Subgroups — C₂×C₂₃×#3043 #19208044

Order 5888.

2858 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#535	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3043	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₂₃×#464	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#460	C₂	1	4	yes	maximal, complemented
8	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#418	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#137	—	2	8	no	—
15	C₂×C₂₃×#137	C₂²	1	8	yes	complemented
16	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	complemented
17	C₂²×SD₁₆×C₂₃	—	2	8	no	—
18	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
19	C₂²×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
20	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
21	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#535	—	2	4	no	—
24	C₂₃×#535	C₂²	1	8	yes	—
25	C₂×C₂₃×#127	C₂²	1	2	yes	—
26	C₂×C₂₃×#124	C₂²	1	2	yes	—
27	C₂×C₂₃×#121	—	2	2	no	—
28	C₂×C₂₃×#120	—	2	2	no	—
29	C₂₃×#519	—	2	4	no	—
30	C₂₃×#519	C₂²	1	8	yes	complemented
31	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
32	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
33	C₂×C₂₃×#117	—	2	2	no	—
34	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
36	C₂₃×#464	C₂²	1	32	yes	complemented
37	C₂₃×#460	C₂²	1	32	yes	complemented
38	C₂₃×#424	C₂²	1	8	yes	complemented
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
40	C₂₃×#418	C₂²	1	8	yes	complemented
41	C₂₃×#416	C₂²	1	8	yes	complemented
42	C₂×C₂₃×#103	C₂²	1	2	yes	complemented
43	C₂×C₂₃×#102	C₂²	1	4	yes	—
44	C₂×C₂₃×#102	C₂²	1	2	yes	complemented
45	C₂⁵×C₂₃	—	2	1	no	—
46	C₂×#41×C₂₃	—	2	2	no	—
47	C₂×#41×C₂₃	—	2	4	no	—
48	C₂×#41×C₂₃	—	2	4	no	—
49	C₂×#41×C₂₃	C₂³	1	4	yes	—
50	C₂²×Q₈×C₂₃	—	2	1	no	—
51	C₂²×Q₈×C₂₃	—	2	2	no	—
52	C₂²×Q₈×C₂₃	—	2	4	no	—
53	C₂²×Q₈×C₂₃	D₄	1	8	yes	complemented
54	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
56	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
57	C₂²×D₄×C₂₃	—	2	1	no	—
58	C₂²×D₄×C₂₃	—	2	4	no	—
59	C₂²×D₄×C₂₃	—	2	1	no	—
60	C₂²×D₄×C₂₃	—	2	2	no	—
61	C₂²×D₄×C₂₃	—	2	4	no	—
62	C₂²×D₄×C₂₃	—	2	4	no	—
63	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
64	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂₃×#137	—	2	32	no	—
70	C₂×Q₁₆×C₂₃	—	2	8	no	—
71	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	complemented
72	C₂×SD₁₆×C₂₃	—	2	16	no	—
73	C₂×SD₁₆×C₂₃	—	2	8	no	—
74	C₂×SD₁₆×C₂₃	—	2	32	no	—
75	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
76	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
77	C₂×M₁₆×C₂₃	—	2	4	no	—
78	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
79	C₂²×C₈×C₂₃	—	2	4	no	—
80	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
81	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
82	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
83	C₂₃×#127	—	2	4	no	—
84	C₂₃×#127	C₂³	1	8	yes	—
85	C₂₃×#124	—	2	4	no	—
86	C₂₃×#124	C₂³	1	8	yes	—
87	C₂₃×#121	—	2	16	no	—
88	C₂₃×#120	—	2	16	no	—
89	C₄×Q₈×C₂₃	—	2	16	no	—
90	C₄×Q₈×C₂₃	—	2	8	no	—
