Subgroups — C₂₃×#60428 #19208054

Order 5888.

2866 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#576	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3045	C₂	1	6	yes	maximal, complemented
5	C₂₃×#3041	C₂	1	6	yes	maximal, complemented
6	C₂₃×#3038	C₂	1	3	yes	maximal, complemented
7	C₂₃×#3036	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3035	C₂	1	3	yes	maximal, complemented
9	C₂×C₂₃×#459	C₂	1	3	yes	maximal, complemented
10	C₂₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2897	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	2	yes	—
13	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
14	C₂₃×#576	—	2	16	no	—
15	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
16	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
17	C₂×C₂₃×#135	C₂²	1	6	yes	complemented
18	C₂²×D₈×C₂₃	C₂²	1	6	yes	complemented
19	C₂×C₂₃×#131	C₂²	1	2	yes	—
20	C₂₃×#550	—	2	1	no	—
21	C₂₃×#550	—	2	1	no	—
22	D₄²×C₂₃	—	2	3	no	—
23	D₄²×C₂₃	—	2	3	no	—
24	C₂₃×#535	C₂²	1	6	yes	complemented
25	C₂₃×#532	C₂²	1	3	yes	complemented
26	C₂×C₂₃×#127	C₂²	1	3	yes	complemented
27	C₂₃×#520	—	2	1	no	—
28	C₂₃×#520	C₂²	1	1	yes	characteristic
29	C₂₃×#520	C₂²	1	3	yes	complemented
30	C₂₃×#518	—	2	3	no	—
31	C₂₃×#518	C₂²	1	3	yes	complemented
32	C₄×#41×C₂₃	—	2	1	no	—
33	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×#41×C₂₃	C₂²	1	3	yes	complemented
35	C₂×C₄×D₄×C₂₃	—	2	3	no	—
36	C₂×C₄×D₄×C₂₃	C₂²	1	3	yes	complemented
37	C₂₃×#464	C₂²	1	6	yes	complemented
38	C₂₃×#463	C₂²	1	6	yes	complemented
39	C₂₃×#463	C₂²	1	12	yes	complemented
40	C₂₃×#462	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×#461	C₂²	1	3	yes	complemented
42	C₂₃×#460	C₂²	1	3	yes	complemented
43	C₂₃×#460	C₂²	1	6	yes	complemented
44	C₂₃×#459	C₂²	1	3	yes	complemented
45	C₂₃×#459	C₂²	1	6	yes	complemented
46	C₂₃×#459	C₂²	1	12	yes	complemented
47	C₂₃×#459	C₂²	1	6	yes	complemented
48	C₂₃×#424	C₂²	1	6	yes	complemented
49	C₂₃×#423	C₂²	1	3	yes	complemented
50	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	complemented
51	C₂₃×#418	C₂²	1	6	yes	complemented
52	C₂₃×#417	C₂²	1	6	yes	complemented
53	C₂₃×#416	C₂²	1	6	yes	complemented
54	C₂×C₂₃×#102	C₂²	1	6	yes	complemented
55	C₂₃×H(5,𝔽₂)	—	2	4	no	—
56	C₂₃×H(5,𝔽₂)	—	2	8	no	—
57	C₂₃×H(5,𝔽₂)	—	2	8	no	—
58	C₂₃×H(5,𝔽₂)	D₄	1	8	yes	complemented
59	C₂×#41×C₂₃	—	2	1	no	—
60	C₂×#41×C₂₃	—	2	2	no	—
61	C₂×#41×C₂₃	C₂³	1	6	yes	—
62	C₂×#41×C₂₃	C₂³	1	3	yes	—
63	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	—	2	3	no	—
65	C₂²×D₄×C₂₃	—	2	6	no	—
66	C₂²×D₄×C₂₃	—	2	3	no	—
67	C₂²×D₄×C₂₃	—	2	3	no	—
68	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
69	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
