Subgroups — C₂₃×#60305 #19207931

Order 5888.

2004 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3481	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2994	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2990	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2979	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2977	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2974	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2966	C₂	1	4	yes	maximal, complemented
8	C₈×#41×C₂₃	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2929	C₂	1	4	yes	maximal, complemented
11	C₂×C₈×D₄×C₂₃	C₂	1	4	yes	maximal, complemented
12	C₂×C₂₃×#131	C₂²	1	2	yes	complemented
13	C₂³×C₈×C₂₃	C₂²	1	4	yes	complemented
14	C₂₃×#559	C₄	1	1	yes	characteristic
15	C₂₃×#554	C₄	1	2	yes	—
16	C₂₃×#553	C₄	1	4	yes	—
17	C₂₃×#551	C₄	1	4	yes	—
18	C₂₃×#546	C₄	1	4	yes	—
19	D₄²×C₂₃	C₄	1	1	yes	characteristic
20	C₂₃×#532	C₂²	1	1	yes	characteristic
21	C₂₃×#529	C₂²	1	4	yes	—
22	C₂₃×#525	C₂²	1	4	yes	—
23	C₄×#41×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
25	C₂₃×#446	C₂²	1	4	yes	complemented
26	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
27	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
28	C₂₃×#436	C₂²	1	4	yes	complemented
29	C₂₃×#436	C₂²	1	4	yes	complemented
30	C₂₃×#435	C₂²	1	8	yes	complemented
31	C₂₃×#435	C₂²	1	8	yes	complemented
32	C₂₃×#435	C₂²	1	8	yes	complemented
33	C₈×D₄×C₂₃	C₂²	1	2	yes	—
34	C₈×D₄×C₂₃	C₂²	1	4	yes	—
35	C₈×D₄×C₂₃	C₂²	1	2	yes	—
36	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
37	C₈×D₄×C₂₃	C₂²	1	4	yes	complemented
38	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
39	C₈×D₄×C₂₃	C₂²	1	4	yes	complemented
40	C₈×D₄×C₂₃	C₂²	1	4	yes	complemented
41	C₈×D₄×C₂₃	C₂²	1	4	yes	complemented
42	C₂₃×#433	C₂²	1	8	yes	complemented
43	C₂₃×#432	C₂²	1	4	yes	complemented
44	C₂₃×#431	C₂²	1	4	yes	complemented
45	C₂₃×#431	C₂²	1	2	yes	complemented
46	C₂₃×#424	C₂²	1	4	yes	complemented
47	C₂₃×#423	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
49	C₂₃×#408	C₂²	1	4	yes	complemented
50	C₂₃×#408	C₂²	1	2	yes	complemented
51	C₂₃×#407	C₂²	1	4	yes	complemented
52	C₂×C₂₃×C₂²⋊C₈	C₂²	1	8	yes	complemented
53	C₂₃×#405	C₂²	1	4	yes	complemented
54	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	complemented
55	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
56	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
57	C₂×#41×C₂₃	C₂³	1	2	yes	—
58	C₂²×D₄×C₂₃	C₂×C₄	1	4	yes	—
59	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
60	C₂₃×#131	—	4	4	no	—
61	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
62	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
64	C₂²×C₈×C₂₃	—	4	8	no	—
65	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
66	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
69	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
70	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
73	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
74	C₂₃×#126	C₂×C₄	1	8	yes	—
75	C₂₃×#125	C₂×C₄	1	4	yes	—
76	C₂₃×#124	C₂×C₄	1	2	yes	—
77	C₂₃×#124	C₂×C₄	1	4	yes	—
78	C₂₃×#123	C₂×C₄	1	4	yes	—
79	C₂₃×#123	C₂×C₄	1	8	yes	—
80	C₂₃×#123	C₂×C₄	1	4	yes	—
81	C₂₃×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
82	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
83	C₂₃×#121	C₂×C₄	1	8	yes	—
84	C₂₃×#121	C₂×C₄	1	4	yes	—
85	C₂₃×#121	C₂×C₄	1	4	yes	—
86	C₂₃×#120	C₂×C₄	1	4	yes	—
87	C₂₃×#120	C₂×C₄	1	4	yes	—
88	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
89	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
90	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
91	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
92	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
93	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
94	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
95	C₄×D₄×C₂₃	C₂³	1	2	yes	—
96	C₄×D₄×C₂₃	C₂³	1	8	yes	—
97	C₄×D₄×C₂₃	C₂³	1	4	yes	—
98	C₄×D₄×C₂₃	C₂³	1	4	yes	—
99	C₂₃×#117	C₂×C₄	1	4	yes	—
100	C₂₃×#117	C₂×C₄	1	2	yes	—
101	C₂₃×#117	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
103	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
104	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
107	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
108	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
109	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
