Subgroups — C₂₃×#20439 #19168065

Order 5888.

1130 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1771	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1771	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1770	C₂	1	2	yes	maximal, complemented
9	C₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic
11	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic
12	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#424	C₄	1	8	yes	complemented
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂₃×#405	—	2	4	no	—
23	C₂₃×#405	C₄	1	8	yes	complemented
24	C₄×M₁₆×C₂₃	—	2	4	no	—
25	C₂×C₂₃×#97	C₂²	1	2	yes	—
26	C₂×C₂₃×#97	C₂²	1	2	yes	—
27	C₂×C₂₃×#97	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
31	C₂₃×#389	C₄	1	2	yes	—
32	C₂₃×#389	C₄	1	2	yes	—
33	C₂₃×#385	C₄	1	4	yes	—
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
36	C₂₃×#336	C₂²	1	1	yes	characteristic
37	C₂₃×#336	C₂²	1	1	yes	characteristic
38	C₂₃×#336	C₂²	1	1	yes	characteristic
39	C₂₃×#336	C₂²	1	1	yes	characteristic
40	C₂₃×#336	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	2	yes	—
42	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₂₃	—	2	2	no	—
45	C₂×M₁₆×C₂₃	—	2	2	no	—
46	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
48	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
49	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
50	C₂²×C₈×C₂₃	—	2	2	no	—
51	C₂²×C₈×C₂₃	—	2	2	no	—
52	C₂²×C₈×C₂₃	—	2	2	no	—
53	C₂²×C₈×C₂₃	—	2	2	no	—
54	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
56	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂₃×#117	—	2	2	no	—
63	C₂₃×#117	—	2	2	no	—
64	C₂₃×#117	C₂×C₄	1	4	yes	—
65	C₂₃×#117	C₂×C₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
67	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
69	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₂₃	—	2	2	no	—
74	C₂×C₄²×C₂₃	—	2	2	no	—
75	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
76	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
78	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
80	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
81	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
82	C₂₃×#97	—	2	4	no	—
83	C₂₃×#97	—	2	4	no	—
84	C₂₃×#97	C₂×C₄	1	8	yes	complemented
85	C₄×C₈×C₂₃	—	2	4	no	—
86	C₄×C₈×C₂₃	—	2	4	no	—
87	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
88	C₂₃×#95	C₂×C₄	1	4	yes	—
89	C₂₃×#95	C₂×C₄	1	4	yes	—
90	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	—	2	4	no	—
92	C₂²×C₄×C₂₃	—	2	4	no	—
93	C₂²×C₄×C₂₃	—	4	2	no	—
94	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
104	C₂²×C₄×C₂₃	#41	1	2	yes	—
105	C₂²×C₄×C₂₃	#41	1	2	yes	—
106	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
107	M₁₆×C₂₃	—	2	8	no	—
108	M₁₆×C₂₃	C₄⋊C₄	1	16	yes	complemented
109	C₂×C₈×C₂₃	—	2	8	no	—
110	C₂×C₈×C₂₃	—	2	8	no	—
111	C₂×C₈×C₂₃	—	2	1	no	—
112	C₂×C₈×C₂₃	—	2	1	no	—
113	C₂×C₈×C₂₃	—	2	2	no	—
114	C₂×C₈×C₂₃	—	2	1	no	—
115	C₂×C₈×C₂₃	—	2	1	no	—
116	C₂×C₈×C₂₃	—	2	1	no	—
117	C₂×C₈×C₂₃	—	2	1	no	—
118	C₂×C₈×C₂₃	—	2	1	no	—
119	C₂×C₈×C₂₃	—	2	1	no	—
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
127	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
128	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
129	C₄⋊C₄×C₂₃	—	2	4	no	—
130	C₄⋊C₄×C₂₃	—	2	8	no	—
131	C₄⋊C₄×C₂₃	—	4	4	no	—
132	C₄⋊C₄×C₂₃	C₄²	1	4	yes	—
133	C₄⋊C₄×C₂₃	C₄²	1	4	yes	—
134	C₂²⋊C₄×C₂₃	—	2	4	no	—
135	C₂²⋊C₄×C₂₃	C₄²	1	8	yes	—
136	C₄²×C₂₃	—	2	4	no	—
137	C₄²×C₂₃	—	2	1	no	—
138	C₄²×C₂₃	—	2	1	no	—
139	C₄²×C₂₃	—	2	2	no	—
140	C₄²×C₂₃	—	2	1	no	—
141	C₄²×C₂₃	—	2	1	no	—
142	C₄²×C₂₃	—	2	2	no	—
143	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
144	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
145	#20439	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₂₃	—	2	2	no	—
147	C₂³×C₂₃	—	2	2	no	—
148	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
149	C₂³×C₂₃	C₄×D₄	1	2	yes	—
150	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
151	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₂₃	—	2	2	no	—
153	C₂×C₄×C₂₃	—	2	4	no	—
154	C₂×C₄×C₂₃	—	2	4	no	—
155	C₂×C₄×C₂₃	—	2	2	no	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	2	4	no	—
158	C₂×C₄×C₂₃	—	2	4	no	—
159	C₂×C₄×C₂₃	—	2	2	no	—
160	C₂×C₄×C₂₃	—	2	2	no	—
161	C₂×C₄×C₂₃	—	2	4	no	—
162	C₂×C₄×C₂₃	—	4	8	no	—
163	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
164	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
165	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
166	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
167	C₂×C₄×C₂₃	#117	1	4	yes	—
168	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
169	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
171	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
172	C₂×C₄×C₂₃	#131	1	4	yes	—
173	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
