Subgroups — C₂₃×#24797 #19172423

Order 5888.

894 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#472	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3058	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3045	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2913	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2063	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2037	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1970	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1883	C₂	1	2	yes	maximal, complemented
11	C₂₃×#1816	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
17	C₂₃×#535	C₂²	1	2	yes	—
18	C₂₃×#520	C₂²	1	2	yes	—
19	C₂₃×#472	—	2	4	no	—
20	C₂₃×#469	—	2	2	no	—
21	C₂₃×#466	—	2	1	no	—
22	C₂₃×#466	—	2	1	no	—
23	C₂₃×#464	—	2	2	no	—
24	C₂₃×#463	—	2	2	no	—
25	C₂₃×#460	—	2	2	no	—
26	C₂₃×#459	—	2	2	no	—
27	C₂₃×#443	—	2	4	no	—
28	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
30	C₂₃×#428	—	2	2	no	—
31	C₂₃×#428	C₂²	1	2	yes	—
32	C₂₃×#427	—	2	1	no	—
33	C₂₃×#427	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#107	—	2	1	no	—
35	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
36	C₂₃×#424	C₂²	1	2	yes	—
37	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
39	C₂₃×#418	C₂²	1	2	yes	—
40	C₂₃×#417	C₂²	1	2	yes	—
41	C₂×C₂₃×#102	C₂²	1	2	yes	—
42	C₂×C₂₃×#100	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
44	C₂₃×#390	C₂²	1	2	yes	—
45	C₂₃×#343	C₂²	1	2	yes	—
46	C₂₃×#339	C₂²	1	2	yes	—
47	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
50	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
51	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
52	C₂₃×#135	—	2	2	no	—
53	C₂₃×#135	—	2	2	no	—
54	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₂₃	—	2	2	no	—
56	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
57	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
58	C₂×D₈×C₂₃	—	2	2	no	—
59	C₂×D₈×C₂₃	—	2	1	no	—
60	C₂×D₈×C₂₃	—	2	1	no	—
61	C₂×D₈×C₂₃	—	2	2	no	—
62	C₂×D₈×C₂₃	—	2	2	no	—
63	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
64	C₂₃×#131	—	2	2	no	—
65	C₂₃×#131	—	2	2	no	—
66	C₂₃×#131	—	2	2	no	—
67	C₂₃×#131	D₄	1	4	yes	complemented
68	C₂×M₁₆×C₂₃	—	2	1	no	—
69	C₂×M₁₆×C₂₃	—	2	1	no	—
70	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
72	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	—	2	1	no	—
74	C₂²×C₈×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂₃×#127	—	2	2	no	—
78	C₂₃×#124	—	2	2	no	—
79	C₂₃×#121	—	2	2	no	—
80	C₂₃×#121	—	2	2	no	—
81	C₂₃×#120	—	4	2	no	—
82	C₄×Q₈×C₂₃	—	2	2	no	—
83	C₄×D₄×C₂₃	—	2	2	no	—
84	C₄×D₄×C₂₃	—	4	2	no	—
85	C₂₃×#117	—	2	2	no	—
86	C₂₃×#117	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
89	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂₃×#108	—	2	2	no	—
92	C₂₃×#108	—	2	2	no	—
93	C₂₃×#107	—	2	1	no	—
94	C₂₃×#107	—	2	1	no	—
95	C₂₃×#107	—	2	2	no	—
96	C₂₃×#107	D₄	1	1	yes	characteristic
97	C₂₃×#107	D₄	1	1	yes	characteristic
98	C₂₃×#106	—	2	2	no	—
99	C₂₃×#106	D₄	1	2	yes	—
100	C₂₃×C₄⋊C₈	—	2	2	no	—
101	C₂₃×C₄⋊C₈	—	2	2	no	—
102	C₂₃×#104	—	2	2	no	—