110	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
111	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
112	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
113	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
114	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
115	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
116	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
117	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
118	C₂₃×#97	C₂³	1	8	yes	—
119	C₄×C₈×C₂₃	C₂³	1	4	yes	—
120	C₄×C₈×C₂₃	C₂³	1	2	yes	—
121	C₄×C₈×C₂₃	C₂³	1	4	yes	—
122	C₄×C₈×C₂₃	C₂³	1	2	yes	—
123	C₄×C₈×C₂₃	C₂³	1	4	yes	—
124	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
125	#41×C₂₃	—	2	8	no	—
126	#41×C₂₃	—	4	2	no	—
127	#41×C₂₃	—	4	2	no	—
128	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₂₃	—	2	16	no	—
131	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
132	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
133	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
134	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
135	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
136	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₂₃	—	4	4	no	—
138	C₂²×C₄×C₂₃	—	4	4	no	—
139	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
141	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
143	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
147	M₁₆×C₂₃	—	4	4	no	—
148	M₁₆×C₂₃	—	4	2	no	—
149	M₁₆×C₂₃	—	4	4	no	—
150	C₂×C₈×C₂₃	—	2	8	no	—
151	C₂×C₈×C₂₃	—	2	8	no	—
152	C₂×C₈×C₂₃	—	2	2	no	—
153	C₂×C₈×C₂₃	—	4	4	no	—
154	C₂×C₈×C₂₃	—	4	8	no	—
155	C₂×C₈×C₂₃	—	4	1	no	—
156	C₂×C₈×C₂₃	—	4	4	no	—
157	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
158	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
159	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
160	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
161	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
162	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
163	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
164	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
165	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
166	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
167	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
170	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
171	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
173	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
174	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
175	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
176	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
177	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
178	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
180	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
181	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
182	C₄²×C₂₃	C₂⁴	1	2	yes	—
183	C₄²×C₂₃	C₂⁴	1	4	yes	—
184	#60305	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₂₃	—	2	16	no	—
186	C₂³×C₂₃	—	4	4	no	—
187	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
188	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
189	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
190	Q₈×C₂₃	—	2	4	no	—
191	Q₈×C₂₃	—	4	2	no	—
192	D₄×C₂₃	—	2	16	no	—
193	D₄×C₂₃	—	2	8	no	—
194	D₄×C₂₃	—	2	8	no	—
195	D₄×C₂₃	—	4	4	no	—
196	D₄×C₂₃	—	4	2	no	—
197	D₄×C₂₃	#131	1	8	yes	complemented
198	C₂×C₄×C₂₃	—	2	8	no	—
199	C₂×C₄×C₂₃	—	2	8	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	1	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	1	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	4	4	no	—
208	C₂×C₄×C₂₃	—	4	8	no	—
209	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
210	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
211	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
213	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
214	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
215	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
216	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
217	C₈×C₂₃	—	2	4	no	—
218	C₈×C₂₃	—	2	4	no	—
219	C₈×C₂₃	—	2	1	no	—
220	C₈×C₂₃	—	2	2	no	—
221	C₈×C₂₃	—	2	4	no	—
222	C₈×C₂₃	H(5,𝔽₂)	1	2	yes	—
223	#3481	C₂×C₂₃	1	1	yes	characteristic
224	#2994	C₂×C₂₃	1	4	yes	complemented
225	#2990	C₂×C₂₃	1	2	yes	complemented
226	#2979	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#2977	C₂×C₂₃	1	4	yes	complemented
