Subgroups — C₂₃×#22284 #19169910

Order 5888.

796 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×#3154	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2949	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1977	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1912	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1911	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1910	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1909	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1905	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1904	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1903	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1902	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#568	C₂²	1	1	yes	characteristic
21	C₂₃×#568	C₂²	1	1	yes	characteristic
22	C₂₃×#536	C₂²	1	1	yes	characteristic
23	C₂₃×#535	C₂²	1	1	yes	characteristic
24	C₂₃×#519	C₂²	1	1	yes	characteristic
25	C₂₃×#518	C₂²	1	1	yes	characteristic
26	C₂₃×#497	C₄	1	2	yes	complemented
27	C₂₃×#496	C₄	1	2	yes	complemented
28	C₂₃×#495	C₄	1	2	yes	complemented
29	C₂₃×#492	C₄	1	2	yes	complemented
30	C₂₃×#444	—	2	2	no	—
31	C₂₃×#443	—	2	2	no	—
32	C₂₃×#440	—	2	2	no	—
33	C₄×SD₁₆×C₂₃	—	2	2	no	—
34	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
35	C₂₃×#427	C₂²	1	1	yes	characteristic
36	C₂₃×#421	C₂²	1	1	yes	characteristic
37	C₂₃×#421	C₂²	1	1	yes	characteristic
38	C₂₃×#421	C₂²	1	1	yes	characteristic
39	C₂₃×#421	C₂²	1	1	yes	characteristic
40	C₂₃×#419	C₂²	1	1	yes	characteristic
41	C₂₃×#419	C₂²	1	1	yes	characteristic
42	C₂₃×#418	C₂²	1	1	yes	characteristic
43	C₂₃×#418	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
46	C₂₃×#410	C₂²	1	1	yes	characteristic
47	C₂₃×#410	C₂²	1	1	yes	characteristic
48	C₂₃×#405	C₂²	1	1	yes	characteristic
49	C₂₃×#405	C₂²	1	1	yes	characteristic
50	C₂₃×#405	C₂²	1	1	yes	characteristic
51	C₂₃×#344	C₂²	1	1	yes	characteristic
52	C₂₃×#344	C₂²	1	1	yes	characteristic
53	C₂₃×#338	C₂²	1	1	yes	characteristic
54	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
55	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
56	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
57	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
58	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
59	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂₃×#137	—	4	1	no	—
64	C₂₃×#137	—	4	1	no	—
65	C₂₃×AGL(1,ℤ/8)	—	4	1	no	—
66	C₂₃×AGL(1,ℤ/8)	—	4	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	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
78	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
79	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
80	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
81	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
82	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
83	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
84	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
85	C₂₃×#131	—	2	2	no	—
86	C₂₃×#131	—	2	2	no	—
87	C₂₃×#131	—	4	1	no	—
88	C₂₃×#131	—	4	1	no	—
89	C₂×M₁₆×C₂₃	—	4	1	no	—
90	C₂×M₁₆×C₂₃	—	4	1	no	—
91	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	complemented
98	C₂₃×#127	C₂×C₄	1	2	yes	—
99	C₂₃×#124	C₂×C₄	1	2	yes	—
100	C₂₃×#124	C₂×C₄	1	2	yes	complemented
101	C₂₃×#123	—	4	1	no	—
102	C₂₃×C₂²⋊Q₈	—	4	1	no	—
103	C₂₃×#121	—	4	1	no	—
104	C₂₃×#120	—	4	1	no	—
105	C₄×Q₈×C₂₃	—	2	2	no	—
106	C₄×Q₈×C₂₃	—	4	1	no	—
107	C₄×D₄×C₂₃	—	2	2	no	—
108	C₄×D₄×C₂₃	—	4	1	no	—
109	C₂₃×#117	—	4	1	no	—
