Subgroups — C₂₃×#33538 #19181164

Order 5888.

1088 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3414	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3067	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2195	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2194	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2193	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2192	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2148	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#359	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#585	—	2	1	no	—
13	C₂₃×#578	—	2	1	no	—
14	C₂₃×#578	C₂²	1	2	yes	—
15	C₂₃×#577	—	2	2	no	—
16	C₂₃×#577	—	2	2	no	—
17	C₂₃×#576	—	2	1	no	—
18	C₂₃×#576	C₂²	1	2	yes	—
19	C₂₃×#575	—	2	2	no	—
20	C₂₃×#575	—	2	1	no	—
21	C₂₃×#575	—	2	1	no	—
22	C₂×C₂₃×#135	—	2	2	no	—
23	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
27	C₂₃×#510	—	2	1	no	—
28	C₂₃×#510	C₂²	1	2	yes	complemented
29	C₂₃×#509	—	2	1	no	—
30	C₂₃×#509	C₂²	1	2	yes	complemented
31	C₂₃×#508	—	2	2	no	—
32	C₂₃×#508	—	2	2	no	—
33	C₂₃×#508	—	2	2	no	—
34	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₂₃×SD₃₂	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₂₃×SD₃₂	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
39	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#473	C₂²	1	2	yes	complemented
41	C₂₃×#472	C₂²	1	2	yes	complemented
42	C₂₃×#471	—	2	2	no	—
43	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
44	C₂₃×#413	—	2	1	no	—
45	C₂₃×#362	C₂²	1	2	yes	complemented
46	C₂₃×#361	C₂²	1	2	yes	complemented
47	C₂₃×#359	C₂²	1	2	yes	complemented
48	C₂₃×#359	C₂²	1	2	yes	complemented
49	C₂₃×#350	C₂²	1	2	yes	complemented
50	C₂₃×#350	C₂²	1	2	yes	complemented
51	C₂₃×2₋¹⁺⁴	—	2	2	no	—
52	C₂₃×2₋¹⁺⁴	—	2	2	no	—
53	C₂₃×2₋¹⁺⁴	—	4	1	no	—
54	C₂₃×H(5,𝔽₂)	—	2	2	no	—
55	C₂₃×H(5,𝔽₂)	—	2	2	no	—
56	C₂₃×H(5,𝔽₂)	—	4	2	no	—
57	C₂₃×H(5,𝔽₂)	—	4	1	no	—
58	C₂×#41×C₂₃	—	2	4	no	—
59	C₂×#41×C₂₃	—	2	1	no	—
60	C₂×#41×C₂₃	—	2	1	no	—
61	C₂×#41×C₂₃	—	2	1	no	—
62	C₂×#41×C₂₃	—	4	2	no	—
63	C₂×#41×C₂₃	—	4	2	no	—
64	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
65	C₂₃×#137	—	2	2	no	—
66	C₂₃×#137	—	2	4	no	—
67	C₂₃×#137	—	2	4	no	—
68	C₂₃×#137	—	2	2	no	—
69	C₂₃×#137	—	2	2	no	—
70	C₂₃×#137	—	4	1	no	—
71	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
72	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
73	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
74	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
75	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
76	C₂₃×AGL(1,ℤ/8)	—	4	1	no	—
