Subgroups — C₂×C₂₃×#3041 #19208042

Order 5888.

3074 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3041	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₂₃×#463	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#459	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×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×AGL(1,ℤ/8)	—	2	8	no	—
15	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
16	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
17	C₂²×D₈×C₂₃	—	2	8	no	—
18	C₂²×D₈×C₂₃	C₂²	1	4	yes	—
19	C₂²×D₈×C₂₃	C₂²	1	8	yes	complemented
20	C₂²×D₈×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	complemented
25	C₂×C₂₃×#127	C₂²	1	2	yes	complemented
26	C₂×C₂₃×#124	C₂²	1	2	yes	complemented
27	C₂×C₂₃×#121	—	2	2	no	—
28	C₂×C₂₃×#120	—	2	2	no	—
29	C₂₃×#518	—	2	4	no	—
30	C₂₃×#518	C₂²	1	8	yes	complemented
31	C₂×C₄×D₄×C₂₃	—	2	2	no	—
32	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
33	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
35	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
36	C₂₃×#463	C₂²	1	32	yes	complemented
37	C₂₃×#459	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	2	yes	complemented
44	C₂×C₂₃×#102	C₂²	1	4	yes	complemented
45	C₂⁵×C₂₃	—	2	1	no	—
46	C₂⁵×C₂₃	—	2	1	no	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	—	2	4	no	—
49	C₂×#41×C₂₃	—	2	4	no	—
50	C₂×#41×C₂₃	C₂³	1	4	yes	—
51	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₂₃	—	2	1	no	—
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	—	2	2	no	—
55	C₂²×D₄×C₂₃	—	2	4	no	—
56	C₂²×D₄×C₂₃	—	2	1	no	—
57	C₂²×D₄×C₂₃	—	2	2	no	—
58	C₂²×D₄×C₂₃	—	2	4	no	—
59	C₂²×D₄×C₂₃	—	2	4	no	—
60	C₂²×D₄×C₂₃	—	2	4	no	—
61	C₂²×D₄×C₂₃	D₄	1	8	yes	complemented
62	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
63	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
64	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
66	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	—	2	4	no	—
69	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×AGL(1,ℤ/8)	—	2	32	no	—
71	C₂×SD₁₆×C₂₃	—	2	8	no	—
72	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
73	C₂×D₈×C₂₃	—	2	16	no	—
74	C₂×D₈×C₂₃	—	2	8	no	—
75	C₂×D₈×C₂₃	—	2	32	no	—
76	C₂×D₈×C₂₃	C₂³	1	16	yes	—
77	C₂×D₈×C₂₃	C₂³	1	8	yes	complemented
78	C₂×M₁₆×C₂₃	—	2	4	no	—
79	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
80	C₂²×C₈×C₂₃	—	2	4	no	—
81	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
82	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
83	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
84	C₂₃×#127	—	2	4	no	—
85	C₂₃×#127	C₂³	1	8	yes	complemented
86	C₂₃×#124	—	2	4	no	—
87	C₂₃×#124	C₂³	1	8	yes	complemented
88	C₂₃×#121	—	2	16	no	—
89	C₂₃×#120	—	2	16	no	—
90	C₄×D₄×C₂₃	—	2	16	no	—
91	C₄×D₄×C₂₃	—	2	8	no	—
92	C₄×D₄×C₂₃	C₂³	1	16	yes	complemented
93	C₂₃×#117	—	2	2	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	D₄	1	2	yes	—
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
103	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
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	8	yes	complemented
109	C₂₃×#102	C₂³	1	16	yes	complemented
110	C₂⁴×C₂₃	—	2	2	no	—
111	C₂⁴×C₂₃	—	2	8	no	—
112	C₂⁴×C₂₃	—	2	4	no	—
113	C₂⁴×C₂₃	—	2	2	no	—
114	C₂⁴×C₂₃	—	2	1	no	—
115	C₂⁴×C₂₃	—	2	4	no	—
116	C₂⁴×C₂₃	—	4	4	no	—
117	C₂⁴×C₂₃	—	8	4	no	—
118	C₂⁴×C₂₃	—	8	4	no	—
119	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
120	#41×C₂₃	—	2	8	no	—
121	#41×C₂₃	—	2	8	no	—
122	#41×C₂₃	—	4	8	no	—
123	C₂×Q₈×C₂₃	—	2	1	no	—
124	C₂×Q₈×C₂₃	—	2	2	no	—
125	C₂×Q₈×C₂₃	—	2	2	no	—