91	C₄×Q₈×C₂₃	C₂³	1	16	yes	complemented
92	C₂₃×#117	—	2	2	no	—
93	C₂₃×#117	—	2	16	no	—
94	C₂₃×#117	D₄	1	4	yes	—
95	C₂₃×#117	C₂³	1	4	yes	—
96	C₂₃×#117	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
98	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
102	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
103	C₂×C₄²×C₂₃	—	2	4	no	—
104	C₂×C₄²×C₂₃	—	2	1	no	—
105	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
106	C₂₃×C₄⋊C₈	C₂³	1	16	yes	complemented
107	C₂₃×#103	C₂³	1	8	yes	complemented
108	C₂₃×#102	C₂³	1	16	yes	—
109	C₂₃×#102	C₂³	1	8	yes	complemented
110	C₂⁴×C₂₃	—	2	2	no	—
111	C₂⁴×C₂₃	—	2	1	no	—
112	C₂⁴×C₂₃	—	2	4	no	—
113	C₂⁴×C₂₃	—	4	4	no	—
114	C₂⁴×C₂₃	—	8	4	no	—
115	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
116	#41×C₂₃	—	2	8	no	—
117	#41×C₂₃	—	2	8	no	—
118	#41×C₂₃	—	4	8	no	—
119	C₂×Q₈×C₂₃	—	2	8	no	—
120	C₂×Q₈×C₂₃	—	2	16	no	—
121	C₂×Q₈×C₂₃	—	2	4	no	—
122	C₂×Q₈×C₂₃	—	2	2	no	—
123	C₂×Q₈×C₂₃	—	2	1	no	—
124	C₂×Q₈×C₂₃	—	2	4	no	—
125	C₂×Q₈×C₂₃	—	2	8	no	—
126	C₂×Q₈×C₂₃	—	2	2	no	—
127	C₂×Q₈×C₂₃	—	2	2	no	—
128	C₂×Q₈×C₂₃	C₂×D₄	1	8	yes	—
129	C₂×Q₈×C₂₃	C₂×D₄	1	16	yes	complemented
130	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
131	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
132	C₂×D₄×C₂₃	—	2	4	no	—
133	C₂×D₄×C₂₃	—	2	4	no	—
134	C₂×D₄×C₂₃	—	2	8	no	—
135	C₂×D₄×C₂₃	—	2	2	no	—
136	C₂×D₄×C₂₃	—	2	4	no	—
137	C₂×D₄×C₂₃	—	2	2	no	—
138	C₂×D₄×C₂₃	—	2	1	no	—
139	C₂×D₄×C₂₃	—	2	16	no	—
140	C₂×D₄×C₂₃	—	2	4	no	—
141	C₂×D₄×C₂₃	—	2	8	no	—
142	C₂×D₄×C₂₃	—	2	2	no	—
143	C₂×D₄×C₂₃	—	4	4	no	—
144	C₂×D₄×C₂₃	—	4	4	no	—
145	C₂×D₄×C₂₃	—	4	8	no	—
146	C₂×D₄×C₂₃	—	8	8	no	—
147	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
148	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₂₃	—	2	8	no	—
150	C₂²×C₄×C₂₃	—	2	4	no	—
151	C₂²×C₄×C₂₃	—	2	2	no	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	4	no	—
155	C₂²×C₄×C₂₃	—	4	4	no	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₂₃	#41	1	4	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
163	Q₁₆×C₂₃	—	2	16	no	—
164	SD₁₆×C₂₃	—	2	32	no	—
165	SD₁₆×C₂₃	—	2	16	no	—
166	M₁₆×C₂₃	—	2	8	no	—
167	C₂×C₈×C₂₃	—	2	4	no	—
168	C₂×C₈×C₂₃	—	2	8	no	—
169	C₂×C₈×C₂₃	—	2	4	no	—
170	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
171	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
172	C₄⋊C₄×C₂₃	—	2	16	no	—
173	C₄⋊C₄×C₂₃	—	2	2	no	—
174	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
175	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
176	C₂²⋊C₄×C₂₃	—	2	4	no	—
177	C₂²⋊C₄×C₂₃	—	2	16	no	—
178	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
179	C₄²×C₂₃	—	2	16	no	—
180	C₄²×C₂₃	—	2	4	no	—
181	C₄²×C₂₃	C₂×D₄	1	4	yes	—
182	C₄²×C₂₃	C₂⁴	1	4	yes	—
183	C₂×#3043	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₂₃	—	2	4	no	—
185	C₂³×C₂₃	—	2	4	no	—
186	C₂³×C₂₃	—	2	2	no	—
187	C₂³×C₂₃	—	4	2	no	—
188	C₂³×C₂₃	—	4	8	no	—
189	C₂³×C₂₃	—	4	4	no	—
190	C₂³×C₂₃	—	4	4	no	—
191	C₂³×C₂₃	—	4	8	no	—
192	C₂³×C₂₃	—	8	8	no	—
193	C₂³×C₂₃	—	8	16	no	—
194	C₂³×C₂₃	#121	1	4	yes	—
195	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