110	C₂₃×#117	C₂³	1	1	yes	characteristic
111	C₂₃×#117	C₂³	1	1	yes	characteristic
112	C₂₃×#117	C₂³	1	1	yes	characteristic
113	C₂₃×#117	C₂³	1	1	yes	characteristic
114	C₂₃×#117	C₂³	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
116	C₂₃×#108	—	2	1	no	—
117	C₂₃×#108	—	2	1	no	—
118	C₂₃×#107	—	2	1	no	—
119	C₂₃×#106	—	2	1	no	—
120	C₂₃×#104	—	2	2	no	—
121	C₂₃×#104	—	2	2	no	—
122	C₂₃×#104	—	4	1	no	—
123	C₂₃×#104	—	4	1	no	—
124	C₂₃×#103	—	2	2	no	—
125	C₂₃×#103	—	4	1	no	—
126	C₂₃×#102	—	2	2	no	—
127	C₂₃×#102	—	4	1	no	—
128	C₂₃×#101	—	4	1	no	—
129	C₂₃×#100	—	4	1	no	—
130	C₂₃×#99	—	4	1	no	—
131	C₂₃×#99	—	4	1	no	—
132	C₂₃×#97	—	2	1	no	—
133	C₂₃×#97	C₂×C₄	1	2	yes	complemented
134	C₂₃×#97	D₄	1	2	yes	complemented
135	C₄×C₈×C₂₃	—	2	1	no	—
136	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	complemented
137	C₄×C₈×C₂₃	D₄	1	2	yes	complemented
138	C₂⁴×C₂₃	—	2	1	no	—
139	#41×C₂₃	—	2	1	no	—
140	#41×C₂₃	—	2	1	no	—
141	#41×C₂₃	—	2	1	no	—
142	#41×C₂₃	—	2	1	no	—
143	#41×C₂₃	—	2	1	no	—
144	#41×C₂₃	—	2	1	no	—
145	#41×C₂₃	—	2	1	no	—
146	#41×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	—	2	1	no	—
152	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
154	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	—	2	1	no	—
160	C₂×D₄×C₂₃	—	2	1	no	—
161	C₂×D₄×C₂₃	—	2	1	no	—
162	C₂×D₄×C₂₃	—	4	2	no	—
163	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
164	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
165	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	4	2	no	—
170	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
171	Q₁₆×C₂₃	—	2	1	no	—
172	Q₁₆×C₂₃	—	2	1	no	—
173	Q₁₆×C₂₃	—	4	2	no	—
174	SD₁₆×C₂₃	—	2	2	no	—
175	SD₁₆×C₂₃	—	2	2	no	—
176	SD₁₆×C₂₃	—	2	2	no	—
177	SD₁₆×C₂₃	—	2	2	no	—
178	SD₁₆×C₂₃	—	2	1	no	—
179	SD₁₆×C₂₃	—	2	1	no	—
180	SD₁₆×C₂₃	—	2	1	no	—
181	SD₁₆×C₂₃	—	2	1	no	—
182	SD₁₆×C₂₃	—	4	2	no	—
183	SD₁₆×C₂₃	—	4	2	no	—
184	D₈×C₂₃	—	2	1	no	—
185	D₈×C₂₃	—	2	1	no	—
186	D₈×C₂₃	—	4	2	no	—
187	M₁₆×C₂₃	—	2	1	no	—
188	M₁₆×C₂₃	—	2	1	no	—
189	M₁₆×C₂₃	—	2	1	no	—
190	M₁₆×C₂₃	—	2	1	no	—
191	M₁₆×C₂₃	—	2	1	no	—
192	M₁₆×C₂₃	—	2	1	no	—
193	M₁₆×C₂₃	—	2	1	no	—
194	M₁₆×C₂₃	—	2	1	no	—
195	M₁₆×C₂₃	—	2	1	no	—
196	M₁₆×C₂₃	—	2	1	no	—
197	M₁₆×C₂₃	—	4	2	no	—
198	M₁₆×C₂₃	—	4	2	no	—
199	C₂×C₈×C₂₃	—	2	1	no	—
200	C₂×C₈×C₂₃	—	2	1	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	4	2	no	—
203	C₂×C₈×C₂₃	—	4	2	no	—
204	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
205	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
206	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
207	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
208	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
209	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
211	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
212	C₄⋊C₄×C₂₃	—	2	1	no	—
213	C₄⋊C₄×C₂₃	—	2	1	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	4	2	no	—
218	C₄⋊C₄×C₂₃	—	4	2	no	—
219	C₂²⋊C₄×C₂₃	—	2	1	no	—
220	C₂²⋊C₄×C₂₃	—	2	1	no	—
221	C₂²⋊C₄×C₂₃	—	2	1	no	—
222	C₂²⋊C₄×C₂₃	—	2	1	no	—
223	C₂²⋊C₄×C₂₃	—	2	1	no	—
224	C₂²⋊C₄×C₂₃	—	4	2	no	—
225	C₂²⋊C₄×C₂₃	—	4	2	no	—
226	C₂²⋊C₄×C₂₃	—	4	2	no	—
227	C₄²×C₂₃	—	2	1	no	—
228	C₄²×C₂₃	—	2	1	no	—
229	C₄²×C₂₃	—	2	1	no	—
230	C₄²×C₂₃	—	4	2	no	—
231	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
232	C₄²×C₂₃	C₂×D₄	1	2	yes	—
233	#22284	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂³×C₂₃	—	2	1	no	—
235	C₂³×C₂₃	—	2	2	no	—
236	C₂³×C₂₃	—	2	1	no	—
237	C₂³×C₂₃	—	4	1	no	—
238	C₂³×C₂₃	—	4	1	no	—
239	C₂³×C₂₃	—	16	1	no	—
