Subgroups — C₂₃×#33540 #19181166

Order 5888.

1078 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3414	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#510	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3067	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2195	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2193	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2143	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2117	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
11	C₂₃×#578	—	2	2	no	—
12	C₂₃×#578	—	2	2	no	—
13	C₂₃×#578	—	2	2	no	—
14	C₂₃×#578	C₂²	1	4	yes	—
15	C₂₃×#575	—	2	1	no	—
16	C₂×C₂₃×#137	—	2	2	no	—
17	C₂×C₂₃×#137	—	2	1	no	—
18	C₂×C₂₃×#135	—	2	1	no	—
19	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
21	C₂²×Q₁₆×C₂₃	—	2	1	no	—
22	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
24	C₂₃×#510	—	2	2	no	—
25	C₂₃×#510	—	2	2	no	—
26	C₂₃×#510	—	2	1	no	—
27	C₂₃×#510	C₂²	1	2	yes	complemented
28	C₂₃×#509	—	2	1	no	—
29	C₂₃×#509	C₂²	1	2	yes	complemented
30	C₂₃×#508	—	2	2	no	—
31	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	complemented
32	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	complemented
33	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
35	C₂₃×#473	C₂²	1	2	yes	complemented
36	C₂₃×#472	C₂²	1	2	yes	complemented
37	C₂₃×#471	—	2	2	no	—
38	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
39	C₂₃×#413	—	2	1	no	—
40	C₂₃×#362	C₂²	1	4	yes	complemented
41	C₂₃×#359	C₂²	1	4	yes	complemented
42	C₂₃×#350	C₂²	1	4	yes	complemented
43	C₂₃×2₋¹⁺⁴	—	2	4	no	—
44	C₂₃×2₋¹⁺⁴	—	2	4	no	—
45	C₂₃×2₋¹⁺⁴	—	4	2	no	—
46	C₂₃×2₋¹⁺⁴	—	4	2	no	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	—	2	1	no	—
49	C₂×#41×C₂₃	—	2	1	no	—
50	C₂×#41×C₂₃	—	4	2	no	—
51	C₂×#41×C₂₃	—	4	1	no	—
52	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×Q₈×C₂₃	—	2	2	no	—
54	C₂²×Q₈×C₂₃	—	2	1	no	—
55	C₂²×Q₈×C₂₃	—	4	1	no	—
56	C₂₃×#137	—	2	4	no	—
57	C₂₃×#137	—	2	8	no	—
58	C₂₃×#137	—	2	8	no	—
59	C₂₃×#137	—	2	2	no	—
60	C₂₃×#137	—	2	4	no	—
61	C₂₃×#137	—	4	1	no	—
62	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
63	C₂₃×AGL(1,ℤ/8)	—	4	1	no	—
64	C₂₃×#135	—	2	4	no	—
65	C₂₃×#135	—	2	4	no	—
66	C₂₃×#135	—	2	1	no	—
67	C₂₃×#135	—	2	1	no	—
68	C₂₃×#135	—	2	1	no	—
69	C₂₃×#135	—	2	1	no	—
70	C₂₃×#135	—	2	1	no	—
71	C₂₃×#135	—	2	1	no	—
72	C₂₃×#135	—	4	2	no	—
73	C₂₃×#135	D₄	1	2	yes	—
74	C₂₃×#135	C₂³	1	2	yes	—
75	C₂×Q₁₆×C₂₃	—	2	2	no	—
76	C₂×Q₁₆×C₂₃	—	2	4	no	—
77	C₂×Q₁₆×C₂₃	—	2	4	no	—
78	C₂×Q₁₆×C₂₃	—	2	1	no	—
79	C₂×Q₁₆×C₂₃	—	2	1	no	—
80	C₂×Q₁₆×C₂₃	—	2	1	no	—
81	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₂₃	D₄	1	2	yes	—
83	C₂×Q₁₆×C₂₃	D₄	1	2	yes	—
84	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
86	C₂×SD₁₆×C₂₃	—	2	2	no	—
