Subgroups — C₂₃×#23015 #19170641

Order 5888.

892 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3035	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#443	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1962	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1937	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1876	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1866	C₂	1	2	yes	maximal, complemented
10	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×#104	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	2	yes	—
13	C₂×C₂₃×#131	C₂²	1	2	yes	—
14	C₂×C₂₃×#131	C₂²	1	2	yes	—
15	C₂₃×#532	C₂²	1	1	yes	characteristic
16	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#464	C₄	1	2	yes	complemented
18	C₂₃×#463	C₄	1	2	yes	complemented
19	C₂₃×#462	C₄	1	1	yes	characteristic, complemented
20	C₂₃×#461	C₄	1	1	yes	characteristic, complemented
21	C₂₃×#460	C₄	1	1	yes	characteristic, complemented
22	C₂₃×#459	C₄	1	1	yes	characteristic, complemented
23	C₂₃×#443	—	2	8	no	—
24	C₂₃×#436	—	2	1	no	—
25	C₂₃×#435	—	2	2	no	—
26	C₈×D₄×C₂₃	—	2	1	no	—
27	C₂×C₂₃×#108	C₂²	1	2	yes	—
28	C₂₃×#424	—	2	2	no	—
29	C₂₃×#424	C₂²	1	2	yes	—
30	C₂₃×#423	—	2	1	no	—
31	C₂₃×#423	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
33	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#104	C₂²	1	2	yes	—
36	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#104	C₂²	1	2	yes	—
38	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
39	C₂₃×#416	C₂²	1	2	yes	—
40	C₂₃×#408	C₂²	1	2	yes	—
41	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
43	C₄³×C₂₃	C₂²	1	1	yes	characteristic
44	C₂₃×#341	C₂²	1	2	yes	—
45	C₂₃×#339	C₂²	1	1	yes	characteristic
46	C₂₃×#339	C₂²	1	1	yes	characteristic
47	C₂₃×#336	C₂²	1	2	yes	—
48	C₂×#41×C₂₃	C₂³	1	2	yes	—
49	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#135	—	2	4	no	—
51	C₂₃×#135	—	2	4	no	—
52	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
53	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
54	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
56	C₂₃×#131	—	2	4	no	—
57	C₂₃×#131	—	2	4	no	—
58	C₂₃×#131	—	4	2	no	—
59	C₂₃×#131	—	4	2	no	—
60	C₂×M₁₆×C₂₃	—	2	2	no	—
61	C₂×M₁₆×C₂₃	—	2	2	no	—
62	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	—	2	2	no	—
66	C₂²×C₈×C₂₃	—	2	2	no	—
67	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
69	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
70	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
71	C₂₃×#124	C₂×C₄	1	2	yes	—
72	C₂₃×#121	—	2	1	no	—
73	C₂₃×#121	—	2	1	no	—
74	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₂₃	—	2	1	no	—
77	C₄×D₄×C₂₃	—	2	1	no	—
78	C₄×D₄×C₂₃	—	2	1	no	—
79	C₄×D₄×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
81	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
82	C₂₃×#117	—	2	1	no	—
83	C₂₃×#117	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₂₃	—	2	1	no	—
85	C₂×C₄²×C₂₃	—	2	1	no	—
86	C₂×C₄²×C₂₃	—	2	1	no	—
87	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
90	C₂₃×#108	—	2	4	no	—
91	C₂₃×C₄⋊C₈	—	2	2	no	—
92	C₂₃×C₄⋊C₈	—	2	2	no	—
93	C₂₃×C₄⋊C₈	—	2	1	no	—
94	C₂₃×C₄⋊C₈	—	2	1	no	—
95	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
97	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
98	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
99	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
