Subgroups — C₂₃×#20441 #19168067

Order 5888.

1138 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#405	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1771	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1770	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1729	C₂	1	2	yes	maximal
9	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×#117	C₂²	1	2	yes	—
13	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₄	1	8	yes	complemented
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂₃×#405	—	2	8	no	—
19	C₄×M₁₆×C₂₃	C₄	1	8	yes	complemented
20	C₂×C₂₃×#97	C₂²	1	4	yes	—
21	C₂×C₂₃×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
24	C₂₃×#384	C₄	1	2	yes	—
25	C₂₃×#382	C₄	1	2	yes	—
26	C₂₃×#380	C₄	1	2	yes	—
27	C₂₃×#380	C₄	1	2	yes	—
28	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
30	C₂₃×#336	C₂²	1	4	yes	—
31	C₂₃×#336	C₂²	1	4	yes	—
32	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₂₃	—	2	2	no	—
36	C₂×M₁₆×C₂₃	—	2	2	no	—
37	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
39	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
40	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
41	C₂²×C₈×C₂₃	—	2	4	no	—
42	C₂²×C₈×C₂₃	—	2	4	no	—
43	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂₃×#117	—	2	4	no	—
49	C₂₃×#117	—	2	4	no	—
50	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
52	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
59	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
64	C₂₃×#97	—	2	8	no	—
65	C₂₃×#97	C₂×C₄	1	8	yes	—
66	C₄×C₈×C₂₃	—	2	8	no	—
67	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
68	C₂₃×#95	C₂×C₄	1	4	yes	—
69	C₂₃×#95	C₂×C₄	1	4	yes	—
70	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	—	2	4	no	—
72	C₂²×C₄×C₂₃	—	4	2	no	—
73	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
74	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
75	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₂₃	#41	1	2	yes	—
86	C₂²×C₄×C₂₃	#41	1	2	yes	—
87	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
88	M₁₆×C₂₃	—	2	8	no	—
89	M₁₆×C₂₃	C₄⋊C₄	1	16	yes	complemented
90	C₂×C₈×C₂₃	—	2	16	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	2	2	no	—
94	C₂×C₈×C₂₃	—	2	2	no	—
95	C₂×C₈×C₂₃	—	2	2	no	—
96	C₂×C₈×C₂₃	—	2	2	no	—
97	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
102	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
103	C₄⋊C₄×C₂₃	—	2	8	no	—
104	C₄⋊C₄×C₂₃	—	2	8	no	—
105	C₄⋊C₄×C₂₃	—	4	4	no	—
106	C₂²⋊C₄×C₂₃	—	2	8	no	—
107	C₄²×C₂₃	—	2	4	no	—
108	C₄²×C₂₃	—	2	4	no	—
109	C₄²×C₂₃	C₄²	1	4	yes	—
110	C₄²×C₂₃	C₄²	1	4	yes	—
111	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
112	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
114	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
115	#20441	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₂₃	—	2	2	no	—
117	C₂³×C₂₃	—	2	2	no	—
118	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
119	C₂³×C₂₃	C₄×D₄	1	2	yes	—
120	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
121	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	8	no	—
126	C₂×C₄×C₂₃	—	2	8	no	—
127	C₂×C₄×C₂₃	—	2	8	no	—
128	C₂×C₄×C₂₃	—	2	8	no	—
129	C₂×C₄×C₂₃	—	4	4	no	—
130	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×C₄×C₂₃	#117	1	4	yes	—
137	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
138	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
140	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₂₃	#131	1	4	yes	—
142	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₂₃	—	2	4	no	—
149	C₈×C₂₃	—	2	4	no	—
150	C₈×C₂₃	—	2	4	no	—
151	C₈×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
152	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
153	C₂×#405	C₂×C₂₃	1	2	yes	—
154	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
155	#2285	C₂×C₂₃	1	1	yes	characteristic
156	#1771	C₂×C₂₃	1	4	yes	complemented
157	#1770	C₂×C₂₃	1	2	yes	complemented
158	C₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
159	#1729	C₂×C₂₃	1	2	yes	—
160	C₂²×C₂₃	—	2	2	no	—
161	C₂²×C₂₃	—	2	2	no	—
162	C₂²×C₂₃	—	2	4	no	—
163	C₂²×C₂₃	—	2	4	no	—
164	C₂²×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
165	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
166	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂²×C₂₃	C₂×#131	1	2	yes	—
169	C₂²×C₂₃	#568	1	2	yes	—
170	C₄×C₂₃	—	2	2	no	—
171	C₄×C₂₃	—	2	2	no	—
172	C₄×C₂₃	—	2	2	no	—
173	C₄×C₂₃	—	4	8	no	—
174	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
175	C₄×C₂₃	#405	1	4	yes	complemented
176	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
178	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
179	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
181	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
182	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
183	C₂×#117	C₂²×C₂₃	1	2	yes	—
184	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×C₄⋊C₈	C₄×C₂₃	1	8	yes	complemented
186	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
187	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
188	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
189	#405	—	2	8	no	—
190	C₄×M₁₆	C₄×C₂₃	1	8	yes	complemented
191	C₂×#97	C₂²×C₂₃	1	4	yes	—
192	C₂×#97	C₂²×C₂₃	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
194	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
195	#384	C₄×C₂₃	1	2	yes	—
196	#382	C₄×C₂₃	1	2	yes	—
197	#380	C₄×C₂₃	1	2	yes	—
198	#380	C₄×C₂₃	1	2	yes	—
199	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
200	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
201	#336	C₂²×C₂₃	1	4	yes	—
202	#336	C₂²×C₂₃	1	4	yes	—
203	C₂×C₂₃	—	2	4	no	—
204	C₂×C₂₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₂₃	C₂×#405	1	1	yes	characteristic
206	C₂×C₂₃	#2873	1	1	yes	characteristic
207	C₂×C₂₃	#2935	1	2	yes	—
208	C₂×C₂₃	#2962	1	2	yes	—
209	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
210	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
211	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
212	C₂×M₁₆	—	2	2	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
