Subgroups — C₂₃×#60821 #19208447

Order 5888.

2002 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3499	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3223	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3210	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3209	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3044	C₂	1	4	yes	maximal, complemented
11	C₂₃×#3011	C₂	1	4	yes	maximal, complemented
12	C₂₃×#2922	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	—
14	C₂×C₂₃×#137	C₂²	1	4	yes	—
15	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	—
16	C₂₃×#557	—	2	2	no	—
17	C₂₃×#556	—	2	2	no	—
18	C₂₃×#552	—	2	4	no	—
19	D₄×Q₈×C₂₃	—	2	2	no	—
20	C₂₃×#547	—	2	4	no	—
21	C₂₃×#547	—	2	2	no	—
22	C₂₃×#541	C₂²	1	4	yes	—
23	C₂₃×#536	C₂²	1	4	yes	—
24	C₂₃×#536	C₂²	1	2	yes	—
25	C₂₃×#536	C₂²	1	2	yes	—
26	C₂₃×#533	C₂²	1	4	yes	—
27	C₂₃×#533	C₂²	1	1	yes	characteristic
28	C₂₃×#532	C₂²	1	2	yes	—
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
31	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
33	C₂₃×#519	C₂²	1	2	yes	—
34	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#497	C₂²	1	4	yes	—
36	C₂₃×#494	C₂²	1	2	yes	—
37	C₂₃×#494	C₂²	1	4	yes	—
38	C₂₃×#494	C₂²	1	4	yes	—
39	C₂₃×#492	C₂²	1	2	yes	—
40	C₂₃×#491	C₂²	1	4	yes	—
41	C₂₃×#488	C₂²	1	4	yes	—
42	C₂₃×#487	C₂²	1	2	yes	—
43	C₂₃×#487	C₂²	1	4	yes	—
44	C₂₃×#486	C₂²	1	2	yes	—
45	C₂₃×#464	C₂²	1	16	yes	—
46	C₂₃×#462	C₂²	1	16	yes	—
47	C₂₃×#452	C₂²	1	16	yes	—
48	C₂₃×#451	C₂²	1	16	yes	—
49	C₂₃×#433	C₂²	1	4	yes	—
50	C₂₃×#431	C₂²	1	4	yes	—
51	C₂₃×#424	C₂²	1	2	yes	—
52	C₂₃×#419	C₂²	1	4	yes	—
53	C₂₃×#416	C₂²	1	4	yes	—
54	C₂₃×#408	C₂²	1	2	yes	—
55	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
57	C₂₃×2₋¹⁺⁴	—	4	8	no	—
58	C₂×#41×C₂₃	—	2	4	no	—
59	C₂×#41×C₂₃	—	2	2	no	—
60	C₂×#41×C₂₃	C₂³	1	2	yes	—
61	C₂×#41×C₂₃	C₂³	1	4	yes	—
62	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	—	2	2	no	—
64	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
65	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
66	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
67	C₂₃×#137	—	4	8	no	—
68	C₂×Q₁₆×C₂₃	—	4	8	no	—
69	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
70	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
71	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
72	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
73	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
74	C₂₃×C₄⋊Q₈	—	2	2	no	—
75	C₂₃×C₄⋊Q₈	—	2	4	no	—
76	C₂₃×C₄⋊Q₈	—	2	2	no	—
77	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
81	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
85	C₂₃×#127	C₂³	1	1	yes	characteristic
86	C₂₃×#126	—	2	8	no	—
87	C₂₃×#125	—	2	8	no	—
88	C₂₃×#125	—	2	4	no	—
89	C₂₃×#125	C₂³	1	2	yes	—
90	C₂₃×#124	—	2	2	no	—
91	C₂₃×#124	C₂³	1	4	yes	—
92	C₂₃×#124	C₂³	1	2	yes	—
93	C₂₃×#123	—	2	4	no	—
94	C₂₃×#123	—	2	4	no	—
95	C₂₃×#123	—	2	8	no	—
96	C₂₃×C₂²⋊Q₈	—	2	2	no	—
97	C₂₃×C₂²⋊Q₈	—	2	4	no	—
98	C₂₃×C₂²⋊Q₈	—	2	4	no	—
99	C₂₃×C₂²⋊Q₈	—	2	8	no	—
100	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
101	C₂₃×#121	—	2	2	no	—
102	C₂₃×#121	—	2	4	no	—
103	C₄×Q₈×C₂₃	—	2	4	no	—