228	#2974	C₂×C₂₃	1	4	yes	complemented
229	#2966	C₂×C₂₃	1	4	yes	complemented
230	C₈×#41	C₂×C₂₃	1	2	yes	complemented
231	#2931	C₂×C₂₃	1	1	yes	characteristic, complemented
232	#2929	C₂×C₂₃	1	4	yes	complemented
233	C₂×C₈×D₄	C₂×C₂₃	1	4	yes	complemented
234	C₂²×C₂₃	—	2	4	no	—
235	C₂²×C₂₃	—	2	4	no	—
236	C₂²×C₂₃	—	2	4	no	—
237	C₂²×C₂₃	—	2	1	no	—
238	C₂²×C₂₃	—	2	2	no	—
239	C₂²×C₂₃	—	2	8	no	—
240	C₂²×C₂₃	—	4	16	no	—
241	C₂²×C₂₃	C₂×#131	1	8	yes	—
242	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
243	C₄×C₂₃	—	2	4	no	—
244	C₄×C₂₃	—	2	4	no	—
245	C₄×C₂₃	—	2	1	no	—
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	2	4	no	—
248	C₄×C₂₃	—	2	4	no	—
249	C₄×C₂₃	C₂×#131	1	4	yes	—
250	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
251	C₄×C₂₃	#585	1	1	yes	characteristic
252	C₂×#131	C₂²×C₂₃	1	2	yes	complemented
253	C₂³×C₈	C₂²×C₂₃	1	4	yes	complemented
254	#559	C₄×C₂₃	1	1	yes	characteristic
255	#554	C₄×C₂₃	1	2	yes	—
256	#553	C₄×C₂₃	1	4	yes	—
257	#551	C₄×C₂₃	1	4	yes	—
258	#546	C₄×C₂₃	1	4	yes	—
259	D₄²	C₄×C₂₃	1	1	yes	characteristic
260	#532	C₂²×C₂₃	1	1	yes	characteristic
261	#529	C₂²×C₂₃	1	4	yes	—
262	#525	C₂²×C₂₃	1	4	yes	—
263	C₄×#41	C₂²×C₂₃	1	2	yes	—
264	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
265	#446	C₂²×C₂₃	1	4	yes	complemented
266	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
267	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
268	#436	C₂²×C₂₃	1	4	yes	complemented
269	#436	C₂²×C₂₃	1	4	yes	complemented
270	#435	C₂²×C₂₃	1	8	yes	complemented
271	#435	C₂²×C₂₃	1	8	yes	complemented
272	#435	C₂²×C₂₃	1	8	yes	complemented
273	C₈×D₄	C₂²×C₂₃	1	2	yes	—
274	C₈×D₄	C₂²×C₂₃	1	4	yes	—
275	C₈×D₄	C₂²×C₂₃	1	2	yes	—
276	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
277	C₈×D₄	C₂²×C₂₃	1	4	yes	complemented
278	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
279	C₈×D₄	C₂²×C₂₃	1	4	yes	complemented
280	C₈×D₄	C₂²×C₂₃	1	4	yes	complemented
281	C₈×D₄	C₂²×C₂₃	1	4	yes	complemented
282	#433	C₂²×C₂₃	1	8	yes	complemented
283	#432	C₂²×C₂₃	1	4	yes	complemented
284	#431	C₂²×C₂₃	1	4	yes	complemented
285	#431	C₂²×C₂₃	1	2	yes	complemented
286	#424	C₂²×C₂₃	1	4	yes	complemented
287	#423	C₂²×C₂₃	1	1	yes	characteristic, complemented
288	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
289	#408	C₂²×C₂₃	1	4	yes	complemented
290	#408	C₂²×C₂₃	1	2	yes	complemented
291	#407	C₂²×C₂₃	1	4	yes	complemented
292	C₂×C₂²⋊C₈	C₂²×C₂₃	1	8	yes	complemented
293	#405	C₂²×C₂₃	1	4	yes	complemented
294	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	complemented
295	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
296	C₂×C₂₃	—	2	8	no	—
297	C₂×C₂₃	—	4	1	no	—
298	C₂×C₂₃	—	4	4	no	—
299	C₂×C₂₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂×C₂₃	C₂²×#131	1	2	yes	—
301	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
302	C₂×#41	C₂³×C₂₃	1	2	yes	—
303	C₂²×D₄	C₂×C₄×C₂₃	1	4	yes	—
304	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
305	#131	—	4	4	no	—
306	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
307	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
308	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
309	C₂²×C₈	—	4	8	no	—
310	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
311	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
312	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
314	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
315	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
316	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
317	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
318	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
319	#126	C₂×C₄×C₂₃	1	8	yes	—
320	#125	C₂×C₄×C₂₃	1	4	yes	—
321	#124	C₂×C₄×C₂₃	1	4	yes	—
322	#124	C₂×C₄×C₂₃	1	2	yes	—
323	#123	C₂×C₄×C₂₃	1	4	yes	—
324	#123	C₂×C₄×C₂₃	1	4	yes	—
325	#123	C₂×C₄×C₂₃	1	8	yes	—
326	C₂²⋊Q₈	C₂×C₄×C₂₃	1	8	yes	—
327	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
328	#121	C₂×C₄×C₂₃	1	8	yes	—
329	#121	C₂×C₄×C₂₃	1	4	yes	—
330	#121	C₂×C₄×C₂₃	1	4	yes	—
331	#120	C₂×C₄×C₂₃	1	4	yes	—
332	#120	C₂×C₄×C₂₃	1	4	yes	—
333	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
334	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
335	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
336	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
337	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
