²⋊C₄×C₂₃	—	2	4	no	—
203	C₂²⋊C₄×C₂₃	—	2	2	no	—
204	C₂²⋊C₄×C₂₃	—	4	1	no	—
205	C₂²⋊C₄×C₂₃	—	4	2	no	—
206	C₂²⋊C₄×C₂₃	—	4	2	no	—
207	C₂²⋊C₄×C₂₃	—	4	2	no	—
208	C₂²⋊C₄×C₂₃	—	8	1	no	—
209	C₂²⋊C₄×C₂₃	—	8	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	4	2	no	—
215	C₄²×C₂₃	—	4	2	no	—
216	#26354	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂³×C₂₃	—	2	1	no	—
218	C₂³×C₂₃	—	2	1	no	—
219	C₂³×C₂₃	—	2	1	no	—
220	C₂³×C₂₃	—	2	1	no	—
221	C₂³×C₂₃	—	2	1	no	—
222	C₂³×C₂₃	—	2	1	no	—
223	C₂³×C₂₃	—	2	2	no	—
224	C₂³×C₂₃	—	4	1	no	—
225	C₂³×C₂₃	—	4	1	no	—
226	C₂³×C₂₃	—	4	1	no	—
227	C₂³×C₂₃	—	4	1	no	—
228	C₂³×C₂₃	—	4	1	no	—
229	C₂³×C₂₃	—	4	1	no	—
230	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
231	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₂₃	—	8	4	no	—
233	D₄×C₂₃	—	2	2	no	—
234	D₄×C₂₃	—	2	2	no	—
235	D₄×C₂₃	—	4	8	no	—
236	D₄×C₂₃	—	4	1	no	—
237	D₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	1	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	2	1	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	8	4	no	—
277	C₂×C₄×C₂₃	—	8	4	no	—
278	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
279	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
280	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
282	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
285	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
286	C₈×C₂₃	—	2	2	no	—
287	C₈×C₂₃	—	2	2	no	—
288	C₈×C₂₃	—	4	1	no	—
289	C₈×C₂₃	—	4	1	no	—
290	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
291	C₂×#428	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#2439	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#2361	C₂×C₂₃	1	1	yes	characteristic
294	#2092	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#2086	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#2065	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#2058	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#1919	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#1895	C₂×C₂₃	1	1	yes	characteristic
301	#1875	C₂×C₂₃	1	1	yes	characteristic
302	#1862	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#1853	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#1851	C₂×C₂₃	1	1	yes	characteristic, complemented
305	C₂²×C₂₃	—	2	2	no	—
306	C₂²×C₂₃	—	2	2	no	—
307	C₂²×C₂₃	—	2	2	no	—
308	C₂²×C₂₃	—	4	1	no	—
309	C₂²×C₂₃	—	4	1	no	—
310	C₂²×C₂₃	—	4	1	no	—
311	C₂²×C₂₃	—	4	1	no	—
312	C₂²×C₂₃	—	4	2	no	—
313	C₂²×C₂₃	—	4	2	no	—
314	C₂²×C₂₃	—	4	2	no	—
315	C₂²×C₂₃	—	8	1	no	—
316	C₂²×C₂₃	—	8	1	no	—
317	C₂²×C₂₃	—	8	1	no	—
318	C₂²×C₂₃	—	8	1	no	—
319	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
320	C₂²×C₂₃	D₄²	1	1	yes	characteristic
321	C₂²×C₂₃	#547	1	1	yes	characteristic
322	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
323	C₂²×C₂₃	#575	1	1	yes	characteristic
324	C₂²×C₂₃	#577	1	1	yes	characteristic
325	C₂²×C₂₃	#577	1	1	yes	characteristic
326	C₄×C₂₃	—	2	2	no	—
327	C₄×C₂₃	—	2	2	no	—
328	C₄×C₂₃	—	4	2	no	—
329	C₄×C₂₃	—	4	1	no	—
330	C₄×C₂₃	—	4	1	no	—
331	C₄×C₂₃	—	4	2	no	—
332	C₄×C₂₃	—	4	1	no	—
333	C₄×C₂₃	—	4	2	no	—
334	C₄×C₂₃	—	8	1	no	—
335	C₄×C₂₃	—	8	1	no	—
336	C₄×C₂₃	—	8	1	no	—
337	C₄×C₂₃	—	8	1	no	—
338	C₄×C₂₃	#525	1	1	yes	characteristic
339	C₄×C₂₃	#539	1	1	yes	characteristic
340	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
341	C₄×C₂₃	#554	1	1	yes	characteristic
342	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×C₄×D₄	—	2	1	no	—
346	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
348	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
349	#476	—	2	4	no	—
350	#428	—	2	4	no	—
351	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
363	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