100	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
101	C₂₃×#104	—	2	2	no	—
102	C₂₃×#104	—	2	4	no	—
103	C₂₃×#104	—	4	2	no	—
104	C₂₃×#104	D₄	1	4	yes	complemented
105	C₂₃×#103	C₂×C₄	1	2	yes	—
106	C₂₃×#103	C₂×C₄	1	2	yes	—
107	C₂₃×#102	C₂×C₄	1	2	yes	—
108	C₂₃×#102	C₂×C₄	1	2	yes	—
109	C₂₃×C₂²⋊C₈	—	2	2	no	—
110	C₂₃×C₂²⋊C₈	—	2	2	no	—
111	C₂₃×#97	—	2	1	no	—
112	C₄×C₈×C₂₃	—	2	4	no	—
113	C₄×C₈×C₂₃	—	2	1	no	—
114	#41×C₂₃	—	2	1	no	—
115	#41×C₂₃	—	2	1	no	—
116	#41×C₂₃	—	2	2	no	—
117	#41×C₂₃	—	2	2	no	—
118	#41×C₂₃	—	2	2	no	—
119	#41×C₂₃	—	2	2	no	—
120	#41×C₂₃	C₂×D₄	1	2	yes	—
121	#41×C₂₃	#41	1	2	yes	—
122	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
123	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×D₄×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	2	2	no	—
126	C₂×D₄×C₂₃	—	2	1	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	—	2	1	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	2	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₂₃	—	2	4	no	—
140	SD₁₆×C₂₃	—	2	4	no	—
141	SD₁₆×C₂₃	—	2	4	no	—
142	D₈×C₂₃	—	2	4	no	—
143	M₁₆×C₂₃	—	2	1	no	—
144	M₁₆×C₂₃	—	2	2	no	—
145	M₁₆×C₂₃	—	2	1	no	—
146	M₁₆×C₂₃	—	2	2	no	—
147	M₁₆×C₂₃	—	2	1	no	—
148	M₁₆×C₂₃	—	4	4	no	—
149	M₁₆×C₂₃	—	4	4	no	—
150	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₂₃	—	2	2	no	—
152	C₂×C₈×C₂₃	—	2	2	no	—
153	C₂×C₈×C₂₃	—	2	1	no	—
154	C₂×C₈×C₂₃	—	2	2	no	—
155	C₂×C₈×C₂₃	—	2	2	no	—
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	4	4	no	—
159	C₂×C₈×C₂₃	—	4	4	no	—
160	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
161	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
169	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
170	C₂²⋊C₄×C₂₃	—	2	1	no	—
171	C₂²⋊C₄×C₂₃	—	2	1	no	—
172	C₂²⋊C₄×C₂₃	—	2	1	no	—
173	C₂²⋊C₄×C₂₃	—	2	2	no	—
174	C₂²⋊C₄×C₂₃	—	2	2	no	—
175	C₄²×C₂₃	—	2	4	no	—
176	C₄²×C₂₃	—	2	1	no	—
177	C₄²×C₂₃	—	2	1	no	—
178	C₄²×C₂₃	—	2	1	no	—
179	C₄²×C₂₃	—	2	1	no	—
180	C₄²×C₂₃	—	2	1	no	—
181	C₄²×C₂₃	—	2	1	no	—
182	C₄²×C₂₃	—	4	2	no	—
183	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
184	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
185	C₄²×C₂₃	C₂×D₄	1	2	yes	—
186	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
188	#23015	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₂₃	—	2	1	no	—
190	C₂³×C₂₃	—	2	2	no	—
191	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
192	Q₈×C₂₃	—	2	1	no	—
193	Q₈×C₂₃	—	2	2	no	—
194	Q₈×C₂₃	—	2	2	no	—
195	Q₈×C₂₃	C₄×D₄	1	2	yes	—
196	D₄×C₂₃	—	2	2	no	—
197	D₄×C₂₃	—	2	2	no	—
198	D₄×C₂₃	—	2	2	no	—
199	D₄×C₂₃	—	2	1	no	—
200	D₄×C₂₃	—	2	2	no	—
201	D₄×C₂₃	—	4	2	no	—
202	D₄×C₂₃	—	4	2	no	—
203	D₄×C₂₃	—	4	2	no	—
204	D₄×C₂₃	—	8	1	no	—
205	D₄×C₂₃	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	1	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	1	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	4	1	no	—
218	C₂×C₄×C₂₃	—	4	1	no	—
219	C₂×C₄×C₂₃	—	4	1	no	—
220	C₂×C₄×C₂₃	—	4	2	no	—
221	C₂×C₄×C₂₃	—	4	2	no	—
222	C₂×C₄×C₂₃	—	4	2	no	—
223	C₂×C₄×C₂₃	—	8	1	no	—
224	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
225	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
226	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
227	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
228	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