77	C₂₃×#135	—	2	4	no	—
78	C₂₃×#135	—	2	4	no	—
79	C₂₃×#135	—	2	2	no	—
80	C₂₃×#135	—	2	2	no	—
81	C₂₃×#135	—	2	1	no	—
82	C₂₃×#135	—	2	1	no	—
83	C₂₃×#135	—	2	1	no	—
84	C₂₃×#135	—	2	1	no	—
85	C₂₃×#135	—	2	1	no	—
86	C₂₃×#135	—	2	1	no	—
87	C₂₃×#135	—	2	1	no	—
88	C₂₃×#135	—	2	1	no	—
89	C₂₃×#135	—	4	2	no	—
90	C₂₃×#135	D₄	1	2	yes	—
91	C₂₃×#135	D₄	1	2	yes	—
92	C₂₃×#135	D₄	1	2	yes	—
93	C₂₃×#135	C₂³	1	2	yes	—
94	C₂×Q₁₆×C₂₃	—	2	2	no	—
95	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
96	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×SD₁₆×C₂₃	—	2	4	no	—
99	C₂×SD₁₆×C₂₃	—	2	1	no	—
100	C₂×SD₁₆×C₂₃	—	2	1	no	—
101	C₂×SD₁₆×C₂₃	—	2	1	no	—
102	C₂×D₈×C₂₃	—	2	2	no	—
103	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
104	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
106	C₂₃×#131	—	2	1	no	—
107	C₂₃×#131	—	2	1	no	—
108	C₂₃×#131	D₄	1	2	yes	—
109	C₂₃×#131	C₂³	1	2	yes	—
110	C₂×M₁₆×C₂₃	—	2	1	no	—
111	C₂×M₁₆×C₂₃	—	2	1	no	—
112	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
113	C₂²×C₈×C₂₃	—	2	1	no	—
114	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
115	C₂₃×Q₃₂	—	2	2	no	—
116	C₂₃×Q₃₂	—	2	2	no	—
117	C₂₃×SD₃₂	—	2	2	no	—
118	C₂₃×SD₃₂	—	2	2	no	—
119	C₂₃×SD₃₂	—	2	2	no	—
120	C₂₃×SD₃₂	—	2	2	no	—
121	C₂₃×D₁₆	—	2	2	no	—
122	C₂₃×D₁₆	—	2	2	no	—
123	C₂₃×M₃₂	—	2	1	no	—
124	C₂₃×M₃₂	C₂³	1	2	yes	—
125	C₂×C₁₆×C₂₃	—	2	1	no	—
126	C₂×C₁₆×C₂₃	—	2	1	no	—
127	C₂×C₁₆×C₂₃	—	2	1	no	—
128	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
129	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
130	C₂₃×#108	—	2	1	no	—
131	C₂₃×#108	C₂³	1	2	yes	—
132	C₂₃×#101	—	2	2	no	—
133	C₂₃×#100	—	2	2	no	—
134	#41×C₂₃	—	2	1	no	—
135	#41×C₂₃	—	2	1	no	—
136	#41×C₂₃	—	2	4	no	—
137	#41×C₂₃	—	2	4	no	—
138	#41×C₂₃	—	2	2	no	—
139	#41×C₂₃	—	2	2	no	—
140	#41×C₂₃	—	2	2	no	—
141	#41×C₂₃	—	2	2	no	—
142	#41×C₂₃	—	4	4	no	—
143	#41×C₂₃	—	4	4	no	—
144	#41×C₂₃	—	4	4	no	—
145	#41×C₂₃	—	4	4	no	—
146	#41×C₂₃	—	4	2	no	—
147	#41×C₂₃	—	4	2	no	—
148	#41×C₂₃	—	4	2	no	—
149	#41×C₂₃	—	4	2	no	—
150	#41×C₂₃	—	4	1	no	—
151	#41×C₂₃	—	4	1	no	—
152	#41×C₂₃	—	4	1	no	—
153	#41×C₂₃	—	4	1	no	—
154	#41×C₂₃	—	4	1	no	—
155	#41×C₂₃	—	4	1	no	—
156	#41×C₂₃	—	4	1	no	—
157	#41×C₂₃	—	4	1	no	—
158	#41×C₂₃	D₈	1	2	yes	—
159	#41×C₂₃	C₂×D₄	1	2	yes	—
160	C₂×Q₈×C₂₃	—	2	4	no	—
161	C₂×Q₈×C₂₃	—	2	1	no	—
162	C₂×Q₈×C₂₃	—	2	1	no	—
163	C₂×Q₈×C₂₃	—	2	1	no	—
164	C₂×Q₈×C₂₃	—	4	2	no	—
165	C₂×Q₈×C₂₃	—	4	2	no	—
166	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
167	C₂×D₄×C₂₃	—	2	1	no	—
168	C₂×D₄×C₂₃	—	2	4	no	—
169	C₂×D₄×C₂₃	—	2	1	no	—
170	C₂×D₄×C₂₃	—	2	1	no	—
