Subgroups — C₂₃×#60237 #19207863

Order 5888.

2168 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2981	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2980	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2974	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2973	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2935	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2932	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2880	C₂	1	4	yes	maximal, complemented
11	C₂²×#41×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×#131	C₂²	1	8	yes	complemented
13	C₂₃×#532	C₄	1	2	yes	—
14	C₂₃×#532	C₄	1	2	yes	—
15	C₂₃×#532	C₄	1	2	yes	—
16	C₂₃×#532	C₄	1	2	yes	—
17	C₂₃×#532	C₂²	1	2	yes	—
18	C₂₃×#532	C₂²	1	2	yes	—
19	C₂₃×#532	C₂²	1	2	yes	—
20	C₂₃×#532	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
22	C₂×C₂₃×#127	C₄	1	1	yes	characteristic
23	C₂×C₂₃×#124	C₄	1	2	yes	—
24	C₂×C₂₃×#121	C₄	1	2	yes	—
25	C₂×C₂₃×#121	C₄	1	2	yes	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#436	C₂²	1	8	yes	complemented
30	C₂₃×#435	C₂²	1	16	yes	complemented
31	C₂₃×#435	C₂²	1	16	yes	complemented
32	C₂₃×#435	C₂²	1	16	yes	complemented
33	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
34	C₂₃×#432	C₂²	1	2	yes	—
35	C₂₃×#432	C₂²	1	2	yes	—
36	C₂₃×#432	C₂²	1	2	yes	—
37	C₂₃×#432	C₂²	1	2	yes	complemented
38	C₂₃×#432	C₂²	1	2	yes	complemented
39	C₂₃×#432	C₂²	1	2	yes	complemented
40	C₂₃×#431	C₂²	1	2	yes	—
41	C₂₃×#431	C₂²	1	2	yes	complemented
42	C₂₃×#423	C₂²	1	2	yes	complemented
43	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
44	C₂₃×#408	C₂²	1	8	yes	complemented
45	C₂₃×#408	C₂²	1	16	yes	complemented
46	C₂₃×#407	C₂²	1	2	yes	—
47	C₂₃×#407	C₂²	1	8	yes	—
48	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
49	C₂×C₂₃×C₂²⋊C₈	C₂²	1	8	yes	—
50	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
51	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
52	C₂×#41×C₂₃	—	4	8	no	—
53	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
54	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
55	C₂×#41×C₂₃	C₂³	1	4	yes	—
56	C₂×#41×C₂₃	C₂³	1	4	yes	—
57	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
58	C₂²×D₄×C₂₃	C₂×C₄	1	4	yes	—
59	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
60	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
61	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
62	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
66	C₂₃×#131	—	4	16	no	—
67	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
68	C₂×M₁₆×C₂₃	C₂³	1	16	yes	—
69	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
70	C₂²×C₈×C₂₃	C₂³	1	16	yes	—
71	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
72	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
73	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
74	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
75	C₂₃×#127	C₂×C₄	1	2	yes	—
76	C₂₃×#127	C₂×C₄	1	2	yes	—
77	C₂₃×#127	C₂×C₄	1	2	yes	—
78	C₂₃×#127	C₂×C₄	1	2	yes	—
79	C₂₃×#124	C₂×C₄	1	4	yes	—
80	C₂₃×#124	C₂×C₄	1	4	yes	—
81	C₂₃×#124	C₂×C₄	1	4	yes	—
82	C₂₃×#124	C₂×C₄	1	4	yes	—
83	C₂₃×#121	C₂×C₄	1	4	yes	—
84	C₂₃×#121	C₂×C₄	1	4	yes	—
85	C₂₃×#121	C₂×C₄	1	4	yes	—
86	C₂₃×#121	C₂×C₄	1	4	yes	—
87	C₂₃×#121	C₂×C₄	1	4	yes	—
88	C₂₃×#121	C₂×C₄	1	4	yes	—
89	C₂₃×#121	C₂×C₄	1	4	yes	—
90	C₂₃×#121	C₂×C₄	1	4	yes	—
91	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
92	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
93	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
94	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
95	C₄×D₄×C₂₃	C₂³	1	4	yes	—
96	C₄×D₄×C₂₃	C₂³	1	4	yes	—
97	C₄×D₄×C₂₃	C₂³	1	4	yes	—
98	C₄×D₄×C₂₃	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
101	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
102	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
103	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
115	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
116	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
117	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
118	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
119	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
120	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
121	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
122	C₂₃×C₂²⋊C₈	C₂³	1	16	yes	—
123	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
