Subgroups — C₂₃×#60306 #19207932

Order 5888.

1902 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3460	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2993	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2989	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2981	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2976	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2966	C₂	1	4	yes	maximal, complemented
8	C₈×#41×C₂₃	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2931	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2929	C₂	1	4	yes	maximal, complemented
11	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#131	C₂²	1	4	yes	complemented
13	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#544	C₄	1	1	yes	characteristic
15	C₂₃×#542	C₄	1	4	yes	—
16	C₂₃×#540	C₄	1	4	yes	—
17	C₂₃×#538	C₄	1	4	yes	—
18	C₂₃×#537	C₄	1	1	yes	characteristic
19	C₂₃×#535	C₄	1	2	yes	—
20	C₂₃×#532	C₂²	1	4	yes	—
21	C₂₃×#529	C₂²	1	4	yes	—
22	C₂₃×#525	C₂²	1	4	yes	—
23	C₄×#41×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#446	C₂²	1	4	yes	complemented
26	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
27	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
28	C₂₃×#436	C₂²	1	4	yes	—
29	C₂₃×#436	C₂²	1	4	yes	—
30	C₂₃×#436	C₂²	1	8	yes	complemented
31	C₂₃×#436	C₂²	1	4	yes	complemented
32	C₂₃×#435	C₂²	1	8	yes	—
33	C₂₃×#435	C₂²	1	8	yes	complemented
34	C₂₃×#435	C₂²	1	8	yes	complemented
35	C₈×D₄×C₂₃	C₂²	1	4	yes	—
36	C₈×D₄×C₂₃	C₂²	1	4	yes	—
37	C₈×D₄×C₂₃	C₂²	1	4	yes	—
38	C₈×D₄×C₂₃	C₂²	1	4	yes	—
39	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
40	C₈×D₄×C₂₃	C₂²	1	2	yes	complemented
41	C₈×D₄×C₂₃	C₂²	1	2	yes	complemented
42	C₂₃×#433	C₂²	1	8	yes	complemented
43	C₂₃×#432	C₂²	1	4	yes	complemented
44	C₂₃×#431	C₂²	1	2	yes	—
45	C₂₃×#431	C₂²	1	4	yes	complemented
46	C₂₃×#424	C₂²	1	4	yes	complemented
47	C₂₃×#423	C₂²	1	4	yes	complemented
48	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
49	C₂₃×#408	C₂²	1	4	yes	complemented
50	C₂₃×#408	C₂²	1	8	yes	complemented
51	C₂₃×#407	C₂²	1	4	yes	—
52	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
53	C₂₃×#405	C₂²	1	4	yes	complemented
54	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	complemented
56	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
57	C₂×#41×C₂₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
60	C₂₃×#131	—	4	8	no	—
61	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
62	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
63	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
64	C₂²×C₈×C₂₃	—	4	4	no	—
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
67	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
68	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
69	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
72	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
73	C₂₃×#127	C₂×C₄	1	4	yes	—
74	C₂₃×#126	C₂×C₄	1	8	yes	—
75	C₂₃×#125	C₂×C₄	1	4	yes	—
76	C₂₃×#124	C₂×C₄	1	4	yes	—
77	C₂₃×#124	C₂×C₄	1	8	yes	—
78	C₂₃×#123	C₂×C₄	1	8	yes	—
79	C₂₃×#123	C₂×C₄	1	8	yes	—
80	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
81	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
82	C₂₃×C₂²⋊Q₈	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	8	yes	—
89	C₂₃×#120	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
91	C₄×Q₈×C₂₃	C₂³	1	2	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	2	yes	—
95	C₄×D₄×C₂₃	C₂³	1	4	yes	—
96	C₄×D₄×C₂₃	C₂³	1	4	yes	—
97	C₄×D₄×C₂₃	C₂³	1	2	yes	—
98	C₄×D₄×C₂₃	C₂³	1	8	yes	—
99	C₂₃×#117	C₂×C₄	1	2	yes	—
100	C₂₃×#117	C₂×C₄	1	4	yes	—
101	C₂₃×#117	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
106	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
107	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
108	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
109	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
110	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
111	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
112	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
113	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