70	C₂³×C₄×C₂₃	—	2	3	no	—
71	C₂₃×AGL(1,ℤ/8)	—	2	24	no	—
72	C₂₃×AGL(1,ℤ/8)	—	2	24	no	—
73	C₂₃×#135	—	2	24	no	—
74	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
75	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
76	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
77	C₂×D₈×C₂₃	—	2	24	no	—
78	C₂×D₈×C₂₃	C₂³	1	6	yes	—
79	C₂×D₈×C₂₃	C₂³	1	12	yes	—
80	C₂₃×#131	—	2	8	no	—
81	C₂×M₁₆×C₂₃	C₂³	1	6	yes	—
82	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×#127	—	2	3	no	—
85	C₂₃×#127	—	2	3	no	—
86	C₂₃×#127	C₂³	1	3	yes	—
87	C₂₃×#127	C₂³	1	6	yes	—
88	C₂₃×#124	C₂³	1	6	yes	—
89	C₂₃×#121	—	2	3	no	—
90	C₂₃×#121	—	2	3	no	—
91	C₂₃×#121	—	2	6	no	—
92	C₂₃×#121	—	2	6	no	—
93	C₂₃×#120	—	2	12	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
97	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
99	C₄×D₄×C₂₃	—	2	3	no	—
100	C₄×D₄×C₂₃	—	2	3	no	—
101	C₄×D₄×C₂₃	—	2	3	no	—
102	C₄×D₄×C₂₃	—	2	3	no	—
103	C₄×D₄×C₂₃	—	2	3	no	—
104	C₄×D₄×C₂₃	—	2	6	no	—
105	C₄×D₄×C₂₃	—	2	3	no	—
106	C₄×D₄×C₂₃	—	2	3	no	—
107	C₄×D₄×C₂₃	—	2	3	no	—
108	C₄×D₄×C₂₃	—	2	3	no	—
109	C₄×D₄×C₂₃	D₄	1	3	yes	—
110	C₄×D₄×C₂₃	D₄	1	3	yes	—
111	C₄×D₄×C₂₃	C₂³	1	3	yes	—
112	C₄×D₄×C₂₃	C₂³	1	6	yes	—
113	C₄×D₄×C₂₃	C₂³	1	3	yes	—
114	C₂₃×#117	—	2	3	no	—
115	C₂₃×#117	—	2	3	no	—
116	C₂₃×#117	C₂³	1	3	yes	—
117	C₂₃×#117	C₂³	1	3	yes	—
118	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
119	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
120	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	6	no	—
122	C₂×C₄²×C₂₃	—	2	3	no	—
123	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
124	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
125	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
126	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
127	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
128	C₂₃×#103	C₂³	1	2	yes	—
129	C₂₃×#103	C₂³	1	6	yes	—
130	C₂₃×#102	C₂³	1	6	yes	—
131	C₂₃×#102	C₂³	1	12	yes	—
132	C₂₃×#102	C₂³	1	6	yes	—
133	C₂⁴×C₂₃	—	2	3	no	—
134	C₂⁴×C₂₃	—	2	6	no	—
135	#41×C₂₃	—	2	8	no	—
136	#41×C₂₃	—	2	6	no	—
137	#41×C₂₃	—	2	12	no	—
138	#41×C₂₃	—	2	12	no	—
139	#41×C₂₃	—	4	8	no	—
140	#41×C₂₃	C₂×D₄	1	4	yes	—
141	#41×C₂₃	C₂×D₄	1	12	yes	complemented
142	#41×C₂₃	#41	1	4	yes	—
143	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
145	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
146	C₂×D₄×C₂₃	—	2	3	no	—
147	C₂×D₄×C₂₃	—	2	24	no	—
148	C₂×D₄×C₂₃	—	2	6	no	—
149	C₂×D₄×C₂₃	—	2	3	no	—
150	C₂×D₄×C₂₃	—	2	12	no	—
151	C₂×D₄×C₂₃	—	2	3	no	—
152	C₂×D₄×C₂₃	—	2	3	no	—
153	C₂×D₄×C₂₃	—	2	6	no	—
154	C₂×D₄×C₂₃	—	2	6	no	—
155	C₂×D₄×C₂₃	—	2	6	no	—
156	C₂×D₄×C₂₃	—	2	6	no	—
157	C₂×D₄×C₂₃	—	2	12	no	—
158	C₂×D₄×C₂₃	—	4	24	no	—
