Subgroups — C₂₃×#21476 #19169102

Order 5888.

1322 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#456	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#137	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2327	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1886	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1882	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1860	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1857	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1793	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1759	C₂	1	2	yes	maximal, complemented
11	C₄×C₂₃×#101	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#137	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×#520	—	2	2	no	—
16	C₂₃×#520	—	2	2	no	—
17	C₂₃×#520	C₂²	1	2	yes	—
18	C₂₃×#519	—	2	2	no	—
19	C₂₃×#519	C₂²	1	1	yes	characteristic
20	C₄×#41×C₂₃	—	2	2	no	—
21	C₄×#41×C₂₃	—	2	2	no	—
22	C₄×#41×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
24	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#456	C₄	1	8	yes	complemented
27	C₂₃×#441	—	2	2	no	—
28	C₂₃×#440	—	2	2	no	—
29	C₄×Q₁₆×C₂₃	—	2	2	no	—
30	C₄×SD₁₆×C₂₃	—	2	2	no	—
31	C₂₃×#428	C₂²	1	2	yes	—
32	C₂×C₂₃×#104	C₂²	1	2	yes	—
33	C₂×C₂₃×#104	C₂²	1	2	yes	—
34	C₂₃×#419	C₂²	1	2	yes	—
35	C₂₃×#417	C₂²	1	2	yes	—
36	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
39	C₂₃×#391	C₂²	1	1	yes	characteristic
40	C₂₃×#391	C₂²	1	1	yes	characteristic
41	C₂₃×#387	—	2	2	no	—
42	C₂₃×#387	—	2	2	no	—
43	C₂₃×#382	C₂²	1	1	yes	characteristic
44	C₂₃×#376	C₂²	1	1	yes	characteristic
45	C₂₃×#343	C₂²	1	2	yes	—
46	C₂₃×#339	C₂²	1	2	yes	—
47	C₂₃×#339	C₂²	1	2	yes	—
48	C₂₃×2₋¹⁺⁴	—	2	4	no	—
49	C₂₃×2₋¹⁺⁴	—	2	2	no	—
50	C₂₃×2₋¹⁺⁴	C₂×C₄	1	4	yes	—
51	C₂×#41×C₂₃	—	2	2	no	—
52	C₂×#41×C₂₃	—	2	2	no	—
53	C₂×#41×C₂₃	C₂³	1	2	yes	—
54	C₂²×Q₈×C₂₃	—	2	2	no	—
55	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂₃×#137	—	2	4	no	—
58	C₂₃×#137	C₂×C₄	1	8	yes	complemented
59	C₂×Q₁₆×C₂₃	—	2	2	no	—
60	C₂×SD₁₆×C₂₃	—	2	2	no	—
61	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂₃×C₄⋊Q₈	—	2	2	no	—
64	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
65	C₄×Q₈×C₂₃	—	2	2	no	—
66	C₄×Q₈×C₂₃	—	2	4	no	—
67	C₄×Q₈×C₂₃	—	2	2	no	—
68	C₄×Q₈×C₂₃	—	2	2	no	—
69	C₄×Q₈×C₂₃	—	2	2	no	—
70	C₄×Q₈×C₂₃	—	2	2	no	—
71	C₄×Q₈×C₂₃	—	2	2	no	—
72	C₄×Q₈×C₂₃	—	2	2	no	—
73	C₄×Q₈×C₂₃	—	2	2	no	—
74	C₄×Q₈×C₂₃	—	2	2	no	—
75	C₄×Q₈×C₂₃	D₄	1	2	yes	—
76	C₄×Q₈×C₂₃	D₄	1	2	yes	—
77	C₄×D₄×C₂₃	—	2	2	no	—
78	C₄×D₄×C₂₃	—	2	4	no	—
79	C₄×D₄×C₂₃	—	2	2	no	—
80	C₄×D₄×C₂₃	—	2	2	no	—
81	C₄×D₄×C₂₃	—	2	2	no	—
82	C₄×D₄×C₂₃	—	2	4	no	—
83	C₄×D₄×C₂₃	—	2	2	no	—
84	C₄×D₄×C₂₃	D₄	1	2	yes	—
85	C₄×D₄×C₂₃	D₄	1	2	yes	—
86	C₂₃×#117	—	2	4	no	—
87	C₂₃×#117	—	2	4	no	—
88	C₂₃×#117	—	2	2	no	—
89	C₂₃×#117	—	2	2	no	—
90	C₂₃×#117	—	2	2	no	—
91	C₂₃×#117	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	—	2	4	no	—
100	C₂×C₄²×C₂₃	—	2	2	no	—
101	C₂×C₄²×C₂₃	—	2	1	no	—
102	C₂×C₄²×C₂₃	D₄	1	2	yes	—