240	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
241	Q₈×C₂₃	—	2	2	no	—
242	Q₈×C₂₃	—	2	1	no	—
243	Q₈×C₂₃	—	2	1	no	—
244	Q₈×C₂₃	—	2	1	no	—
245	Q₈×C₂₃	—	2	2	no	—
246	Q₈×C₂₃	—	2	1	no	—
247	Q₈×C₂₃	—	4	1	no	—
248	Q₈×C₂₃	—	4	1	no	—
249	D₄×C₂₃	—	2	1	no	—
250	D₄×C₂₃	—	2	1	no	—
251	D₄×C₂₃	—	2	2	no	—
252	D₄×C₂₃	—	2	1	no	—
253	D₄×C₂₃	—	2	2	no	—
254	D₄×C₂₃	—	2	1	no	—
255	D₄×C₂₃	—	4	1	no	—
256	D₄×C₂₃	—	4	1	no	—
257	D₄×C₂₃	—	4	1	no	—
258	D₄×C₂₃	—	4	1	no	—
259	D₄×C₂₃	—	4	1	no	—
260	D₄×C₂₃	—	4	1	no	—
261	D₄×C₂₃	—	16	2	no	—
262	C₂×C₄×C₂₃	—	2	2	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	2	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	16	2	no	—
277	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
280	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
282	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
283	C₈×C₂₃	—	2	2	no	—
284	C₈×C₂₃	—	2	2	no	—
285	C₈×C₂₃	—	4	1	no	—
286	C₈×C₂₃	—	4	1	no	—
287	C₈×C₂₃	—	4	1	no	—
288	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
289	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
290	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
291	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
292	#3154	C₂×C₂₃	1	1	yes	characteristic
293	#2957	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#2949	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#1977	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#1976	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#1912	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1911	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#1910	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#1909	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1905	C₂×C₂₃	1	1	yes	characteristic, complemented
302	#1904	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#1903	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#1902	C₂×C₂₃	1	1	yes	characteristic, complemented
305	#1833	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#1747	C₂×C₂₃	1	1	yes	characteristic, complemented
307	C₂²×C₂₃	—	4	1	no	—
308	C₂²×C₂₃	—	4	1	no	—
309	C₂²×C₂₃	—	4	1	no	—
310	C₂²×C₂₃	—	4	2	no	—
311	C₂²×C₂₃	—	8	1	no	—
312	C₂²×C₂₃	—	8	2	no	—
313	C₂²×C₂₃	—	8	1	no	—
314	C₂²×C₂₃	—	8	2	no	—
315	C₂²×C₂₃	—	16	1	no	—
316	C₂²×C₂₃	#518	1	1	yes	characteristic
317	C₂²×C₂₃	#535	1	1	yes	characteristic
318	C₂²×C₂₃	#540	1	1	yes	characteristic
319	C₄×C₂₃	—	4	2	no	—
320	C₄×C₂₃	—	4	2	no	—
321	C₄×C₂₃	—	4	1	no	—
322	C₄×C₂₃	—	4	1	no	—
323	C₄×C₂₃	—	4	2	no	—
324	C₄×C₂₃	—	8	1	no	—
325	C₄×C₂₃	—	8	1	no	—
326	C₄×C₂₃	—	8	1	no	—
327	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
328	C₄×C₂₃	#518	1	1	yes	characteristic
329	C₄×C₂₃	#535	1	1	yes	characteristic
330	C₄×C₂₃	#540	1	1	yes	characteristic
331	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	1	yes	characteristic
333	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
334	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
335	#568	C₂²×C₂₃	1	1	yes	characteristic
336	#568	C₂²×C₂₃	1	1	yes	characteristic
337	#536	C₂²×C₂₃	1	1	yes	characteristic
338	#535	C₂²×C₂₃	1	1	yes	characteristic
339	#519	C₂²×C₂₃	1	1	yes	characteristic
340	#518	C₂²×C₂₃	1	1	yes	characteristic
341	#497	C₄×C₂₃	1	2	yes	complemented
342	#496	C₄×C₂₃	1	2	yes	complemented
343	#495	C₄×C₂₃	1	2	yes	complemented
344	#492	C₄×C₂₃	1	2	yes	complemented
345	#444	—	2	2	no	—
346	#443	—	2	2	no	—
347	#440	—	2	2	no	—
348	C₄×SD₁₆	—	2	2	no	—
349	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
350	#427	C₂²×C₂₃	1	1	yes	characteristic
