Subgroups — C₂₃×#24987 #19172613

Order 5888.

638 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#471	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2949	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2082	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2081	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2049	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2048	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1977	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1905	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1904	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1830	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1829	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1757	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	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
20	C₂₃×#519	C₂²	1	1	yes	characteristic
21	C₂₃×#518	C₂²	1	1	yes	characteristic
22	C₂₃×#471	—	2	2	no	—
23	C₂₃×#471	—	2	2	no	—
24	C₂₃×#440	—	2	2	no	—
25	C₄×SD₁₆×C₂₃	—	2	2	no	—
26	C₂₃×#430	C₂²	1	1	yes	characteristic
27	C₂₃×#430	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
30	C₂₃×#427	C₂²	1	1	yes	characteristic
31	C₂₃×#421	C₂²	1	1	yes	characteristic
32	C₂₃×#421	C₂²	1	1	yes	characteristic
33	C₂₃×#419	C₂²	1	1	yes	characteristic
34	C₂₃×#419	C₂²	1	1	yes	characteristic
35	C₂₃×#418	C₂²	1	1	yes	characteristic
36	C₂₃×#418	C₂²	1	1	yes	characteristic
37	C₂₃×#413	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
42	C₂₃×#410	C₂²	1	1	yes	characteristic
43	C₂₃×#405	C₂²	1	1	yes	characteristic
44	C₂₃×#405	C₂²	1	1	yes	characteristic
45	C₂₃×#344	C₂²	1	1	yes	characteristic
46	C₂₃×#344	C₂²	1	1	yes	characteristic
47	C₂₃×#344	C₂²	1	1	yes	characteristic
48	C₂₃×#344	C₂²	1	1	yes	characteristic
49	C₂₃×#344	C₂²	1	1	yes	characteristic
50	C₂₃×#344	C₂²	1	1	yes	characteristic
51	C₂₃×#344	C₂²	1	1	yes	characteristic
52	C₂₃×#344	C₂²	1	1	yes	characteristic
53	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
54	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
55	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂₃×#137	—	2	2	no	—
59	C₂₃×#137	—	4	1	no	—
60	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
61	C₂₃×AGL(1,ℤ/8)	—	4	1	no	—
62	C₂×Q₁₆×C₂₃	—	2	1	no	—
63	C₂×SD₁₆×C₂₃	—	2	1	no	—
64	C₂×SD₁₆×C₂₃	—	2	1	no	—
65	C₂×SD₁₆×C₂₃	—	2	1	no	—
66	C₂×SD₁₆×C₂₃	—	2	1	no	—
67	C₂×SD₁₆×C₂₃	—	2	1	no	—
68	C₂×SD₁₆×C₂₃	—	2	1	no	—
69	C₂×SD₁₆×C₂₃	D₄	1	2	yes	—
70	C₂×SD₁₆×C₂₃	D₄	1	2	yes	—
71	C₂×D₈×C₂₃	—	2	1	no	—
72	C₂₃×#131	—	2	1	no	—
73	C₂₃×#131	—	2	1	no	—
74	C₂₃×#131	—	2	1	no	—
75	C₂₃×#131	—	2	1	no	—
76	C₂×M₁₆×C₂₃	—	2	1	no	—
77	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₂₃	—	2	1	no	—
85	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₄×Q₈×C₂₃	—	2	2	no	—
87	C₄×Q₈×C₂₃	—	4	1	no	—
88	C₄×D₄×C₂₃	—	2	2	no	—
89	C₄×D₄×C₂₃	—	4	1	no	—
90	C₂₃×#117	—	2	2	no	—
91	C₂₃×#117	C₂³	1	1	yes	characteristic
92	C₂₃×#117	C₂³	1	1	yes	characteristic
93	C₂₃×#117	C₂³	1	1	yes	characteristic
94	C₂₃×#117	C₂³	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₂₃×#108	—	2	1	no	—
97	C₂₃×#108	—	2	1	no	—
98	C₂₃×#108	—	2	1	no	—
99	C₂₃×#108	—	2	1	no	—
100	C₂₃×#108	—	4	1	no	—
101	C₂₃×#108	—	4	1	no	—
102	C₂₃×#107	—	2	1	no	—