104	C₄×Q₈×C₂₃	—	2	4	no	—
105	C₄×Q₈×C₂₃	—	2	2	no	—
106	C₄×Q₈×C₂₃	D₄	1	2	yes	—
107	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
108	C₄×D₄×C₂₃	—	2	4	no	—
109	C₄×D₄×C₂₃	—	2	2	no	—
110	C₄×D₄×C₂₃	D₄	1	4	yes	—
111	C₄×D₄×C₂₃	D₄	1	2	yes	—
112	C₂₃×#117	—	2	4	no	—
113	C₂₃×#117	—	2	4	no	—
114	C₂₃×#117	C₂³	1	2	yes	—
115	C₂₃×#117	C₂³	1	1	yes	characteristic
116	C₂₃×#117	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
118	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
119	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
122	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
123	C₂₃×#103	C₂³	1	8	yes	—
124	C₂₃×#103	C₂³	1	16	yes	—
125	C₂₃×#102	C₂³	1	8	yes	—
126	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
127	C₂₃×#97	C₂³	1	2	yes	—
128	C₄×C₈×C₂₃	C₂³	1	2	yes	—
129	C₄×C₈×C₂₃	C₂³	1	2	yes	—
130	C₄×C₈×C₂₃	C₂³	1	2	yes	—
131	#41×C₂₃	—	2	4	no	—
132	#41×C₂₃	—	2	8	no	—
133	#41×C₂₃	—	4	8	no	—
134	#41×C₂₃	—	4	16	no	—
135	#41×C₂₃	—	4	8	no	—
136	C₂×Q₈×C₂₃	—	2	2	no	—
137	C₂×Q₈×C₂₃	—	2	2	no	—
138	C₂×Q₈×C₂₃	—	2	4	no	—
139	C₂×Q₈×C₂₃	—	2	2	no	—
140	C₂×Q₈×C₂₃	—	2	8	no	—
141	C₂×Q₈×C₂₃	—	2	8	no	—
142	C₂×Q₈×C₂₃	—	4	8	no	—
143	C₂×Q₈×C₂₃	—	4	8	no	—
144	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
146	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
147	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
148	C₂×D₄×C₂₃	—	2	4	no	—
149	C₂×D₄×C₂₃	—	2	2	no	—
150	C₂×D₄×C₂₃	—	2	2	no	—
151	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₂₃	—	2	2	no	—
155	C₂²×C₄×C₂₃	—	2	4	no	—
156	C₂²×C₄×C₂₃	—	2	2	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	2	2	no	—
159	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
163	Q₁₆×C₂₃	—	4	16	no	—
164	Q₁₆×C₂₃	—	4	8	no	—
165	SD₁₆×C₂₃	—	4	8	no	—
166	M₁₆×C₂₃	—	2	4	no	—
167	C₂×C₈×C₂₃	—	2	4	no	—
168	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
169	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
170	C₄⋊C₄×C₂₃	—	2	4	no	—
171	C₄⋊C₄×C₂₃	—	2	2	no	—
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	8	no	—
174	C₄⋊C₄×C₂₃	—	2	4	no	—
175	C₄⋊C₄×C₂₃	—	2	4	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	—	2	8	no	—
178	C₄⋊C₄×C₂₃	—	2	4	no	—
179	C₄⋊C₄×C₂₃	—	2	8	no	—
180	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
183	C₂²⋊C₄×C₂₃	—	2	2	no	—
184	C₂²⋊C₄×C₂₃	—	2	8	no	—
185	C₂²⋊C₄×C₂₃	—	2	8	no	—
186	C₂²⋊C₄×C₂₃	—	2	4	no	—
187	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
188	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
189	C₄²×C₂₃	—	2	8	no	—
190	C₄²×C₂₃	—	2	2	no	—
191	C₄²×C₂₃	C₂×D₄	1	4	yes	—
192	C₄²×C₂₃	C₂×D₄	1	2	yes	—
193	C₄²×C₂₃	C₂⁴	1	2	yes	—
194	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
195	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
196	#60821	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₂₃	—	2	2	no	—
198	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
199	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
200	Q₈×C₂₃	—	2	2	no	—
201	Q₈×C₂₃	—	2	8	no	—
202	Q₈×C₂₃	—	2	8	no	—
203	Q₈×C₂₃	—	2	4	no	—
204	Q₈×C₂₃	—	4	8	no	—
205	Q₈×C₂₃	—	4	4	no	—
206	Q₈×C₂₃	—	4	4	no	—
207	Q₈×C₂₃	—	4	4	no	—
208	D₄×C₂₃	—	2	4	no	—
209	D₄×C₂₃	—	2	2	no	—