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂₃×#107	—	2	2	no	—
107	C₂₃×#106	—	2	2	no	—
108	C₂₃×#104	—	2	4	no	—
109	C₂₃×#104	C₂×C₄	1	8	yes	complemented
110	C₂₃×#103	—	2	2	no	—
111	C₂₃×#103	—	2	2	no	—
112	C₂₃×#103	—	2	2	no	—
113	C₂₃×#102	—	2	2	no	—
114	C₂₃×#101	—	2	2	no	—
115	C₂₃×#101	C₂×C₄	1	4	yes	complemented
116	C₂₃×#97	—	2	2	no	—
117	C₄×C₈×C₂₃	—	2	2	no	—
118	C₂₃×#95	—	2	1	no	—
119	C₂₃×#95	—	2	1	no	—
120	C₂₃×#95	—	2	2	no	—
121	C₂₃×#95	—	2	2	no	—
122	C₂₃×#95	—	2	2	no	—
123	#41×C₂₃	—	2	8	no	—
124	#41×C₂₃	—	2	8	no	—
125	#41×C₂₃	—	2	8	no	—
126	#41×C₂₃	—	2	8	no	—
127	#41×C₂₃	—	2	2	no	—
128	#41×C₂₃	—	2	2	no	—
129	#41×C₂₃	C₂²×C₄	1	4	yes	—
130	#41×C₂₃	#41	1	4	yes	—
131	C₂×Q₈×C₂₃	—	2	2	no	—
132	C₂×Q₈×C₂₃	—	2	8	no	—
133	C₂×Q₈×C₂₃	—	2	8	no	—
134	C₂×Q₈×C₂₃	—	2	2	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	C₂²×C₄	1	2	yes	—
140	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
141	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
142	C₂×Q₈×C₂₃	#41	1	2	yes	—
143	C₂×D₄×C₂₃	—	2	2	no	—
144	C₂×D₄×C₂₃	—	2	2	no	—
145	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₂₃	—	2	4	no	—
147	C₂²×C₄×C₂₃	—	2	2	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	—	2	4	no	—
151	C₂²×C₄×C₂₃	—	2	2	no	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
160	Q₁₆×C₂₃	—	2	8	no	—
161	SD₁₆×C₂₃	—	2	8	no	—
162	M₁₆×C₂₃	—	2	2	no	—
163	M₁₆×C₂₃	—	2	2	no	—
164	M₁₆×C₂₃	—	2	2	no	—
165	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
166	C₂×C₈×C₂₃	—	2	2	no	—
167	C₂×C₈×C₂₃	—	2	2	no	—
168	C₄⋊C₄×C₂₃	—	2	4	no	—
169	C₄⋊C₄×C₂₃	—	2	2	no	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	8	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	4	no	—
174	C₄⋊C₄×C₂₃	—	2	2	no	—
175	C₄⋊C₄×C₂₃	—	2	2	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	—	4	1	no	—
178	C₄⋊C₄×C₂₃	—	4	1	no	—
179	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
181	C₂²⋊C₄×C₂₃	—	2	4	no	—
182	C₂²⋊C₄×C₂₃	—	2	2	no	—
183	C₂²⋊C₄×C₂₃	—	2	2	no	—
184	C₂²⋊C₄×C₂₃	—	2	2	no	—
185	C₄²×C₂₃	—	2	4	no	—
186	C₄²×C₂₃	—	2	4	no	—
187	C₄²×C₂₃	—	2	2	no	—
188	C₄²×C₂₃	—	2	2	no	—
189	C₄²×C₂₃	—	2	2	no	—
190	C₄²×C₂₃	—	2	2	no	—
191	C₄²×C₂₃	—	2	2	no	—
192	C₄²×C₂₃	—	2	1	no	—
193	C₄²×C₂₃	—	2	2	no	—
194	C₄²×C₂₃	—	4	2	no	—
195	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
196	C₄²×C₂₃	C₂×D₄	1	2	yes	—
197	C₄²×C₂₃	C₂×D₄	1	2	yes	—
198	#21476	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₂₃	—	2	2	no	—
200	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₂₃	—	2	8	no	—
202	Q₈×C₂₃	—	2	4	no	—
203	Q₈×C₂₃	—	2	2	no	—
204	Q₈×C₂₃	—	2	4	no	—
205	Q₈×C₂₃	—	2	4	no	—
206	Q₈×C₂₃	—	4	2	no	—
207	Q₈×C₂₃	C₄×D₄	1	4	yes	—
208	D₄×C₂₃	—	2	8	no	—
209	D₄×C₂₃	—	2	4	no	—
210	D₄×C₂₃	—	2	4	no	—
211	D₄×C₂₃	—	2	2	no	—
212	D₄×C₂₃	C₄×D₄	1	4	yes	—
213	C₂×C₄×C₂₃	—	2	16	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	4	no	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	4	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	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	2	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	2	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	8	2	no	—