196	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
197	Q₈×C₂₃	—	2	8	no	—
198	Q₈×C₂₃	—	2	16	no	—
199	Q₈×C₂₃	—	2	4	no	—
200	Q₈×C₂₃	—	2	2	no	—
201	Q₈×C₂₃	—	2	2	no	—
202	Q₈×C₂₃	—	4	2	no	—
203	Q₈×C₂₃	C₂²×D₄	1	8	yes	—
204	D₄×C₂₃	—	2	4	no	—
205	D₄×C₂₃	—	2	2	no	—
206	D₄×C₂₃	—	2	4	no	—
207	D₄×C₂₃	—	2	2	no	—
208	D₄×C₂₃	—	4	8	no	—
209	D₄×C₂₃	—	4	8	no	—
210	D₄×C₂₃	—	4	4	no	—
211	D₄×C₂₃	—	4	2	no	—
212	D₄×C₂₃	—	8	16	no	—
213	C₂×C₄×C₂₃	—	2	8	no	—
214	C₂×C₄×C₂₃	—	2	16	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	2	no	—
219	C₂×C₄×C₂₃	—	4	2	no	—
220	C₂×C₄×C₂₃	—	4	4	no	—
221	C₂×C₄×C₂₃	—	4	8	no	—
222	C₂×C₄×C₂₃	#121	1	4	yes	—
223	C₂×C₄×C₂₃	#121	1	8	yes	—
224	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
225	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
229	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
230	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
231	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
232	C₈×C₂₃	—	2	4	no	—
233	C₈×C₂₃	—	2	4	no	—
234	C₂²×#137	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂³×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂×#535	C₂×C₂₃	1	1	yes	characteristic
237	C₂×#519	C₂×C₂₃	1	1	yes	characteristic, complemented
238	#3043	C₂×C₂₃	1	16	yes	direct product factor
239	C₂×#464	C₂×C₂₃	1	4	yes	complemented
240	C₂×#460	C₂×C₂₃	1	4	yes	complemented
241	C₂×#424	C₂×C₂₃	1	1	yes	characteristic, complemented
242	C₂×#418	C₂×C₂₃	1	1	yes	characteristic, complemented
243	C₂×#416	C₂×C₂₃	1	1	yes	characteristic, complemented
244	C₂²×C₂₃	—	2	8	no	—
245	C₂²×C₂₃	—	2	4	no	—
246	C₂²×C₂₃	—	4	2	no	—
247	C₂²×C₂₃	—	4	4	no	—
248	C₂²×C₂₃	—	4	4	no	—
249	C₂²×C₂₃	—	4	8	no	—
250	C₂²×C₂₃	—	8	16	no	—
251	C₂²×C₂₃	—	8	4	no	—
252	C₂²×C₂₃	—	8	8	no	—
253	C₂²×C₂₃	C₂×#121	1	4	yes	—
254	C₂²×C₂₃	C₂×#121	1	2	yes	—
255	C₂²×C₂₃	#577	1	4	yes	—
256	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₂₃	—	2	8	no	—
258	C₄×C₂₃	—	2	8	no	—
259	C₄×C₂₃	—	4	2	no	—
260	C₄×C₂₃	—	4	4	no	—
261	C₄×C₂₃	C₂×#121	1	4	yes	—
262	C₄×C₂₃	C₂×#121	1	2	yes	—
263	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
264	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
265	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
266	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
267	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
268	C₂×#137	—	2	8	no	—
269	C₂×#137	C₂²×C₂₃	1	8	yes	complemented
270	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	complemented
271	C₂²×SD₁₆	—	2	8	no	—
272	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
273	C₂²×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
274	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
275	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
276	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
277	#535	—	2	4	no	—
278	#535	C₂²×C₂₃	1	8	yes	—
279	C₂×#127	C₂²×C₂₃	1	2	yes	—
280	C₂×#124	C₂²×C₂₃	1	2	yes	—
281	C₂×#121	—	2	2	no	—
282	C₂×#120	—	2	2	no	—
283	#519	—	2	4	no	—
284	#519	C₂²×C₂₃	1	8	yes	complemented
285	C₂×C₄×Q₈	—	2	2	no	—