210	D₄×C₂₃	—	2	4	no	—
211	D₄×C₂₃	—	4	8	no	—
212	D₄×C₂₃	—	4	4	no	—
213	D₄×C₂₃	—	4	4	no	—
214	C₂×C₄×C₂₃	—	2	4	no	—
215	C₂×C₄×C₂₃	—	2	8	no	—
216	C₂×C₄×C₂₃	—	2	4	no	—
217	C₂×C₄×C₂₃	—	2	4	no	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	4	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	4	8	no	—
222	C₂×C₄×C₂₃	—	4	8	no	—
223	C₂×C₄×C₂₃	—	4	4	no	—
224	C₂×C₄×C₂₃	—	4	4	no	—
225	C₂×C₄×C₂₃	—	4	4	no	—
226	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
232	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
233	C₈×C₂₃	—	2	4	no	—
234	C₈×C₂₃	—	2	4	no	—
235	#3509	C₂×C₂₃	1	1	yes	characteristic, complemented
236	#3499	C₂×C₂₃	1	1	yes	characteristic
237	#3223	C₂×C₂₃	1	8	yes	complemented
238	#3210	C₂×C₂₃	1	4	yes	complemented
239	#3209	C₂×C₂₃	1	4	yes	complemented
240	#3152	C₂×C₂₃	1	1	yes	characteristic, complemented
241	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
242	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
243	#3131	C₂×C₂₃	1	1	yes	characteristic, complemented
244	#3044	C₂×C₂₃	1	4	yes	complemented
245	#3011	C₂×C₂₃	1	4	yes	complemented
246	#2922	C₂×C₂₃	1	1	yes	characteristic, complemented
247	C₂²×C₂₃	—	2	2	no	—
248	C₂²×C₂₃	—	2	2	no	—
249	C₂²×C₂₃	—	2	4	no	—
250	C₂²×C₂₃	—	4	2	no	—
251	C₂²×C₂₃	—	4	4	no	—
252	C₂²×C₂₃	#535	1	2	yes	—
253	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	2	4	no	—
256	C₄×C₂₃	—	2	2	no	—
257	C₄×C₂₃	—	4	4	no	—
258	C₄×C₂₃	—	4	2	no	—
259	C₄×C₂₃	—	4	4	no	—
260	C₄×C₂₃	—	4	4	no	—
261	C₄×C₂₃	#535	1	2	yes	—
262	C₄×C₂₃	#578	1	4	yes	—
263	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
264	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	—
265	C₂×#137	C₂²×C₂₃	1	4	yes	—
266	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	—
267	#557	—	2	2	no	—
268	#556	—	2	2	no	—
269	#552	—	2	4	no	—
270	D₄×Q₈	—	2	2	no	—
271	#547	—	2	4	no	—
272	#547	—	2	2	no	—
273	#541	C₂²×C₂₃	1	4	yes	—
274	#536	C₂²×C₂₃	1	4	yes	—
275	#536	C₂²×C₂₃	1	2	yes	—
276	#536	C₂²×C₂₃	1	2	yes	—
277	#533	C₂²×C₂₃	1	4	yes	—
278	#533	C₂²×C₂₃	1	1	yes	characteristic
279	#532	C₂²×C₂₃	1	1	yes	characteristic
280	#532	C₂²×C₂₃	1	2	yes	—
281	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
282	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
283	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
284	#519	C₂²×C₂₃	1	2	yes	—
285	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
286	#497	C₂²×C₂₃	1	4	yes	—
287	#494	C₂²×C₂₃	1	4	yes	—
288	#494	C₂²×C₂₃	1	4	yes	—
289	#494	C₂²×C₂₃	1	2	yes	—
290	#492	C₂²×C₂₃	1	2	yes	—
291	#491	C₂²×C₂₃	1	4	yes	—
292	#488	C₂²×C₂₃	1	4	yes	—
293	#487	C₂²×C₂₃	1	2	yes	—
294	#487	C₂²×C₂₃	1	4	yes	—
295	#486	C₂²×C₂₃	1	2	yes	—
296	#464	C₂²×C₂₃	1	16	yes	—
297	#462	C₂²×C₂₃	1	16	yes	—
298	#452	C₂²×C₂₃	1	16	yes	—
299	#451	C₂²×C₂₃	1	16	yes	—
300	#433	C₂²×C₂₃	1	4	yes	—
301	#431	C₂²×C₂₃	1	4	yes	—
302	#424	C₂²×C₂₃	1	2	yes	—
303	#419	C₂²×C₂₃	1	4	yes	—
304	#416	C₂²×C₂₃	1	4	yes	—
305	#408	C₂²×C₂₃	1	2	yes	—
306	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×C₂₃	—	2	2	no	—
309	C₂×C₂₃	—	4	2	no	—
310	C₂×C₂₃	—	8	2	no	—
311	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
312	C₂×C₂₃	C₂×#578	1	2	yes	—
313	2₋¹⁺⁴	—	4	8	no	—