238	C₂×C₄×C₂₃	—	8	4	no	—
239	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
240	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
241	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
242	C₂×C₄×C₂₃	#120	1	2	yes	—
243	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
244	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
246	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
247	C₈×C₂₃	—	2	4	no	—
248	C₈×C₂₃	—	4	2	no	—
249	C₄×2₋¹⁺⁴	C₂×C₂₃	1	1	yes	characteristic
250	C₂×#456	C₂×C₂₃	1	1	yes	characteristic
251	C₄×#137	C₂×C₂₃	1	2	yes	complemented
252	#2327	C₂×C₂₃	1	1	yes	characteristic, complemented
253	#1886	C₂×C₂₃	1	1	yes	characteristic, complemented
254	#1882	C₂×C₂₃	1	2	yes	complemented
255	#1860	C₂×C₂₃	1	2	yes	complemented
256	#1857	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#1793	C₂×C₂₃	1	1	yes	characteristic
258	#1759	C₂×C₂₃	1	2	yes	complemented
259	C₄×#101	C₂×C₂₃	1	1	yes	characteristic, complemented
260	C₂²×C₂₃	—	2	1	no	—
261	C₂²×C₂₃	—	2	1	no	—
262	C₂²×C₂₃	—	2	4	no	—
263	C₂²×C₂₃	—	4	2	no	—
264	C₂²×C₂₃	—	4	2	no	—
265	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
266	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
267	C₂²×C₂₃	#525	1	1	yes	characteristic
268	C₄×C₂₃	—	2	1	no	—
269	C₄×C₂₃	—	2	1	no	—
270	C₄×C₂₃	—	2	2	no	—
271	C₄×C₂₃	—	2	4	no	—
272	C₄×C₂₃	—	2	4	no	—
273	C₄×C₂₃	—	4	2	no	—
274	C₄×C₂₃	—	4	2	no	—
275	C₄×C₂₃	—	4	2	no	—
276	C₄×C₂₃	—	4	2	no	—
277	C₄×C₂₃	—	4	2	no	—
278	C₄×C₂₃	—	8	1	no	—
279	C₄×C₂₃	—	8	1	no	—
280	C₄×C₂₃	C₂×C₄×D₄	1	2	yes	—
281	C₄×C₂₃	#525	1	2	yes	—
282	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
283	C₂×#137	C₂²×C₂₃	1	2	yes	—
284	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
285	#520	—	2	2	no	—
286	#520	—	2	2	no	—
287	#520	C₂²×C₂₃	1	2	yes	—
288	#519	—	2	2	no	—
289	#519	C₂²×C₂₃	1	1	yes	characteristic
290	C₄×#41	—	2	2	no	—
291	C₄×#41	—	2	2	no	—
292	C₄×#41	C₂²×C₂₃	1	2	yes	—
293	C₂×C₄×Q₈	—	2	2	no	—
294	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
296	#456	C₄×C₂₃	1	8	yes	complemented
297	#441	—	2	2	no	—
298	#440	—	2	2	no	—
299	C₄×Q₁₆	—	2	2	no	—
300	C₄×SD₁₆	—	2	2	no	—
301	#428	C₂²×C₂₃	1	2	yes	—
302	C₂×#104	C₂²×C₂₃	1	2	yes	—
303	C₂×#104	C₂²×C₂₃	1	2	yes	—
304	#419	C₂²×C₂₃	1	2	yes	—
305	#417	C₂²×C₂₃	1	2	yes	—
306	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
308	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
309	#391	C₂²×C₂₃	1	1	yes	characteristic
310	#391	C₂²×C₂₃	1	1	yes	characteristic
311	#387	—	2	2	no	—
312	#387	—	2	2	no	—
313	#382	C₂²×C₂₃	1	1	yes	characteristic
314	#376	C₂²×C₂₃	1	1	yes	characteristic
315	#343	C₂²×C₂₃	1	2	yes	—
316	#339	C₂²×C₂₃	1	2	yes	—
317	#339	C₂²×C₂₃	1	2	yes	—
318	C₂×C₂₃	—	2	1	no	—
319	C₂×C₂₃	—	2	1	no	—
320	C₂×C₂₃	—	4	4	no	—
321	C₂×C₂₃	C₄×#120	1	1	yes	characteristic