114	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
115	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
116	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
117	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
118	C₂₃×#97	C₂³	1	8	yes	—
119	C₄×C₈×C₂₃	C₂³	1	2	yes	—
120	C₄×C₈×C₂₃	C₂³	1	4	yes	—
121	C₄×C₈×C₂₃	C₂³	1	4	yes	—
122	C₄×C₈×C₂₃	C₂³	1	2	yes	—
123	C₄×C₈×C₂₃	C₂³	1	4	yes	—
124	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
125	#41×C₂₃	—	2	8	no	—
126	#41×C₂₃	—	4	4	no	—
127	#41×C₂₃	—	4	4	no	—
128	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₂₃	—	2	4	no	—
131	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
132	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
133	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
134	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
135	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
136	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₂₃	—	4	2	no	—
138	C₂²×C₄×C₂₃	—	4	2	no	—
139	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
142	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
148	M₁₆×C₂₃	—	4	4	no	—
149	M₁₆×C₂₃	—	4	8	no	—
150	M₁₆×C₂₃	—	4	4	no	—
151	C₂×C₈×C₂₃	—	2	2	no	—
152	C₂×C₈×C₂₃	—	2	8	no	—
153	C₂×C₈×C₂₃	—	2	8	no	—
154	C₂×C₈×C₂₃	—	4	4	no	—
155	C₂×C₈×C₂₃	—	4	2	no	—
156	C₂×C₈×C₂₃	—	4	1	no	—
157	C₂×C₈×C₂₃	—	4	4	no	—
158	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
159	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
160	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
161	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
162	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
163	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
164	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
165	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
166	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
167	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
168	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
170	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
171	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
173	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
174	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
175	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
176	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
177	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
178	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
179	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
180	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
181	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
182	C₄²×C₂₃	C₂⁴	1	2	yes	—
183	C₄²×C₂₃	C₂⁴	1	4	yes	—
184	#60306	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₂₃	—	2	4	no	—
186	C₂³×C₂₃	—	4	2	no	—
187	C₂³×C₂₃	—	4	2	no	—
188	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
189	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
190	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
191	Q₈×C₂₃	—	2	4	no	—
192	Q₈×C₂₃	—	4	4	no	—
193	D₄×C₂₃	—	2	8	no	—
194	D₄×C₂₃	—	2	8	no	—
195	D₄×C₂₃	—	2	4	no	—
196	D₄×C₂₃	—	4	8	no	—
197	D₄×C₂₃	—	4	4	no	—
198	D₄×C₂₃	—	4	8	no	—
199	C₂×C₄×C₂₃	—	2	8	no	—
200	C₂×C₄×C₂₃	—	2	1	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	1	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	4	4	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	4	1	no	—
209	C₂×C₄×C₂₃	—	4	4	no	—
210	C₂×C₄×C₂₃	#131	1	4	yes	complemented
211	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
212	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
214	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
215	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
216	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
217	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
218	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
219	C₈×C₂₃	—	2	1	no	—
220	C₈×C₂₃	—	2	4	no	—
221	C₈×C₂₃	—	2	4	no	—
222	C₈×C₂₃	—	2	2	no	—
223	C₈×C₂₃	—	2	4	no	—
224	C₈×C₂₃	H(5,𝔽₂)	1	2	yes	—
225	#3460	C₂×C₂₃	1	1	yes	characteristic
226	#2993	C₂×C₂₃	1	4	yes	complemented