314	C₂×#41	—	2	4	no	—
315	C₂×#41	—	2	2	no	—
316	C₂×#41	C₂³×C₂₃	1	2	yes	—
317	C₂×#41	C₂³×C₂₃	1	4	yes	—
318	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
319	C₂²×Q₈	—	2	2	no	—
320	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
321	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
322	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
323	#137	—	4	8	no	—
324	C₂×Q₁₆	—	4	8	no	—
325	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
326	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
327	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
328	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
329	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	—	2	4	no	—
332	C₄⋊Q₈	—	2	2	no	—
333	C₄⋊Q₈	—	2	2	no	—
334	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
335	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
336	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
337	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
338	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
339	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
340	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
341	#127	C₂³×C₂₃	1	1	yes	characteristic
342	#126	—	2	8	no	—
343	#125	—	2	8	no	—
344	#125	—	2	4	no	—
345	#125	C₂³×C₂₃	1	2	yes	—
346	#124	—	2	2	no	—
347	#124	C₂³×C₂₃	1	4	yes	—
348	#124	C₂³×C₂₃	1	2	yes	—
349	#123	—	2	4	no	—
350	#123	—	2	4	no	—
351	#123	—	2	8	no	—
352	C₂²⋊Q₈	—	2	2	no	—
353	C₂²⋊Q₈	—	2	4	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	8	no	—
356	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
357	#121	—	2	2	no	—
358	#121	—	2	4	no	—
359	C₄×Q₈	—	2	4	no	—
360	C₄×Q₈	—	2	4	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	D₄×C₂₃	1	2	yes	—
363	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	D₄×C₂₃	1	4	yes	—
367	C₄×D₄	D₄×C₂₃	1	2	yes	—
368	#117	—	2	4	no	—
369	#117	—	2	4	no	—
370	#117	C₂³×C₂₃	1	2	yes	—
371	#117	C₂³×C₂₃	1	1	yes	characteristic
372	#117	C₂³×C₂₃	1	2	yes	—
373	C₂×C₄⋊C₄	—	2	2	no	—
374	C₂×C₄⋊C₄	—	2	4	no	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
378	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
379	#103	C₂³×C₂₃	1	8	yes	—
380	#103	C₂³×C₂₃	1	16	yes	—
381	#102	C₂³×C₂₃	1	8	yes	—
382	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
383	#97	C₂³×C₂₃	1	2	yes	—
384	C₄×C₈	C₂³×C₂₃	1	2	yes	—
385	C₄×C₈	C₂³×C₂₃	1	2	yes	—
386	C₄×C₈	C₂³×C₂₃	1	2	yes	—
387	C₂₃	#60821	1	1	yes	characteristic, direct product factor, Sylow
388	#41	—	2	4	no	—
389	#41	—	2	8	no	—
390	#41	—	4	8	no	—
391	#41	—	4	16	no	—
392	#41	—	4	8	no	—
393	C₂×Q₈	—	2	2	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	2	8	no	—
398	C₂×Q₈	—	2	8	no	—
399	C₂×Q₈	—	4	8	no	—
400	C₂×Q₈	—	4	8	no	—
401	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
402	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
403	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
404	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
409	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
410	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
417	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