87	C₂×SD₁₆×C₂₃	—	2	1	no	—
88	C₂×SD₁₆×C₂₃	—	2	1	no	—
89	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
90	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
91	C₂₃×#131	—	2	1	no	—
92	C₂₃×#131	—	2	1	no	—
93	C₂₃×#131	D₄	1	2	yes	—
94	C₂₃×#131	C₂³	1	2	yes	—
95	C₂×M₁₆×C₂₃	—	2	1	no	—
96	C₂×M₁₆×C₂₃	—	2	1	no	—
97	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
98	C₂²×C₈×C₂₃	—	2	1	no	—
99	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₂₃×Q₃₂	—	2	4	no	—
101	C₂₃×Q₃₂	—	2	2	no	—
102	C₂₃×SD₃₂	—	2	4	no	—
103	C₂₃×SD₃₂	—	2	2	no	—
104	C₂₃×SD₃₂	—	2	2	no	—
105	C₂₃×D₁₆	—	2	2	no	—
106	C₂₃×M₃₂	—	2	1	no	—
107	C₂₃×M₃₂	—	2	1	no	—
108	C₂₃×M₃₂	—	2	1	no	—
109	C₂₃×M₃₂	C₂³	1	2	yes	—
110	C₂×C₁₆×C₂₃	—	2	1	no	—
111	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
112	C₂₃×#108	—	2	1	no	—
113	C₂₃×#108	C₂³	1	2	yes	—
114	C₂₃×#101	—	2	2	no	—
115	C₂₃×#100	—	2	2	no	—
116	#41×C₂₃	—	2	1	no	—
117	#41×C₂₃	—	2	1	no	—
118	#41×C₂₃	—	2	8	no	—
119	#41×C₂₃	—	2	2	no	—
120	#41×C₂₃	—	2	2	no	—
121	#41×C₂₃	—	4	4	no	—
122	#41×C₂₃	—	4	8	no	—
123	#41×C₂₃	—	4	8	no	—
124	#41×C₂₃	—	4	4	no	—
125	#41×C₂₃	—	4	4	no	—
126	#41×C₂₃	—	4	1	no	—
127	#41×C₂₃	—	4	1	no	—
128	#41×C₂₃	—	4	1	no	—
129	#41×C₂₃	—	4	1	no	—
130	#41×C₂₃	—	4	1	no	—
131	#41×C₂₃	—	4	1	no	—
132	#41×C₂₃	D₈	1	2	yes	—
133	#41×C₂₃	C₂×D₄	1	2	yes	—
134	C₂×Q₈×C₂₃	—	2	8	no	—
135	C₂×Q₈×C₂₃	—	2	2	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	2	no	—
138	C₂×Q₈×C₂₃	—	2	2	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	4	2	no	—
141	C₂×Q₈×C₂₃	—	4	4	no	—
142	C₂×Q₈×C₂₃	—	4	4	no	—
143	C₂×Q₈×C₂₃	—	4	4	no	—
144	C₂×Q₈×C₂₃	—	4	1	no	—
145	C₂×Q₈×C₂₃	—	4	1	no	—
146	C₂×Q₈×C₂₃	—	4	1	no	—
147	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	4	2	no	—
152	C₂×D₄×C₂₃	—	4	1	no	—
153	C₂×D₄×C₂₃	—	4	1	no	—
154	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	4	2	no	—
157	C₂²×C₄×C₂₃	—	4	2	no	—
158	C₂²×C₄×C₂₃	—	4	1	no	—
159	C₂²×C₄×C₂₃	—	4	1	no	—
160	C₂²×C₄×C₂₃	—	4	1	no	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	Q₁₆×C₂₃	—	2	8	no	—
163	Q₁₆×C₂₃	—	2	1	no	—
164	Q₁₆×C₂₃	—	2	2	no	—
165	Q₁₆×C₂₃	—	2	2	no	—
166	Q₁₆×C₂₃	—	2	2	no	—
167	Q₁₆×C₂₃	—	2	1	no	—
168	Q₁₆×C₂₃	—	2	1	no	—
169	Q₁₆×C₂₃	—	4	2	no	—
170	Q₁₆×C₂₃	C₂×D₄	1	2	yes	—
171	Q₁₆×C₂₃	C₂×D₄	1	4	yes	—
172	SD₁₆×C₂₃	—	2	8	no	—
173	SD₁₆×C₂₃	—	2	2	no	—
174	SD₁₆×C₂₃	—	2	2	no	—
175	SD₁₆×C₂₃	—	4	2	no	—
176	SD₁₆×C₂₃	—	4	2	no	—
177	SD₁₆×C₂₃	—	4	1	no	—
178	SD₁₆×C₂₃	—	4	1	no	—
179	D₈×C₂₃	—	2	1	no	—
180	D₈×C₂₃	—	2	1	no	—
181	D₈×C₂₃	—	2	1	no	—
182	D₈×C₂₃	—	4	2	no	—
183	D₈×C₂₃	C₂×D₄	1	2	yes	—
184	M₁₆×C₂₃	—	2	1	no	—
185	M₁₆×C₂₃	—	2	2	no	—
186	M₁₆×C₂₃	—	2	2	no	—
187	M₁₆×C₂₃	—	2	2	no	—
188	M₁₆×C₂₃	—	4	1	no	—
189	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
190	C₂×C₈×C₂₃	—	2	1	no	—
191	C₂×C₈×C₂₃	—	2	2	no	—