286	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
287	C₂×#117	—	2	2	no	—
288	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
289	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
290	#464	C₂²×C₂₃	1	32	yes	complemented
291	#460	C₂²×C₂₃	1	32	yes	complemented
292	#424	C₂²×C₂₃	1	8	yes	complemented
293	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
294	#418	C₂²×C₂₃	1	8	yes	complemented
295	#416	C₂²×C₂₃	1	8	yes	complemented
296	C₂×#103	C₂²×C₂₃	1	2	yes	complemented
297	C₂×#102	C₂²×C₂₃	1	4	yes	—
298	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
299	C₂×C₂₃	—	2	4	no	—
300	C₂×C₂₃	—	8	2	no	—
301	C₂×C₂₃	—	8	4	no	—
302	C₂×C₂₃	#3043	1	4	yes	direct product factor
303	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
304	C₂×C₂₃	C₂×#577	1	2	yes	—
305	C₂⁵	—	2	1	no	—
306	C₂×#41	—	2	2	no	—
307	C₂×#41	—	2	4	no	—
308	C₂×#41	—	2	4	no	—
309	C₂×#41	C₂³×C₂₃	1	4	yes	—
310	C₂²×Q₈	—	2	1	no	—
311	C₂²×Q₈	—	2	2	no	—
312	C₂²×Q₈	—	2	4	no	—
313	C₂²×Q₈	D₄×C₂₃	1	8	yes	complemented
314	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
315	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
316	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	—	2	4	no	—
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	—	2	2	no	—
321	C₂²×D₄	—	2	4	no	—
322	C₂²×D₄	—	2	4	no	—
323	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
324	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
325	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
326	C₂³×C₄	—	2	1	no	—
327	C₂³×C₄	—	2	1	no	—
328	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
329	#137	—	2	32	no	—
330	C₂×Q₁₆	—	2	8	no	—
331	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	complemented
332	C₂×SD₁₆	—	2	16	no	—
333	C₂×SD₁₆	—	2	8	no	—
334	C₂×SD₁₆	—	2	32	no	—
335	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
336	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
337	C₂×M₁₆	—	2	4	no	—
338	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
339	C₂²×C₈	—	2	4	no	—
340	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
341	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
342	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
343	#127	—	2	4	no	—
344	#127	C₂³×C₂₃	1	8	yes	—
345	#124	—	2	4	no	—
346	#124	C₂³×C₂₃	1	8	yes	—
347	#121	—	2	16	no	—
348	#120	—	2	16	no	—
349	C₄×Q₈	—	2	16	no	—
350	C₄×Q₈	—	2	8	no	—
351	C₄×Q₈	C₂³×C₂₃	1	16	yes	complemented
352	#117	—	2	2	no	—
353	#117	—	2	16	no	—
354	#117	D₄×C₂₃	1	4	yes	—
355	#117	C₂³×C₂₃	1	4	yes	—
356	#117	C₂³×C₂₃	1	4	yes	—
357	C₂×C₄⋊C₄	—	2	4	no	—
358	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₂²⋊C₄	—	2	4	no	—
362	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
363	C₂×C₄²	—	2	4	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
366	C₄⋊C₈	C₂³×C₂₃	1	16	yes	complemented
367	#103	C₂³×C₂₃	1	8	yes	complemented
368	#102	C₂³×C₂₃	1	16	yes	—
369	#102	C₂³×C₂₃	1	8	yes	complemented
370	C₂₃	C₂×#3043	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	2	2	no	—
372	C₂⁴	—	2	1	no	—
373	C₂⁴	—	2	4	no	—
374	C₂⁴	—	4	4	no	—
375	C₂⁴	—	8	4	no	—