171	C₂×D₄×C₂₃	—	2	1	no	—
172	C₂×D₄×C₂₃	—	4	4	no	—
173	C₂×D₄×C₂₃	—	4	4	no	—
174	C₂×D₄×C₂₃	—	4	2	no	—
175	C₂×D₄×C₂₃	—	4	4	no	—
176	C₂×D₄×C₂₃	—	4	1	no	—
177	C₂×D₄×C₂₃	—	4	1	no	—
178	C₂×D₄×C₂₃	—	4	2	no	—
179	C₂×D₄×C₂₃	—	4	1	no	—
180	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	—	2	1	no	—
182	C₂²×C₄×C₂₃	—	4	4	no	—
183	C₂²×C₄×C₂₃	—	4	1	no	—
184	C₂²×C₄×C₂₃	—	4	1	no	—
185	C₂²×C₄×C₂₃	—	4	1	no	—
186	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	Q₁₆×C₂₃	—	2	4	no	—
188	Q₁₆×C₂₃	—	2	1	no	—
189	Q₁₆×C₂₃	—	2	1	no	—
190	Q₁₆×C₂₃	—	2	1	no	—
191	Q₁₆×C₂₃	—	2	1	no	—
192	Q₁₆×C₂₃	—	4	2	no	—
193	Q₁₆×C₂₃	C₂×D₄	1	2	yes	—
194	Q₁₆×C₂₃	C₂×D₄	1	2	yes	—
195	SD₁₆×C₂₃	—	2	4	no	—
196	SD₁₆×C₂₃	—	2	4	no	—
197	SD₁₆×C₂₃	—	2	2	no	—
198	SD₁₆×C₂₃	—	2	2	no	—
199	SD₁₆×C₂₃	—	2	2	no	—
200	SD₁₆×C₂₃	—	2	2	no	—
201	SD₁₆×C₂₃	—	4	2	no	—
202	SD₁₆×C₂₃	—	4	2	no	—
203	SD₁₆×C₂₃	—	4	1	no	—
204	SD₁₆×C₂₃	—	4	1	no	—
205	D₈×C₂₃	—	2	4	no	—
206	D₈×C₂₃	—	2	1	no	—
207	D₈×C₂₃	—	2	1	no	—
208	D₈×C₂₃	—	2	1	no	—
209	D₈×C₂₃	—	2	1	no	—
210	D₈×C₂₃	—	4	2	no	—
211	D₈×C₂₃	C₂×D₄	1	2	yes	—
212	D₈×C₂₃	C₂×D₄	1	2	yes	—
213	M₁₆×C₂₃	—	2	1	no	—
214	M₁₆×C₂₃	—	2	2	no	—
215	M₁₆×C₂₃	—	2	2	no	—
216	M₁₆×C₂₃	—	2	2	no	—
217	M₁₆×C₂₃	—	4	1	no	—
218	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
219	C₂×C₈×C₂₃	—	2	1	no	—
220	C₂×C₈×C₂₃	—	2	2	no	—
221	C₂×C₈×C₂₃	—	2	2	no	—
222	C₂×C₈×C₂₃	—	4	1	no	—
223	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
224	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
225	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
226	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
227	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
228	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
229	C₁₆×C₂₃	—	2	1	no	—
230	C₁₆×C₂₃	—	2	1	no	—
231	C₁₆×C₂₃	—	2	2	no	—
232	#33538	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂³×C₂₃	—	2	1	no	—
234	C₂³×C₂₃	—	4	4	no	—
235	C₂³×C₂₃	—	4	1	no	—
236	C₂³×C₂₃	—	4	1	no	—
237	C₂³×C₂₃	—	4	1	no	—
238	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
239	Q₈×C₂₃	—	2	1	no	—
240	Q₈×C₂₃	—	2	2	no	—
241	Q₈×C₂₃	—	2	2	no	—
242	Q₈×C₂₃	—	4	4	no	—
243	Q₈×C₂₃	—	4	1	no	—
244	Q₈×C₂₃	—	4	1	no	—
245	Q₈×C₂₃	—	4	1	no	—
246	Q₈×C₂₃	—	4	1	no	—
247	Q₈×C₂₃	C₂×D₈	1	2	yes	—
248	D₄×C₂₃	—	2	2	no	—
249	D₄×C₂₃	—	2	2	no	—
250	D₄×C₂₃	—	2	2	no	—
251	D₄×C₂₃	—	2	2	no	—
252	D₄×C₂₃	—	2	1	no	—
253	D₄×C₂₃	—	4	4	no	—
254	D₄×C₂₃	—	4	4	no	—
255	D₄×C₂₃	—	4	4	no	—
256	D₄×C₂₃	—	4	2	no	—
257	D₄×C₂₃	—	4	2	no	—
258	D₄×C₂₃	—	4	2	no	—
259	D₄×C₂₃	—	4	2	no	—
260	D₄×C₂₃	—	4	1	no	—
261	D₄×C₂₃	—	4	1	no	—
262	D₄×C₂₃	—	4	1	no	—