351	#421	C₂²×C₂₃	1	1	yes	characteristic
352	#421	C₂²×C₂₃	1	1	yes	characteristic
353	#421	C₂²×C₂₃	1	1	yes	characteristic
354	#421	C₂²×C₂₃	1	1	yes	characteristic
355	#419	C₂²×C₂₃	1	1	yes	characteristic
356	#419	C₂²×C₂₃	1	1	yes	characteristic
357	#418	C₂²×C₂₃	1	1	yes	characteristic
358	#418	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
361	#410	C₂²×C₂₃	1	1	yes	characteristic
362	#410	C₂²×C₂₃	1	1	yes	characteristic
363	#405	C₂²×C₂₃	1	1	yes	characteristic
364	#405	C₂²×C₂₃	1	1	yes	characteristic
365	#405	C₂²×C₂₃	1	1	yes	characteristic
366	#344	C₂²×C₂₃	1	1	yes	characteristic
367	#344	C₂²×C₂₃	1	1	yes	characteristic
368	#338	C₂²×C₂₃	1	1	yes	characteristic
369	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
370	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
371	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
372	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
373	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×C₂₃	—	2	1	no	—
375	C₂×C₂₃	—	2	1	no	—
376	C₂×C₂₃	—	2	1	no	—
377	C₂×C₂₃	—	8	1	no	—
378	C₂×C₂₃	—	8	1	no	—
379	C₂×C₂₃	—	8	2	no	—
380	C₂×C₂₃	#2328	1	1	yes	characteristic
381	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
382	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
383	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
385	#137	—	4	1	no	—
386	#137	—	4	1	no	—
387	AGL(1,ℤ/8)	—	4	1	no	—
388	AGL(1,ℤ/8)	—	4	1	no	—
389	#135	—	2	1	no	—
390	#135	—	2	1	no	—
391	#135	—	2	1	no	—
392	#135	—	2	1	no	—
393	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
394	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
395	C₂×SD₁₆	—	2	1	no	—
396	C₂×SD₁₆	—	2	1	no	—
397	C₂×SD₁₆	—	2	1	no	—
398	C₂×SD₁₆	—	2	1	no	—
399	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
400	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
401	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
402	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
403	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
404	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
405	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
406	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
407	#131	—	2	2	no	—
408	#131	—	2	2	no	—
409	#131	—	4	1	no	—
410	#131	—	4	1	no	—
411	C₂×M₁₆	—	4	1	no	—
412	C₂×M₁₆	—	4	1	no	—
413	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
415	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
417	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
418	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
419	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	complemented
420	#127	C₂×C₄×C₂₃	1	2	yes	—
421	#124	C₂×C₄×C₂₃	1	2	yes	—
422	#124	C₂×C₄×C₂₃	1	2	yes	complemented
423	#123	—	4	1	no	—
424	C₂²⋊Q₈	—	4	1	no	—
425	#121	—	4	1	no	—
426	#120	—	4	1	no	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×Q₈	—	4	1	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	4	1	no	—
431	#117	—	4	1	no	—
432	#117	C₂³×C₂₃	1	1	yes	characteristic
433	#117	C₂³×C₂₃	1	1	yes	characteristic
434	#117	C₂³×C₂₃	1	1	yes	characteristic
435	#117	C₂³×C₂₃	1	1	yes	characteristic
436	#117	C₂³×C₂₃	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	—	4	1	no	—
438	#108	—	2	1	no	—
439	#108	—	2	1	no	—
440	#107	—	2	1	no	—
441	#106	—	2	1	no	—
442	#104	—	2	2	no	—
443	#104	—	2	2	no	—
444	#104	—	4	1	no	—
445	#104	—	4	1	no	—
446	#103	—	2	2	no	—
447	#103	—	4	1	no	—
448	#102	—	2	2	no	—
449	#102	—	4	1	no	—
450	#101	—	4	1	no	—
451	#100	—	4	1	no	—
452	#99	—	4	1	no	—
453	#99	—	4	1	no	—