322	C₂×C₂₃	#3020	1	1	yes	characteristic
323	C₂×C₂₃	#3022	1	1	yes	characteristic
324	2₋¹⁺⁴	—	2	4	no	—
325	2₋¹⁺⁴	—	2	2	no	—
326	2₋¹⁺⁴	C₂×C₄×C₂₃	1	4	yes	—
327	C₂×#41	—	2	2	no	—
328	C₂×#41	—	2	2	no	—
329	C₂×#41	C₂³×C₂₃	1	2	yes	—
330	C₂²×Q₈	—	2	2	no	—
331	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
332	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
333	#137	—	2	4	no	—
334	#137	C₂×C₄×C₂₃	1	8	yes	complemented
335	C₂×Q₁₆	—	2	2	no	—
336	C₂×SD₁₆	—	2	2	no	—
337	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
338	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	—	2	4	no	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	—	2	2	no	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	D₄×C₂₃	1	2	yes	—
352	C₄×Q₈	D₄×C₂₃	1	2	yes	—
353	C₄×D₄	—	2	2	no	—
354	C₄×D₄	—	2	4	no	—
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	4	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	D₄×C₂₃	1	2	yes	—
361	C₄×D₄	D₄×C₂₃	1	2	yes	—
362	#117	—	2	4	no	—
363	#117	—	2	4	no	—
364	#117	—	2	2	no	—
365	#117	—	2	2	no	—
366	#117	—	2	2	no	—
367	#117	C₂³×C₂₃	1	2	yes	—
368	C₂×C₄⋊C₄	—	2	4	no	—
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	—	2	2	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
375	C₂×C₄²	—	2	4	no	—
376	C₂×C₄²	—	2	2	no	—
377	C₂×C₄²	—	2	1	no	—
378	C₂×C₄²	D₄×C₂₃	1	2	yes	—
379	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
381	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
382	#107	—	2	2	no	—
383	#106	—	2	2	no	—
384	#104	—	2	4	no	—
385	#104	C₂×C₄×C₂₃	1	8	yes	complemented
386	#103	—	2	2	no	—
387	#103	—	2	2	no	—
388	#103	—	2	2	no	—
389	#102	—	2	2	no	—
390	#101	—	2	2	no	—
391	#101	C₂×C₄×C₂₃	1	4	yes	complemented
392	#97	—	2	2	no	—
393	C₄×C₈	—	2	2	no	—
394	#95	—	2	1	no	—
395	#95	—	2	1	no	—
396	#95	—	2	2	no	—
397	#95	—	2	2	no	—
398	#95	—	2	2	no	—
399	C₂₃	#21476	1	1	yes	characteristic, direct product factor, Sylow
400	#41	—	2	8	no	—
401	#41	—	2	8	no	—
402	#41	—	2	8	no	—
403	#41	—	2	8	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	C₂²×C₄×C₂₃	1	4	yes	—
407	#41	#41×C₂₃	1	4	yes	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	8	no	—
410	C₂×Q₈	—	2	8	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
417	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
418	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
419	C₂×Q₈	#41×C₂₃	1	2	yes	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
433	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
434	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
437	Q₁₆	—	2	8	no	—
438	SD₁₆	—	2	8	no	—
439	M₁₆	—	2	2	no	—
440	M₁₆	—	2	2	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	2	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	4	1	no	—
455	C₄⋊C₄	—	4	1	no	—
456	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
457	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	2	no	—
460	C₂²⋊C₄	—	2	2	no	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₄²	—	2	4	no	—