263	D₄×C₂₃	—	4	1	no	—
264	D₄×C₂₃	—	8	1	no	—
265	D₄×C₂₃	—	8	1	no	—
266	D₄×C₂₃	C₂×D₈	1	2	yes	—
267	C₂×C₄×C₂₃	—	2	1	no	—
268	C₂×C₄×C₂₃	—	2	2	no	—
269	C₂×C₄×C₂₃	—	2	2	no	—
270	C₂×C₄×C₂₃	—	4	4	no	—
271	C₂×C₄×C₂₃	—	4	4	no	—
272	C₂×C₄×C₂₃	—	4	4	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	2	no	—
276	C₂×C₄×C₂₃	—	4	2	no	—
277	C₂×C₄×C₂₃	—	4	2	no	—
278	C₂×C₄×C₂₃	—	4	2	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	8	1	no	—
281	C₂×C₄×C₂₃	—	8	1	no	—
282	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
283	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
284	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
285	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
286	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
287	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
288	C₈×C₂₃	—	2	2	no	—
289	C₈×C₂₃	—	4	2	no	—
290	C₈×C₂₃	#120	1	1	yes	characteristic
291	C₈×C₂₃	#120	1	1	yes	characteristic
292	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
293	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
294	#3584	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#3414	C₂×C₂₃	1	1	yes	characteristic, complemented
296	C₂×#508	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#3067	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#2195	C₂×C₂₃	1	2	yes	complemented
299	#2194	C₂×C₂₃	1	2	yes	complemented
300	#2193	C₂×C₂₃	1	2	yes	complemented
301	#2192	C₂×C₂₃	1	2	yes	complemented
302	#2148	C₂×C₂₃	1	1	yes	characteristic, complemented
303	C₂×#359	C₂×C₂₃	1	1	yes	characteristic, complemented
304	C₂×#350	C₂×C₂₃	1	1	yes	characteristic, complemented
305	C₂²×C₂₃	—	2	2	no	—
306	C₂²×C₂₃	—	4	1	no	—
307	C₂²×C₂₃	—	4	2	no	—
308	C₂²×C₂₃	—	4	2	no	—
309	C₂²×C₂₃	—	8	1	no	—
310	C₂²×C₂₃	—	8	1	no	—
311	C₂²×C₂₃	—	16	1	no	—
312	C₂²×C₂₃	—	16	1	no	—
313	C₂²×C₂₃	—	16	1	no	—
314	C₂²×C₂₃	—	16	4	no	—
315	C₂²×C₂₃	#447	1	1	yes	characteristic
316	C₂²×C₂₃	#447	1	1	yes	characteristic
317	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₄×C₂₃	—	2	2	no	—
319	C₄×C₂₃	—	4	2	no	—
320	C₄×C₂₃	—	4	2	no	—
321	C₄×C₂₃	—	8	1	no	—
322	C₄×C₂₃	#447	1	1	yes	characteristic
323	C₄×C₂₃	#447	1	1	yes	characteristic
324	C₄×C₂₃	C₂×#120	1	1	yes	characteristic
325	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
326	#585	—	2	1	no	—
327	#578	—	2	1	no	—
328	#578	C₂²×C₂₃	1	2	yes	—
329	#577	—	2	2	no	—
330	#577	—	2	2	no	—
331	#576	—	2	1	no	—
332	#576	C₂²×C₂₃	1	2	yes	—
333	#575	—	2	2	no	—
334	#575	—	2	1	no	—
335	#575	—	2	1	no	—
336	C₂×#135	—	2	2	no	—
337	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic, complemented