454	#97	—	2	1	no	—
455	#97	C₂×C₄×C₂₃	1	2	yes	complemented
456	#97	D₄×C₂₃	1	2	yes	complemented
457	C₄×C₈	—	2	1	no	—
458	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	complemented
459	C₄×C₈	D₄×C₂₃	1	2	yes	complemented
460	C₂₃	#22284	1	1	yes	characteristic, direct product factor, Sylow
461	C₂⁴	—	2	1	no	—
462	#41	—	2	1	no	—
463	#41	—	2	1	no	—
464	#41	—	2	1	no	—
465	#41	—	2	1	no	—
466	#41	—	2	1	no	—
467	#41	—	2	1	no	—
468	#41	—	2	1	no	—
469	#41	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
476	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
477	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	4	2	no	—
486	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
487	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
488	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
494	Q₁₆	—	2	1	no	—
495	Q₁₆	—	2	1	no	—
496	Q₁₆	—	4	2	no	—
497	SD₁₆	—	2	2	no	—
498	SD₁₆	—	2	2	no	—
499	SD₁₆	—	2	2	no	—
500	SD₁₆	—	2	2	no	—
501	SD₁₆	—	2	1	no	—
502	SD₁₆	—	2	1	no	—
503	SD₁₆	—	2	1	no	—
504	SD₁₆	—	2	1	no	—
505	SD₁₆	—	4	2	no	—
506	SD₁₆	—	4	2	no	—
507	D₈	—	2	1	no	—
508	D₈	—	2	1	no	—
509	D₈	—	4	2	no	—
510	M₁₆	—	2	1	no	—
511	M₁₆	—	2	1	no	—
512	M₁₆	—	2	1	no	—
513	M₁₆	—	2	1	no	—
514	M₁₆	—	2	1	no	—
515	M₁₆	—	2	1	no	—
516	M₁₆	—	2	1	no	—
517	M₁₆	—	2	1	no	—
518	M₁₆	—	2	1	no	—
519	M₁₆	—	2	1	no	—
520	M₁₆	—	4	2	no	—
521	M₁₆	—	4	2	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	4	2	no	—
526	C₂×C₈	—	4	2	no	—
527	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
528	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
529	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
530	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
531	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
532	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
533	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
534	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	1	no	—
540	C₄⋊C₄	—	4	2	no	—
541	C₄⋊C₄	—	4	2	no	—
542	C₂²⋊C₄	—	2	1	no	—
543	C₂²⋊C₄	—	2	1	no	—
544	C₂²⋊C₄	—	2	1	no	—
545	C₂²⋊C₄	—	2	1	no	—
546	C₂²⋊C₄	—	2	1	no	—
547	C₂²⋊C₄	—	4	2	no	—
548	C₂²⋊C₄	—	4	2	no	—
549	C₂²⋊C₄	—	4	2	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	2	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	4	2	no	—
554	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
555	C₄²	C₂×D₄×C₂₃	1	2	yes	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	4	1	no	—
561	C₂³	—	16	1	no	—
562	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
563	Q₈	—	2	2	no	—
564	Q₈	—	2	1	no	—
565	Q₈	—	2	1	no	—
566	Q₈	—	2	1	no	—
567	Q₈	—	2	2	no	—
568	Q₈	—	2	1	no	—
569	Q₈	—	4	1	no	—
570	Q₈	—	4	1	no	—
571	D₄	—	2	1	no	—
572	D₄	—	2	1	no	—
573	D₄	—	2	2	no	—
574	D₄	—	2	1	no	—
575	D₄	—	2	2	no	—
576	D₄	—	2	1	no	—
577	D₄	—	4	1	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	D₄	—	4	1	no	—
583	D₄	—	16	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	16	2	no	—
599	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
600	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
601	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
602	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
603	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
604	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
605	C₈	—	2	2	no	—
606	C₈	—	2	2	no	—
607	C₈	—	4	1	no	—
608	C₈	—	4	1	no	—
609	C₈	—	4	1	no	—
610	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
611	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