103	C₂₃×#106	—	2	1	no	—
104	C₂₃×#104	—	4	1	no	—
105	C₂₃×#104	—	4	1	no	—
106	C₂₃×#103	—	2	2	no	—
107	C₂₃×#103	—	4	1	no	—
108	C₂₃×#102	—	2	2	no	—
109	C₂₃×#102	—	4	1	no	—
110	C₂₃×#101	—	2	2	no	—
111	C₂₃×#101	—	2	1	no	—
112	C₂₃×#101	—	4	1	no	—
113	C₂₃×#100	—	2	2	no	—
114	C₂₃×#100	—	2	1	no	—
115	C₂₃×#100	—	4	1	no	—
116	C₂₃×#99	—	2	2	no	—
117	C₂₃×#97	—	2	1	no	—
118	C₂₃×#97	—	2	1	no	—
119	C₄×C₈×C₂₃	—	2	1	no	—
120	C₄×C₈×C₂₃	—	2	1	no	—
121	C₂⁴×C₂₃	—	2	1	no	—
122	#41×C₂₃	—	2	1	no	—
123	#41×C₂₃	—	2	1	no	—
124	#41×C₂₃	—	2	1	no	—
125	#41×C₂₃	—	2	1	no	—
126	C₂×Q₈×C₂₃	—	2	1	no	—
127	C₂×Q₈×C₂₃	—	2	1	no	—
128	C₂×Q₈×C₂₃	—	2	1	no	—
129	C₂×Q₈×C₂₃	—	2	1	no	—
130	C₂×Q₈×C₂₃	—	2	1	no	—
131	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
132	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
133	C₂×D₄×C₂₃	—	2	1	no	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	2	1	no	—
136	C₂×D₄×C₂₃	—	2	1	no	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	2	1	no	—
139	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
140	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	4	2	no	—
144	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
145	Q₁₆×C₂₃	—	4	2	no	—
146	SD₁₆×C₂₃	—	2	2	no	—
147	SD₁₆×C₂₃	—	2	2	no	—
148	SD₁₆×C₂₃	—	2	2	no	—
149	SD₁₆×C₂₃	—	2	2	no	—
150	SD₁₆×C₂₃	—	4	2	no	—
151	SD₁₆×C₂₃	—	4	2	no	—
152	D₈×C₂₃	—	4	2	no	—
153	M₁₆×C₂₃	—	2	1	no	—
154	M₁₆×C₂₃	—	2	1	no	—
155	M₁₆×C₂₃	—	2	1	no	—
156	M₁₆×C₂₃	—	2	1	no	—
157	M₁₆×C₂₃	—	2	1	no	—
158	M₁₆×C₂₃	—	2	1	no	—
159	M₁₆×C₂₃	—	2	1	no	—
160	M₁₆×C₂₃	—	2	1	no	—
161	M₁₆×C₂₃	—	2	1	no	—
162	M₁₆×C₂₃	—	2	1	no	—
163	M₁₆×C₂₃	—	2	1	no	—
164	M₁₆×C₂₃	—	2	1	no	—
165	M₁₆×C₂₃	—	2	1	no	—
166	M₁₆×C₂₃	—	2	1	no	—
167	M₁₆×C₂₃	—	4	1	no	—
168	M₁₆×C₂₃	—	4	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	4	1	no	—
177	C₂×C₈×C₂₃	—	4	1	no	—
178	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
181	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
182	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
183	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	4	2	no	—
189	C₂²⋊C₄×C₂₃	—	2	1	no	—
190	C₂²⋊C₄×C₂₃	—	2	1	no	—
191	C₂²⋊C₄×C₂₃	—	2	1	no	—
192	C₂²⋊C₄×C₂₃	—	4	2	no	—
193	C₂²⋊C₄×C₂₃	#41	1	2	yes	—
194	C₄²×C₂₃	—	2	1	no	—
195	C₄²×C₂₃	—	2	1	no	—
196	C₄²×C₂₃	—	2	1	no	—
197	C₄²×C₂₃	—	2	1	no	—
198	C₄²×C₂₃	—	4	2	no	—
199	#24987	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₂₃	—	2	1	no	—
201	C₂³×C₂₃	—	2	2	no	—
202	C₂³×C₂₃	—	4	1	no	—
203	C₂³×C₂₃	—	4	1	no	—
204	C₂³×C₂₃	—	16	1	no	—
205	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
206	Q₈×C₂₃	—	2	2	no	—
207	Q₈×C₂₃	—	2	1	no	—
208	Q₈×C₂₃	—	2	2	no	—
209	Q₈×C₂₃	—	2	1	no	—
210	Q₈×C₂₃	—	4	1	no	—
211	Q₈×C₂₃	—	4	1	no	—
212	D₄×C₂₃	—	2	2	no	—
213	D₄×C₂₃	—	2	1	no	—
214	D₄×C₂₃	—	2	2	no	—
215	D₄×C₂₃	—	2	1	no	—
216	D₄×C₂₃	—	4	1	no	—
217	D₄×C₂₃	—	4	1	no	—
218	D₄×C₂₃	—	4	1	no	—
219	D₄×C₂₃	—	4	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	4	1	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	1	no	—
227	C₂×C₄×C₂₃	—	4	1	no	—
228	C₂×C₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	4	1	no	—
230	C₂×C₄×C₂₃	—	16	2	no	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
237	C₈×C₂₃	—	4	1	no	—
238	C₈×C₂₃	—	4	1	no	—
239	C₈×C₂₃	—	4	1	no	—
240	C₈×C₂₃	—	4	1	no	—
