Subgroups — C₂₃×#23003 #19170629

Order 5888.

1054 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#460	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#440	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2366	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1957	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1936	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1880	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1862	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₂₃	—	2	1	no	—
17	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#460	C₄	1	8	yes	complemented
19	C₂₃×#440	—	2	8	no	—
20	C₂×C₂₃×#107	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#102	C₂²	1	2	yes	—
25	C₂×C₂₃×#102	C₂²	1	2	yes	—
26	C₂×C₂₃×#102	C₂²	1	2	yes	—
27	C₂×C₂₃×#97	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#390	—	2	1	no	—
30	C₂₃×#390	C₂²	1	2	yes	—
31	C₂₃×#389	—	2	2	no	—
32	C₂₃×#389	C₂²	1	2	yes	—
33	C₂₃×#386	—	2	2	no	—
34	C₂₃×#384	—	2	1	no	—
35	C₂₃×#384	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
37	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#340	C₂²	1	1	yes	characteristic
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#340	C₂²	1	1	yes	characteristic
42	C₂₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₂₃	—	2	1	no	—
45	C₂²×D₄×C₂₃	—	2	1	no	—
46	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
47	C₂³×C₄×C₂₃	—	2	2	no	—
48	C₂×SD₁₆×C₂₃	—	2	4	no	—
49	C₂×SD₁₆×C₂₃	—	2	4	no	—
50	C₂×SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
51	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂₃×#127	—	2	2	no	—
55	C₂₃×#127	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₂₃	—	2	2	no	—
57	C₄×Q₈×C₂₃	—	2	4	no	—
58	C₄×Q₈×C₂₃	—	4	2	no	—
59	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
60	C₄×D₄×C₂₃	—	2	4	no	—
61	C₄×D₄×C₂₃	—	4	2	no	—
62	C₄×D₄×C₂₃	—	4	4	no	—
63	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
66	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₄²×C₂₃	—	2	1	no	—
78	C₂×C₄²×C₂₃	—	2	2	no	—
79	C₂×C₄²×C₂₃	—	2	1	no	—
80	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
81	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₂₃×#107	—	2	4	no	—
83	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
84	C₂₃×#103	—	2	2	no	—
85	C₂₃×#103	D₄	1	4	yes	complemented
86	C₂₃×#102	—	2	4	no	—
87	C₂₃×#102	—	4	2	no	—
88	C₂₃×#102	C₂×C₄	1	8	yes	complemented
89	C₂₃×#97	—	2	4	no	—
90	C₄×C₈×C₂₃	—	2	2	no	—
91	C₂₃×#95	—	2	2	no	—
92	C₂₃×#95	—	2	1	no	—
93	C₂₃×#95	D₄	1	2	yes	—
94	C₂⁴×C₂₃	—	2	2	no	—
95	C₂×Q₈×C₂₃	—	2	1	no	—
96	C₂×Q₈×C₂₃	—	2	1	no	—
97	C₂×Q₈×C₂₃	—	2	1	no	—
98	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
99	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₂₃	#41	1	2	yes	—
101	C₂×D₄×C₂₃	—	2	4	no	—
102	C₂×D₄×C₂₃	—	2	4	no	—
103	C₂×D₄×C₂₃	—	2	2	no	—
104	C₂×D₄×C₂₃	—	2	2	no	—
105	C₂×D₄×C₂₃	—	2	2	no	—
106	C₂×D₄×C₂₃	—	2	2	no	—
107	C₂×D₄×C₂₃	—	2	2	no	—
108	C₂×D₄×C₂₃	—	4	2	no	—
109	C₂×D₄×C₂₃	—	4	2	no	—
110	C₂×D₄×C₂₃	—	8	1	no	—
111	C₂×D₄×C₂₃	—	8	1	no	—
112	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
113	C₂²×C₄×C₂₃	—	2	1	no	—
114	C₂²×C₄×C₂₃	—	2	1	no	—
115	C₂²×C₄×C₂₃	—	2	1	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	2	no	—
118	C₂²×C₄×C₂₃	—	4	4	no	—
119	C₂²×C₄×C₂₃	—	4	4	no	—
120	C₂²×C₄×C₂₃	—	8	2	no	—
121	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
126	SD₁₆×C₂₃	—	2	16	no	—
127	C₂×C₈×C₂₃	—	2	2	no	—
128	C₂×C₈×C₂₃	—	2	2	no	—
129	C₂×C₈×C₂₃	—	2	1	no	—
130	C₂×C₈×C₂₃	—	2	1	no	—
131	C₂×C₈×C₂₃	—	2	1	no	—
132	C₂×C₈×C₂₃	—	2	1	no	—
133	C₂×C₈×C₂₃	—	2	1	no	—