103	C₂₃×#104	—	2	2	no	—
104	C₂₃×#103	—	2	2	no	—
105	C₂₃×#102	—	2	2	no	—
106	C₂₃×#102	—	2	2	no	—
107	C₂₃×#102	—	2	2	no	—
108	C₂₃×#102	—	2	2	no	—
109	C₂₃×#102	—	2	2	no	—
110	C₂₃×#100	—	2	4	no	—
111	C₄×C₈×C₂₃	—	2	2	no	—
112	C₂⁴×C₂₃	—	2	2	no	—
113	#41×C₂₃	—	2	1	no	—
114	#41×C₂₃	—	2	1	no	—
115	#41×C₂₃	—	2	1	no	—
116	#41×C₂₃	—	2	1	no	—
117	#41×C₂₃	—	2	1	no	—
118	#41×C₂₃	—	2	1	no	—
119	#41×C₂₃	C₂×D₄	1	2	yes	—
120	#41×C₂₃	#41	1	2	yes	—
121	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂×D₄×C₂₃	—	2	1	no	—
124	C₂×D₄×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	2	2	no	—
126	C₂×D₄×C₂₃	—	2	2	no	—
127	C₂×D₄×C₂₃	—	2	2	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	2	2	no	—
130	C₂×D₄×C₂₃	—	2	2	no	—
131	C₂×D₄×C₂₃	—	4	2	no	—
132	C₂×D₄×C₂₃	—	4	2	no	—
133	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	4	2	no	—
140	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₂₃	—	2	2	no	—
142	SD₁₆×C₂₃	—	2	2	no	—
143	SD₁₆×C₂₃	—	2	2	no	—
144	SD₁₆×C₂₃	—	4	4	no	—
145	D₈×C₂₃	—	2	2	no	—
146	D₈×C₂₃	—	4	4	no	—
147	D₈×C₂₃	—	4	2	no	—
148	D₈×C₂₃	—	4	2	no	—
149	M₁₆×C₂₃	—	2	1	no	—
150	M₁₆×C₂₃	—	2	4	no	—
151	M₁₆×C₂₃	—	2	2	no	—
152	M₁₆×C₂₃	—	2	1	no	—
153	M₁₆×C₂₃	—	2	1	no	—
154	M₁₆×C₂₃	—	2	2	no	—
155	M₁₆×C₂₃	—	4	2	no	—
156	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	4	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	2	no	—
163	C₂×C₈×C₂₃	—	4	2	no	—
164	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
169	C₂×C₈×C₂₃	#41	1	2	yes	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	2	no	—
172	C₄⋊C₄×C₂₃	—	4	2	no	—
173	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
175	C₂²⋊C₄×C₂₃	—	2	2	no	—
176	C₂²⋊C₄×C₂₃	—	4	2	no	—
177	C₂²⋊C₄×C₂₃	—	4	2	no	—
178	C₂²⋊C₄×C₂₃	—	4	2	no	—
179	C₂²⋊C₄×C₂₃	—	8	4	no	—
180	C₄²×C₂₃	—	2	2	no	—
181	C₄²×C₂₃	—	2	1	no	—
182	C₄²×C₂₃	—	2	1	no	—
183	C₄²×C₂₃	—	4	2	no	—
184	#24797	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₂₃	—	2	1	no	—
186	C₂³×C₂₃	—	2	1	no	—
187	C₂³×C₂₃	—	4	2	no	—
188	C₂³×C₂₃	—	4	2	no	—
189	C₂³×C₂₃	—	4	2	no	—
190	C₂³×C₂₃	—	8	2	no	—
191	C₂³×C₂₃	—	8	2	no	—
192	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
193	Q₈×C₂₃	—	2	1	no	—
194	Q₈×C₂₃	—	2	1	no	—
195	Q₈×C₂₃	—	2	1	no	—
196	Q₈×C₂₃	#121	1	2	yes	—
197	D₄×C₂₃	—	2	2	no	—
198	D₄×C₂₃	—	2	2	no	—
199	D₄×C₂₃	—	2	1	no	—
200	D₄×C₂₃	—	2	1	no	—
201	D₄×C₂₃	—	2	1	no	—
202	D₄×C₂₃	—	4	1	no	—
203	D₄×C₂₃	—	4	1	no	—
204	D₄×C₂₃	—	4	2	no	—
205	D₄×C₂₃	—	4	2	no	—
206	D₄×C₂₃	—	4	2	no	—
207	D₄×C₂₃	—	4	2	no	—
208	D₄×C₂₃	—	16	2	no	—
209	D₄×C₂₃	—	16	2	no	—
210	D₄×C₂₃	#121	1	2	yes	—
211	C₂×C₄×C₂₃	—	2	1	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	2	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	4	1	no	—
219	C₂×C₄×C₂₃	—	4	1	no	—
220	C₂×C₄×C₂₃	—	4	1	no	—
221	C₂×C₄×C₂₃	—	4	1	no	—
222	C₂×C₄×C₂₃	—	4	2	no	—
223	C₂×C₄×C₂₃	—	8	2	no	—
224	C₂×C₄×C₂₃	—	16	2	no	—
225	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
231	C₈×C₂₃	—	2	2	no	—
232	C₈×C₂₃	—	2	1	no	—
233	C₈×C₂₃	—	2	1	no	—
234	C₈×C₂₃	—	4	1	no	—
235	C₈×C₂₃	—	4	2	no	—
236	C₈×C₂₃	#121	1	2	yes	—
237	C₈×C₂₃	#121	1	2	yes	—