241	C₈×C₂₃	—	4	1	no	—
242	C₈×C₂₃	—	4	1	no	—
243	C₈×C₂₃	—	4	1	no	—
244	C₈×C₂₃	#123	1	1	yes	characteristic
245	C₈×C₂₃	#123	1	1	yes	characteristic
246	C₈×C₂₃	#123	1	1	yes	characteristic
247	C₈×C₂₃	#123	1	1	yes	characteristic
248	C₂×#471	C₂×C₂₃	1	1	yes	characteristic
249	#2949	C₂×C₂₃	1	1	yes	characteristic, complemented
250	#2082	C₂×C₂₃	1	1	yes	characteristic, complemented
251	#2081	C₂×C₂₃	1	1	yes	characteristic, complemented
252	#2049	C₂×C₂₃	1	1	yes	characteristic, complemented
253	#2048	C₂×C₂₃	1	1	yes	characteristic
254	#1977	C₂×C₂₃	1	1	yes	characteristic, complemented
255	#1953	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#1905	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#1904	C₂×C₂₃	1	1	yes	characteristic
258	#1833	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#1830	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#1829	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1814	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1757	C₂×C₂₃	1	1	yes	characteristic, complemented
263	C₂²×C₂₃	—	4	1	no	—
264	C₂²×C₂₃	—	4	1	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	—	8	2	no	—
267	C₂²×C₂₃	—	8	1	no	—
268	C₂²×C₂₃	—	16	1	no	—
269	C₂²×C₂₃	—	16	1	no	—
270	C₂²×C₂₃	#529	1	1	yes	characteristic
271	C₂²×C₂₃	#546	1	1	yes	characteristic
272	C₂²×C₂₃	#555	1	1	yes	characteristic
273	C₄×C₂₃	—	4	2	no	—
274	C₄×C₂₃	—	4	1	no	—
275	C₄×C₂₃	—	4	2	no	—
276	C₄×C₂₃	—	8	1	no	—
277	C₄×C₂₃	—	8	1	no	—
278	C₄×C₂₃	—	8	1	no	—
279	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
280	C₄×C₂₃	#540	1	1	yes	characteristic
281	C₄×C₂₃	#546	1	1	yes	characteristic
282	C₄×C₂₃	#553	1	1	yes	characteristic
283	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
284	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	1	yes	characteristic
285	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
286	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
287	#519	C₂²×C₂₃	1	1	yes	characteristic
288	#518	C₂²×C₂₃	1	1	yes	characteristic
289	#471	—	2	2	no	—
290	#471	—	2	2	no	—
291	#440	—	2	2	no	—
292	C₄×SD₁₆	—	2	2	no	—
293	#430	C₂²×C₂₃	1	1	yes	characteristic
294	#430	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
297	#427	C₂²×C₂₃	1	1	yes	characteristic
298	#421	C₂²×C₂₃	1	1	yes	characteristic
299	#421	C₂²×C₂₃	1	1	yes	characteristic
300	#419	C₂²×C₂₃	1	1	yes	characteristic
301	#419	C₂²×C₂₃	1	1	yes	characteristic
302	#418	C₂²×C₂₃	1	1	yes	characteristic
303	#418	C₂²×C₂₃	1	1	yes	characteristic
304	#413	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
306	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
309	#410	C₂²×C₂₃	1	1	yes	characteristic
310	#405	C₂²×C₂₃	1	1	yes	characteristic
311	#405	C₂²×C₂₃	1	1	yes	characteristic
312	#344	C₂²×C₂₃	1	1	yes	characteristic
313	#344	C₂²×C₂₃	1	1	yes	characteristic
314	#344	C₂²×C₂₃	1	1	yes	characteristic
315	#344	C₂²×C₂₃	1	1	yes	characteristic
316	#344	C₂²×C₂₃	1	1	yes	characteristic
317	#344	C₂²×C₂₃	1	1	yes	characteristic
318	#344	C₂²×C₂₃	1	1	yes	characteristic
319	#344	C₂²×C₂₃	1	1	yes	characteristic
320	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
321	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×C₂₃	—	2	1	no	—
323	C₂×C₂₃	—	2	1	no	—
324	C₂×C₂₃	—	2	1	no	—