134	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
135	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₂₃	—	2	4	no	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	—	2	2	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	2	1	no	—
141	C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₄⋊C₄×C₂₃	—	2	1	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₄⋊C₄×C₂₃	—	4	4	no	—
146	C₄⋊C₄×C₂₃	—	4	2	no	—
147	C₄⋊C₄×C₂₃	—	8	1	no	—
148	C₄⋊C₄×C₂₃	—	8	2	no	—
149	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
150	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₂₃	—	4	4	no	—
152	C₂²⋊C₄×C₂₃	—	4	4	no	—
153	C₂²⋊C₄×C₂₃	—	8	2	no	—
154	C₄²×C₂₃	—	2	4	no	—
155	C₄²×C₂₃	—	2	2	no	—
156	C₄²×C₂₃	—	2	2	no	—
157	C₄²×C₂₃	—	4	4	no	—
158	C₄²×C₂₃	—	4	2	no	—
159	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
160	C₄²×C₂₃	#41	1	2	yes	—
161	#23003	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₂₃	—	2	4	no	—
163	C₂³×C₂₃	—	4	2	no	—
164	C₂³×C₂₃	—	4	2	no	—
165	C₂³×C₂₃	—	4	2	no	—
166	C₂³×C₂₃	—	4	2	no	—
167	C₂³×C₂₃	—	4	2	no	—
168	C₂³×C₂₃	—	4	2	no	—
169	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
170	Q₈×C₂₃	—	2	2	no	—
171	Q₈×C₂₃	—	2	2	no	—
172	Q₈×C₂₃	—	2	2	no	—
173	Q₈×C₂₃	C₄×D₄	1	4	yes	complemented
174	D₄×C₂₃	—	2	4	no	—
175	D₄×C₂₃	—	2	4	no	—
176	D₄×C₂₃	—	4	2	no	—
177	D₄×C₂₃	—	4	2	no	—
178	D₄×C₂₃	—	4	8	no	—
179	D₄×C₂₃	—	8	4	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	1	no	—
184	C₂×C₄×C₂₃	—	2	1	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	2	no	—
189	C₂×C₄×C₂₃	—	2	2	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	4	1	no	—
194	C₂×C₄×C₂₃	—	4	1	no	—
195	C₂×C₄×C₂₃	—	4	1	no	—
196	C₂×C₄×C₂₃	—	4	1	no	—
197	C₂×C₄×C₂₃	—	4	2	no	—
198	C₂×C₄×C₂₃	—	4	1	no	—
199	C₂×C₄×C₂₃	—	4	1	no	—
200	C₂×C₄×C₂₃	—	4	2	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	8	8	no	—
204	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
205	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
207	C₂×C₄×C₂₃	#137	1	2	yes	—
208	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
214	C₈×C₂₃	—	2	4	no	—
215	C₈×C₂₃	—	2	2	no	—
216	C₈×C₂₃	—	4	1	no	—
217	C₂×#460	C₂×C₂₃	1	1	yes	characteristic
218	C₂×#440	C₂×C₂₃	1	2	yes	complemented
219	#2366	C₂×C₂₃	1	1	yes	characteristic
220	Q₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#1957	C₂×C₂₃	1	1	yes	characteristic
222	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
223	#1936	C₂×C₂₃	1	2	yes	complemented
224	#1880	C₂×C₂₃	1	2	yes	complemented
225	#1862	C₂×C₂₃	1	2	yes	complemented
226	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
227	C₄×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂²×C₂₃	—	4	4	no	—
229	C₂²×C₂₃	—	4	4	no	—
230	C₂²×C₂₃	—	4	4	no	—
231	C₂²×C₂₃	—	8	2	no	—
232	C₂²×C₂₃	—	8	2	no	—
233	C₂²×C₂₃	—	8	2	no	—
234	C₂²×C₂₃	—	8	2	no	—
235	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂²×C₂₃	D₄²	1	1	yes	characteristic
237	C₂²×C₂₃	#548	1	1	yes	characteristic
238	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
240	C₂²×C₂₃	#576	1	1	yes	characteristic
241	C₂²×C₂₃	#577	1	1	yes	characteristic
242	C₄×C₂₃	—	2	4	no	—
243	C₄×C₂₃	—	2	2	no	—
244	C₄×C₂₃	—	4	2	no	—
245	C₄×C₂₃	—	4	2	no	—
246	C₄×C₂₃	—	4	2	no	—
247	C₄×C₂₃	—	4	2	no	—
248	C₄×C₂₃	—	8	1	no	—
249	C₄×C₂₃	—	8	2	no	—
250	C₄×C₂₃	#440	1	4	yes	complemented
251	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
252	C₄×C₂₃	#520	1	1	yes	characteristic
253	C₄×C₂₃	#547	1	1	yes	characteristic
254	C₄×C₂₃	#553	1	1	yes	characteristic
255	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