338	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
339	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
340	C₄×D₄	C₂³×C₂₃	1	2	yes	—
341	C₄×D₄	C₂³×C₂₃	1	8	yes	—
342	C₄×D₄	C₂³×C₂₃	1	4	yes	—
343	C₄×D₄	C₂³×C₂₃	1	4	yes	—
344	#117	C₂×C₄×C₂₃	1	4	yes	—
345	#117	C₂×C₄×C₂₃	1	2	yes	—
346	#117	C₂³×C₂₃	1	4	yes	—
347	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
348	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
349	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
350	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
351	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
352	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
353	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
354	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
355	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
356	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
357	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
358	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
359	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
360	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
361	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
362	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
363	#97	C₂³×C₂₃	1	8	yes	—
364	C₄×C₈	C₂³×C₂₃	1	2	yes	—
365	C₄×C₈	C₂³×C₂₃	1	4	yes	—
366	C₄×C₈	C₂³×C₂₃	1	4	yes	—
367	C₄×C₈	C₂³×C₂₃	1	2	yes	—
368	C₄×C₈	C₂³×C₂₃	1	4	yes	—
369	C₂₃	#60305	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
371	#41	—	2	8	no	—
372	#41	—	4	2	no	—
373	#41	—	4	2	no	—
374	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
375	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
376	C₂×D₄	—	2	16	no	—
377	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
378	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
379	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
380	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
381	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
382	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
386	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
387	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
388	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
389	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
390	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
391	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
393	M₁₆	—	4	2	no	—
394	M₁₆	—	4	4	no	—
395	M₁₆	—	4	4	no	—
396	C₂×C₈	—	2	8	no	—
397	C₂×C₈	—	2	8	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	4	4	no	—
400	C₂×C₈	—	4	8	no	—
401	C₂×C₈	—	4	1	no	—
402	C₂×C₈	—	4	4	no	—
403	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
404	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
405	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
406	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
407	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
408	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
409	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
410	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
411	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
412	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
413	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
414	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
415	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
416	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
417	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
418	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
419	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
420	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
421	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
422	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
423	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
424	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
425	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
426	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
427	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
428	C₄²	C₂⁴×C₂₃	1	2	yes	—
429	C₄²	C₂⁴×C₂₃	1	4	yes	—
430	C₂³	—	2	16	no	—
431	C₂³	—	4	4	no	—
432	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
433	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
434	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
435	Q₈	—	2	4	no	—