340	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
341	#510	—	2	1	no	—
342	#510	C₂²×C₂₃	1	2	yes	complemented
343	#509	—	2	1	no	—
344	#509	C₂²×C₂₃	1	2	yes	complemented
345	#508	—	2	2	no	—
346	#508	—	2	2	no	—
347	#508	—	2	2	no	—
348	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
349	C₂×SD₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
350	C₂×SD₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
351	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
352	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
353	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
354	#473	C₂²×C₂₃	1	2	yes	complemented
355	#472	C₂²×C₂₃	1	2	yes	complemented
356	#471	—	2	2	no	—
357	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic, complemented
358	#413	—	2	1	no	—
359	#362	C₂²×C₂₃	1	2	yes	complemented
360	#361	C₂²×C₂₃	1	2	yes	complemented
361	#359	C₂²×C₂₃	1	2	yes	complemented
362	#359	C₂²×C₂₃	1	2	yes	complemented
363	#350	C₂²×C₂₃	1	2	yes	complemented
364	#350	C₂²×C₂₃	1	2	yes	complemented
365	C₂×C₂₃	—	2	1	no	—
366	C₂×C₂₃	—	2	1	no	—
367	C₂×C₂₃	—	2	1	no	—
368	C₂×C₂₃	—	4	2	no	—
369	C₂×C₂₃	—	8	2	no	—
370	C₂×C₂₃	—	8	2	no	—
371	C₂×C₂₃	—	16	1	no	—
372	C₂×C₂₃	C₂×#447	1	1	yes	characteristic
373	2₋¹⁺⁴	—	2	2	no	—
374	2₋¹⁺⁴	—	2	2	no	—
375	2₋¹⁺⁴	—	4	1	no	—
376	H(5,𝔽₂)	—	2	2	no	—
377	H(5,𝔽₂)	—	2	2	no	—
378	H(5,𝔽₂)	—	4	2	no	—
379	H(5,𝔽₂)	—	4	1	no	—
380	C₂×#41	—	2	4	no	—
381	C₂×#41	—	2	1	no	—
382	C₂×#41	—	2	1	no	—
383	C₂×#41	—	2	1	no	—
384	C₂×#41	—	4	2	no	—
385	C₂×#41	—	4	2	no	—
386	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
387	#137	—	2	2	no	—
388	#137	—	2	4	no	—
389	#137	—	2	4	no	—
390	#137	—	2	2	no	—
391	#137	—	2	2	no	—
392	#137	—	4	1	no	—
393	AGL(1,ℤ/8)	—	2	2	no	—
394	AGL(1,ℤ/8)	—	2	4	no	—
395	AGL(1,ℤ/8)	—	2	4	no	—
396	AGL(1,ℤ/8)	—	2	2	no	—
397	AGL(1,ℤ/8)	—	2	2	no	—
398	AGL(1,ℤ/8)	—	4	1	no	—
399	#135	—	2	4	no	—
400	#135	—	2	4	no	—
401	#135	—	2	2	no	—
402	#135	—	2	2	no	—
403	#135	—	2	1	no	—
404	#135	—	2	1	no	—
405	#135	—	2	1	no	—
406	#135	—	2	1	no	—
407	#135	—	2	1	no	—
408	#135	—	2	1	no	—
409	#135	—	2	1	no	—
410	#135	—	2	1	no	—
411	#135	—	4	2	no	—
412	#135	D₄×C₂₃	1	2	yes	—
413	#135	D₄×C₂₃	1	2	yes	—
414	#135	D₄×C₂₃	1	2	yes	—
415	#135	C₂³×C₂₃	1	2	yes	—
416	C₂×Q₁₆	—	2	2	no	—
417	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
418	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
419	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
420	C₂×SD₁₆	—	2	4	no	—
421	C₂×SD₁₆	—	2	1	no	—
422	C₂×SD₁₆	—	2	1	no	—
423	C₂×SD₁₆	—	2	1	no	—
424	C₂×D₈	—	2	2	no	—