192	C₂×C₈×C₂₃	—	2	2	no	—
193	C₂×C₈×C₂₃	—	4	1	no	—
194	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
200	C₁₆×C₂₃	—	2	2	no	—
201	C₁₆×C₂₃	—	2	2	no	—
202	#33540	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂³×C₂₃	—	2	1	no	—
204	C₂³×C₂₃	—	4	1	no	—
205	C₂³×C₂₃	—	4	1	no	—
206	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
207	Q₈×C₂₃	—	2	1	no	—
208	Q₈×C₂₃	—	2	4	no	—
209	Q₈×C₂₃	—	2	2	no	—
210	Q₈×C₂₃	—	4	8	no	—
211	Q₈×C₂₃	—	4	2	no	—
212	Q₈×C₂₃	—	4	1	no	—
213	Q₈×C₂₃	—	4	2	no	—
214	Q₈×C₂₃	—	4	2	no	—
215	Q₈×C₂₃	—	4	1	no	—
216	Q₈×C₂₃	—	4	1	no	—
217	Q₈×C₂₃	C₂×D₈	1	2	yes	—
218	D₄×C₂₃	—	2	2	no	—
219	D₄×C₂₃	—	2	2	no	—
220	D₄×C₂₃	—	2	2	no	—
221	D₄×C₂₃	—	2	1	no	—
222	D₄×C₂₃	—	4	8	no	—
223	D₄×C₂₃	—	4	2	no	—
224	D₄×C₂₃	—	4	2	no	—
225	D₄×C₂₃	—	4	1	no	—
226	D₄×C₂₃	—	4	1	no	—
227	D₄×C₂₃	—	4	1	no	—
228	D₄×C₂₃	—	8	1	no	—
229	D₄×C₂₃	—	8	1	no	—
230	D₄×C₂₃	C₂×D₈	1	2	yes	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	4	8	no	—
235	C₂×C₄×C₂₃	—	4	8	no	—
236	C₂×C₄×C₂₃	—	4	2	no	—
237	C₂×C₄×C₂₃	—	4	2	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	2	no	—
240	C₂×C₄×C₂₃	—	4	2	no	—
241	C₂×C₄×C₂₃	—	4	2	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	8	1	no	—
244	C₂×C₄×C₂₃	—	8	1	no	—
245	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
246	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
247	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
249	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
251	C₈×C₂₃	—	2	2	no	—
252	C₈×C₂₃	—	4	2	no	—
253	C₈×C₂₃	#120	1	1	yes	characteristic
254	C₈×C₂₃	#120	1	1	yes	characteristic
255	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
256	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
257	C₂×#578	C₂×C₂₃	1	1	yes	characteristic, complemented
258	#3414	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₂×#510	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#3067	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#2195	C₂×C₂₃	1	4	yes	complemented
262	#2193	C₂×C₂₃	1	4	yes	complemented
263	C₂×#362	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#2143	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#2117	C₂×C₂₃	1	1	yes	characteristic, complemented
266	C₂²×C₂₃	—	2	2	no	—
267	C₂²×C₂₃	—	4	1	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	—	4	2	no	—
270	C₂²×C₂₃	—	8	2	no	—
271	C₂²×C₂₃	—	8	1	no	—
272	C₂²×C₂₃	—	16	1	no	—
273	C₂²×C₂₃	#447	1	1	yes	characteristic
274	C₂²×C₂₃	#447	1	1	yes	characteristic
275	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₄×C₂₃	—	2	2	no	—
277	C₄×C₂₃	—	4	4	no	—
278	C₄×C₂₃	—	4	2	no	—
279	C₄×C₂₃	—	8	1	no	—
280	C₄×C₂₃	#447	1	1	yes	characteristic
281	C₄×C₂₃	#447	1	1	yes	characteristic
282	C₄×C₂₃	C₂×#120	1	1	yes	characteristic
283	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