124	C₂₃×C₂²⋊C₈	C₂³	1	16	yes	—
125	C₂₃×#97	C₂³	1	4	yes	—
126	C₂₃×#97	C₂³	1	4	yes	—
127	C₂₃×#97	C₂³	1	4	yes	—
128	C₄×C₈×C₂₃	C₂³	1	4	yes	—
129	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
131	#41×C₂₃	—	2	16	no	—
132	#41×C₂₃	—	4	8	no	—
133	#41×C₂₃	—	4	8	no	—
134	#41×C₂₃	—	4	8	no	—
135	C₂×Q₈×C₂₃	—	4	4	no	—
136	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
137	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
138	C₂×D₄×C₂₃	—	4	4	no	—
139	C₂×D₄×C₂₃	—	4	4	no	—
140	C₂×D₄×C₂₃	—	4	4	no	—
141	C₂×D₄×C₂₃	—	4	8	no	—
142	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
143	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
144	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
145	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
146	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
147	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
148	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
149	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
150	C₂²×C₄×C₂₃	—	2	4	no	—
151	C₂²×C₄×C₂₃	—	4	2	no	—
152	C₂²×C₄×C₂₃	—	4	8	no	—
153	C₂²×C₄×C₂₃	—	4	2	no	—
154	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
171	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
175	M₁₆×C₂₃	—	2	8	no	—
176	M₁₆×C₂₃	—	4	16	no	—
177	C₂×C₈×C₂₃	—	2	8	no	—
178	C₂×C₈×C₂₃	—	4	16	no	—
179	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
180	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
181	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
182	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
183	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
184	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
185	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
186	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
187	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
188	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
189	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
190	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
191	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
192	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
193	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
194	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
195	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
197	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
198	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
199	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
200	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
201	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
202	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
203	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
204	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
205	#60237	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₂₃	—	2	2	no	—
207	C₂³×C₂₃	—	2	2	no	—
208	C₂³×C₂₃	—	2	8	no	—
209	C₂³×C₂₃	—	4	1	no	—
210	C₂³×C₂₃	—	4	4	no	—
211	C₂³×C₂₃	—	4	4	no	—
212	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
213	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
214	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
215	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
216	Q₈×C₂₃	—	2	8	no	—
217	Q₈×C₂₃	—	4	4	no	—
218	D₄×C₂₃	—	2	16	no	—
219	D₄×C₂₃	—	2	8	no	—
220	D₄×C₂₃	—	2	8	no	—
221	D₄×C₂₃	—	4	4	no	—
222	D₄×C₂₃	—	4	4	no	—
223	D₄×C₂₃	—	4	4	no	—
224	D₄×C₂₃	—	4	4	no	—
225	D₄×C₂₃	—	4	8	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	2	16	no	—
235	C₂×C₄×C₂₃	—	2	2	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	4	no	—
245	C₂×C₄×C₂₃	—	4	8	no	—
246	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
255	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
256	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
258	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
259	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
260	C₈×C₂₃	—	2	8	no	—
261	C₈×C₂₃	—	2	8	no	—
262	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
263	#2981	C₂×C₂₃	1	4	yes	complemented
264	#2980	C₂×C₂₃	1	4	yes	complemented
265	#2974	C₂×C₂₃	1	4	yes	complemented
266	#2973	C₂×C₂₃	1	4	yes	complemented
267	#2935	C₂×C₂₃	1	4	yes	complemented
268	#2932	C₂×C₂₃	1	4	yes	complemented
269	#2920	C₂×C₂₃	1	1	yes	characteristic, complemented
270	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#2880	C₂×C₂₃	1	4	yes	complemented