159	C₂×D₄×C₂₃	—	8	6	no	—
160	C₂×D₄×C₂₃	—	8	6	no	—
161	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
162	C₂×D₄×C₂₃	C₂×D₄	1	12	yes	complemented
163	C₂×D₄×C₂₃	C₂⁴	1	6	yes	—
164	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
165	C₂²×C₄×C₂₃	—	2	3	no	—
166	C₂²×C₄×C₂₃	—	2	3	no	—
167	C₂²×C₄×C₂₃	—	2	3	no	—
168	C₂²×C₄×C₂₃	—	2	6	no	—
169	C₂²×C₄×C₂₃	—	2	3	no	—
170	C₂²×C₄×C₂₃	—	2	6	no	—
171	C₂²×C₄×C₂₃	—	8	6	no	—
172	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
173	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
174	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
175	Q₁₆×C₂₃	—	2	4	no	—
176	SD₁₆×C₂₃	—	2	12	no	—
177	SD₁₆×C₂₃	—	2	12	no	—
178	D₈×C₂₃	—	2	12	no	—
179	D₈×C₂₃	—	2	24	no	—
180	M₁₆×C₂₃	—	2	12	no	—
181	C₂×C₈×C₂₃	—	2	12	no	—
182	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
183	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
184	C₄⋊C₄×C₂₃	—	2	3	no	—
185	C₄⋊C₄×C₂₃	—	2	3	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	C₂×D₄	1	3	yes	—
190	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
191	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
192	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
193	C₂²⋊C₄×C₂₃	—	2	3	no	—
194	C₂²⋊C₄×C₂₃	—	2	3	no	—
195	C₂²⋊C₄×C₂₃	—	2	6	no	—
196	C₂²⋊C₄×C₂₃	—	2	3	no	—
197	C₂²⋊C₄×C₂₃	—	2	6	no	—
198	C₂²⋊C₄×C₂₃	—	2	6	no	—
199	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
200	C₄²×C₂₃	—	2	1	no	—
201	C₄²×C₂₃	—	2	3	no	—
202	C₄²×C₂₃	—	2	3	no	—
203	C₄²×C₂₃	C₂×D₄	1	3	yes	—
204	C₄²×C₂₃	C₂×D₄	1	3	yes	—
205	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
206	C₄²×C₂₃	C₂⁴	1	3	yes	—
207	#60428	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂³×C₂₃	—	2	3	no	—
209	C₂³×C₂₃	—	2	12	no	—
210	C₂³×C₂₃	—	2	6	no	—
211	C₂³×C₂₃	—	4	12	no	—
212	C₂³×C₂₃	—	8	3	no	—
213	C₂³×C₂₃	—	8	3	no	—
214	C₂³×C₂₃	—	8	6	no	—
215	C₂³×C₂₃	—	8	6	no	—
216	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
217	Q₈×C₂₃	—	2	1	no	—
218	Q₈×C₂₃	—	2	2	no	—
219	Q₈×C₂₃	—	2	2	no	—
220	Q₈×C₂₃	#121	1	4	yes	—
221	Q₈×C₂₃	C₂²×D₄	1	2	yes	—
222	D₄×C₂₃	—	2	12	no	—
223	D₄×C₂₃	—	2	12	no	—
224	D₄×C₂₃	—	2	6	no	—
225	D₄×C₂₃	—	2	3	no	—
226	D₄×C₂₃	—	2	6	no	—
227	D₄×C₂₃	—	4	6	no	—
228	D₄×C₂₃	—	4	12	no	—
229	D₄×C₂₃	—	4	12	no	—
230	D₄×C₂₃	—	4	6	no	—
231	D₄×C₂₃	—	8	6	no	—
232	D₄×C₂₃	—	8	6	no	—
233	D₄×C₂₃	#121	1	12	yes	—
234	D₄×C₂₃	C₂²×D₄	1	6	yes	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	12	no	—
237	C₂×C₄×C₂₃	—	2	6	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	3	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	4	3	no	—
242	C₂×C₄×C₂₃	—	4	12	no	—
243	C₂×C₄×C₂₃	—	4	6	no	—
244	C₂×C₄×C₂₃	—	8	3	no	—
245	C₂×C₄×C₂₃	—	8	6	no	—