376	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
377	#41	—	2	8	no	—
378	#41	—	2	8	no	—
379	#41	—	4	8	no	—
380	C₂×Q₈	—	2	8	no	—
381	C₂×Q₈	—	2	16	no	—
382	C₂×Q₈	—	2	4	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	2	8	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	C₂×D₄×C₂₃	1	8	yes	—
390	C₂×Q₈	C₂×D₄×C₂₃	1	16	yes	complemented
391	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
392	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	2	4	no	—
395	C₂×D₄	—	2	8	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	16	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	4	4	no	—
405	C₂×D₄	—	4	4	no	—
406	C₂×D₄	—	4	8	no	—
407	C₂×D₄	—	8	8	no	—
408	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
409	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	4	4	no	—
417	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
418	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
421	C₂²×C₄	#41×C₂₃	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
423	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
424	Q₁₆	—	2	16	no	—
425	SD₁₆	—	2	32	no	—
426	SD₁₆	—	2	16	no	—
427	M₁₆	—	2	8	no	—
428	C₂×C₈	—	2	4	no	—
429	C₂×C₈	—	2	8	no	—
430	C₂×C₈	—	2	4	no	—
431	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
432	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
433	C₄⋊C₄	—	2	16	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
436	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	16	no	—
439	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
440	C₄²	—	2	16	no	—
441	C₄²	—	2	4	no	—
442	C₄²	C₂×D₄×C₂₃	1	4	yes	—
443	C₄²	C₂⁴×C₂₃	1	4	yes	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	4	4	no	—
448	C₂³	—	4	8	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	8	no	—
451	C₂³	—	4	4	no	—
452	C₂³	—	8	8	no	—
453	C₂³	—	8	16	no	—
454	C₂³	C₂₃×#121	1	4	yes	—
455	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
456	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
457	Q₈	—	2	4	no	—
458	Q₈	—	2	8	no	—
459	Q₈	—	2	16	no	—
460	Q₈	—	2	2	no	—
461	Q₈	—	2	2	no	—
462	Q₈	—	4	2	no	—
463	Q₈	C₂²×D₄×C₂₃	1	8	yes	—
464	D₄	—	2	4	no	—
465	D₄	—	2	2	no	—
466	D₄	—	2	4	no	—
467	D₄	—	2	2	no	—
468	D₄	—	4	8	no	—
469	D₄	—	4	8	no	—
470	D₄	—	4	4	no	—
471	D₄	—	4	2	no	—
472	D₄	—	8	16	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	8	no	—
476	C₂×C₄	—	2	16	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	8	no	—
482	C₂×C₄	C₂₃×#121	1	4	yes	—
483	C₂×C₄	C₂₃×#121	1	8	yes	—
484	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
485	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
487	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
488	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
490	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
491	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
492	C₈	—	2	4	no	—
493	C₈	—	2	4	no	—
494	C₂²	—	2	8	no	—
495	C₂²	—	2	4	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	8	no	—
500	C₂²	—	8	16	no	—
501	C₂²	—	8	8	no	—
502	C₂²	—	8	4	no	—
503	C₂²	C₂×C₂₃×#121	1	4	yes	—