284	C₂×2₋¹⁺⁴	—	2	1	no	—
285	#578	—	2	2	no	—
286	#578	—	2	2	no	—
287	#578	—	2	2	no	—
288	#578	C₂²×C₂₃	1	4	yes	—
289	#575	—	2	1	no	—
290	C₂×#137	—	2	2	no	—
291	C₂×#137	—	2	1	no	—
292	C₂×#135	—	2	1	no	—
293	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
294	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
295	C₂²×Q₁₆	—	2	1	no	—
296	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
298	#510	—	2	2	no	—
299	#510	—	2	2	no	—
300	#510	—	2	1	no	—
301	#510	C₂²×C₂₃	1	2	yes	complemented
302	#509	—	2	1	no	—
303	#509	C₂²×C₂₃	1	2	yes	complemented
304	#508	—	2	2	no	—
305	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	complemented
306	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	complemented
307	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
309	#473	C₂²×C₂₃	1	2	yes	complemented
310	#472	C₂²×C₂₃	1	2	yes	complemented
311	#471	—	2	2	no	—
312	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
313	#413	—	2	1	no	—
314	#362	C₂²×C₂₃	1	4	yes	complemented
315	#359	C₂²×C₂₃	1	4	yes	complemented
316	#350	C₂²×C₂₃	1	4	yes	complemented
317	C₂×C₂₃	—	2	1	no	—
318	C₂×C₂₃	—	2	1	no	—
319	C₂×C₂₃	—	2	1	no	—
320	C₂×C₂₃	—	4	2	no	—
321	C₂×C₂₃	—	8	2	no	—
322	C₂×C₂₃	—	16	1	no	—
323	C₂×C₂₃	C₂×#447	1	1	yes	characteristic
324	2₋¹⁺⁴	—	2	4	no	—
325	2₋¹⁺⁴	—	2	4	no	—
326	2₋¹⁺⁴	—	4	2	no	—
327	2₋¹⁺⁴	—	4	2	no	—
328	C₂×#41	—	2	2	no	—
329	C₂×#41	—	2	1	no	—
330	C₂×#41	—	2	1	no	—
331	C₂×#41	—	4	2	no	—
332	C₂×#41	—	4	1	no	—
333	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
334	C₂²×Q₈	—	2	2	no	—
335	C₂²×Q₈	—	2	1	no	—
336	C₂²×Q₈	—	4	1	no	—
337	#137	—	2	4	no	—
338	#137	—	2	8	no	—
339	#137	—	2	8	no	—
340	#137	—	2	2	no	—
341	#137	—	2	4	no	—
342	#137	—	4	1	no	—
343	AGL(1,ℤ/8)	—	2	2	no	—
344	AGL(1,ℤ/8)	—	4	1	no	—
345	#135	—	2	4	no	—
346	#135	—	2	4	no	—
347	#135	—	2	1	no	—
348	#135	—	2	1	no	—
349	#135	—	2	1	no	—
350	#135	—	2	1	no	—
351	#135	—	2	1	no	—
352	#135	—	2	1	no	—
353	#135	—	4	2	no	—
354	#135	D₄×C₂₃	1	2	yes	—
355	#135	C₂³×C₂₃	1	2	yes	—
356	C₂×Q₁₆	—	2	2	no	—
357	C₂×Q₁₆	—	2	4	no	—
358	C₂×Q₁₆	—	2	4	no	—
359	C₂×Q₁₆	—	2	1	no	—
360	C₂×Q₁₆	—	2	1	no	—
361	C₂×Q₁₆	—	2	1	no	—
362	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
363	C₂×Q₁₆	D₄×C₂₃	1	2	yes	—
364	C₂×Q₁₆	D₄×C₂₃	1	2	yes	—
365	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
366	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
367	C₂×SD₁₆	—	2	2	no	—
368	C₂×SD₁₆	—	2	1	no	—
369	C₂×SD₁₆	—	2	1	no	—
370	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
371	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
372	#131	—	2	1	no	—
373	#131	—	2	1	no	—
374	#131	D₄×C₂₃	1	2	yes	—
375	#131	C₂³×C₂₃	1	2	yes	—
376	C₂×M₁₆	—	2	1	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
379	C₂²×C₈	—	2	1	no	—