246	C₂×C₄×C₂₃	—	8	3	no	—
247	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
248	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
250	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
251	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
252	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
254	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
255	C₈×C₂₃	—	2	2	no	—
256	C₈×C₂₃	—	2	6	no	—
257	C₂×#576	C₂×C₂₃	1	2	yes	complemented
258	#3498	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₄×H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#3045	C₂×C₂₃	1	6	yes	complemented
261	#3041	C₂×C₂₃	1	6	yes	complemented
262	#3038	C₂×C₂₃	1	3	yes	complemented
263	#3036	C₂×C₂₃	1	3	yes	complemented
264	#3035	C₂×C₂₃	1	3	yes	complemented
265	C₂×#459	C₂×C₂₃	1	3	yes	complemented
266	#2906	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#2897	C₂×C₂₃	1	2	yes	complemented
268	C₂²×C₂₃	—	2	3	no	—
269	C₂²×C₂₃	—	2	3	no	—
270	C₂²×C₂₃	—	2	6	no	—
271	C₂²×C₂₃	—	4	3	no	—
272	C₂²×C₂₃	—	4	3	no	—
273	C₂²×C₂₃	—	4	6	no	—
274	C₂²×C₂₃	—	4	6	no	—
275	C₂²×C₂₃	—	4	6	no	—
276	C₂²×C₂₃	—	8	12	no	—
277	C₂²×C₂₃	—	16	12	no	—
278	C₂²×C₂₃	C₂×#121	1	6	yes	—
279	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
280	C₄×C₂₃	—	2	1	no	—
281	C₄×C₂₃	—	2	3	no	—
282	C₄×C₂₃	—	2	2	no	—
283	C₄×C₂₃	—	2	3	no	—
284	C₄×C₂₃	—	4	1	no	—
285	C₄×C₂₃	—	4	2	no	—
286	C₄×C₂₃	—	4	3	no	—
287	C₄×C₂₃	C₂×#121	1	6	yes	—
288	C₄×C₂₃	#576	1	2	yes	complemented
289	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
290	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
291	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	2	yes	—
292	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	1	yes	characteristic
293	#576	—	2	16	no	—
294	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	6	yes	complemented
295	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	6	yes	complemented
296	C₂×#135	C₂²×C₂₃	1	6	yes	complemented
297	C₂²×D₈	C₂²×C₂₃	1	6	yes	complemented
298	C₂×#131	C₂²×C₂₃	1	2	yes	—
299	#550	—	2	1	no	—
300	#550	—	2	1	no	—
301	D₄²	—	2	3	no	—
302	D₄²	—	2	3	no	—
303	#535	C₂²×C₂₃	1	6	yes	complemented
304	#532	C₂²×C₂₃	1	3	yes	complemented
305	C₂×#127	C₂²×C₂₃	1	3	yes	complemented
306	#520	—	2	1	no	—
307	#520	C₂²×C₂₃	1	1	yes	characteristic
308	#520	C₂²×C₂₃	1	3	yes	complemented
309	#518	—	2	3	no	—
310	#518	C₂²×C₂₃	1	3	yes	complemented
311	C₄×#41	—	2	1	no	—
312	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
313	C₄×#41	C₂²×C₂₃	1	3	yes	complemented
314	C₂×C₄×D₄	—	2	3	no	—
315	C₂×C₄×D₄	C₂²×C₂₃	1	3	yes	complemented
316	#464	C₂²×C₂₃	1	6	yes	complemented
317	#463	C₂²×C₂₃	1	6	yes	complemented
318	#463	C₂²×C₂₃	1	12	yes	complemented