364	#396	—	2	1	no	—
365	#393	—	2	1	no	—
366	#391	—	2	1	no	—
367	#390	C₂²×C₂₃	1	1	yes	characteristic
368	#389	C₂²×C₂₃	1	1	yes	characteristic
369	#389	C₂²×C₂₃	1	1	yes	characteristic
370	#389	C₂²×C₂₃	1	1	yes	characteristic
371	#388	—	2	1	no	—
372	#387	—	2	1	no	—
373	#386	—	2	1	no	—
374	#385	—	2	1	no	—
375	#382	C₂²×C₂₃	1	1	yes	characteristic
376	#380	C₂²×C₂₃	1	1	yes	characteristic
377	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
378	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
379	#340	C₂²×C₂₃	1	1	yes	characteristic
380	#340	C₂²×C₂₃	1	1	yes	characteristic
381	#340	C₂²×C₂₃	1	1	yes	characteristic
382	#340	C₂²×C₂₃	1	1	yes	characteristic
383	#340	C₂²×C₂₃	1	1	yes	characteristic
384	#340	C₂²×C₂₃	1	1	yes	characteristic
385	#336	C₂²×C₂₃	1	1	yes	characteristic
386	#336	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×C₂₃	—	4	2	no	—
388	C₂×C₂₃	—	8	2	no	—
389	C₂×C₂₃	#2449	1	1	yes	characteristic
390	C₂×C₂₃	#3077	1	1	yes	characteristic
391	C₂×C₂₃	#3084	1	1	yes	characteristic
392	C₂×C₂₃	#3265	1	1	yes	characteristic
393	C₂×C₂₃	#3275	1	1	yes	characteristic
394	C₂×C₂₃	#3350	1	1	yes	characteristic
395	C₂×C₂₃	#3354	1	1	yes	characteristic
396	C₂²×Q₈	—	2	1	no	—
397	C₂²×D₄	—	2	1	no	—
398	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
399	C₂³×C₄	—	2	1	no	—
400	C₂³×C₄	—	2	1	no	—
401	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
402	C₂×M₁₆	—	2	2	no	—
403	C₂×M₁₆	—	2	2	no	—
404	C₂×M₁₆	—	2	2	no	—
405	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
408	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₄⋊Q₈	—	2	2	no	—
410	C₄⋊Q₈	—	4	1	no	—
411	#121	—	2	2	no	—
412	#121	Q₈×C₂₃	1	4	yes	complemented
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	4	1	no	—
415	C₄×D₄	—	4	4	no	—
416	#117	—	2	2	no	—
417	#117	—	4	1	no	—
418	C₂×C₄⋊C₄	—	2	2	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	4	1	no	—
425	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₂²⋊C₄	—	2	1	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
444	#107	—	2	2	no	—
445	#107	—	2	2	no	—
446	#106	—	2	2	no	—
447	#106	—	2	2	no	—
448	#106	—	4	1	no	—
449	#106	D₄×C₂₃	1	4	yes	complemented
450	C₄⋊C₈	—	2	2	no	—
451	C₄⋊C₈	—	2	2	no	—
452	#102	—	2	2	no	—
453	#102	—	2	2	no	—
454	#102	—	2	2	no	—
455	#102	—	4	1	no	—
456	C₂²⋊C₈	—	2	2	no	—
457	#95	—	2	1	no	—
458	#95	—	2	1	no	—
459	#95	—	2	1	no	—
460	#95	—	2	1	no	—
461	#95	—	2	1	no	—
462	#95	—	2	1	no	—
463	#95	—	2	1	no	—
464	#95	D₄×C₂₃	1	1	yes	characteristic
465	#95	D₄×C₂₃	1	1	yes	characteristic
466	C₂₃	#26354	1	1	yes	characteristic, direct product factor, Sylow
467	C₂⁴	—	2	1	no	—
468	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
469	C₂×Q₈	—	4	2	no	—
470	C₂×Q₈	—	4	2	no	—
471	C₂×Q₈	—	8	1	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	4	2	no	—
479	C₂×D₄	—	4	2	no	—
480	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	—	4	2	no	—
494	C₂²×C₄	—	4	2	no	—
495	C₂²×C₄	—	4	2	no	—
496	C₂²×C₄	—	8	1	no	—
497	C₂²×C₄	—	8	1	no	—
498	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
501	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
504	C₂²×C₄	#41×C₂₃	1	2	yes	—
505	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
506	M₁₆	—	2	8	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
518	C₂×C₈	#41×C₂₃	1	2	yes	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	—	4	2	no	—
532	C₄⋊C₄	—	4	2	no	—
533	C₄⋊C₄	—	4	2	no	—
534	C₄⋊C₄	—	8	1	no	—
535	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
536	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
537	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	4	1	no	—
541	C₂²⋊C₄	—	4	2	no	—
542	C₂²⋊C₄	—	4	2	no	—
543	C₂²⋊C₄	—	4	2	no	—
544	C₂²⋊C₄	—	8	1	no	—
