Subgroups — C₂₃×#20421 #19168047

Order 5888.

1490 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#568	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#104	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2896	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1765	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1764	C₂	1	4	yes	maximal, complemented
8	C₂₃×#1738	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#339	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#568	—	2	4	no	—
12	C₂₃×#568	C₄	1	8	yes	complemented
13	C₂×C₂₃×#131	—	2	2	no	—
14	C₂×C₂₃×#131	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₂₃	—	2	2	no	—
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#104	C₄	1	8	yes	complemented
23	C₂×C₂₃×#104	C₂²	1	2	yes	—
24	C₂×C₂₃×#104	C₂²	1	2	yes	—
25	C₂₃×#419	C₄	1	2	yes	complemented
26	C₂₃×#418	C₄	1	2	yes	complemented
27	C₂₃×#417	C₄	1	4	yes	complemented
28	C₂₃×#416	C₂²	1	4	yes	—
29	C₂₃×#405	—	2	2	no	—
30	C₂₃×#405	C₂²	1	4	yes	complemented
31	C₄×M₁₆×C₂₃	—	2	4	no	—
32	C₄×M₁₆×C₂₃	—	2	2	no	—
33	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
34	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
35	C₂₃×#341	C₂²	1	4	yes	complemented
36	C₂₃×#339	C₂²	1	2	yes	complemented
37	C₂₃×#339	C₂²	1	2	yes	complemented
38	C₂×#41×C₂₃	—	2	1	no	—
39	C₂×#41×C₂₃	—	2	2	no	—
40	C₂×#41×C₂₃	—	2	1	no	—
41	C₂×#41×C₂₃	—	2	2	no	—
42	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
43	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
44	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
45	C₂×#41×C₂₃	C₂³	1	2	yes	—
46	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₄×C₂₃	—	2	1	no	—
48	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₂₃	—	2	1	no	—
50	C₂³×C₄×C₂₃	—	2	1	no	—
51	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#131	—	2	16	no	—
53	C₂₃×#131	—	2	8	no	—
54	C₂₃×#131	C₂×C₄	1	16	yes	complemented
55	C₂×M₁₆×C₂₃	—	2	1	no	—
56	C₂×M₁₆×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	—	2	2	no	—
58	C₂×M₁₆×C₂₃	—	2	1	no	—
59	C₂×M₁₆×C₂₃	—	2	2	no	—
60	C₂×M₁₆×C₂₃	—	2	2	no	—
61	C₂×M₁₆×C₂₃	—	2	16	no	—
62	C₂×M₁₆×C₂₃	—	2	4	no	—
63	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
64	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
65	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
66	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
67	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
68	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
69	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
70	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
71	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
72	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
73	C₂²×C₈×C₂₃	—	2	1	no	—
74	C₂²×C₈×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	4	no	—
76	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
78	C₂₃×#117	—	2	1	no	—
79	C₂₃×#117	—	2	1	no	—
80	C₂₃×#117	C₂×C₄	1	2	yes	—
81	C₂₃×#117	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄²×C₂₃	—	2	2	no	—
85	C₂×C₄²×C₂₃	—	2	1	no	—
86	C₂×C₄²×C₂₃	—	2	1	no	—
87	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
88	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
90	C₂₃×#104	C₂×C₄	1	8	yes	complemented
91	C₂₃×#104	C₂×C₄	1	8	yes	complemented
92	C₂₃×#103	C₂×C₄	1	8	yes	complemented
93	C₂₃×#102	C₂×C₄	1	8	yes	complemented
94	C₂₃×#97	—	2	4	no	—
95	C₂₃×#97	—	2	4	no	—
96	C₄×C₈×C₂₃	—	2	4	no	—
97	C₄×C₈×C₂₃	—	2	4	no	—
98	C₂⁴×C₂₃	—	2	1	no	—
99	C₂⁴×C₂₃	#41	1	1	yes	characteristic
100	#41×C₂₃	—	2	4	no	—
101	#41×C₂₃	—	2	4	no	—
102	#41×C₂₃	—	2	4	no	—
103	#41×C₂₃	—	2	4	no	—
104	#41×C₂₃	—	2	4	no	—
105	#41×C₂₃	—	2	2	no	—
106	#41×C₂₃	—	2	2	no	—
107	#41×C₂₃	C₄²	1	8	yes	—
108	#41×C₂₃	C₂²×C₄	1	4	yes	—
109	#41×C₂₃	C₂²×C₄	1	4	yes	—
110	C₂×Q₈×C₂₃	—	2	1	no	—
111	C₂×Q₈×C₂₃	—	2	2	no	—
112	C₂×Q₈×C₂₃	C₄²	1	4	yes	—
113	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
114	C₂×D₄×C₂₃	—	2	4	no	—
115	C₂×D₄×C₂₃	—	2	2	no	—
116	C₂×D₄×C₂₃	—	2	4	no	—
117	C₂×D₄×C₂₃	—	2	2	no	—
118	C₂×D₄×C₂₃	—	2	2	no	—
119	C₂×D₄×C₂₃	—	2	1	no	—
120	C₂×D₄×C₂₃	C₄²	1	4	yes	—
121	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	1	no	—
124	C₂²×C₄×C₂₃	—	2	4	no	—
125	C₂²×C₄×C₂₃	—	2	2	no	—
126	C₂²×C₄×C₂₃	—	2	4	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	—	2	4	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	4	4	no	—