238	C₂×#472	C₂×C₂₃	1	1	yes	characteristic, complemented
239	#3058	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#3045	C₂×C₂₃	1	2	yes	complemented
241	C₂×#443	C₂×C₂₃	1	1	yes	characteristic, complemented
242	#2913	C₂×C₂₃	1	1	yes	characteristic, complemented
243	#2063	C₂×C₂₃	1	2	yes	complemented
244	#2037	C₂×C₂₃	1	2	yes	complemented
245	#1970	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#1945	C₂×C₂₃	1	1	yes	characteristic, complemented
247	#1883	C₂×C₂₃	1	2	yes	complemented
248	#1816	C₂×C₂₃	1	1	yes	characteristic, complemented
249	C₂²×C₂₃	—	2	1	no	—
250	C₂²×C₂₃	—	2	1	no	—
251	C₂²×C₂₃	—	2	2	no	—
252	C₂²×C₂₃	—	4	1	no	—
253	C₂²×C₂₃	—	4	1	no	—
254	C₂²×C₂₃	—	4	1	no	—
255	C₂²×C₂₃	—	4	1	no	—
256	C₂²×C₂₃	—	4	1	no	—
257	C₂²×C₂₃	—	8	4	no	—
258	C₂²×C₂₃	—	8	2	no	—
259	C₂²×C₂₃	—	8	2	no	—
260	C₂²×C₂₃	—	8	2	no	—
261	C₂²×C₂₃	—	16	2	no	—
262	C₂²×C₂₃	#537	1	1	yes	characteristic
263	C₂²×C₂₃	D₄²	1	1	yes	characteristic
264	C₂²×C₂₃	#553	1	1	yes	characteristic
265	C₄×C₂₃	—	2	2	no	—
266	C₄×C₂₃	—	4	2	no	—
267	C₄×C₂₃	—	4	1	no	—
268	C₄×C₂₃	—	4	2	no	—
269	C₄×C₂₃	—	4	2	no	—
270	C₄×C₂₃	—	8	2	no	—
271	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
272	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
273	C₄×C₂₃	#547	1	1	yes	characteristic
274	C₄×C₂₃	#548	1	1	yes	characteristic
275	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	—
276	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
277	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
278	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
279	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
280	#535	C₂²×C₂₃	1	2	yes	—
281	#520	C₂²×C₂₃	1	2	yes	—
282	#472	—	2	4	no	—
283	#469	—	2	2	no	—
284	#466	—	2	1	no	—
285	#466	—	2	1	no	—
286	#464	—	2	2	no	—
287	#463	—	2	2	no	—
288	#460	—	2	2	no	—
289	#459	—	2	2	no	—
290	#443	—	2	4	no	—
291	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
292	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
293	#428	—	2	2	no	—
294	#428	C₂²×C₂₃	1	2	yes	—
295	#427	—	2	1	no	—
296	#427	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×#107	—	2	1	no	—
298	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
299	#424	C₂²×C₂₃	1	2	yes	—
300	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
301	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
302	#418	C₂²×C₂₃	1	2	yes	—
303	#417	C₂²×C₂₃	1	2	yes	—
304	C₂×#102	C₂²×C₂₃	1	2	yes	—
305	C₂×#100	C₂²×C₂₃	1	2	yes	—
306	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
307	#390	C₂²×C₂₃	1	2	yes	—
308	#343	C₂²×C₂₃	1	2	yes	—
309	#339	C₂²×C₂₃	1	2	yes	—
310	C₂×C₂₃	—	2	1	no	—
311	C₂×C₂₃	—	2	1	no	—
312	C₂×C₂₃	—	4	2	no	—
313	C₂×C₂₃	—	8	1	no	—
314	C₂×C₂₃	—	16	2	no	—
315	C₂×C₂₃	#2416	1	1	yes	characteristic
316	C₂×C₂₃	#3290	1	1	yes	characteristic
317	C₂×C₂₃	#3292	1	1	yes	characteristic
318	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
319	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
320	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
321	AGL(1,ℤ/8)	—	2	4	no	—
322	AGL(1,ℤ/8)	—	2	4	no	—
323	#135	—	2	2	no	—
324	#135	—	2	2	no	—