256	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
258	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×C₄×D₄	—	2	1	no	—
260	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
261	#460	C₄×C₂₃	1	8	yes	complemented
262	#440	—	2	8	no	—
263	C₂×#107	C₂²×C₂₃	1	2	yes	—
264	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
267	C₂×#102	C₂²×C₂₃	1	2	yes	—
268	C₂×#102	C₂²×C₂₃	1	2	yes	—
269	C₂×#102	C₂²×C₂₃	1	2	yes	—
270	C₂×#97	C₂²×C₂₃	1	2	yes	—
271	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
272	#390	—	2	1	no	—
273	#390	C₂²×C₂₃	1	2	yes	—
274	#389	—	2	2	no	—
275	#389	C₂²×C₂₃	1	2	yes	—
276	#386	—	2	2	no	—
277	#384	—	2	1	no	—
278	#384	C₂²×C₂₃	1	1	yes	characteristic
279	C₄×C₄⋊C₄	—	2	1	no	—
280	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
281	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
282	#340	C₂²×C₂₃	1	1	yes	characteristic
283	#340	C₂²×C₂₃	1	2	yes	—
284	#340	C₂²×C₂₃	1	1	yes	characteristic
285	#336	C₂²×C₂₃	1	2	yes	—
286	C₂×C₂₃	—	8	4	no	—
287	C₂×C₂₃	#2361	1	1	yes	characteristic
288	C₂×C₂₃	C₂×#440	1	1	yes	characteristic
289	C₂×C₂₃	#2946	1	1	yes	characteristic
290	C₂×C₂₃	#3274	1	1	yes	characteristic
291	C₂×C₂₃	#3278	1	1	yes	characteristic
292	C₂×C₂₃	#3368	1	1	yes	characteristic
293	C₂×C₂₃	#3375	1	1	yes	characteristic
294	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
295	C₂²×D₄	—	2	1	no	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
298	C₂³×C₄	—	2	2	no	—
299	C₂×SD₁₆	—	2	4	no	—
300	C₂×SD₁₆	—	2	4	no	—
301	C₂×SD₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
302	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
304	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
305	#127	—	2	2	no	—
306	#127	C₂×C₄×C₂₃	1	4	yes	—
307	C₄×Q₈	—	2	2	no	—
308	C₄×Q₈	—	2	4	no	—
309	C₄×Q₈	—	4	2	no	—
310	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
311	C₄×D₄	—	2	4	no	—
312	C₄×D₄	—	4	2	no	—
313	C₄×D₄	—	4	4	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
320	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
321	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
322	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	—	2	2	no	—
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	—	2	2	no	—
331	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
332	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
333	#107	—	2	4	no	—
334	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
335	#103	—	2	2	no	—
336	#103	D₄×C₂₃	1	4	yes	complemented
337	#102	—	2	4	no	—
338	#102	—	4	2	no	—
339	#102	C₂×C₄×C₂₃	1	8	yes	complemented
340	#97	—	2	4	no	—
341	C₄×C₈	—	2	2	no	—
342	#95	—	2	2	no	—
343	#95	—	2	1	no	—
344	#95	D₄×C₂₃	1	2	yes	—
345	C₂₃	#23003	1	1	yes	characteristic, direct product factor, Sylow
346	C₂⁴	—	2	2	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
351	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
352	C₂×Q₈	#41×C₂₃	1	2	yes	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	4	2	no	—
361	C₂×D₄	—	4	2	no	—
362	C₂×D₄	—	8	1	no	—
363	C₂×D₄	—	8	1	no	—
364	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	4	4	no	—
371	C₂²×C₄	—	4	4	no	—
372	C₂²×C₄	—	8	2	no	—
373	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
374	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
375	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
378	SD₁₆	—	2	16	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
387	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	4	4	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	—	8	1	no	—
400	C₄⋊C₄	—	8	2	no	—
401	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