132	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	#41	1	2	yes	—
141	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
142	M₁₆×C₂₃	—	2	4	no	—
143	M₁₆×C₂₃	—	2	4	no	—
144	M₁₆×C₂₃	—	2	4	no	—
145	M₁₆×C₂₃	—	2	4	no	—
146	M₁₆×C₂₃	—	2	2	no	—
147	M₁₆×C₂₃	—	2	2	no	—
148	M₁₆×C₂₃	—	2	16	no	—
149	M₁₆×C₂₃	C₂²⋊C₄	1	8	yes	—
150	M₁₆×C₂₃	C₂²⋊C₄	1	8	yes	—
151	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
152	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
153	C₂×C₈×C₂₃	—	2	4	no	—
154	C₂×C₈×C₂₃	—	2	4	no	—
155	C₂×C₈×C₂₃	—	2	2	no	—
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	2	2	no	—
158	C₂×C₈×C₂₃	—	2	2	no	—
159	C₂×C₈×C₂₃	—	2	16	no	—
160	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
161	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
162	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
163	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
166	C₂²⋊C₄×C₂₃	—	2	2	no	—
167	C₂²⋊C₄×C₂₃	—	2	2	no	—
168	C₄²×C₂₃	—	2	4	no	—
169	C₄²×C₂₃	—	2	2	no	—
170	C₄²×C₂₃	—	2	1	no	—
171	C₄²×C₂₃	—	2	1	no	—
172	C₄²×C₂₃	—	2	2	no	—
173	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
174	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
175	#20421	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₂₃	—	2	2	no	—
177	C₂³×C₂₃	—	2	4	no	—
178	C₂³×C₂₃	—	4	2	no	—
179	C₂³×C₂₃	—	8	2	no	—
180	C₂³×C₂₃	#117	1	2	yes	—
181	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
182	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
183	C₂³×C₂₃	C₄×D₄	1	2	yes	—
184	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
185	Q₈×C₂₃	—	2	2	no	—
186	Q₈×C₂₃	—	2	4	no	—
187	Q₈×C₂₃	C₂×C₄²	1	4	yes	—
188	D₄×C₂₃	—	2	4	no	—
189	D₄×C₂₃	—	2	4	no	—
190	D₄×C₂₃	—	2	8	no	—
191	D₄×C₂₃	—	2	2	no	—
192	D₄×C₂₃	—	2	4	no	—
193	D₄×C₂₃	C₂×C₄²	1	4	yes	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	8	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	4	8	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	8	2	no	—
206	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
207	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
208	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄×C₂₃	#117	1	2	yes	—
211	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
212	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
215	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
216	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
217	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
224	C₈×C₂₃	—	2	4	no	—
225	C₈×C₂₃	—	2	4	no	—
226	C₈×C₂₃	—	2	4	no	—
227	C₈×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
228	C₂×#568	C₂×C₂₃	1	1	yes	characteristic
229	C₂²×#104	C₂×C₂₃	1	1	yes	characteristic
230	#2896	C₂×C₂₃	1	1	yes	characteristic
231	#2873	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#1765	C₂×C₂₃	1	4	yes	complemented
234	#1764	C₂×C₂₃	1	4	yes	complemented
235	#1738	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂×#339	C₂×C₂₃	1	1	yes	characteristic, complemented
237	C₂²×C₂₃	—	2	2	no	—
238	C₂²×C₂₃	—	2	2	no	—
239	C₂²×C₂₃	—	2	2	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	4	2	no	—
242	C₂²×C₂₃	—	4	2	no	—
243	C₂²×C₂₃	—	4	4	no	—
244	C₂²×C₂₃	—	8	4	no	—
245	C₂²×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
246	C₂²×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
247	C₂²×C₂₃	C₂×#117	1	1	yes	characteristic
248	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
249	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
250	C₄×C₂₃	—	2	4	no	—
251	C₄×C₂₃	—	2	4	no	—
252	C₄×C₂₃	—	4	2	no	—
253	C₄×C₂₃	—	4	4	no	—
254	C₄×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
255	C₄×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
256	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
257	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
258	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
259	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
260	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
261	#568	—	2	4	no	—
262	#568	C₄×C₂₃	1	8	yes	complemented
263	C₂×#131	—	2	2	no	—
264	C₂×#131	C₂²×C₂₃	1	4	yes	—
265	C₂²×M₁₆	—	2	2	no	—
266	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
267	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
268	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
269	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