325	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
326	C₂×SD₁₆	—	2	2	no	—
327	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
328	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
329	C₂×D₈	—	2	2	no	—
330	C₂×D₈	—	2	1	no	—
331	C₂×D₈	—	2	1	no	—
332	C₂×D₈	—	2	2	no	—
333	C₂×D₈	—	2	2	no	—
334	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
335	#131	—	2	2	no	—
336	#131	—	2	2	no	—
337	#131	—	2	2	no	—
338	#131	D₄×C₂₃	1	4	yes	complemented
339	C₂×M₁₆	—	2	1	no	—
340	C₂×M₁₆	—	2	1	no	—
341	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
342	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
343	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
344	C₂²×C₈	—	2	1	no	—
345	C₂²×C₈	—	2	1	no	—
346	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
347	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
348	#127	—	2	2	no	—
349	#124	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#120	—	4	2	no	—
353	C₄×Q₈	—	2	2	no	—
354	C₄×D₄	—	2	2	no	—
355	C₄×D₄	—	4	2	no	—
356	#117	—	2	2	no	—
357	#117	C₂³×C₂₃	1	2	yes	—
358	C₂×C₄⋊C₄	—	2	2	no	—
359	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
360	C₂×C₂²⋊C₄	—	4	2	no	—
361	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
362	#108	—	2	2	no	—
363	#108	—	2	2	no	—
364	#107	—	2	1	no	—
365	#107	—	2	1	no	—
366	#107	—	2	2	no	—
367	#107	D₄×C₂₃	1	1	yes	characteristic
368	#107	D₄×C₂₃	1	1	yes	characteristic
369	#106	—	2	2	no	—
370	#106	D₄×C₂₃	1	2	yes	—
371	C₄⋊C₈	—	2	2	no	—
372	C₄⋊C₈	—	2	2	no	—
373	#104	—	2	2	no	—
374	#104	—	2	2	no	—
375	#103	—	2	2	no	—
376	#102	—	2	2	no	—
377	#102	—	2	2	no	—
378	#102	—	2	2	no	—
379	#102	—	2	2	no	—
380	#102	—	2	2	no	—
381	#100	—	2	4	no	—
382	C₄×C₈	—	2	2	no	—
383	C₂₃	#24797	1	1	yes	characteristic, direct product factor, Sylow
384	C₂⁴	—	2	2	no	—
385	#41	—	2	1	no	—
386	#41	—	2	1	no	—
387	#41	—	2	1	no	—
388	#41	—	2	1	no	—
389	#41	—	2	1	no	—
390	#41	—	2	1	no	—
391	#41	C₂×D₄×C₂₃	1	2	yes	—
392	#41	#41×C₂₃	1	2	yes	—
393	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
394	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂×D₄	—	4	2	no	—
405	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
406	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	4	2	no	—
412	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
413	Q₁₆	—	2	2	no	—
414	SD₁₆	—	2	2	no	—
415	SD₁₆	—	2	2	no	—
416	SD₁₆	—	4	4	no	—
417	D₈	—	2	2	no	—
418	D₈	—	4	4	no	—
419	D₈	—	4	2	no	—
420	D₈	—	4	2	no	—
421	M₁₆	—	2	1	no	—
422	M₁₆	—	2	4	no	—
423	M₁₆	—	2	2	no	—
424	M₁₆	—	2	1	no	—
425	M₁₆	—	2	1	no	—
426	M₁₆	—	2	2	no	—
427	M₁₆	—	4	2	no	—
428	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	4	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	4	2	no	—
436	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
439	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
440	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₂×C₈	#41×C₂₃	1	2	yes	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	4	2	no	—
445	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
446	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	4	2	no	—