319	#462	C₂²×C₂₃	1	1	yes	characteristic, complemented
320	#461	C₂²×C₂₃	1	3	yes	complemented
321	#460	C₂²×C₂₃	1	3	yes	complemented
322	#460	C₂²×C₂₃	1	6	yes	complemented
323	#459	C₂²×C₂₃	1	3	yes	complemented
324	#459	C₂²×C₂₃	1	6	yes	complemented
325	#459	C₂²×C₂₃	1	12	yes	complemented
326	#459	C₂²×C₂₃	1	6	yes	complemented
327	#424	C₂²×C₂₃	1	6	yes	complemented
328	#423	C₂²×C₂₃	1	3	yes	complemented
329	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	complemented
330	#418	C₂²×C₂₃	1	6	yes	complemented
331	#417	C₂²×C₂₃	1	6	yes	complemented
332	#416	C₂²×C₂₃	1	6	yes	complemented
333	C₂×#102	C₂²×C₂₃	1	6	yes	complemented
334	C₂×C₂₃	—	2	6	no	—
335	C₂×C₂₃	—	4	6	no	—
336	C₂×C₂₃	—	8	6	no	—
337	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
338	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
339	C₂×C₂₃	#3585	1	1	yes	characteristic
340	H(5,𝔽₂)	—	2	4	no	—
341	H(5,𝔽₂)	—	2	8	no	—
342	H(5,𝔽₂)	—	2	8	no	—
343	H(5,𝔽₂)	D₄×C₂₃	1	8	yes	complemented
344	C₂×#41	—	2	1	no	—
345	C₂×#41	—	2	2	no	—
346	C₂×#41	C₂³×C₂₃	1	6	yes	—
347	C₂×#41	C₂³×C₂₃	1	3	yes	—
348	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
349	C₂²×D₄	—	2	3	no	—
350	C₂²×D₄	—	2	6	no	—
351	C₂²×D₄	—	2	3	no	—
352	C₂²×D₄	—	2	3	no	—
353	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
354	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
355	C₂³×C₄	—	2	3	no	—
356	AGL(1,ℤ/8)	—	2	24	no	—
357	AGL(1,ℤ/8)	—	2	24	no	—
358	#135	—	2	24	no	—
359	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
360	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
361	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
362	C₂×D₈	—	2	24	no	—
363	C₂×D₈	C₂³×C₂₃	1	6	yes	—
364	C₂×D₈	C₂³×C₂₃	1	12	yes	—
365	#131	—	2	8	no	—
366	C₂×M₁₆	C₂³×C₂₃	1	6	yes	—
367	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
368	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
369	#127	—	2	3	no	—
370	#127	—	2	3	no	—
371	#127	C₂³×C₂₃	1	3	yes	—
372	#127	C₂³×C₂₃	1	6	yes	—
373	#124	C₂³×C₂₃	1	6	yes	—
374	#121	—	2	3	no	—
375	#121	—	2	3	no	—
376	#121	—	2	6	no	—
377	#121	—	2	6	no	—
378	#120	—	2	12	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
381	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
382	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
384	C₄×D₄	—	2	3	no	—
385	C₄×D₄	—	2	3	no	—
386	C₄×D₄	—	2	3	no	—
387	C₄×D₄	—	2	3	no	—
388	C₄×D₄	—	2	3	no	—
389	C₄×D₄	—	2	6	no	—
390	C₄×D₄	—	2	3	no	—
391	C₄×D₄	—	2	3	no	—
392	C₄×D₄	—	2	3	no	—
393	C₄×D₄	—	2	3	no	—
394	C₄×D₄	D₄×C₂₃	1	3	yes	—
395	C₄×D₄	D₄×C₂₃	1	3	yes	—
396	C₄×D₄	C₂³×C₂₃	1	3	yes	—
397	C₄×D₄	C₂³×C₂₃	1	6	yes	—
398	C₄×D₄	C₂³×C₂₃	1	3	yes	—
399	#117	—	2	3	no	—
400	#117	—	2	3	no	—
401	#117	C₂³×C₂₃	1	3	yes	—