227	#2989	C₂×C₂₃	1	2	yes	complemented
228	#2981	C₂×C₂₃	1	4	yes	complemented
229	#2976	C₂×C₂₃	1	4	yes	complemented
230	#2975	C₂×C₂₃	1	1	yes	characteristic, complemented
231	#2966	C₂×C₂₃	1	4	yes	complemented
232	C₈×#41	C₂×C₂₃	1	2	yes	complemented
233	#2931	C₂×C₂₃	1	4	yes	complemented
234	#2929	C₂×C₂₃	1	4	yes	complemented
235	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	1	no	—
238	C₂²×C₂₃	—	2	4	no	—
239	C₂²×C₂₃	—	2	4	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	2	4	no	—
242	C₂²×C₂₃	—	2	4	no	—
243	C₂²×C₂₃	—	2	4	no	—
244	C₂²×C₂₃	—	4	8	no	—
245	C₂²×C₂₃	C₂×#131	1	2	yes	—
246	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
247	C₄×C₂₃	—	2	1	no	—
248	C₄×C₂₃	—	2	4	no	—
249	C₄×C₂₃	—	2	4	no	—
250	C₄×C₂₃	—	2	4	no	—
251	C₄×C₂₃	—	2	2	no	—
252	C₄×C₂₃	—	2	4	no	—
253	C₄×C₂₃	C₂×#131	1	4	yes	—
254	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
255	C₄×C₂₃	#585	1	1	yes	characteristic
256	C₂×#131	C₂²×C₂₃	1	4	yes	complemented
257	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
258	#544	C₄×C₂₃	1	1	yes	characteristic
259	#542	C₄×C₂₃	1	4	yes	—
260	#540	C₄×C₂₃	1	4	yes	—
261	#538	C₄×C₂₃	1	4	yes	—
262	#537	C₄×C₂₃	1	1	yes	characteristic
263	#535	C₄×C₂₃	1	2	yes	—
264	#532	C₂²×C₂₃	1	4	yes	—
265	#529	C₂²×C₂₃	1	4	yes	—
266	#525	C₂²×C₂₃	1	4	yes	—
267	C₄×#41	C₂²×C₂₃	1	2	yes	—
268	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
269	#446	C₂²×C₂₃	1	4	yes	complemented
270	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
271	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
272	#436	C₂²×C₂₃	1	4	yes	—
273	#436	C₂²×C₂₃	1	4	yes	—
274	#436	C₂²×C₂₃	1	8	yes	complemented
275	#436	C₂²×C₂₃	1	4	yes	complemented
276	#435	C₂²×C₂₃	1	8	yes	—
277	#435	C₂²×C₂₃	1	8	yes	complemented
278	#435	C₂²×C₂₃	1	8	yes	complemented
279	C₈×D₄	C₂²×C₂₃	1	4	yes	—
280	C₈×D₄	C₂²×C₂₃	1	4	yes	—
281	C₈×D₄	C₂²×C₂₃	1	4	yes	—
282	C₈×D₄	C₂²×C₂₃	1	4	yes	—
283	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
284	C₈×D₄	C₂²×C₂₃	1	2	yes	complemented
285	C₈×D₄	C₂²×C₂₃	1	2	yes	complemented
286	#433	C₂²×C₂₃	1	8	yes	complemented
287	#432	C₂²×C₂₃	1	4	yes	complemented
288	#431	C₂²×C₂₃	1	2	yes	—
289	#431	C₂²×C₂₃	1	4	yes	complemented
290	#424	C₂²×C₂₃	1	4	yes	complemented
291	#423	C₂²×C₂₃	1	4	yes	complemented
292	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
293	#408	C₂²×C₂₃	1	4	yes	complemented
294	#408	C₂²×C₂₃	1	8	yes	complemented
295	#407	C₂²×C₂₃	1	4	yes	—
296	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
297	#405	C₂²×C₂₃	1	4	yes	complemented
298	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
299	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	complemented
300	C₂×C₂₃	—	2	2	no	—
301	C₂×C₂₃	—	4	4	no	—
302	C₂×C₂₃	—	4	4	no	—
303	C₂×C₂₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂×C₂₃	C₂²×#131	1	1	yes	characteristic
305	C₂×C₂₃	#3573	1	1	yes	characteristic
306	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
307	C₂×#41	C₂³×C₂₃	1	4	yes	—
308	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
309	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
310	#131	—	4	8	no	—
311	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
312	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
313	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
314	C₂²×C₈	—	4	4	no	—
315	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
317	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
318	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
319	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
320	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
322	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
323	#127	C₂×C₄×C₂₃	1	4	yes	—
324	#126	C₂×C₄×C₂₃	1	8	yes	—
325	#125	C₂×C₄×C₂₃	1	4	yes	—
326	#124	C₂×C₄×C₂₃	1	4	yes	—
327	#124	C₂×C₄×C₂₃	1	8	yes	—
328	#123	C₂×C₄×C₂₃	1	8	yes	—
329	#123	C₂×C₄×C₂₃	1	8	yes	—
330	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
331	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
332	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
333	#121	C₂×C₄×C₂₃	1	4	yes	—
334	#121	C₂×C₄×C₂₃	1	4	yes	—
335	#121	C₂×C₄×C₂₃	1	4	yes	—