325	C₂×C₂₃	—	8	1	no	—
326	C₂×C₂₃	—	8	2	no	—
327	C₂×C₂₃	#2420	1	1	yes	characteristic
328	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
329	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
330	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
331	#137	—	2	2	no	—
332	#137	—	4	1	no	—
333	AGL(1,ℤ/8)	—	2	2	no	—
334	AGL(1,ℤ/8)	—	4	1	no	—
335	C₂×Q₁₆	—	2	1	no	—
336	C₂×SD₁₆	—	2	1	no	—
337	C₂×SD₁₆	—	2	1	no	—
338	C₂×SD₁₆	—	2	1	no	—
339	C₂×SD₁₆	—	2	1	no	—
340	C₂×SD₁₆	—	2	1	no	—
341	C₂×SD₁₆	—	2	1	no	—
342	C₂×SD₁₆	D₄×C₂₃	1	2	yes	—
343	C₂×SD₁₆	D₄×C₂₃	1	2	yes	—
344	C₂×D₈	—	2	1	no	—
345	#131	—	2	1	no	—
346	#131	—	2	1	no	—
347	#131	—	2	1	no	—
348	#131	—	2	1	no	—
349	C₂×M₁₆	—	2	1	no	—
350	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
351	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
352	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
353	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
354	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
355	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
356	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
357	C₂²×C₈	—	2	1	no	—
358	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
359	C₄×Q₈	—	2	2	no	—
360	C₄×Q₈	—	4	1	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	4	1	no	—
363	#117	—	2	2	no	—
364	#117	C₂³×C₂₃	1	1	yes	characteristic
365	#117	C₂³×C₂₃	1	1	yes	characteristic
366	#117	C₂³×C₂₃	1	1	yes	characteristic
367	#117	C₂³×C₂₃	1	1	yes	characteristic
368	C₂×C₂²⋊C₄	—	2	2	no	—
369	#108	—	2	1	no	—
370	#108	—	2	1	no	—
371	#108	—	2	1	no	—
372	#108	—	2	1	no	—
373	#108	—	4	1	no	—
374	#108	—	4	1	no	—
375	#107	—	2	1	no	—
376	#106	—	2	1	no	—
377	#104	—	4	1	no	—
378	#104	—	4	1	no	—
379	#103	—	2	2	no	—
380	#103	—	4	1	no	—
381	#102	—	2	2	no	—
382	#102	—	4	1	no	—
383	#101	—	2	2	no	—
384	#101	—	2	1	no	—
385	#101	—	4	1	no	—
386	#100	—	2	2	no	—
387	#100	—	2	1	no	—
388	#100	—	4	1	no	—
389	#99	—	2	2	no	—
390	#97	—	2	1	no	—
391	#97	—	2	1	no	—
392	C₄×C₈	—	2	1	no	—
393	C₄×C₈	—	2	1	no	—
394	C₂₃	#24987	1	1	yes	characteristic, direct product factor, Sylow
395	C₂⁴	—	2	1	no	—
396	#41	—	2	1	no	—
397	#41	—	2	1	no	—
398	#41	—	2	1	no	—
399	#41	—	2	1	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
406	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
414	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	4	2	no	—
418	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
419	Q₁₆	—	4	2	no	—
420	SD₁₆	—	2	2	no	—
421	SD₁₆	—	2	2	no	—
422	SD₁₆	—	2	2	no	—
423	SD₁₆	—	2	2	no	—
424	SD₁₆	—	4	2	no	—
425	SD₁₆	—	4	2	no	—
426	D₈	—	4	2	no	—
427	M₁₆	—	2	1	no	—
428	M₁₆	—	2	1	no	—
429	M₁₆	—	2	1	no	—
430	M₁₆	—	2	1	no	—
431	M₁₆	—	2	1	no	—
432	M₁₆	—	2	1	no	—
433	M₁₆	—	2	1	no	—
434	M₁₆	—	2	1	no	—
435	M₁₆	—	2	1	no	—
436	M₁₆	—	2	1	no	—
437	M₁₆	—	2	1	no	—
438	M₁₆	—	2	1	no	—
439	M₁₆	—	2	1	no	—
440	M₁₆	—	2	1	no	—
441	M₁₆	—	4	1	no	—
442	M₁₆	—	4	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	4	1	no	—
451	C₂×C₈	—	4	1	no	—
452	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
455	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
456	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
457	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	4	2	no	—
463	C₂²⋊C₄	—	2	1	no	—