504	C₂²	C₂×C₂₃×#121	1	2	yes	—
505	C₂²	C₂₃×#577	1	4	yes	—
506	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
507	C₄	—	2	8	no	—
508	C₄	—	2	8	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	4	no	—
511	C₄	C₂×C₂₃×#121	1	4	yes	—
512	C₄	C₂×C₂₃×#121	1	2	yes	—
513	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
514	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
515	C₂	—	2	4	no	—
516	C₂	—	8	4	no	—
517	C₂	—	8	2	no	—
518	C₂	C₂₃×#3043	1	4	yes	direct product factor
519	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
520	C₂	C₂×C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×#137	C₂	semidirect
2. C₂³×SD₁₆×C₂₃	C₂	semidirect
4. C₂×C₂₃×#519	C₂	semidirect
5. C₂₃×#3043	C₂	direct (N × H)
6. C₂×C₂₃×#464	C₂	semidirect
7. C₂×C₂₃×#460	C₂	semidirect
8. C₂×C₂₃×#424	C₂	semidirect
9. C₂×C₂₃×#418	C₂	semidirect
10. C₂×C₂₃×#416	C₂	semidirect
15. C₂×C₂₃×#137	C₂²	semidirect
16. C₂²×Q₁₆×C₂₃	C₂²	semidirect
19. C₂²×SD₁₆×C₂₃	C₂²	semidirect
20. C₂²×SD₁₆×C₂₃	C₂²	semidirect
30. C₂₃×#519	C₂²	semidirect
32. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
36. C₂₃×#464	C₂²	semidirect
37. C₂₃×#460	C₂²	semidirect
38. C₂₃×#424	C₂²	semidirect
39. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
40. C₂₃×#418	C₂²	semidirect
41. C₂₃×#416	C₂²	semidirect
42. C₂×C₂₃×#103	C₂²	semidirect
44. C₂×C₂₃×#102	C₂²	semidirect
53. C₂²×Q₈×C₂₃	D₄	semidirect
71. C₂×Q₁₆×C₂₃	C₂³	semidirect
76. C₂×SD₁₆×C₂₃	C₂³	semidirect
91. C₄×Q₈×C₂₃	C₂³	semidirect
106. C₂₃×C₄⋊C₈	C₂³	semidirect
107. C₂₃×#103	C₂³	semidirect
109. C₂₃×#102	C₂³	semidirect
129. C₂×Q₈×C₂₃	C₂×D₄	semidirect
183. C₂×#3043	C₂₃	direct (N × H)
234. C₂²×#137	C₂×C₂₃	semidirect
235. C₂³×SD₁₆	C₂×C₂₃	semidirect
237. C₂×#519	C₂×C₂₃	semidirect
238. #3043	C₂×C₂₃	direct (N × H)
239. C₂×#464	C₂×C₂₃	semidirect
240. C₂×#460	C₂×C₂₃	semidirect
241. C₂×#424	C₂×C₂₃	semidirect
242. C₂×#418	C₂×C₂₃	semidirect
243. C₂×#416	C₂×C₂₃	semidirect
269. C₂×#137	C₂²×C₂₃	semidirect
270. C₂²×Q₁₆	C₂²×C₂₃	semidirect
273. C₂²×SD₁₆	C₂²×C₂₃	semidirect
274. C₂²×SD₁₆	C₂²×C₂₃	semidirect
284. #519	C₂²×C₂₃	semidirect
286. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
290. #464	C₂²×C₂₃	semidirect
291. #460	C₂²×C₂₃	semidirect
292. #424	C₂²×C₂₃	semidirect
293. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
294. #418	C₂²×C₂₃	semidirect
295. #416	C₂²×C₂₃	semidirect
296. C₂×#103	C₂²×C₂₃	semidirect
298. C₂×#102	C₂²×C₂₃	semidirect
302. C₂×C₂₃	#3043	direct (N × H)
313. C₂²×Q₈	D₄×C₂₃	semidirect
331. C₂×Q₁₆	C₂³×C₂₃	semidirect
336. C₂×SD₁₆	C₂³×C₂₃	semidirect
351. C₄×Q₈	C₂³×C₂₃	semidirect
366. C₄⋊C₈	C₂³×C₂₃	semidirect
367. #103	C₂³×C₂₃	semidirect
369. #102	C₂³×C₂₃	semidirect
370. C₂₃	C₂×#3043	direct (N × H)
390. C₂×Q₈	C₂×D₄×C₂₃	semidirect
518. C₂	C₂₃×#3043	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#137	2
C₂³×SD₁₆×C₂₃	2
C₂×C₂₃×#535	2
C₂×C₂₃×#519	2
C₂₃×#3043	2
C₂×C₂₃×#464	2
C₂×C₂₃×#460	2
C₂×C₂₃×#424	2
C₂×C₂₃×#418	2
C₂×C₂₃×#416	2
C₂×#3043	23

Subgroups — C2×C23×#3043 #19208044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3043 #19208044