336	#121	C₂×C₄×C₂₃	1	4	yes	—
337	#121	C₂×C₄×C₂₃	1	4	yes	—
338	#121	C₂×C₄×C₂₃	1	8	yes	—
339	#120	C₂×C₄×C₂₃	1	8	yes	—
340	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
341	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
342	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
343	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
344	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
345	C₄×D₄	C₂³×C₂₃	1	4	yes	—
346	C₄×D₄	C₂³×C₂₃	1	4	yes	—
347	C₄×D₄	C₂³×C₂₃	1	2	yes	—
348	C₄×D₄	C₂³×C₂₃	1	8	yes	—
349	#117	C₂×C₄×C₂₃	1	2	yes	—
350	#117	C₂×C₄×C₂₃	1	4	yes	—
351	#117	C₂³×C₂₃	1	4	yes	—
352	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
353	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
354	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
356	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
357	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
358	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
359	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
360	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
361	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
362	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
363	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
364	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
365	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
366	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
367	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
368	#97	C₂³×C₂₃	1	8	yes	—
369	C₄×C₈	C₂³×C₂₃	1	2	yes	—
370	C₄×C₈	C₂³×C₂₃	1	2	yes	—
371	C₄×C₈	C₂³×C₂₃	1	4	yes	—
372	C₄×C₈	C₂³×C₂₃	1	4	yes	—
373	C₄×C₈	C₂³×C₂₃	1	4	yes	—
374	C₂₃	#60306	1	1	yes	characteristic, direct product factor, Sylow
375	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
376	#41	—	2	8	no	—
377	#41	—	4	4	no	—
378	#41	—	4	4	no	—
379	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
380	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
383	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
384	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
385	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
386	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
387	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
388	C₂²×C₄	—	4	2	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
391	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
393	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
394	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
395	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
398	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
399	M₁₆	—	4	8	no	—
400	M₁₆	—	4	4	no	—
401	M₁₆	—	4	4	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	8	no	—
404	C₂×C₈	—	2	8	no	—
405	C₂×C₈	—	4	4	no	—
406	C₂×C₈	—	4	2	no	—
407	C₂×C₈	—	4	1	no	—
408	C₂×C₈	—	4	4	no	—
409	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
410	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
411	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
412	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
413	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
414	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
415	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
416	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
417	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
418	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
419	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
420	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
421	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
422	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
423	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
424	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
425	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
426	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
427	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
428	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
429	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