229	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
232	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
233	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
234	C₈×C₂₃	—	2	2	no	—
235	C₈×C₂₃	—	2	2	no	—
236	C₈×C₂₃	—	2	2	no	—
237	C₈×C₂₃	—	2	2	no	—
238	C₈×C₂₃	—	2	2	no	—
239	C₈×C₂₃	—	4	2	no	—
240	C₈×C₂₃	—	4	1	no	—
241	#3035	C₂×C₂₃	1	1	yes	characteristic
242	C₂×#443	C₂×C₂₃	1	2	yes	complemented
243	#2935	C₂×C₂₃	1	1	yes	characteristic, complemented
244	#2906	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#1962	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#1950	C₂×C₂₃	1	1	yes	characteristic, complemented
247	#1937	C₂×C₂₃	1	2	yes	complemented
248	#1876	C₂×C₂₃	1	2	yes	complemented
249	#1866	C₂×C₂₃	1	2	yes	complemented
250	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
251	C₄×#104	C₂×C₂₃	1	1	yes	characteristic, complemented
252	C₂²×C₂₃	—	2	1	no	—
253	C₂²×C₂₃	—	2	1	no	—
254	C₂²×C₂₃	—	2	2	no	—
255	C₂²×C₂₃	—	4	1	no	—
256	C₂²×C₂₃	—	4	1	no	—
257	C₂²×C₂₃	—	4	2	no	—
258	C₂²×C₂₃	—	4	2	no	—
259	C₂²×C₂₃	—	4	2	no	—
260	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
261	C₂²×C₂₃	#547	1	1	yes	characteristic
262	C₂²×C₂₃	#553	1	1	yes	characteristic
263	C₄×C₂₃	—	2	4	no	—
264	C₄×C₂₃	—	2	2	no	—
265	C₄×C₂₃	—	2	1	no	—
266	C₄×C₂₃	—	2	1	no	—
267	C₄×C₂₃	—	2	1	no	—
268	C₄×C₂₃	—	4	2	no	—
269	C₄×C₂₃	—	4	2	no	—
270	C₄×C₂₃	—	4	1	no	—
271	C₄×C₂₃	—	4	1	no	—
272	C₄×C₂₃	#443	1	2	yes	complemented
273	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
274	C₄×C₂₃	#520	1	1	yes	characteristic
275	C₄×C₂₃	D₄²	1	1	yes	characteristic
276	C₄×C₂₃	#548	1	1	yes	characteristic
277	C₂×#135	C₂²×C₂₃	1	2	yes	—
278	C₂×#131	C₂²×C₂₃	1	2	yes	—
279	C₂×#131	C₂²×C₂₃	1	2	yes	—
280	#532	C₂²×C₂₃	1	1	yes	characteristic
281	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
282	#464	C₄×C₂₃	1	2	yes	complemented
283	#463	C₄×C₂₃	1	2	yes	complemented
284	#462	C₄×C₂₃	1	1	yes	characteristic, complemented
285	#461	C₄×C₂₃	1	1	yes	characteristic, complemented
286	#460	C₄×C₂₃	1	1	yes	characteristic, complemented
287	#459	C₄×C₂₃	1	1	yes	characteristic, complemented
288	#443	—	2	8	no	—
289	#436	—	2	1	no	—
290	#435	—	2	2	no	—
291	C₈×D₄	—	2	1	no	—
292	C₂×#108	C₂²×C₂₃	1	2	yes	—
293	#424	—	2	2	no	—
294	#424	C₂²×C₂₃	1	2	yes	—
295	#423	—	2	1	no	—
296	#423	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×C₄⋊C₈	—	2	1	no	—
298	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
300	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
301	C₂×#104	C₂²×C₂₃	1	2	yes	—
302	C₂×#104	C₂²×C₂₃	1	2	yes	—
303	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
304	#416	C₂²×C₂₃	1	2	yes	—
305	#408	C₂²×C₂₃	1	2	yes	—
306	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
308	C₄³	C₂²×C₂₃	1	1	yes	characteristic
309	#341	C₂²×C₂₃	1	2	yes	—
310	#339	C₂²×C₂₃	1	1	yes	characteristic
311	#339	C₂²×C₂₃	1	1	yes	characteristic
312	#336	C₂²×C₂₃	1	2	yes	—
313	C₂×C₂₃	—	2	1	no	—
314	C₂×C₂₃	—	2	1	no	—
315	C₂×C₂₃	—	4	2	no	—
316	C₂×C₂₃	—	8	2	no	—
317	C₂×C₂₃	#2361	1	1	yes	characteristic
318	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
319	C₂×C₂₃	#2956	1	1	yes	characteristic
320	C₂×#41	C₂³×C₂₃	1	2	yes	—
321	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
322	#135	—	2	4	no	—
323	#135	—	2	4	no	—
324	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