425	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
426	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
428	#131	—	2	1	no	—
429	#131	—	2	1	no	—
430	#131	D₄×C₂₃	1	2	yes	—
431	#131	C₂³×C₂₃	1	2	yes	—
432	C₂×M₁₆	—	2	1	no	—
433	C₂×M₁₆	—	2	1	no	—
434	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
435	C₂²×C₈	—	2	1	no	—
436	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
437	Q₃₂	—	2	2	no	—
438	Q₃₂	—	2	2	no	—
439	SD₃₂	—	2	2	no	—
440	SD₃₂	—	2	2	no	—
441	SD₃₂	—	2	2	no	—
442	SD₃₂	—	2	2	no	—
443	D₁₆	—	2	2	no	—
444	D₁₆	—	2	2	no	—
445	M₃₂	—	2	1	no	—
446	M₃₂	C₂³×C₂₃	1	2	yes	—
447	C₂×C₁₆	—	2	1	no	—
448	C₂×C₁₆	—	2	1	no	—
449	C₂×C₁₆	—	2	1	no	—
450	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
451	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
452	#108	—	2	1	no	—
453	#108	C₂³×C₂₃	1	2	yes	—
454	#101	—	2	2	no	—
455	#100	—	2	2	no	—
456	C₂₃	#33538	1	1	yes	characteristic, direct product factor, Sylow
457	#41	—	2	1	no	—
458	#41	—	2	1	no	—
459	#41	—	2	4	no	—
460	#41	—	2	4	no	—
461	#41	—	2	2	no	—
462	#41	—	2	2	no	—
463	#41	—	2	2	no	—
464	#41	—	2	2	no	—
465	#41	—	4	4	no	—
466	#41	—	4	4	no	—
467	#41	—	4	4	no	—
468	#41	—	4	4	no	—
469	#41	—	4	2	no	—
470	#41	—	4	2	no	—
471	#41	—	4	2	no	—
472	#41	—	4	2	no	—
473	#41	—	4	1	no	—
474	#41	—	4	1	no	—
475	#41	—	4	1	no	—
476	#41	—	4	1	no	—
477	#41	—	4	1	no	—
478	#41	—	4	1	no	—
479	#41	—	4	1	no	—
480	#41	—	4	1	no	—
481	#41	D₈×C₂₃	1	2	yes	—
482	#41	C₂×D₄×C₂₃	1	2	yes	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	4	2	no	—
488	C₂×Q₈	—	4	2	no	—
489	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	4	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	4	4	no	—
496	C₂×D₄	—	4	4	no	—
497	C₂×D₄	—	4	2	no	—
498	C₂×D₄	—	4	4	no	—
499	C₂×D₄	—	4	1	no	—
500	C₂×D₄	—	4	1	no	—
501	C₂×D₄	—	4	2	no	—
502	C₂×D₄	—	4	1	no	—
503	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	4	4	no	—
506	C₂²×C₄	—	4	1	no	—
507	C₂²×C₄	—	4	1	no	—
508	C₂²×C₄	—	4	1	no	—
509	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
510	Q₁₆	—	2	4	no	—
511	Q₁₆	—	2	1	no	—
512	Q₁₆	—	2	1	no	—
513	Q₁₆	—	2	1	no	—
514	Q₁₆	—	2	1	no	—
515	Q₁₆	—	4	2	no	—
516	Q₁₆	C₂×D₄×C₂₃	1	2	yes	—
517	Q₁₆	C₂×D₄×C₂₃	1	2	yes	—
518	SD₁₆	—	2	4	no	—
519	SD₁₆	—	2	4	no	—
520	SD₁₆	—	2	2	no	—
521	SD₁₆	—	2	2	no	—
522	SD₁₆	—	2	2	no	—
523	SD₁₆	—	2	2	no	—
524	SD₁₆	—	4	2	no	—
525	SD₁₆	—	4	2	no	—
526	SD₁₆	—	4	1	no	—
527	SD₁₆	—	4	1	no	—
528	D₈	—	2	4	no	—
529	D₈	—	2	1	no	—
530	D₈	—	2	1	no	—
531	D₈	—	2	1	no	—
532	D₈	—	2	1	no	—
533	D₈	—	4	2	no	—
534	D₈	C₂×D₄×C₂₃	1	2	yes	—
535	D₈	C₂×D₄×C₂₃	1	2	yes	—
536	M₁₆	—	2	1	no	—