430	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
431	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
432	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
433	C₄²	C₂⁴×C₂₃	1	2	yes	—
434	C₄²	C₂⁴×C₂₃	1	4	yes	—
435	C₂³	—	2	4	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
439	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
440	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
441	Q₈	—	2	4	no	—
442	Q₈	—	4	4	no	—
443	D₄	—	2	8	no	—
444	D₄	—	2	8	no	—
445	D₄	—	2	4	no	—
446	D₄	—	4	4	no	—
447	D₄	—	4	8	no	—
448	D₄	—	4	8	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	C₂₃×#131	1	4	yes	complemented
461	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
462	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
463	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
464	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
465	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
466	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
467	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
468	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
469	C₈	—	2	1	no	—
470	C₈	—	2	4	no	—
471	C₈	—	2	4	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	4	no	—
474	C₈	C₂₃×H(5,𝔽₂)	1	2	yes	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	4	8	no	—
484	C₂²	C₂×C₂₃×#131	1	2	yes	—
485	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
486	C₄	—	2	1	no	—
487	C₄	—	2	4	no	—
488	C₄	—	2	4	no	—
489	C₄	—	2	4	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	4	no	—
492	C₄	C₂×C₂₃×#131	1	4	yes	—
493	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
494	C₄	C₂₃×#585	1	1	yes	characteristic
495	C₂	—	2	2	no	—
496	C₂	—	4	4	no	—
497	C₂	—	4	4	no	—
498	C₂	C₄×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
499	C₂	C₂²×C₂₃×#131	1	1	yes	characteristic
500	C₂	C₂₃×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂₃×#3460	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2993	C₂	semidirect
3. C₂₃×#2989	C₂	semidirect
4. C₂₃×#2981	C₂	semidirect
5. C₂₃×#2976	C₂	semidirect
6. C₂₃×#2975	C₂	semidirect
7. C₂₃×#2966	C₂	semidirect
8. C₈×#41×C₂₃	C₂	semidirect
9. C₂₃×#2931	C₂	semidirect
10. C₂₃×#2929	C₂	semidirect
11. C₂×C₈×D₄×C₂₃	C₂	semidirect
12. C₂×C₂₃×#131	C₂²	semidirect
25. C₂₃×#446	C₂²	semidirect
26. C₈×Q₈×C₂₃	C₂²	semidirect
27. C₈×Q₈×C₂₃	C₂²	semidirect
30. C₂₃×#436	C₂²	semidirect
31. C₂₃×#436	C₂²	semidirect
33. C₂₃×#435	C₂²	semidirect
34. C₂₃×#435	C₂²	semidirect
39. C₈×D₄×C₂₃	C₂²	semidirect
40. C₈×D₄×C₂₃	C₂²	semidirect
41. C₈×D₄×C₂₃	C₂²	semidirect
42. C₂₃×#433	C₂²	semidirect
43. C₂₃×#432	C₂²	semidirect
45. C₂₃×#431	C₂²	semidirect
46. C₂₃×#424	C₂²	semidirect
47. C₂₃×#423	C₂²	semidirect
49. C₂₃×#408	C₂²	semidirect
50. C₂₃×#408	C₂²	semidirect
53. C₂₃×#405	C₂²	semidirect
55. C₂×C₄×C₈×C₂₃	C₂²	semidirect
184. #60306	C₂₃	direct (N × H)
210. C₂×C₄×C₂₃	#131	semidirect
226. #2993	C₂×C₂₃	semidirect
227. #2989	C₂×C₂₃	semidirect
228. #2981	C₂×C₂₃	semidirect
229. #2976	C₂×C₂₃	semidirect
230. #2975	C₂×C₂₃	semidirect
231. #2966	C₂×C₂₃	semidirect
232. C₈×#41	C₂×C₂₃	semidirect
233. #2931	C₂×C₂₃	semidirect
234. #2929	C₂×C₂₃	semidirect
235. C₂×C₈×D₄	C₂×C₂₃	semidirect
256. C₂×#131	C₂²×C₂₃	semidirect
269. #446	C₂²×C₂₃	semidirect
270. C₈×Q₈	C₂²×C₂₃	semidirect
271. C₈×Q₈	C₂²×C₂₃	semidirect
274. #436	C₂²×C₂₃	semidirect
275. #436	C₂²×C₂₃	semidirect
277. #435	C₂²×C₂₃	semidirect
278. #435	C₂²×C₂₃	semidirect
283. C₈×D₄	C₂²×C₂₃	semidirect
284. C₈×D₄	C₂²×C₂₃	semidirect
285. C₈×D₄	C₂²×C₂₃	semidirect
286. #433	C₂²×C₂₃	semidirect
287. #432	C₂²×C₂₃	semidirect
289. #431	C₂²×C₂₃	semidirect
290. #424	C₂²×C₂₃	semidirect
291. #423	C₂²×C₂₃	semidirect
293. #408	C₂²×C₂₃	semidirect
294. #408	C₂²×C₂₃	semidirect
297. #405	C₂²×C₂₃	semidirect
299. C₂×C₄×C₈	C₂²×C₂₃	semidirect
374. C₂₃	#60306	direct (N × H)
460. C₂×C₄	C₂₃×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3460	2
C₂₃×#2993	2
C₂₃×#2989	2
C₂₃×#2981	2
C₂₃×#2976	2
C₂₃×#2975	2
C₂₃×#2966	2
C₈×#41×C₂₃	2
C₂₃×#2931	2
C₂₃×#2929	2
C₂×C₈×D₄×C₂₃	2
#60306	23

Subgroups — C23×#60306 #19207932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60306 #19207932