449	C₂²⋊C₄	—	4	2	no	—
450	C₂²⋊C₄	—	4	2	no	—
451	C₂²⋊C₄	—	8	4	no	—
452	C₄²	—	2	2	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	2	1	no	—
455	C₄²	—	4	2	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	1	no	—
458	C₂³	—	4	2	no	—
459	C₂³	—	4	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	8	2	no	—
462	C₂³	—	8	2	no	—
463	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	2	1	no	—
466	Q₈	—	2	1	no	—
467	Q₈	C₂₃×#121	1	2	yes	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	1	no	—
471	D₄	—	2	1	no	—
472	D₄	—	2	1	no	—
473	D₄	—	4	1	no	—
474	D₄	—	4	1	no	—
475	D₄	—	4	2	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	16	2	no	—
480	D₄	—	16	2	no	—
481	D₄	C₂₃×#121	1	2	yes	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	8	2	no	—
495	C₂×C₄	—	16	2	no	—
496	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
499	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
500	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
501	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
502	C₈	—	2	2	no	—
503	C₈	—	2	1	no	—
504	C₈	—	2	1	no	—
505	C₈	—	4	1	no	—
506	C₈	—	4	2	no	—
507	C₈	C₂₃×#121	1	2	yes	—
508	C₈	C₂₃×#121	1	2	yes	—
509	C₂²	—	2	1	no	—
510	C₂²	—	2	1	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	4	1	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	8	4	no	—
518	C₂²	—	8	2	no	—
519	C₂²	—	8	2	no	—
520	C₂²	—	8	2	no	—
521	C₂²	—	16	2	no	—
522	C₂²	C₂₃×#537	1	1	yes	characteristic
523	C₂²	D₄²×C₂₃	1	1	yes	characteristic
524	C₂²	C₂₃×#553	1	1	yes	characteristic
525	C₄	—	2	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	8	2	no	—
531	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
532	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
533	C₄	C₂₃×#547	1	1	yes	characteristic
534	C₄	C₂₃×#548	1	1	yes	characteristic
535	C₂	—	2	1	no	—
536	C₂	—	2	1	no	—
537	C₂	—	4	2	no	—
538	C₂	—	8	1	no	—
539	C₂	—	16	2	no	—
540	C₂	C₂₃×#2416	1	1	yes	characteristic
541	C₂	C₂₃×#3290	1	1	yes	characteristic
542	C₂	C₂₃×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#472	C₂	semidirect
2. C₂₃×#3058	C₂	semidirect
3. C₂₃×#3045	C₂	semidirect
4. C₂×C₂₃×#443	C₂	semidirect
5. C₂₃×#2913	C₂	semidirect
6. C₂₃×#2063	C₂	semidirect
7. C₂₃×#2037	C₂	semidirect
8. C₂₃×#1970	C₂	semidirect
9. C₂₃×#1945	C₂	semidirect
10. C₂₃×#1883	C₂	semidirect
11. C₂₃×#1816	C₂	semidirect
67. C₂₃×#131	D₄	semidirect
184. #24797	C₂₃	direct (N × H)
238. C₂×#472	C₂×C₂₃	semidirect
239. #3058	C₂×C₂₃	semidirect
240. #3045	C₂×C₂₃	semidirect
241. C₂×#443	C₂×C₂₃	semidirect
242. #2913	C₂×C₂₃	semidirect
243. #2063	C₂×C₂₃	semidirect
244. #2037	C₂×C₂₃	semidirect
245. #1970	C₂×C₂₃	semidirect
246. #1945	C₂×C₂₃	semidirect
247. #1883	C₂×C₂₃	semidirect
248. #1816	C₂×C₂₃	semidirect
338. #131	D₄×C₂₃	semidirect
383. C₂₃	#24797	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×#472	2
C₂₃×#3058	2
C₂₃×#3045	2
C₂×C₂₃×#443	2
C₂₃×#2913	2
C₂₃×#2063	2
C₂₃×#2037	2
C₂₃×#1970	2
C₂₃×#1945	2
C₂₃×#1883	2
C₂₃×#1816	2
#24797	23

Subgroups — C23×#24797 #19172423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24797 #19172423