215	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
216	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
217	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
218	C₂²×C₈	—	2	4	no	—
219	C₂²×C₈	—	2	4	no	—
220	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
221	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
222	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
223	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
224	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
225	#117	—	2	4	no	—
226	#117	—	2	4	no	—
227	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
228	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
229	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
230	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
231	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
233	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
234	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
235	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
236	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
238	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
239	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
240	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
241	#97	—	2	8	no	—
242	#97	C₂×C₄×C₂₃	1	8	yes	—
243	C₄×C₈	—	2	8	no	—
244	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
245	#95	C₂×C₄×C₂₃	1	4	yes	—
246	#95	C₂×C₄×C₂₃	1	4	yes	—
247	C₂₃	#20441	1	1	yes	characteristic, direct product factor, Sylow
248	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
252	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
253	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
254	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
256	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
258	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
260	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
262	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
263	C₂²×C₄	#41×C₂₃	1	2	yes	—
264	C₂²×C₄	#41×C₂₃	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
266	M₁₆	—	2	8	no	—
267	M₁₆	C₄⋊C₄×C₂₃	1	16	yes	complemented
268	C₂×C₈	—	2	16	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
276	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
277	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
278	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
279	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
280	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
281	C₄⋊C₄	—	2	8	no	—
282	C₄⋊C₄	—	2	8	no	—
283	C₄⋊C₄	—	4	4	no	—
284	C₂²⋊C₄	—	2	8	no	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	4	no	—
287	C₄²	C₄²×C₂₃	1	4	yes	—
288	C₄²	C₄²×C₂₃	1	4	yes	—
289	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
290	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
291	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
292	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
293	C₂³	—	2	2	no	—
294	C₂³	—	2	2	no	—
295	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
296	C₂³	C₄×D₄×C₂₃	1	2	yes	—
297	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
298	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	8	no	—
305	C₂×C₄	—	2	8	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
308	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
309	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
310	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
311	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
313	C₂×C₄	C₂₃×#117	1	4	yes	—
314	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
315	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
316	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
317	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
318	C₂×C₄	C₂₃×#131	1	4	yes	—
319	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
320	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
322	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
323	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
324	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	2	4	no	—
327	C₈	—	2	4	no	—
328	C₈	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
329	C₂²	—	2	4	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
334	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
335	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
336	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
337	C₂²	C₂×C₂₃×#131	1	2	yes	—
338	C₂²	C₂₃×#568	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	8	no	—
343	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
344	C₄	C₂₃×#405	1	4	yes	complemented
345	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
346	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
347	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
348	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
349	C₂	—	2	4	no	—
350	C₂	C₂×C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
351	C₂	C₂×C₂₃×#405	1	1	yes	characteristic
352	C₂	C₂₃×#2873	1	1	yes	characteristic
353	C₂	C₂₃×#2935	1	2	yes	—
354	C₂	C₂₃×#2962	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1771	C₂	semidirect
6. C₂₃×#1770	C₂	semidirect
7. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
14. C₂×C₂₃×C₄⋊C₈	C₄	semidirect
19. C₄×M₁₆×C₂₃	C₄	semidirect
89. M₁₆×C₂₃	C₄⋊C₄	semidirect
115. #20441	C₂₃	direct (N × H)
156. #1771	C₂×C₂₃	semidirect
157. #1770	C₂×C₂₃	semidirect
158. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
175. C₄×C₂₃	#405	semidirect
185. C₂×C₄⋊C₈	C₄×C₂₃	semidirect
190. C₄×M₁₆	C₄×C₂₃	semidirect
247. C₂₃	#20441	direct (N × H)
267. M₁₆	C₄⋊C₄×C₂₃	semidirect
344. C₄	C₂₃×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×C₄⋊C₈	2
C₂×C₂₃×#405	2
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#2285	2
C₂₃×#1771	2
C₂₃×#1770	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1729	2
#20441	23

Subgroups — C23×#20441 #19168067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20441 #19168067