545	C₂²⋊C₄	—	8	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	1	no	—
549	C₄²	—	2	1	no	—
550	C₄²	—	4	2	no	—
551	C₄²	—	4	2	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	4	1	no	—
561	C₂³	—	4	1	no	—
562	C₂³	—	4	1	no	—
563	C₂³	—	4	1	no	—
564	C₂³	—	4	1	no	—
565	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
566	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
567	Q₈	—	8	4	no	—
568	D₄	—	2	2	no	—
569	D₄	—	2	2	no	—
570	D₄	—	4	8	no	—
571	D₄	—	4	1	no	—
572	D₄	—	4	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	8	4	no	—
612	C₂×C₄	—	8	4	no	—
613	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
614	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
615	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
616	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
617	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
618	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
619	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
620	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
621	C₈	—	2	2	no	—
622	C₈	—	2	2	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	1	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	4	1	no	—
629	C₂²	—	4	1	no	—
630	C₂²	—	4	1	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	8	1	no	—
636	C₂²	—	8	1	no	—
637	C₂²	—	8	1	no	—
638	C₂²	—	8	1	no	—
639	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
640	C₂²	D₄²×C₂₃	1	1	yes	characteristic
641	C₂²	C₂₃×#547	1	1	yes	characteristic
642	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
643	C₂²	C₂₃×#575	1	1	yes	characteristic
644	C₂²	C₂₃×#577	1	1	yes	characteristic
645	C₂²	C₂₃×#577	1	1	yes	characteristic
646	C₄	—	2	2	no	—
647	C₄	—	2	2	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	1	no	—
650	C₄	—	4	2	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	1	no	—
653	C₄	—	4	1	no	—
654	C₄	—	8	1	no	—
655	C₄	—	8	1	no	—
656	C₄	—	8	1	no	—
657	C₄	—	8	1	no	—
658	C₄	C₂₃×#525	1	1	yes	characteristic
659	C₄	C₂₃×#539	1	1	yes	characteristic
660	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
661	C₄	C₂₃×#554	1	1	yes	characteristic
662	C₂	—	4	2	no	—
663	C₂	—	8	2	no	—
664	C₂	C₂₃×#2449	1	1	yes	characteristic
665	C₂	C₂₃×#3077	1	1	yes	characteristic
666	C₂	C₂₃×#3084	1	1	yes	characteristic
667	C₂	C₂₃×#3265	1	1	yes	characteristic
668	C₂	C₂₃×#3275	1	1	yes	characteristic
669	C₂	C₂₃×#3350	1	1	yes	characteristic
670	C₂	C₂₃×#3354	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#476	C₂	semidirect
2. C₂×C₂₃×#428	C₂	semidirect
3. C₂₃×#2439	C₂	semidirect
5. C₂₃×#2092	C₂	semidirect
6. C₂₃×#2086	C₂	semidirect
7. C₂₃×#2065	C₂	semidirect
8. C₂₃×#2058	C₂	semidirect
9. C₂₃×#1949	C₂	semidirect
10. C₂₃×#1919	C₂	semidirect
13. C₂₃×#1862	C₂	semidirect
14. C₂₃×#1853	C₂	semidirect
15. C₂₃×#1851	C₂	semidirect
77. C₂₃×#121	Q₈	semidirect
114. C₂₃×#106	D₄	semidirect
216. #26354	C₂₃	direct (N × H)
290. C₂×#476	C₂×C₂₃	semidirect
291. C₂×#428	C₂×C₂₃	semidirect
292. #2439	C₂×C₂₃	semidirect
294. #2092	C₂×C₂₃	semidirect
295. #2086	C₂×C₂₃	semidirect
296. #2065	C₂×C₂₃	semidirect
297. #2058	C₂×C₂₃	semidirect
298. #1949	C₂×C₂₃	semidirect
299. #1919	C₂×C₂₃	semidirect
302. #1862	C₂×C₂₃	semidirect
303. #1853	C₂×C₂₃	semidirect
304. #1851	C₂×C₂₃	semidirect
412. #121	Q₈×C₂₃	semidirect
449. #106	D₄×C₂₃	semidirect
466. C₂₃	#26354	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×#476	2
C₂×C₂₃×#428	2
C₂₃×#2439	2
C₂₃×#2361	2
C₂₃×#2092	2
C₂₃×#2086	2
C₂₃×#2065	2
C₂₃×#2058	2
C₂₃×#1949	2
C₂₃×#1919	2
C₂₃×#1895	2
C₂₃×#1875	2
C₂₃×#1862	2
C₂₃×#1853	2
C₂₃×#1851	2
#26354	23

Subgroups — C23×#26354 #19173980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26354 #19173980