402	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	—	4	4	no	—
405	C₂²⋊C₄	—	8	2	no	—
406	C₄²	—	2	4	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	4	4	no	—
410	C₄²	—	4	2	no	—
411	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
412	C₄²	#41×C₂₃	1	2	yes	—
413	C₂³	—	2	4	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
421	Q₈	—	2	2	no	—
422	Q₈	—	2	2	no	—
423	Q₈	—	2	2	no	—
424	Q₈	C₄×D₄×C₂₃	1	4	yes	complemented
425	D₄	—	2	4	no	—
426	D₄	—	2	4	no	—
427	D₄	—	4	2	no	—
428	D₄	—	4	2	no	—
429	D₄	—	4	8	no	—
430	D₄	—	8	4	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	8	no	—
455	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
456	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
457	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
458	C₂×C₄	C₂₃×#137	1	2	yes	—
459	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
463	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
464	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
465	C₈	—	2	2	no	—
466	C₈	—	2	4	no	—
467	C₈	—	4	1	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	8	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
476	C₂²	D₄²×C₂₃	1	1	yes	characteristic
477	C₂²	C₂₃×#548	1	1	yes	characteristic
478	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
479	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
480	C₂²	C₂₃×#576	1	1	yes	characteristic
481	C₂²	C₂₃×#577	1	1	yes	characteristic
482	C₄	—	2	2	no	—
483	C₄	—	2	4	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	2	no	—
488	C₄	—	8	2	no	—
489	C₄	—	8	1	no	—
490	C₄	C₂₃×#440	1	4	yes	complemented
491	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
492	C₄	C₂₃×#520	1	1	yes	characteristic
493	C₄	C₂₃×#547	1	1	yes	characteristic
494	C₄	C₂₃×#553	1	1	yes	characteristic
495	C₂	—	8	4	no	—
496	C₂	C₂₃×#2361	1	1	yes	characteristic
497	C₂	C₂×C₂₃×#440	1	1	yes	characteristic
498	C₂	C₂₃×#2946	1	1	yes	characteristic
499	C₂	C₂₃×#3274	1	1	yes	characteristic
500	C₂	C₂₃×#3278	1	1	yes	characteristic
501	C₂	C₂₃×#3368	1	1	yes	characteristic
502	C₂	C₂₃×#3375	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#440	C₂	semidirect
4. Q₈×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#1947	C₂	semidirect
7. C₂₃×#1936	C₂	semidirect
8. C₂₃×#1880	C₂	semidirect
9. C₂₃×#1862	C₂	semidirect
10. C₂₃×#1770	C₂	semidirect
11. C₄×C₂₃×#103	C₂	semidirect
18. C₂₃×#460	C₄	semidirect
50. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
59. C₄×Q₈×C₂₃	C₂×C₄	semidirect
83. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
85. C₂₃×#103	D₄	semidirect
88. C₂₃×#102	C₂×C₄	semidirect
161. #23003	C₂₃	direct (N × H)
173. Q₈×C₂₃	C₄×D₄	semidirect
218. C₂×#440	C₂×C₂₃	semidirect
220. Q₈×C₄⋊C₄	C₂×C₂₃	semidirect
222. #1947	C₂×C₂₃	semidirect
223. #1936	C₂×C₂₃	semidirect
224. #1880	C₂×C₂₃	semidirect
225. #1862	C₂×C₂₃	semidirect
226. #1770	C₂×C₂₃	semidirect
227. C₄×#103	C₂×C₂₃	semidirect
250. C₄×C₂₃	#440	semidirect
261. #460	C₄×C₂₃	semidirect
301. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
310. C₄×Q₈	C₂×C₄×C₂₃	semidirect
334. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
336. #103	D₄×C₂₃	semidirect
339. #102	C₂×C₄×C₂₃	semidirect
345. C₂₃	#23003	direct (N × H)
424. Q₈	C₄×D₄×C₂₃	semidirect
490. C₄	C₂₃×#440	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#460	2
C₂×C₂₃×#440	2
C₂₃×#2366	2
Q₈×C₄⋊C₄×C₂₃	2
C₂₃×#1957	2
C₂₃×#1947	2
C₂₃×#1936	2
C₂₃×#1880	2
C₂₃×#1862	2
C₂₃×#1770	2
C₄×C₂₃×#103	2
#23003	23

Subgroups — C23×#23003 #19170629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23003 #19170629