418	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
419	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
420	Q₁₆	—	4	16	no	—
421	Q₁₆	—	4	8	no	—
422	SD₁₆	—	4	8	no	—
423	M₁₆	—	2	4	no	—
424	C₂×C₈	—	2	4	no	—
425	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
426	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	8	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
438	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
439	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
440	C₂²⋊C₄	—	2	2	no	—
441	C₂²⋊C₄	—	2	8	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
445	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
446	C₄²	—	2	8	no	—
447	C₄²	—	2	2	no	—
448	C₄²	C₂×D₄×C₂₃	1	4	yes	—
449	C₄²	C₂×D₄×C₂₃	1	2	yes	—
450	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
451	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
452	C₄²	C₂⁴×C₂₃	1	2	yes	—
453	C₂³	—	2	2	no	—
454	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
455	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
456	Q₈	—	2	2	no	—
457	Q₈	—	2	8	no	—
458	Q₈	—	2	8	no	—
459	Q₈	—	2	4	no	—
460	Q₈	—	4	8	no	—
461	Q₈	—	4	4	no	—
462	Q₈	—	4	4	no	—
463	Q₈	—	4	4	no	—
464	D₄	—	2	4	no	—
465	D₄	—	2	4	no	—
466	D₄	—	2	2	no	—
467	D₄	—	4	8	no	—
468	D₄	—	4	4	no	—
469	D₄	—	4	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	4	4	no	—
481	C₂×C₄	—	4	4	no	—
482	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
483	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
487	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
488	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
489	C₈	—	2	4	no	—
490	C₈	—	2	4	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	4	no	—
496	C₂²	C₂₃×#535	1	2	yes	—
497	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	2	no	—
501	C₄	—	4	4	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	4	no	—
505	C₄	C₂₃×#535	1	2	yes	—
506	C₄	C₂₃×#578	1	4	yes	—
507	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
508	C₂	—	2	2	no	—
509	C₂	—	4	2	no	—
510	C₂	—	8	2	no	—
511	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
512	C₂	C₂×C₂₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3509	C₂	semidirect
3. C₂₃×#3223	C₂	semidirect
4. C₂₃×#3210	C₂	semidirect
5. C₂₃×#3209	C₂	semidirect
6. C₂₃×#3152	C₂	semidirect
7. C₂×C₂₃×#494	C₂	semidirect
8. C₂₃×#3139	C₂	semidirect
9. C₂₃×#3131	C₂	semidirect
10. C₂₃×#3044	C₂	semidirect
11. C₂₃×#3011	C₂	semidirect
12. C₂₃×#2922	C₂	semidirect
196. #60821	C₂₃	direct (N × H)
235. #3509	C₂×C₂₃	semidirect
237. #3223	C₂×C₂₃	semidirect
238. #3210	C₂×C₂₃	semidirect
239. #3209	C₂×C₂₃	semidirect
240. #3152	C₂×C₂₃	semidirect
241. C₂×#494	C₂×C₂₃	semidirect
242. #3139	C₂×C₂₃	semidirect
243. #3131	C₂×C₂₃	semidirect
244. #3044	C₂×C₂₃	semidirect
245. #3011	C₂×C₂₃	semidirect
246. #2922	C₂×C₂₃	semidirect
387. C₂₃	#60821	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3509	2
C₂₃×#3499	2
C₂₃×#3223	2
C₂₃×#3210	2
C₂₃×#3209	2
C₂₃×#3152	2
C₂×C₂₃×#494	2
C₂₃×#3139	2
C₂₃×#3131	2
C₂₃×#3044	2
C₂₃×#3011	2
C₂₃×#2922	2
#60821	23

Subgroups — C23×#60821 #19208447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60821 #19208447