537	M₁₆	—	2	2	no	—
538	M₁₆	—	2	2	no	—
539	M₁₆	—	2	2	no	—
540	M₁₆	—	4	1	no	—
541	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	2	no	—
544	C₂×C₈	—	2	2	no	—
545	C₂×C₈	—	4	1	no	—
546	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
547	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
548	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
549	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
550	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
551	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
552	C₁₆	—	2	1	no	—
553	C₁₆	—	2	1	no	—
554	C₁₆	—	2	2	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	4	4	no	—
557	C₂³	—	4	1	no	—
558	C₂³	—	4	1	no	—
559	C₂³	—	4	1	no	—
560	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
561	Q₈	—	2	1	no	—
562	Q₈	—	2	2	no	—
563	Q₈	—	2	2	no	—
564	Q₈	—	4	4	no	—
565	Q₈	—	4	1	no	—
566	Q₈	—	4	1	no	—
567	Q₈	—	4	1	no	—
568	Q₈	—	4	1	no	—
569	Q₈	C₂×D₈×C₂₃	1	2	yes	—
570	D₄	—	2	2	no	—
571	D₄	—	2	2	no	—
572	D₄	—	2	1	no	—
573	D₄	—	2	2	no	—
574	D₄	—	2	2	no	—
575	D₄	—	4	4	no	—
576	D₄	—	4	4	no	—
577	D₄	—	4	4	no	—
578	D₄	—	4	2	no	—
579	D₄	—	4	2	no	—
580	D₄	—	4	2	no	—
581	D₄	—	4	2	no	—
582	D₄	—	4	1	no	—
583	D₄	—	4	1	no	—
584	D₄	—	4	1	no	—
585	D₄	—	4	1	no	—
586	D₄	—	8	1	no	—
587	D₄	—	8	1	no	—
588	D₄	C₂×D₈×C₂₃	1	2	yes	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	4	4	no	—
593	C₂×C₄	—	4	4	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	2	no	—
600	C₂×C₄	—	4	2	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	8	1	no	—
603	C₂×C₄	—	8	1	no	—
604	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
605	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
606	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
607	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
608	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
610	C₈	—	2	2	no	—
611	C₈	—	4	2	no	—
612	C₈	C₂₃×#120	1	1	yes	characteristic
613	C₈	C₂₃×#120	1	1	yes	characteristic
614	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
615	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
616	C₂²	—	2	2	no	—
617	C₂²	—	4	1	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	8	1	no	—
621	C₂²	—	8	1	no	—
622	C₂²	—	16	1	no	—
623	C₂²	—	16	1	no	—
624	C₂²	—	16	1	no	—
625	C₂²	—	16	4	no	—
626	C₂²	C₂₃×#447	1	1	yes	characteristic
627	C₂²	C₂₃×#447	1	1	yes	characteristic
628	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
629	C₄	—	2	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	2	no	—
632	C₄	—	8	1	no	—
633	C₄	C₂₃×#447	1	1	yes	characteristic
634	C₄	C₂₃×#447	1	1	yes	characteristic