271	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×#104	C₄×C₂₃	1	8	yes	complemented
273	C₂×#104	C₂²×C₂₃	1	2	yes	—
274	C₂×#104	C₂²×C₂₃	1	2	yes	—
275	#419	C₄×C₂₃	1	2	yes	complemented
276	#418	C₄×C₂₃	1	2	yes	complemented
277	#417	C₄×C₂₃	1	4	yes	complemented
278	#416	C₂²×C₂₃	1	4	yes	—
279	#405	—	2	2	no	—
280	#405	C₂²×C₂₃	1	4	yes	complemented
281	C₄×M₁₆	—	2	4	no	—
282	C₄×M₁₆	—	2	2	no	—
283	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
284	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
285	#341	C₂²×C₂₃	1	4	yes	complemented
286	#339	C₂²×C₂₃	1	2	yes	complemented
287	#339	C₂²×C₂₃	1	2	yes	complemented
288	C₂×C₂₃	—	2	1	no	—
289	C₂×C₂₃	—	2	2	no	—
290	C₂×C₂₃	—	2	1	no	—
291	C₂×C₂₃	—	2	2	no	—
292	C₂×C₂₃	—	4	4	no	—
293	C₂×C₂₃	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
294	C₂×C₂₃	#2956	1	2	yes	—
295	C₂×#41	—	2	1	no	—
296	C₂×#41	—	2	2	no	—
297	C₂×#41	—	2	1	no	—
298	C₂×#41	—	2	2	no	—
299	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
300	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
301	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
302	C₂×#41	C₂³×C₂₃	1	2	yes	—
303	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
304	C₂²×D₄	—	2	1	no	—
305	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
306	C₂³×C₄	—	2	1	no	—
307	C₂³×C₄	—	2	1	no	—
308	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
309	#131	—	2	16	no	—
310	#131	—	2	8	no	—
311	#131	C₂×C₄×C₂₃	1	16	yes	complemented
312	C₂×M₁₆	—	2	1	no	—
313	C₂×M₁₆	—	2	2	no	—
314	C₂×M₁₆	—	2	1	no	—
315	C₂×M₁₆	—	2	2	no	—
316	C₂×M₁₆	—	2	2	no	—
317	C₂×M₁₆	—	2	2	no	—
318	C₂×M₁₆	—	2	16	no	—
319	C₂×M₁₆	—	2	4	no	—
320	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
321	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
322	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
323	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
324	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
325	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
326	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
327	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
328	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
329	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
330	C₂²×C₈	—	2	1	no	—
331	C₂²×C₈	—	2	1	no	—
332	C₂²×C₈	—	2	4	no	—
333	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
334	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
335	#117	—	2	1	no	—
336	#117	—	2	1	no	—
337	#117	C₂×C₄×C₂₃	1	2	yes	—
338	#117	C₂³×C₂₃	1	2	yes	—
339	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₄²	—	2	1	no	—
342	C₂×C₄²	—	2	2	no	—
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
345	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
347	#104	C₂×C₄×C₂₃	1	8	yes	complemented
348	#104	C₂×C₄×C₂₃	1	8	yes	complemented
349	#103	C₂×C₄×C₂₃	1	8	yes	complemented
350	#102	C₂×C₄×C₂₃	1	8	yes	complemented
351	#97	—	2	4	no	—
352	#97	—	2	4	no	—
353	C₄×C₈	—	2	4	no	—
354	C₄×C₈	—	2	4	no	—
355	C₂₃	#20421	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	—	2	1	no	—
357	C₂⁴	#41×C₂₃	1	1	yes	characteristic
358	#41	—	2	4	no	—
359	#41	—	2	4	no	—
360	#41	—	2	4	no	—
361	#41	—	2	4	no	—
362	#41	—	2	4	no	—
363	#41	—	2	2	no	—
364	#41	—	2	2	no	—
365	#41	C₄²×C₂₃	1	8	yes	—
366	#41	C₂²×C₄×C₂₃	1	4	yes	—
367	#41	C₂²×C₄×C₂₃	1	4	yes	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	C₄²×C₂₃	1	4	yes	—
371	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	C₄²×C₂₃	1	4	yes	—
379	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
391	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
392	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	#41×C₂₃	1	2	yes	—
399	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
400	M₁₆	—	2	4	no	—
401	M₁₆	—	2	4	no	—
402	M₁₆	—	2	4	no	—
403	M₁₆	—	2	4	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	—	2	16	no	—
407	M₁₆	C₂²⋊C₄×C₂₃	1	8	yes	—
408	M₁₆	C₂²⋊C₄×C₂₃	1	8	yes	—
409	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
410	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
411	C₂×C₈	—	2	4	no	—
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	16	no	—
418	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
419	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
420	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
421	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