463	C₄²	—	2	4	no	—
464	C₄²	—	2	2	no	—
465	C₄²	—	2	2	no	—
466	C₄²	—	2	2	no	—
467	C₄²	—	2	2	no	—
468	C₄²	—	2	2	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	2	no	—
471	C₄²	—	4	2	no	—
472	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
473	C₄²	C₂×D₄×C₂₃	1	2	yes	—
474	C₄²	C₂×D₄×C₂₃	1	2	yes	—
475	C₂³	—	2	2	no	—
476	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
477	Q₈	—	2	8	no	—
478	Q₈	—	2	4	no	—
479	Q₈	—	2	2	no	—
480	Q₈	—	2	4	no	—
481	Q₈	—	2	4	no	—
482	Q₈	—	4	2	no	—
483	Q₈	C₄×D₄×C₂₃	1	4	yes	—
484	D₄	—	2	8	no	—
485	D₄	—	2	4	no	—
486	D₄	—	2	4	no	—
487	D₄	—	2	2	no	—
488	D₄	C₄×D₄×C₂₃	1	4	yes	—
489	C₂×C₄	—	2	16	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	8	2	no	—
514	C₂×C₄	—	8	4	no	—
515	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
516	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
517	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
518	C₂×C₄	C₂₃×#120	1	2	yes	—
519	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
521	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
522	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
523	C₈	—	2	4	no	—
524	C₈	—	4	2	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
531	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
532	C₂²	C₂₃×#525	1	1	yes	characteristic
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	2	no	—
536	C₄	—	2	4	no	—
537	C₄	—	2	4	no	—
538	C₄	—	4	2	no	—
539	C₄	—	4	2	no	—
540	C₄	—	4	2	no	—
541	C₄	—	4	2	no	—
542	C₄	—	4	2	no	—
543	C₄	—	8	1	no	—
544	C₄	—	8	1	no	—
545	C₄	C₂×C₄×D₄×C₂₃	1	2	yes	—
546	C₄	C₂₃×#525	1	2	yes	—
547	C₂	—	2	1	no	—
548	C₂	—	2	1	no	—
549	C₂	—	4	4	no	—
550	C₂	C₄×C₂₃×#120	1	1	yes	characteristic
551	C₂	C₂₃×#3020	1	1	yes	characteristic
552	C₂	C₂₃×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₂₃×#137	C₂	semidirect
4. C₂₃×#2327	C₂	semidirect
5. C₂₃×#1886	C₂	semidirect
6. C₂₃×#1882	C₂	semidirect
7. C₂₃×#1860	C₂	semidirect
8. C₂₃×#1857	C₂	semidirect
10. C₂₃×#1759	C₂	semidirect
11. C₄×C₂₃×#101	C₂	semidirect
26. C₂₃×#456	C₄	semidirect
58. C₂₃×#137	C₂×C₄	semidirect
64. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
109. C₂₃×#104	C₂×C₄	semidirect
115. C₂₃×#101	C₂×C₄	semidirect
198. #21476	C₂₃	direct (N × H)
251. C₄×#137	C₂×C₂₃	semidirect
252. #2327	C₂×C₂₃	semidirect
253. #1886	C₂×C₂₃	semidirect
254. #1882	C₂×C₂₃	semidirect
255. #1860	C₂×C₂₃	semidirect
256. #1857	C₂×C₂₃	semidirect
258. #1759	C₂×C₂₃	semidirect
259. C₄×#101	C₂×C₂₃	semidirect
296. #456	C₄×C₂₃	semidirect
334. #137	C₂×C₄×C₂₃	semidirect
340. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
385. #104	C₂×C₄×C₂₃	semidirect
391. #101	C₂×C₄×C₂₃	semidirect
399. C₂₃	#21476	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×2₋¹⁺⁴	2
C₂×C₂₃×#456	2
C₄×C₂₃×#137	2
C₂₃×#2327	2
C₂₃×#1886	2
C₂₃×#1882	2
C₂₃×#1860	2
C₂₃×#1857	2
C₂₃×#1793	2
C₂₃×#1759	2
C₄×C₂₃×#101	2
#21476	23

Subgroups — C23×#21476 #19169102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21476 #19169102