436	Q₈	—	4	2	no	—
437	D₄	—	2	16	no	—
438	D₄	—	2	8	no	—
439	D₄	—	2	8	no	—
440	D₄	—	4	4	no	—
441	D₄	—	4	2	no	—
442	D₄	C₂₃×#131	1	8	yes	complemented
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	8	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	8	no	—
454	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
455	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
456	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
457	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
458	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
459	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
460	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
461	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
462	C₈	—	2	4	no	—
463	C₈	—	2	4	no	—
464	C₈	—	2	1	no	—
465	C₈	—	2	2	no	—
466	C₈	—	2	4	no	—
467	C₈	C₂₃×H(5,𝔽₂)	1	2	yes	—
468	C₂²	—	2	4	no	—
469	C₂²	—	2	4	no	—
470	C₂²	—	2	4	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	8	no	—
474	C₂²	—	4	16	no	—
475	C₂²	C₂×C₂₃×#131	1	8	yes	—
476	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
477	C₄	—	2	4	no	—
478	C₄	—	2	4	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	4	no	—
483	C₄	C₂×C₂₃×#131	1	4	yes	—
484	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
485	C₄	C₂₃×#585	1	1	yes	characteristic
486	C₂	—	2	8	no	—
487	C₂	—	4	4	no	—
488	C₂	—	4	1	no	—
489	C₂	C₄×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
490	C₂	C₂²×C₂₃×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂₃×#3481	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2994	C₂	semidirect
3. C₂₃×#2990	C₂	semidirect
4. C₂₃×#2979	C₂	semidirect
5. C₂₃×#2977	C₂	semidirect
6. C₂₃×#2974	C₂	semidirect
7. C₂₃×#2966	C₂	semidirect
8. C₈×#41×C₂₃	C₂	semidirect
9. C₂₃×#2931	C₂	semidirect
10. C₂₃×#2929	C₂	semidirect
11. C₂×C₈×D₄×C₂₃	C₂	semidirect
12. C₂×C₂₃×#131	C₂²	semidirect
13. C₂³×C₈×C₂₃	C₂²	semidirect
25. C₂₃×#446	C₂²	semidirect
26. C₈×Q₈×C₂₃	C₂²	semidirect
27. C₈×Q₈×C₂₃	C₂²	semidirect
28. C₂₃×#436	C₂²	semidirect
29. C₂₃×#436	C₂²	semidirect
30. C₂₃×#435	C₂²	semidirect
31. C₂₃×#435	C₂²	semidirect
32. C₂₃×#435	C₂²	semidirect
36. C₈×D₄×C₂₃	C₂²	semidirect
37. C₈×D₄×C₂₃	C₂²	semidirect
38. C₈×D₄×C₂₃	C₂²	semidirect
39. C₈×D₄×C₂₃	C₂²	semidirect
40. C₈×D₄×C₂₃	C₂²	semidirect
41. C₈×D₄×C₂₃	C₂²	semidirect
42. C₂₃×#433	C₂²	semidirect
43. C₂₃×#432	C₂²	semidirect
44. C₂₃×#431	C₂²	semidirect
45. C₂₃×#431	C₂²	semidirect
46. C₂₃×#424	C₂²	semidirect
47. C₂₃×#423	C₂²	semidirect
48. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
49. C₂₃×#408	C₂²	semidirect
50. C₂₃×#408	C₂²	semidirect
51. C₂₃×#407	C₂²	semidirect
52. C₂×C₂₃×C₂²⋊C₈	C₂²	semidirect
53. C₂₃×#405	C₂²	semidirect
54. C₂×C₄×C₈×C₂₃	C₂²	semidirect
55. C₂×C₄×C₈×C₂₃	C₂²	semidirect
184. #60305	C₂₃	direct (N × H)
197. D₄×C₂₃	#131	semidirect
224. #2994	C₂×C₂₃	semidirect
225. #2990	C₂×C₂₃	semidirect
226. #2979	C₂×C₂₃	semidirect
227. #2977	C₂×C₂₃	semidirect
228. #2974	C₂×C₂₃	semidirect
229. #2966	C₂×C₂₃	semidirect
230. C₈×#41	C₂×C₂₃	semidirect
231. #2931	C₂×C₂₃	semidirect
232. #2929	C₂×C₂₃	semidirect
233. C₂×C₈×D₄	C₂×C₂₃	semidirect
252. C₂×#131	C₂²×C₂₃	semidirect
253. C₂³×C₈	C₂²×C₂₃	semidirect
265. #446	C₂²×C₂₃	semidirect
266. C₈×Q₈	C₂²×C₂₃	semidirect
267. C₈×Q₈	C₂²×C₂₃	semidirect
268. #436	C₂²×C₂₃	semidirect
269. #436	C₂²×C₂₃	semidirect
270. #435	C₂²×C₂₃	semidirect
271. #435	C₂²×C₂₃	semidirect
272. #435	C₂²×C₂₃	semidirect
276. C₈×D₄	C₂²×C₂₃	semidirect
277. C₈×D₄	C₂²×C₂₃	semidirect
278. C₈×D₄	C₂²×C₂₃	semidirect
279. C₈×D₄	C₂²×C₂₃	semidirect
280. C₈×D₄	C₂²×C₂₃	semidirect
281. C₈×D₄	C₂²×C₂₃	semidirect
282. #433	C₂²×C₂₃	semidirect
283. #432	C₂²×C₂₃	semidirect
284. #431	C₂²×C₂₃	semidirect
285. #431	C₂²×C₂₃	semidirect
286. #424	C₂²×C₂₃	semidirect
287. #423	C₂²×C₂₃	semidirect
288. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
289. #408	C₂²×C₂₃	semidirect
290. #408	C₂²×C₂₃	semidirect
291. #407	C₂²×C₂₃	semidirect
292. C₂×C₂²⋊C₈	C₂²×C₂₃	semidirect
293. #405	C₂²×C₂₃	semidirect
294. C₂×C₄×C₈	C₂²×C₂₃	semidirect
295. C₂×C₄×C₈	C₂²×C₂₃	semidirect
369. C₂₃	#60305	direct (N × H)
442. D₄	C₂₃×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3481	2
C₂₃×#2994	2
C₂₃×#2990	2
C₂₃×#2979	2
C₂₃×#2977	2
C₂₃×#2974	2
C₂₃×#2966	2
C₈×#41×C₂₃	2
C₂₃×#2931	2
C₂₃×#2929	2
C₂×C₈×D₄×C₂₃	2
#60305	23

Subgroups — C23×#60305 #19207931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60305 #19207931