424	C₂²⋊C₄	—	2	2	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₄²	—	2	4	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	1	no	—
431	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
432	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	4	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	8	2	no	—
437	C₂³	C₂₃×#117	1	2	yes	—
438	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
439	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
440	C₂³	C₄×D₄×C₂₃	1	2	yes	—
441	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
442	Q₈	—	2	4	no	—
443	Q₈	—	2	2	no	—
444	Q₈	C₂×C₄²×C₂₃	1	4	yes	—
445	D₄	—	2	8	no	—
446	D₄	—	2	4	no	—
447	D₄	—	2	4	no	—
448	D₄	—	2	4	no	—
449	D₄	—	2	2	no	—
450	D₄	C₂×C₄²×C₂₃	1	4	yes	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	8	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	8	2	no	—
463	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
464	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
465	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
467	C₂×C₄	C₂₃×#117	1	2	yes	—
468	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
469	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
470	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
471	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
472	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
473	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
474	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
475	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
476	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
477	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
481	C₈	—	2	4	no	—
482	C₈	—	2	4	no	—
483	C₈	—	2	4	no	—
484	C₈	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	2	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	8	4	no	—
493	C₂²	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
494	C₂²	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
495	C₂²	C₂×C₂₃×#117	1	1	yes	characteristic
496	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
497	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	4	no	—
502	C₄	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
503	C₄	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
504	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
505	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
506	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
507	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
508	C₂	—	2	1	no	—
509	C₂	—	2	2	no	—
510	C₂	—	2	2	no	—
511	C₂	—	2	1	no	—
512	C₂	—	4	4	no	—
513	C₂	C₂×C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
514	C₂	C₂₃×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#2873	C₂	semidirect
5. C₂×C₄×M₁₆×C₂₃	C₂	semidirect
6. C₂₃×#1765	C₂	semidirect
7. C₂₃×#1764	C₂	semidirect
8. C₂₃×#1738	C₂	semidirect
9. C₂×C₂₃×#339	C₂	semidirect
12. C₂₃×#568	C₄	semidirect
22. C₂×C₂₃×#104	C₄	semidirect
25. C₂₃×#419	C₄	semidirect
26. C₂₃×#418	C₄	semidirect
27. C₂₃×#417	C₄	semidirect
30. C₂₃×#405	C₂²	semidirect
33. C₂×C₂₃×#97	C₂²	semidirect
34. C₂×C₄×C₈×C₂₃	C₂²	semidirect
35. C₂₃×#341	C₂²	semidirect
36. C₂₃×#339	C₂²	semidirect
37. C₂₃×#339	C₂²	semidirect
54. C₂₃×#131	C₂×C₄	semidirect
90. C₂₃×#104	C₂×C₄	semidirect
91. C₂₃×#104	C₂×C₄	semidirect
92. C₂₃×#103	C₂×C₄	semidirect
93. C₂₃×#102	C₂×C₄	semidirect
175. #20421	C₂₃	direct (N × H)
231. #2873	C₂×C₂₃	semidirect
232. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
233. #1765	C₂×C₂₃	semidirect
234. #1764	C₂×C₂₃	semidirect
235. #1738	C₂×C₂₃	semidirect
236. C₂×#339	C₂×C₂₃	semidirect
262. #568	C₄×C₂₃	semidirect
272. C₂×#104	C₄×C₂₃	semidirect
275. #419	C₄×C₂₃	semidirect
276. #418	C₄×C₂₃	semidirect
277. #417	C₄×C₂₃	semidirect
280. #405	C₂²×C₂₃	semidirect
283. C₂×#97	C₂²×C₂₃	semidirect
284. C₂×C₄×C₈	C₂²×C₂₃	semidirect
285. #341	C₂²×C₂₃	semidirect
286. #339	C₂²×C₂₃	semidirect
287. #339	C₂²×C₂₃	semidirect
311. #131	C₂×C₄×C₂₃	semidirect
347. #104	C₂×C₄×C₂₃	semidirect
348. #104	C₂×C₄×C₂₃	semidirect
349. #103	C₂×C₄×C₂₃	semidirect
350. #102	C₂×C₄×C₂₃	semidirect
355. C₂₃	#20421	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×#568	2
C₂²×C₂₃×#104	2
C₂₃×#2896	2
C₂₃×#2873	2
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#1765	2
C₂₃×#1764	2
C₂₃×#1738	2
C₂×C₂₃×#339	2
#20421	23

Subgroups — C23×#20421 #19168047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20421 #19168047