402	#117	C₂³×C₂₃	1	3	yes	—
403	C₂×C₄⋊C₄	—	2	3	no	—
404	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
405	C₂×C₂²⋊C₄	—	2	3	no	—
406	C₂×C₂²⋊C₄	—	2	6	no	—
407	C₂×C₄²	—	2	3	no	—
408	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
409	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
411	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
412	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
413	#103	C₂³×C₂₃	1	2	yes	—
414	#103	C₂³×C₂₃	1	6	yes	—
415	#102	C₂³×C₂₃	1	6	yes	—
416	#102	C₂³×C₂₃	1	12	yes	—
417	#102	C₂³×C₂₃	1	6	yes	—
418	C₂₃	#60428	1	1	yes	characteristic, direct product factor, Sylow
419	C₂⁴	—	2	3	no	—
420	C₂⁴	—	2	6	no	—
421	#41	—	2	8	no	—
422	#41	—	2	6	no	—
423	#41	—	2	12	no	—
424	#41	—	2	12	no	—
425	#41	—	4	8	no	—
426	#41	C₂×D₄×C₂₃	1	4	yes	—
427	#41	C₂×D₄×C₂₃	1	12	yes	complemented
428	#41	#41×C₂₃	1	4	yes	—
429	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
430	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
431	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
432	C₂×D₄	—	2	3	no	—
433	C₂×D₄	—	2	24	no	—
434	C₂×D₄	—	2	6	no	—
435	C₂×D₄	—	2	3	no	—
436	C₂×D₄	—	2	12	no	—
437	C₂×D₄	—	2	3	no	—
438	C₂×D₄	—	2	3	no	—
439	C₂×D₄	—	2	6	no	—
440	C₂×D₄	—	2	6	no	—
441	C₂×D₄	—	2	6	no	—
442	C₂×D₄	—	2	6	no	—
443	C₂×D₄	—	2	12	no	—
444	C₂×D₄	—	4	24	no	—
445	C₂×D₄	—	8	6	no	—
446	C₂×D₄	—	8	6	no	—
447	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
448	C₂×D₄	C₂×D₄×C₂₃	1	12	yes	complemented
449	C₂×D₄	C₂⁴×C₂₃	1	6	yes	—
450	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
451	C₂²×C₄	—	2	3	no	—
452	C₂²×C₄	—	2	3	no	—
453	C₂²×C₄	—	2	3	no	—
454	C₂²×C₄	—	2	6	no	—
455	C₂²×C₄	—	2	3	no	—
456	C₂²×C₄	—	2	6	no	—
457	C₂²×C₄	—	8	6	no	—
458	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
459	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
460	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
461	Q₁₆	—	2	4	no	—
462	SD₁₆	—	2	12	no	—
463	SD₁₆	—	2	12	no	—
464	D₈	—	2	12	no	—
465	D₈	—	2	24	no	—
466	M₁₆	—	2	12	no	—
467	C₂×C₈	—	2	12	no	—
468	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
469	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
470	C₄⋊C₄	—	2	3	no	—
471	C₄⋊C₄	—	2	3	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
476	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
477	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
478	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
479	C₂²⋊C₄	—	2	3	no	—
480	C₂²⋊C₄	—	2	3	no	—
481	C₂²⋊C₄	—	2	6	no	—
482	C₂²⋊C₄	—	2	3	no	—
483	C₂²⋊C₄	—	2	6	no	—
484	C₂²⋊C₄	—	2	6	no	—
485	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	3	no	—
488	C₄²	—	2	3	no	—
489	C₄²	C₂×D₄×C₂₃	1	3	yes	—