612	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
613	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
614	C₂²	—	4	1	no	—
615	C₂²	—	4	1	no	—
616	C₂²	—	4	1	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	8	1	no	—
619	C₂²	—	8	2	no	—
620	C₂²	—	8	1	no	—
621	C₂²	—	8	2	no	—
622	C₂²	—	16	1	no	—
623	C₂²	C₂₃×#518	1	1	yes	characteristic
624	C₂²	C₂₃×#535	1	1	yes	characteristic
625	C₂²	C₂₃×#540	1	1	yes	characteristic
626	C₄	—	4	2	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	2	no	—
631	C₄	—	8	1	no	—
632	C₄	—	8	1	no	—
633	C₄	—	8	1	no	—
634	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
635	C₄	C₂₃×#518	1	1	yes	characteristic
636	C₄	C₂₃×#535	1	1	yes	characteristic
637	C₄	C₂₃×#540	1	1	yes	characteristic
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	2	1	no	—
641	C₂	—	8	2	no	—
642	C₂	—	8	1	no	—
643	C₂	—	8	1	no	—
644	C₂	C₂₃×#2328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2328
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#3154	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2957	C₂	semidirect
3. C₂₃×#2949	C₂	semidirect
4. C₂₃×#1977	C₂	semidirect
5. C₂₃×#1976	C₂	semidirect
6. C₂₃×#1912	C₂	semidirect
7. C₂₃×#1911	C₂	semidirect
8. C₂₃×#1910	C₂	semidirect
9. C₂₃×#1909	C₂	semidirect
10. C₂₃×#1905	C₂	semidirect
11. C₂₃×#1904	C₂	semidirect
12. C₂₃×#1903	C₂	semidirect
13. C₂₃×#1902	C₂	semidirect
14. C₂₃×#1833	C₂	semidirect
15. C₂₃×#1747	C₂	semidirect
26. C₂₃×#497	C₄	semidirect
27. C₂₃×#496	C₄	semidirect
28. C₂₃×#495	C₄	semidirect
29. C₂₃×#492	C₄	semidirect
71. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
72. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
80. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
81. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
82. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
84. C₂×D₈×C₂₃	C₂×C₄	semidirect
97. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
100. C₂₃×#124	C₂×C₄	semidirect
133. C₂₃×#97	C₂×C₄	semidirect
134. C₂₃×#97	D₄	semidirect
136. C₄×C₈×C₂₃	C₂×C₄	semidirect
137. C₄×C₈×C₂₃	D₄	semidirect
233. #22284	C₂₃	direct (N × H)
293. #2957	C₂×C₂₃	semidirect
294. #2949	C₂×C₂₃	semidirect
295. #1977	C₂×C₂₃	semidirect
296. #1976	C₂×C₂₃	semidirect
297. #1912	C₂×C₂₃	semidirect
298. #1911	C₂×C₂₃	semidirect
299. #1910	C₂×C₂₃	semidirect
300. #1909	C₂×C₂₃	semidirect
301. #1905	C₂×C₂₃	semidirect
302. #1904	C₂×C₂₃	semidirect
303. #1903	C₂×C₂₃	semidirect
304. #1902	C₂×C₂₃	semidirect
305. #1833	C₂×C₂₃	semidirect
306. #1747	C₂×C₂₃	semidirect
341. #497	C₄×C₂₃	semidirect
342. #496	C₄×C₂₃	semidirect
343. #495	C₄×C₂₃	semidirect
344. #492	C₄×C₂₃	semidirect
393. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
394. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
402. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
403. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
404. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
406. C₂×D₈	C₂×C₄×C₂₃	semidirect
419. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
422. #124	C₂×C₄×C₂₃	semidirect
455. #97	C₂×C₄×C₂₃	semidirect
456. #97	D₄×C₂₃	semidirect
458. C₄×C₈	C₂×C₄×C₂₃	semidirect
459. C₄×C₈	D₄×C₂₃	semidirect
460. C₂₃	#22284	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×#3154	2
C₂₃×#2957	2
C₂₃×#2949	2
C₂₃×#1977	2
C₂₃×#1976	2
C₂₃×#1912	2
C₂₃×#1911	2
C₂₃×#1910	2
C₂₃×#1909	2
C₂₃×#1905	2
C₂₃×#1904	2
C₂₃×#1903	2
C₂₃×#1902	2
C₂₃×#1833	2
C₂₃×#1747	2
#22284	23

Subgroups — C23×#22284 #19169910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22284 #19169910