272	C₂²×C₂₃	—	2	1	no	—
273	C₂²×C₂₃	—	2	1	no	—
274	C₂²×C₂₃	—	2	1	no	—
275	C₂²×C₂₃	—	2	1	no	—
276	C₂²×C₂₃	—	2	1	no	—
277	C₂²×C₂₃	—	2	8	no	—
278	C₂²×C₂₃	—	2	1	no	—
279	C₂²×C₂₃	—	2	1	no	—
280	C₂²×C₂₃	—	2	8	no	—
281	C₂²×C₂₃	—	2	8	no	—
282	C₂²×C₂₃	—	4	4	no	—
283	C₂²×C₂₃	—	8	8	no	—
284	C₂²×C₂₃	#518	1	2	yes	—
285	C₂²×C₂₃	#568	1	2	yes	—
286	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
287	C₄×C₂₃	—	2	1	no	—
288	C₄×C₂₃	—	2	1	no	—
289	C₄×C₂₃	—	2	8	no	—
290	C₄×C₂₃	—	2	1	no	—
291	C₄×C₂₃	—	2	1	no	—
292	C₄×C₂₃	—	2	1	no	—
293	C₄×C₂₃	—	2	1	no	—
294	C₄×C₂₃	—	2	1	no	—
295	C₄×C₂₃	—	2	1	no	—
296	C₄×C₂₃	—	2	1	no	—
297	C₄×C₂₃	#518	1	2	yes	—
298	C₄×C₂₃	#568	1	2	yes	—
299	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
300	C₂²×#41	C₂²×C₂₃	1	2	yes	—
301	C₂×#131	C₂²×C₂₃	1	8	yes	complemented
302	#532	C₄×C₂₃	1	2	yes	—
303	#532	C₄×C₂₃	1	2	yes	—
304	#532	C₄×C₂₃	1	2	yes	—
305	#532	C₄×C₂₃	1	2	yes	—
306	#532	C₂²×C₂₃	1	2	yes	—
307	#532	C₂²×C₂₃	1	2	yes	—
308	#532	C₂²×C₂₃	1	2	yes	—
309	#532	C₂²×C₂₃	1	2	yes	—
310	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
311	C₂×#127	C₄×C₂₃	1	1	yes	characteristic
312	C₂×#124	C₄×C₂₃	1	2	yes	—
313	C₂×#121	C₄×C₂₃	1	2	yes	—
314	C₂×#121	C₄×C₂₃	1	2	yes	—
315	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
316	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
317	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
318	#436	C₂²×C₂₃	1	8	yes	complemented
319	#435	C₂²×C₂₃	1	16	yes	complemented
320	#435	C₂²×C₂₃	1	16	yes	complemented
321	#435	C₂²×C₂₃	1	16	yes	complemented
322	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
323	#432	C₂²×C₂₃	1	2	yes	—
324	#432	C₂²×C₂₃	1	2	yes	—
325	#432	C₂²×C₂₃	1	2	yes	—
326	#432	C₂²×C₂₃	1	2	yes	complemented
327	#432	C₂²×C₂₃	1	2	yes	complemented
328	#432	C₂²×C₂₃	1	2	yes	complemented
329	#431	C₂²×C₂₃	1	2	yes	—
330	#431	C₂²×C₂₃	1	2	yes	complemented
331	#423	C₂²×C₂₃	1	2	yes	complemented
332	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
333	#408	C₂²×C₂₃	1	8	yes	complemented
334	#408	C₂²×C₂₃	1	16	yes	complemented
335	#407	C₂²×C₂₃	1	2	yes	—
336	#407	C₂²×C₂₃	1	8	yes	—
337	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
338	C₂×C₂²⋊C₈	C₂²×C₂₃	1	8	yes	—
339	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
340	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
341	C₂×C₂₃	—	2	2	no	—
342	C₂×C₂₃	—	2	2	no	—
343	C₂×C₂₃	—	4	1	no	—
344	C₂×C₂₃	—	4	8	no	—
345	C₂×C₂₃	C₂×#518	1	1	yes	characteristic
346	C₂×C₂₃	C₂×#568	1	1	yes	characteristic
347	C₂×C₂₃	C₂×#568	1	1	yes	characteristic
348	C₂×#41	—	4	8	no	—
349	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
350	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
351	C₂×#41	C₂³×C₂₃	1	4	yes	—
352	C₂×#41	C₂³×C₂₃	1	4	yes	—
353	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
354	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
355	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
356	C₂²×D₄	C₂×C₄×C₂₃	1	4	yes	—
357	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
358	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
359	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
362	#131	—	4	16	no	—
363	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
364	C₂×M₁₆	C₂³×C₂₃	1	16	yes	—
365	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
366	C₂²×C₈	C₂³×C₂₃	1	16	yes	—
367	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
368	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
369	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
370	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
371	#127	C₂×C₄×C₂₃	1	2	yes	—
372	#127	C₂×C₄×C₂₃	1	2	yes	—
373	#127	C₂×C₄×C₂₃	1	2	yes	—
374	#127	C₂×C₄×C₂₃	1	2	yes	—
375	#124	C₂×C₄×C₂₃	1	4	yes	—
376	#124	C₂×C₄×C₂₃	1	4	yes	—
377	#124	C₂×C₄×C₂₃	1	4	yes	—
378	#124	C₂×C₄×C₂₃	1	4	yes	—
379	#121	C₂×C₄×C₂₃	1	4	yes	—
380	#121	C₂×C₄×C₂₃	1	4	yes	—
381	#121	C₂×C₄×C₂₃	1	4	yes	—
382	#121	C₂×C₄×C₂₃	1	4	yes	—
383	#121	C₂×C₄×C₂₃	1	4	yes	—
384	#121	C₂×C₄×C₂₃	1	4	yes	—
385	#121	C₂×C₄×C₂₃	1	4	yes	—
386	#121	C₂×C₄×C₂₃	1	4	yes	—
387	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
388	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
389	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
390	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
391	C₄×D₄	C₂³×C₂₃	1	4	yes	—
392	C₄×D₄	C₂³×C₂₃	1	4	yes	—
393	C₄×D₄	C₂³×C₂₃	1	4	yes	—
394	C₄×D₄	C₂³×C₂₃	1	4	yes	—