325	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
326	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
327	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
328	#131	—	2	4	no	—
329	#131	—	2	4	no	—
330	#131	—	4	2	no	—
331	#131	—	4	2	no	—
332	C₂×M₁₆	—	2	2	no	—
333	C₂×M₁₆	—	2	2	no	—
334	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
335	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
336	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
337	C₂²×C₈	—	2	2	no	—
338	C₂²×C₈	—	2	2	no	—
339	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
340	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
341	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
342	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
343	#124	C₂×C₄×C₂₃	1	2	yes	—
344	#121	—	2	1	no	—
345	#121	—	2	1	no	—
346	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
347	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
348	C₄×D₄	—	2	1	no	—
349	C₄×D₄	—	2	1	no	—
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
353	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
354	#117	—	2	1	no	—
355	#117	C₂³×C₂₃	1	1	yes	characteristic
356	C₂×C₄²	—	2	1	no	—
357	C₂×C₄²	—	2	1	no	—
358	C₂×C₄²	—	2	1	no	—
359	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
362	#108	—	2	4	no	—
363	C₄⋊C₈	—	2	2	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	1	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
368	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
369	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
370	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
371	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
372	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
373	#104	—	2	2	no	—
374	#104	—	2	4	no	—
375	#104	—	4	2	no	—
376	#104	D₄×C₂₃	1	4	yes	complemented
377	#103	C₂×C₄×C₂₃	1	2	yes	—
378	#103	C₂×C₄×C₂₃	1	2	yes	—
379	#102	C₂×C₄×C₂₃	1	2	yes	—
380	#102	C₂×C₄×C₂₃	1	2	yes	—
381	C₂²⋊C₈	—	2	2	no	—
382	C₂²⋊C₈	—	2	2	no	—
383	#97	—	2	1	no	—
384	C₄×C₈	—	2	4	no	—
385	C₄×C₈	—	2	1	no	—
386	C₂₃	#23015	1	1	yes	characteristic, direct product factor, Sylow
387	#41	—	2	1	no	—
388	#41	—	2	1	no	—
389	#41	—	2	2	no	—
390	#41	—	2	2	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	C₂×D₄×C₂₃	1	2	yes	—
394	#41	#41×C₂₃	1	2	yes	—
395	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
396	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
402	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
410	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
412	Q₁₆	—	2	4	no	—
413	SD₁₆	—	2	4	no	—
414	SD₁₆	—	2	4	no	—
415	D₈	—	2	4	no	—
416	M₁₆	—	2	1	no	—
417	M₁₆	—	2	2	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	4	4	no	—
422	M₁₆	—	4	4	no	—
423	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	4	4	no	—
432	C₂×C₈	—	4	4	no	—
433	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
434	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
435	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
436	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
442	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
443	C₂²⋊C₄	—	2	1	no	—
444	C₂²⋊C₄	—	2	1	no	—
445	C₂²⋊C₄	—	2	1	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₄²	—	2	4	no	—
449	C₄²	—	2	1	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	2	1	no	—
455	C₄²	—	4	2	no	—