464	C₂²⋊C₄	—	2	1	no	—
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	4	2	no	—
467	C₂²⋊C₄	#41×C₂₃	1	2	yes	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	4	2	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	4	1	no	—
476	C₂³	—	4	1	no	—
477	C₂³	—	16	1	no	—
478	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
479	Q₈	—	2	1	no	—
480	Q₈	—	2	2	no	—
481	Q₈	—	2	1	no	—
482	Q₈	—	2	2	no	—
483	Q₈	—	4	1	no	—
484	Q₈	—	4	1	no	—
485	D₄	—	2	1	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	1	no	—
488	D₄	—	2	2	no	—
489	D₄	—	4	1	no	—
490	D₄	—	4	1	no	—
491	D₄	—	4	1	no	—
492	D₄	—	4	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	16	2	no	—
504	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
505	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
506	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
507	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
508	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
509	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
510	C₈	—	4	1	no	—
511	C₈	—	4	1	no	—
512	C₈	—	4	1	no	—
513	C₈	—	4	1	no	—
514	C₈	—	4	1	no	—
515	C₈	—	4	1	no	—
516	C₈	—	4	1	no	—
517	C₈	C₂₃×#123	1	1	yes	characteristic
518	C₈	C₂₃×#123	1	1	yes	characteristic
519	C₈	C₂₃×#123	1	1	yes	characteristic
520	C₈	C₂₃×#123	1	1	yes	characteristic
521	C₂²	—	4	1	no	—
522	C₂²	—	4	1	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	8	2	no	—
525	C₂²	—	8	1	no	—
526	C₂²	—	16	1	no	—
527	C₂²	—	16	1	no	—
528	C₂²	C₂₃×#529	1	1	yes	characteristic
529	C₂²	C₂₃×#546	1	1	yes	characteristic
530	C₂²	C₂₃×#555	1	1	yes	characteristic
531	C₄	—	4	2	no	—
532	C₄	—	4	1	no	—
533	C₄	—	4	2	no	—
534	C₄	—	8	1	no	—
535	C₄	—	8	1	no	—
536	C₄	—	8	1	no	—
537	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
538	C₄	C₂₃×#540	1	1	yes	characteristic
539	C₄	C₂₃×#546	1	1	yes	characteristic
540	C₄	C₂₃×#553	1	1	yes	characteristic
541	C₂	—	2	1	no	—
542	C₂	—	2	1	no	—
543	C₂	—	2	1	no	—
544	C₂	—	8	2	no	—
545	C₂	—	8	1	no	—
546	C₂	C₂₃×#2420	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2420
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×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×#2949	C₂	semidirect
3. C₂₃×#2082	C₂	semidirect
4. C₂₃×#2081	C₂	semidirect
5. C₂₃×#2049	C₂	semidirect
7. C₂₃×#1977	C₂	semidirect
8. C₂₃×#1953	C₂	semidirect
9. C₂₃×#1905	C₂	semidirect
11. C₂₃×#1833	C₂	semidirect
12. C₂₃×#1830	C₂	semidirect
13. C₂₃×#1829	C₂	semidirect
14. C₂₃×#1814	C₂	semidirect
15. C₂₃×#1757	C₂	semidirect
199. #24987	C₂₃	direct (N × H)
249. #2949	C₂×C₂₃	semidirect
250. #2082	C₂×C₂₃	semidirect
251. #2081	C₂×C₂₃	semidirect
252. #2049	C₂×C₂₃	semidirect
254. #1977	C₂×C₂₃	semidirect
255. #1953	C₂×C₂₃	semidirect
256. #1905	C₂×C₂₃	semidirect
258. #1833	C₂×C₂₃	semidirect
259. #1830	C₂×C₂₃	semidirect
260. #1829	C₂×C₂₃	semidirect
261. #1814	C₂×C₂₃	semidirect
262. #1757	C₂×C₂₃	semidirect
394. C₂₃	#24987	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×#471	2
C₂₃×#2949	2
C₂₃×#2082	2
C₂₃×#2081	2
C₂₃×#2049	2
C₂₃×#2048	2
C₂₃×#1977	2
C₂₃×#1953	2
C₂₃×#1905	2
C₂₃×#1904	2
C₂₃×#1833	2
C₂₃×#1830	2
C₂₃×#1829	2
C₂₃×#1814	2
C₂₃×#1757	2
#24987	23

Subgroups — C23×#24987 #19172613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24987 #19172613