395	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
396	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
398	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
399	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
400	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
401	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
402	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
403	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
404	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
411	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
412	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
413	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
414	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
415	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
416	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
417	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
418	C₂²⋊C₈	C₂³×C₂₃	1	16	yes	—
419	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
420	C₂²⋊C₈	C₂³×C₂₃	1	16	yes	—
421	#97	C₂³×C₂₃	1	4	yes	—
422	#97	C₂³×C₂₃	1	4	yes	—
423	#97	C₂³×C₂₃	1	4	yes	—
424	C₄×C₈	C₂³×C₂₃	1	4	yes	—
425	C₂₃	#60237	1	1	yes	characteristic, direct product factor, Sylow
426	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
427	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
428	#41	—	2	16	no	—
429	#41	—	4	8	no	—
430	#41	—	4	8	no	—
431	#41	—	4	8	no	—
432	C₂×Q₈	—	4	4	no	—
433	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
434	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
435	C₂×D₄	—	4	4	no	—
436	C₂×D₄	—	4	4	no	—
437	C₂×D₄	—	4	4	no	—
438	C₂×D₄	—	4	8	no	—
439	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
440	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
441	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
442	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
443	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
444	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
445	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
446	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	4	2	no	—
449	C₂²×C₄	—	4	8	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
458	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
459	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
460	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
461	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
463	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
464	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
465	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
466	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
467	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
468	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
471	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
472	M₁₆	—	2	8	no	—
473	M₁₆	—	4	16	no	—
474	C₂×C₈	—	2	8	no	—
475	C₂×C₈	—	4	16	no	—
476	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
477	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
478	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
479	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
480	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
481	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
482	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
483	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
484	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
485	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
486	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
487	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
488	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
489	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
490	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
491	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
492	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
493	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
494	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
495	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
496	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
497	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
498	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
499	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