380	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
381	Q₃₂	—	2	4	no	—
382	Q₃₂	—	2	2	no	—
383	SD₃₂	—	2	4	no	—
384	SD₃₂	—	2	2	no	—
385	SD₃₂	—	2	2	no	—
386	D₁₆	—	2	2	no	—
387	M₃₂	—	2	1	no	—
388	M₃₂	—	2	1	no	—
389	M₃₂	—	2	1	no	—
390	M₃₂	C₂³×C₂₃	1	2	yes	—
391	C₂×C₁₆	—	2	1	no	—
392	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
393	#108	—	2	1	no	—
394	#108	C₂³×C₂₃	1	2	yes	—
395	#101	—	2	2	no	—
396	#100	—	2	2	no	—
397	C₂₃	#33540	1	1	yes	characteristic, direct product factor, Sylow
398	#41	—	2	1	no	—
399	#41	—	2	1	no	—
400	#41	—	2	8	no	—
401	#41	—	2	2	no	—
402	#41	—	2	2	no	—
403	#41	—	4	4	no	—
404	#41	—	4	8	no	—
405	#41	—	4	8	no	—
406	#41	—	4	4	no	—
407	#41	—	4	4	no	—
408	#41	—	4	1	no	—
409	#41	—	4	1	no	—
410	#41	—	4	1	no	—
411	#41	—	4	1	no	—
412	#41	—	4	1	no	—
413	#41	—	4	1	no	—
414	#41	D₈×C₂₃	1	2	yes	—
415	#41	C₂×D₄×C₂₃	1	2	yes	—
416	C₂×Q₈	—	2	8	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	4	2	no	—
423	C₂×Q₈	—	4	4	no	—
424	C₂×Q₈	—	4	4	no	—
425	C₂×Q₈	—	4	4	no	—
426	C₂×Q₈	—	4	1	no	—
427	C₂×Q₈	—	4	1	no	—
428	C₂×Q₈	—	4	1	no	—
429	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	4	2	no	—
434	C₂×D₄	—	4	1	no	—
435	C₂×D₄	—	4	1	no	—
436	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	C₂²×C₄	—	4	1	no	—
443	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
444	Q₁₆	—	2	8	no	—
445	Q₁₆	—	2	1	no	—
446	Q₁₆	—	2	2	no	—
447	Q₁₆	—	2	2	no	—
448	Q₁₆	—	2	2	no	—
449	Q₁₆	—	2	1	no	—
450	Q₁₆	—	2	1	no	—
451	Q₁₆	—	4	2	no	—
452	Q₁₆	C₂×D₄×C₂₃	1	2	yes	—
453	Q₁₆	C₂×D₄×C₂₃	1	4	yes	—
454	SD₁₆	—	2	8	no	—
455	SD₁₆	—	2	2	no	—
456	SD₁₆	—	2	2	no	—
457	SD₁₆	—	4	2	no	—
458	SD₁₆	—	4	2	no	—
459	SD₁₆	—	4	1	no	—
460	SD₁₆	—	4	1	no	—
461	D₈	—	2	1	no	—
462	D₈	—	2	1	no	—
463	D₈	—	2	1	no	—
464	D₈	—	4	2	no	—
465	D₈	C₂×D₄×C₂₃	1	2	yes	—
466	M₁₆	—	2	1	no	—
467	M₁₆	—	2	2	no	—
468	M₁₆	—	2	2	no	—
469	M₁₆	—	2	2	no	—
470	M₁₆	—	4	1	no	—
471	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	4	1	no	—
476	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
482	C₁₆	—	2	2	no	—
483	C₁₆	—	2	2	no	—
484	C₂³	—	2	1	no	—
485	C₂³	—	4	1	no	—
486	C₂³	—	4	1	no	—
487	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
488	Q₈	—	2	1	no	—
489	Q₈	—	2	4	no	—
490	Q₈	—	2	2	no	—
491	Q₈	—	4	8	no	—
492	Q₈	—	4	2	no	—
493	Q₈	—	4	1	no	—
494	Q₈	—	4	2	no	—
495	Q₈	—	4	2	no	—
496	Q₈	—	4	1	no	—
497	Q₈	—	4	1	no	—
498	Q₈	C₂×D₈×C₂₃	1	2	yes	—
499	D₄	—	2	2	no	—
500	D₄	—	2	2	no	—
501	D₄	—	2	1	no	—
502	D₄	—	2	2	no	—
503	D₄	—	4	8	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	2	no	—
506	D₄	—	4	1	no	—
507	D₄	—	4	1	no	—
508	D₄	—	4	1	no	—
509	D₄	—	8	1	no	—
510	D₄	—	8	1	no	—
511	D₄	C₂×D₈×C₂₃	1	2	yes	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	4	8	no	—
516	C₂×C₄	—	4	8	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	8	1	no	—