456	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
457	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
458	C₄²	C₂×D₄×C₂₃	1	2	yes	—
459	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	2	2	no	—
463	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	2	2	no	—
467	Q₈	C₄×D₄×C₂₃	1	2	yes	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	2	no	—
471	D₄	—	2	1	no	—
472	D₄	—	2	2	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	1	no	—
477	D₄	C₄×D₄×C₂₃	1	2	yes	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	8	1	no	—
496	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
497	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
498	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
499	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
500	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
503	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
504	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
505	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
506	C₈	—	2	2	no	—
507	C₈	—	2	2	no	—
508	C₈	—	2	2	no	—
509	C₈	—	2	2	no	—
510	C₈	—	2	2	no	—
511	C₈	—	4	1	no	—
512	C₈	—	4	2	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
522	C₂²	C₂₃×#547	1	1	yes	characteristic
523	C₂²	C₂₃×#553	1	1	yes	characteristic
524	C₄	—	2	4	no	—
525	C₄	—	2	1	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	2	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	1	no	—
531	C₄	—	4	1	no	—
532	C₄	—	4	2	no	—
533	C₄	C₂₃×#443	1	2	yes	complemented
534	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
535	C₄	C₂₃×#520	1	1	yes	characteristic
536	C₄	D₄²×C₂₃	1	1	yes	characteristic
537	C₄	C₂₃×#548	1	1	yes	characteristic
538	C₂	—	2	1	no	—
539	C₂	—	2	1	no	—
540	C₂	—	4	2	no	—
541	C₂	—	8	2	no	—
542	C₂	C₂₃×#2361	1	1	yes	characteristic
543	C₂	C₂×C₂₃×#443	1	1	yes	characteristic
544	C₂	C₂₃×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#443	C₂	semidirect
3. C₂₃×#2935	C₂	semidirect
4. C₂₃×#2906	C₂	semidirect
5. C₂₃×#1962	C₂	semidirect
6. C₂₃×#1950	C₂	semidirect
7. C₂₃×#1937	C₂	semidirect
8. C₂₃×#1876	C₂	semidirect
9. C₂₃×#1866	C₂	semidirect
10. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
11. C₄×C₂₃×#104	C₂	semidirect
17. C₂₃×#464	C₄	semidirect
18. C₂₃×#463	C₄	semidirect
19. C₂₃×#462	C₄	semidirect
20. C₂₃×#461	C₄	semidirect
21. C₂₃×#460	C₄	semidirect
22. C₂₃×#459	C₄	semidirect
104. C₂₃×#104	D₄	semidirect
188. #23015	C₂₃	direct (N × H)
242. C₂×#443	C₂×C₂₃	semidirect
243. #2935	C₂×C₂₃	semidirect
244. #2906	C₂×C₂₃	semidirect
245. #1962	C₂×C₂₃	semidirect
246. #1950	C₂×C₂₃	semidirect
247. #1937	C₂×C₂₃	semidirect
248. #1876	C₂×C₂₃	semidirect
249. #1866	C₂×C₂₃	semidirect
250. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
251. C₄×#104	C₂×C₂₃	semidirect
272. C₄×C₂₃	#443	semidirect
282. #464	C₄×C₂₃	semidirect
283. #463	C₄×C₂₃	semidirect
284. #462	C₄×C₂₃	semidirect
285. #461	C₄×C₂₃	semidirect
286. #460	C₄×C₂₃	semidirect
287. #459	C₄×C₂₃	semidirect
376. #104	D₄×C₂₃	semidirect
386. C₂₃	#23015	direct (N × H)
533. C₄	C₂₃×#443	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3035	2
C₂×C₂₃×#443	2
C₂₃×#2935	2
C₂₃×#2906	2
C₂₃×#1962	2
C₂₃×#1950	2
C₂₃×#1937	2
C₂₃×#1876	2
C₂₃×#1866	2
C₄×C₂₃×C₄⋊C₈	2
C₄×C₂₃×#104	2
#23015	23

Subgroups — C23×#23015 #19170641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23015 #19170641