500	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
501	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
502	C₂³	—	2	2	no	—
503	C₂³	—	2	2	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	4	1	no	—
506	C₂³	—	4	4	no	—
507	C₂³	—	4	4	no	—
508	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
509	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
510	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
511	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
512	Q₈	—	2	8	no	—
513	Q₈	—	4	4	no	—
514	D₄	—	2	16	no	—
515	D₄	—	2	8	no	—
516	D₄	—	2	8	no	—
517	D₄	—	4	4	no	—
518	D₄	—	4	8	no	—
519	D₄	—	4	4	no	—
520	D₄	—	4	4	no	—
521	D₄	—	4	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	16	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	4	4	no	—
534	C₂×C₄	—	4	8	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
543	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
544	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
545	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
546	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
547	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
548	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
549	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
550	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
551	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
552	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
553	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
554	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
555	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
556	C₈	—	2	8	no	—
557	C₈	—	2	8	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	1	no	—
562	C₂²	—	2	1	no	—
563	C₂²	—	2	8	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	8	no	—
567	C₂²	—	2	8	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	8	8	no	—
570	C₂²	C₂₃×#518	1	2	yes	—
571	C₂²	C₂₃×#568	1	2	yes	—
572	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	8	no	—
576	C₄	—	2	1	no	—
577	C₄	—	2	1	no	—
578	C₄	—	2	1	no	—
579	C₄	—	2	1	no	—
580	C₄	—	2	1	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	1	no	—
583	C₄	C₂₃×#518	1	2	yes	—
584	C₄	C₂₃×#568	1	2	yes	—
585	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
586	C₂	—	2	2	no	—
587	C₂	—	2	2	no	—
588	C₂	—	4	1	no	—
589	C₂	—	4	8	no	—
590	C₂	C₂×C₂₃×#518	1	1	yes	characteristic
591	C₂	C₂×C₂₃×#568	1	1	yes	characteristic
592	C₂	C₂×C₂₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂×C₂₃×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2981	C₂	semidirect
3. C₂₃×#2980	C₂	semidirect
4. C₂₃×#2974	C₂	semidirect
5. C₂₃×#2973	C₂	semidirect
6. C₂₃×#2935	C₂	semidirect
7. C₂₃×#2932	C₂	semidirect
8. C₂₃×#2920	C₂	semidirect
9. C₂×C₂₃×#432	C₂	semidirect
10. C₂₃×#2880	C₂	semidirect
12. C₂×C₂₃×#131	C₂²	semidirect
29. C₂₃×#436	C₂²	semidirect
30. C₂₃×#435	C₂²	semidirect
31. C₂₃×#435	C₂²	semidirect
32. C₂₃×#435	C₂²	semidirect
33. C₈×D₄×C₂₃	C₂²	semidirect
37. C₂₃×#432	C₂²	semidirect
38. C₂₃×#432	C₂²	semidirect
39. C₂₃×#432	C₂²	semidirect
41. C₂₃×#431	C₂²	semidirect
42. C₂₃×#423	C₂²	semidirect
43. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
44. C₂₃×#408	C₂²	semidirect
45. C₂₃×#408	C₂²	semidirect
50. C₄×M₁₆×C₂₃	C₂²	semidirect
51. C₂×C₂₃×#97	C₂²	semidirect
205. #60237	C₂₃	direct (N × H)
263. #2981	C₂×C₂₃	semidirect
264. #2980	C₂×C₂₃	semidirect
265. #2974	C₂×C₂₃	semidirect
266. #2973	C₂×C₂₃	semidirect
267. #2935	C₂×C₂₃	semidirect
268. #2932	C₂×C₂₃	semidirect
269. #2920	C₂×C₂₃	semidirect
270. C₂×#432	C₂×C₂₃	semidirect
271. #2880	C₂×C₂₃	semidirect
301. C₂×#131	C₂²×C₂₃	semidirect
318. #436	C₂²×C₂₃	semidirect
319. #435	C₂²×C₂₃	semidirect
320. #435	C₂²×C₂₃	semidirect
321. #435	C₂²×C₂₃	semidirect
322. C₈×D₄	C₂²×C₂₃	semidirect
326. #432	C₂²×C₂₃	semidirect
327. #432	C₂²×C₂₃	semidirect
328. #432	C₂²×C₂₃	semidirect
330. #431	C₂²×C₂₃	semidirect
331. #423	C₂²×C₂₃	semidirect
332. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
333. #408	C₂²×C₂₃	semidirect
334. #408	C₂²×C₂₃	semidirect
339. C₄×M₁₆	C₂²×C₂₃	semidirect
340. C₂×#97	C₂²×C₂₃	semidirect
425. C₂₃	#60237	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×#532	2
C₂₃×#2981	2
C₂₃×#2980	2
C₂₃×#2974	2
C₂₃×#2973	2
C₂₃×#2935	2
C₂₃×#2932	2
C₂₃×#2920	2
C₂×C₂₃×#432	2
C₂₃×#2880	2
#60237	23

Subgroups — C23×#60237 #19207863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60237 #19207863