490	C₄²	C₂×D₄×C₂₃	1	3	yes	—
491	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
492	C₄²	C₂⁴×C₂₃	1	3	yes	—
493	C₂³	—	2	3	no	—
494	C₂³	—	2	12	no	—
495	C₂³	—	2	6	no	—
496	C₂³	—	4	12	no	—
497	C₂³	—	8	3	no	—
498	C₂³	—	8	3	no	—
499	C₂³	—	8	6	no	—
500	C₂³	—	8	6	no	—
501	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
502	Q₈	—	2	1	no	—
503	Q₈	—	2	2	no	—
504	Q₈	—	2	2	no	—
505	Q₈	C₂₃×#121	1	4	yes	—
506	Q₈	C₂²×D₄×C₂₃	1	2	yes	—
507	D₄	—	2	12	no	—
508	D₄	—	2	6	no	—
509	D₄	—	2	3	no	—
510	D₄	—	2	6	no	—
511	D₄	—	2	12	no	—
512	D₄	—	4	6	no	—
513	D₄	—	4	12	no	—
514	D₄	—	4	6	no	—
515	D₄	—	4	12	no	—
516	D₄	—	8	6	no	—
517	D₄	—	8	6	no	—
518	D₄	C₂₃×#121	1	12	yes	—
519	D₄	C₂²×D₄×C₂₃	1	6	yes	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	6	no	—
522	C₂×C₄	—	2	12	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	3	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	4	3	no	—
527	C₂×C₄	—	4	6	no	—
528	C₂×C₄	—	4	12	no	—
529	C₂×C₄	—	8	6	no	—
530	C₂×C₄	—	8	3	no	—
531	C₂×C₄	—	8	3	no	—
532	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
533	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
534	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
535	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
537	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
538	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
539	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
540	C₈	—	2	2	no	—
541	C₈	—	2	6	no	—
542	C₂²	—	2	3	no	—
543	C₂²	—	2	3	no	—
544	C₂²	—	2	6	no	—
545	C₂²	—	4	3	no	—
546	C₂²	—	4	3	no	—
547	C₂²	—	4	6	no	—
548	C₂²	—	4	6	no	—
549	C₂²	—	4	6	no	—
550	C₂²	—	8	12	no	—
551	C₂²	—	16	12	no	—
552	C₂²	C₂×C₂₃×#121	1	6	yes	—
553	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
554	C₄	—	2	2	no	—
555	C₄	—	2	3	no	—
556	C₄	—	2	1	no	—
557	C₄	—	2	3	no	—
558	C₄	—	4	2	no	—
559	C₄	—	4	3	no	—
560	C₄	—	4	1	no	—
561	C₄	C₂×C₂₃×#121	1	6	yes	—
562	C₄	C₂₃×#576	1	2	yes	complemented
563	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
564	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
565	C₂	—	2	6	no	—
566	C₂	—	4	6	no	—
567	C₂	—	8	6	no	—
568	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
569	C₂	C₂×C₂₃×#576	1	1	yes	characteristic
570	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#576	C₂	semidirect
2. C₂₃×#3498	C₂	semidirect
3. C₄×C₂₃×H(5,𝔽₂)	C₂	semidirect
4. C₂₃×#3045	C₂	semidirect
5. C₂₃×#3041	C₂	semidirect
6. C₂₃×#3038	C₂	semidirect
7. C₂₃×#3036	C₂	semidirect
8. C₂₃×#3035	C₂	semidirect
9. C₂×C₂₃×#459	C₂	semidirect
10. C₂₃×#2906	C₂	semidirect
11. C₂₃×#2897	C₂	semidirect
15. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×C₂₃×#135	C₂²	semidirect