177	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
179	C₈×C₂₃	—	2	2	no	—
180	C₈×C₂₃	—	2	2	no	—
181	C₈×C₂₃	—	2	2	no	—
182	C₈×C₂₃	—	2	2	no	—
183	C₈×C₂₃	—	2	4	no	—
184	C₈×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
185	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
186	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
187	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
188	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
189	#2296	C₂×C₂₃	1	1	yes	characteristic
190	#1771	C₂×C₂₃	1	2	yes	complemented
191	#1771	C₂×C₂₃	1	2	yes	complemented
192	#1770	C₂×C₂₃	1	2	yes	complemented
193	C₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
194	#1729	C₂×C₂₃	1	1	yes	characteristic
195	C₂×#336	C₂×C₂₃	1	1	yes	characteristic
196	C₂²×C₂₃	—	2	2	no	—
197	C₂²×C₂₃	—	2	2	no	—
198	C₂²×C₂₃	—	2	4	no	—
199	C₂²×C₂₃	—	2	4	no	—
200	C₂²×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
201	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
202	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
204	C₂²×C₂₃	C₂×#131	1	2	yes	—
205	C₂²×C₂₃	#568	1	2	yes	—
206	C₄×C₂₃	—	2	4	no	—
207	C₄×C₂₃	—	2	4	no	—
208	C₄×C₂₃	—	2	4	no	—
209	C₄×C₂₃	—	4	4	no	—
210	C₄×C₂₃	—	4	2	no	—
211	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
212	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
213	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
214	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
215	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
216	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
217	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
218	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
219	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
220	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
221	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
222	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
223	#424	C₄×C₂₃	1	8	yes	complemented
224	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
225	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
226	#405	—	2	4	no	—
227	#405	C₄×C₂₃	1	8	yes	complemented
228	C₄×M₁₆	—	2	4	no	—
229	C₂×#97	C₂²×C₂₃	1	2	yes	—
230	C₂×#97	C₂²×C₂₃	1	2	yes	—
231	C₂×#97	C₂²×C₂₃	1	2	yes	—
232	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
233	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
234	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
235	#389	C₄×C₂₃	1	2	yes	—
236	#389	C₄×C₂₃	1	2	yes	—
237	#385	C₄×C₂₃	1	4	yes	—
238	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
239	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
240	#336	C₂²×C₂₃	1	1	yes	characteristic
241	#336	C₂²×C₂₃	1	1	yes	characteristic
242	#336	C₂²×C₂₃	1	1	yes	characteristic
243	#336	C₂²×C₂₃	1	1	yes	characteristic
244	#336	C₂²×C₂₃	1	2	yes	—
245	#336	C₂²×C₂₃	1	2	yes	—
246	C₂×C₂₃	—	2	4	no	—
247	C₂×C₂₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₂₃	C₄×#131	1	1	yes	characteristic
249	C₂×C₂₃	#2875	1	1	yes	characteristic
250	C₂×C₂₃	#2933	1	2	yes	—
251	C₂×C₂₃	#2962	1	2	yes	—
252	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
253	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
257	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
258	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
259	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	—	2	2	no	—
264	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
265	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
266	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
271	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
272	#117	—	2	2	no	—
273	#117	—	2	2	no	—
274	#117	C₂×C₄×C₂₃	1	4	yes	—
275	#117	C₂×C₄×C₂₃	1	4	yes	—
276	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
277	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
278	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
279	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
281	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
282	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
283	C₂×C₄²	—	2	2	no	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
286	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
287	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
290	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
291	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
292	#97	—	2	4	no	—
293	#97	—	2	4	no	—
294	#97	C₂×C₄×C₂₃	1	8	yes	complemented
295	C₄×C₈	—	2	4	no	—
296	C₄×C₈	—	2	4	no	—
297	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
298	#95	C₂×C₄×C₂₃	1	4	yes	—
299	#95	C₂×C₄×C₂₃	1	4	yes	—
300	C₂₃	#20439	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
302	C₂²×C₄	—	2	4	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	4	2	no	—
305	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