635	C₄	C₂×C₂₃×#120	1	1	yes	characteristic
636	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
637	C₂	—	2	1	no	—
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	4	2	no	—
641	C₂	—	8	2	no	—
642	C₂	—	8	2	no	—
643	C₂	—	16	1	no	—
644	C₂	C₂×C₂₃×#447	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#447
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂×#447
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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. C₂₃×#3584	C₂	semidirect
2. C₂₃×#3414	C₂	semidirect
3. C₂×C₂₃×#508	C₂	semidirect
4. C₂₃×#3067	C₂	semidirect
5. C₂₃×#2195	C₂	semidirect
6. C₂₃×#2194	C₂	semidirect
7. C₂₃×#2193	C₂	semidirect
8. C₂₃×#2192	C₂	semidirect
9. C₂₃×#2148	C₂	semidirect
10. C₂×C₂₃×#359	C₂	semidirect
11. C₂×C₂₃×#350	C₂	semidirect
25. C₂×C₂₃×#135	C₂²	semidirect
28. C₂₃×#510	C₂²	semidirect
30. C₂₃×#509	C₂²	semidirect
34. C₂×C₂₃×Q₃₂	C₂²	semidirect
35. C₂×C₂₃×SD₃₂	C₂²	semidirect
36. C₂×C₂₃×SD₃₂	C₂²	semidirect
37. C₂×C₂₃×D₁₆	C₂²	semidirect
38. C₂×C₂₃×M₃₂	C₂²	semidirect
39. C₂²×C₁₆×C₂₃	C₂²	semidirect
40. C₂₃×#473	C₂²	semidirect
41. C₂₃×#472	C₂²	semidirect
43. C₂×C₂₃×#108	C₂²	semidirect
45. C₂₃×#362	C₂²	semidirect
46. C₂₃×#361	C₂²	semidirect
47. C₂₃×#359	C₂²	semidirect
48. C₂₃×#359	C₂²	semidirect
49. C₂₃×#350	C₂²	semidirect
50. C₂₃×#350	C₂²	semidirect
232. #33538	C₂₃	direct (N × H)
294. #3584	C₂×C₂₃	semidirect
295. #3414	C₂×C₂₃	semidirect
296. C₂×#508	C₂×C₂₃	semidirect
297. #3067	C₂×C₂₃	semidirect
298. #2195	C₂×C₂₃	semidirect
299. #2194	C₂×C₂₃	semidirect
300. #2193	C₂×C₂₃	semidirect
301. #2192	C₂×C₂₃	semidirect
302. #2148	C₂×C₂₃	semidirect
303. C₂×#359	C₂×C₂₃	semidirect
304. C₂×#350	C₂×C₂₃	semidirect
339. C₂×#135	C₂²×C₂₃	semidirect
342. #510	C₂²×C₂₃	semidirect
344. #509	C₂²×C₂₃	semidirect
348. C₂×Q₃₂	C₂²×C₂₃	semidirect
349. C₂×SD₃₂	C₂²×C₂₃	semidirect
350. C₂×SD₃₂	C₂²×C₂₃	semidirect
351. C₂×D₁₆	C₂²×C₂₃	semidirect
352. C₂×M₃₂	C₂²×C₂₃	semidirect
353. C₂²×C₁₆	C₂²×C₂₃	semidirect
354. #473	C₂²×C₂₃	semidirect
355. #472	C₂²×C₂₃	semidirect
357. C₂×#108	C₂²×C₂₃	semidirect
359. #362	C₂²×C₂₃	semidirect
360. #361	C₂²×C₂₃	semidirect
361. #359	C₂²×C₂₃	semidirect
362. #359	C₂²×C₂₃	semidirect
363. #350	C₂²×C₂₃	semidirect
364. #350	C₂²×C₂₃	semidirect
456. C₂₃	#33538	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3584	2
C₂₃×#3414	2
C₂×C₂₃×#508	2
C₂₃×#3067	2
C₂₃×#2195	2
C₂₃×#2194	2
C₂₃×#2193	2
C₂₃×#2192	2
C₂₃×#2148	2
C₂×C₂₃×#359	2
C₂×C₂₃×#350	2
#33538	23

Subgroups — C23×#33538 #19181164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33538 #19181164