525	C₂×C₄	—	8	1	no	—
526	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
527	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
528	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
529	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
530	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
531	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
532	C₈	—	2	2	no	—
533	C₈	—	4	2	no	—
534	C₈	C₂₃×#120	1	1	yes	characteristic
535	C₈	C₂₃×#120	1	1	yes	characteristic
536	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
537	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
538	C₂²	—	2	2	no	—
539	C₂²	—	4	1	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	8	2	no	—
543	C₂²	—	8	1	no	—
544	C₂²	—	16	1	no	—
545	C₂²	C₂₃×#447	1	1	yes	characteristic
546	C₂²	C₂₃×#447	1	1	yes	characteristic
547	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	4	4	no	—
550	C₄	—	4	2	no	—
551	C₄	—	8	1	no	—
552	C₄	C₂₃×#447	1	1	yes	characteristic
553	C₄	C₂₃×#447	1	1	yes	characteristic
554	C₄	C₂×C₂₃×#120	1	1	yes	characteristic
555	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
556	C₂	—	2	1	no	—
557	C₂	—	2	1	no	—
558	C₂	—	2	1	no	—
559	C₂	—	4	2	no	—
560	C₂	—	8	2	no	—
561	C₂	—	16	1	no	—
562	C₂	C₂×C₂₃×#447	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	C₂×#447
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#578	C₂	semidirect
2. C₂₃×#3414	C₂	semidirect
3. C₂×C₂₃×#510	C₂	semidirect
4. C₂₃×#3067	C₂	semidirect
5. C₂₃×#2195	C₂	semidirect
6. C₂₃×#2193	C₂	semidirect
7. C₂×C₂₃×#362	C₂	semidirect
8. C₂₃×#2143	C₂	semidirect
9. C₂₃×#2117	C₂	semidirect
27. C₂₃×#510	C₂²	semidirect
29. C₂₃×#509	C₂²	semidirect
31. C₂×C₂₃×Q₃₂	C₂²	semidirect
32. C₂×C₂₃×SD₃₂	C₂²	semidirect
35. C₂₃×#473	C₂²	semidirect
36. C₂₃×#472	C₂²	semidirect
40. C₂₃×#362	C₂²	semidirect
41. C₂₃×#359	C₂²	semidirect
42. C₂₃×#350	C₂²	semidirect
202. #33540	C₂₃	direct (N × H)
257. C₂×#578	C₂×C₂₃	semidirect
258. #3414	C₂×C₂₃	semidirect
259. C₂×#510	C₂×C₂₃	semidirect
260. #3067	C₂×C₂₃	semidirect
261. #2195	C₂×C₂₃	semidirect
262. #2193	C₂×C₂₃	semidirect
263. C₂×#362	C₂×C₂₃	semidirect
264. #2143	C₂×C₂₃	semidirect
265. #2117	C₂×C₂₃	semidirect
301. #510	C₂²×C₂₃	semidirect
303. #509	C₂²×C₂₃	semidirect
305. C₂×Q₃₂	C₂²×C₂₃	semidirect
306. C₂×SD₃₂	C₂²×C₂₃	semidirect
309. #473	C₂²×C₂₃	semidirect
310. #472	C₂²×C₂₃	semidirect
314. #362	C₂²×C₂₃	semidirect
315. #359	C₂²×C₂₃	semidirect
316. #350	C₂²×C₂₃	semidirect
397. C₂₃	#33540	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×#578	2
C₂₃×#3414	2
C₂×C₂₃×#510	2
C₂₃×#3067	2
C₂₃×#2195	2
C₂₃×#2193	2
C₂×C₂₃×#362	2
C₂₃×#2143	2
C₂₃×#2117	2
#33540	23

Subgroups — C23×#33540 #19181166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33540 #19181166