307	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
310	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
312	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
314	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
315	C₂²×C₄	#41×C₂₃	1	2	yes	—
316	C₂²×C₄	#41×C₂₃	1	2	yes	—
317	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
318	M₁₆	—	2	8	no	—
319	M₁₆	C₄⋊C₄×C₂₃	1	16	yes	complemented
320	C₂×C₈	—	2	8	no	—
321	C₂×C₈	—	2	8	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
333	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
334	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
335	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
336	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
337	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
338	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
339	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
340	C₄⋊C₄	—	2	4	no	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	4	4	no	—
343	C₄⋊C₄	C₄²×C₂₃	1	4	yes	—
344	C₄⋊C₄	C₄²×C₂₃	1	4	yes	—
345	C₂²⋊C₄	—	2	4	no	—
346	C₂²⋊C₄	C₄²×C₂₃	1	8	yes	—
347	C₄²	—	2	4	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	2	1	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	2	no	—
354	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
355	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
356	C₂³	—	2	2	no	—
357	C₂³	—	2	2	no	—
358	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
359	C₂³	C₄×D₄×C₂₃	1	2	yes	—
360	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
361	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	4	8	no	—
373	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
374	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
375	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
376	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
377	C₂×C₄	C₂₃×#117	1	4	yes	—
378	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
379	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
380	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
381	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
382	C₂×C₄	C₂₃×#131	1	4	yes	—
383	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
384	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
385	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
386	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
388	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
389	C₈	—	2	2	no	—
390	C₈	—	2	2	no	—
391	C₈	—	2	2	no	—
392	C₈	—	2	2	no	—
393	C₈	—	2	4	no	—
394	C₈	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
395	C₂²	—	2	4	no	—
396	C₂²	—	2	4	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
400	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
401	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
402	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
403	C₂²	C₂×C₂₃×#131	1	2	yes	—
404	C₂²	C₂₃×#568	1	2	yes	—
405	C₄	—	2	4	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	4	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	4	no	—
410	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
411	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
412	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
413	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
414	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
415	C₂	—	2	4	no	—
416	C₂	C₂×C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
417	C₂	C₄×C₂₃×#131	1	1	yes	characteristic
418	C₂	C₂₃×#2875	1	1	yes	characteristic
419	C₂	C₂₃×#2933	1	2	yes	—
420	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
6. C₂₃×#1771	C₂	semidirect
7. C₂₃×#1771	C₂	semidirect
8. C₂₃×#1770	C₂	semidirect
9. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
19. C₂₃×#424	C₄	semidirect
23. C₂₃×#405	C₄	semidirect
80. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
81. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
84. C₂₃×#97	C₂×C₄	semidirect
87. C₄×C₈×C₂₃	C₂×C₄	semidirect
108. M₁₆×C₂₃	C₄⋊C₄	semidirect
145. #20439	C₂₃	direct (N × H)
190. #1771	C₂×C₂₃	semidirect
191. #1771	C₂×C₂₃	semidirect
192. #1770	C₂×C₂₃	semidirect
193. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
223. #424	C₄×C₂₃	semidirect
227. #405	C₄×C₂₃	semidirect
290. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
291. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
294. #97	C₂×C₄×C₂₃	semidirect
297. C₄×C₈	C₂×C₄×C₂₃	semidirect
300. C₂₃	#20439	direct (N × H)
319. M₁₆	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#424	2
C₂×C₂₃×#405	2
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#2296	2
C₂₃×#1771	2
C₂₃×#1770	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1729	2
C₂×C₂₃×#336	2
#20439	23

Subgroups — C23×#20439 #19168065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20439 #19168065