18. C₂²×D₈×C₂₃	C₂²	semidirect
24. C₂₃×#535	C₂²	semidirect
25. C₂₃×#532	C₂²	semidirect
26. C₂×C₂₃×#127	C₂²	semidirect
29. C₂₃×#520	C₂²	semidirect
31. C₂₃×#518	C₂²	semidirect
34. C₄×#41×C₂₃	C₂²	semidirect
36. C₂×C₄×D₄×C₂₃	C₂²	semidirect
37. C₂₃×#464	C₂²	semidirect
38. C₂₃×#463	C₂²	semidirect
39. C₂₃×#463	C₂²	semidirect
40. C₂₃×#462	C₂²	semidirect
41. C₂₃×#461	C₂²	semidirect
42. C₂₃×#460	C₂²	semidirect
43. C₂₃×#460	C₂²	semidirect
44. C₂₃×#459	C₂²	semidirect
45. C₂₃×#459	C₂²	semidirect
46. C₂₃×#459	C₂²	semidirect
47. C₂₃×#459	C₂²	semidirect
48. C₂₃×#424	C₂²	semidirect
49. C₂₃×#423	C₂²	semidirect
50. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
51. C₂₃×#418	C₂²	semidirect
52. C₂₃×#417	C₂²	semidirect
53. C₂₃×#416	C₂²	semidirect
54. C₂×C₂₃×#102	C₂²	semidirect
58. C₂₃×H(5,𝔽₂)	D₄	semidirect
141. #41×C₂₃	C₂×D₄	semidirect
162. C₂×D₄×C₂₃	C₂×D₄	semidirect
207. #60428	C₂₃	direct (N × H)
257. C₂×#576	C₂×C₂₃	semidirect
258. #3498	C₂×C₂₃	semidirect
259. C₄×H(5,𝔽₂)	C₂×C₂₃	semidirect
260. #3045	C₂×C₂₃	semidirect
261. #3041	C₂×C₂₃	semidirect
262. #3038	C₂×C₂₃	semidirect
263. #3036	C₂×C₂₃	semidirect
264. #3035	C₂×C₂₃	semidirect
265. C₂×#459	C₂×C₂₃	semidirect
266. #2906	C₂×C₂₃	semidirect
267. #2897	C₂×C₂₃	semidirect
288. C₄×C₂₃	#576	semidirect
294. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
295. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
296. C₂×#135	C₂²×C₂₃	semidirect
297. C₂²×D₈	C₂²×C₂₃	semidirect
303. #535	C₂²×C₂₃	semidirect
304. #532	C₂²×C₂₃	semidirect
305. C₂×#127	C₂²×C₂₃	semidirect
308. #520	C₂²×C₂₃	semidirect
310. #518	C₂²×C₂₃	semidirect
313. C₄×#41	C₂²×C₂₃	semidirect
315. C₂×C₄×D₄	C₂²×C₂₃	semidirect
316. #464	C₂²×C₂₃	semidirect
317. #463	C₂²×C₂₃	semidirect
318. #463	C₂²×C₂₃	semidirect
319. #462	C₂²×C₂₃	semidirect
320. #461	C₂²×C₂₃	semidirect
321. #460	C₂²×C₂₃	semidirect
322. #460	C₂²×C₂₃	semidirect
323. #459	C₂²×C₂₃	semidirect
324. #459	C₂²×C₂₃	semidirect
325. #459	C₂²×C₂₃	semidirect
326. #459	C₂²×C₂₃	semidirect
327. #424	C₂²×C₂₃	semidirect
328. #423	C₂²×C₂₃	semidirect
329. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
330. #418	C₂²×C₂₃	semidirect
331. #417	C₂²×C₂₃	semidirect
332. #416	C₂²×C₂₃	semidirect
333. C₂×#102	C₂²×C₂₃	semidirect
343. H(5,𝔽₂)	D₄×C₂₃	semidirect
418. C₂₃	#60428	direct (N × H)
427. #41	C₂×D₄×C₂₃	semidirect
448. C₂×D₄	C₂×D₄×C₂₃	semidirect
562. C₄	C₂₃×#576	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#576	2
C₂₃×#3498	2
C₄×C₂₃×H(5,𝔽₂)	2
C₂₃×#3045	2
C₂₃×#3041	2
C₂₃×#3038	2
C₂₃×#3036	2
C₂₃×#3035	2
C₂×C₂₃×#459	2
C₂₃×#2906	2
C₂₃×#2897	2
#60428	23

Subgroups — C23×#60428 #19208054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60428 #19208054