126	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
127	C₂×D₄×C₂₃	—	2	4	no	—
128	C₂×D₄×C₂₃	—	2	8	no	—
129	C₂×D₄×C₂₃	—	2	16	no	—
130	C₂×D₄×C₂₃	—	2	4	no	—
131	C₂×D₄×C₂₃	—	2	2	no	—
132	C₂×D₄×C₂₃	—	2	4	no	—
133	C₂×D₄×C₂₃	—	2	8	no	—
134	C₂×D₄×C₂₃	—	2	2	no	—
135	C₂×D₄×C₂₃	—	2	4	no	—
136	C₂×D₄×C₂₃	—	2	2	no	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	2	16	no	—
139	C₂×D₄×C₂₃	—	2	4	no	—
140	C₂×D₄×C₂₃	—	2	8	no	—
141	C₂×D₄×C₂₃	—	2	4	no	—
142	C₂×D₄×C₂₃	—	2	8	no	—
143	C₂×D₄×C₂₃	—	2	2	no	—
144	C₂×D₄×C₂₃	—	4	4	no	—
145	C₂×D₄×C₂₃	—	4	4	no	—
146	C₂×D₄×C₂₃	—	4	8	no	—
147	C₂×D₄×C₂₃	—	8	8	no	—
148	C₂×D₄×C₂₃	C₂×D₄	1	8	yes	—
149	C₂×D₄×C₂₃	C₂×D₄	1	16	yes	complemented
150	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
151	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
152	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₂₃	—	2	2	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	4	no	—
156	C₂²×C₄×C₂₃	—	2	16	no	—
157	C₂²×C₄×C₂₃	—	4	4	no	—
158	C₂²×C₄×C₂₃	—	8	8	no	—
159	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₂₃	#41	1	4	yes	—
164	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
166	SD₁₆×C₂₃	—	2	16	no	—
167	D₈×C₂₃	—	2	32	no	—
168	D₈×C₂₃	—	2	16	no	—
169	M₁₆×C₂₃	—	2	8	no	—
170	C₂×C₈×C₂₃	—	2	4	no	—
171	C₂×C₈×C₂₃	—	2	8	no	—
172	C₂×C₈×C₂₃	—	2	4	no	—
173	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
174	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
175	C₄⋊C₄×C₂₃	—	2	2	no	—
176	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
177	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
178	C₂²⋊C₄×C₂₃	—	2	4	no	—
179	C₂²⋊C₄×C₂₃	—	2	16	no	—
180	C₂²⋊C₄×C₂₃	—	2	16	no	—
181	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
182	C₄²×C₂₃	—	2	4	no	—
183	C₄²×C₂₃	C₂×D₄	1	4	yes	—
184	C₄²×C₂₃	C₂⁴	1	4	yes	—
185	C₂×#3041	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₂₃	—	2	4	no	—
187	C₂³×C₂₃	—	2	8	no	—
188	C₂³×C₂₃	—	2	4	no	—
189	C₂³×C₂₃	—	2	16	no	—
190	C₂³×C₂₃	—	2	4	no	—
191	C₂³×C₂₃	—	2	2	no	—
192	C₂³×C₂₃	—	4	8	no	—
193	C₂³×C₂₃	—	4	2	no	—
194	C₂³×C₂₃	—	4	8	no	—
195	C₂³×C₂₃	—	4	4	no	—
196	C₂³×C₂₃	—	4	4	no	—
197	C₂³×C₂₃	—	4	8	no	—
198	C₂³×C₂₃	—	8	16	no	—
199	C₂³×C₂₃	—	8	8	no	—
200	C₂³×C₂₃	—	8	8	no	—
201	C₂³×C₂₃	—	8	16	no	—
202	C₂³×C₂₃	#121	1	4	yes	—
203	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
204	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₂₃	—	2	2	no	—
206	Q₈×C₂₃	—	2	2	no	—
207	Q₈×C₂₃	—	4	2	no	—
208	D₄×C₂₃	—	2	8	no	—
209	D₄×C₂₃	—	2	16	no	—
210	D₄×C₂₃	—	2	4	no	—
211	D₄×C₂₃	—	2	2	no	—
212	D₄×C₂₃	—	2	4	no	—
213	D₄×C₂₃	—	2	4	no	—
214	D₄×C₂₃	—	2	2	no	—
215	D₄×C₂₃	—	4	8	no	—
216	D₄×C₂₃	—	4	8	no	—
217	D₄×C₂₃	—	4	4	no	—
218	D₄×C₂₃	—	4	2	no	—
219	D₄×C₂₃	—	8	16	no	—
220	D₄×C₂₃	C₂²×D₄	1	8	yes	—
221	C₂×C₄×C₂₃	—	2	8	no	—
222	C₂×C₄×C₂₃	—	2	4	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	4	2	no	—
225	C₂×C₄×C₂₃	—	4	4	no	—
226	C₂×C₄×C₂₃	—	4	8	no	—
227	C₂×C₄×C₂₃	—	8	16	no	—
228	C₂×C₄×C₂₃	#121	1	4	yes	—
229	C₂×C₄×C₂₃	#121	1	8	yes	—
230	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
236	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
237	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
238	C₈×C₂₃	—	2	4	no	—
239	C₈×C₂₃	—	2	4	no	—
240	C₂²×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic, complemented
241	C₂³×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
242	C₂×#535	C₂×C₂₃	1	1	yes	characteristic, complemented
243	C₂×#518	C₂×C₂₃	1	1	yes	characteristic, complemented
244	#3041	C₂×C₂₃	1	16	yes	direct product factor
245	C₂×#463	C₂×C₂₃	1	4	yes	complemented
246	C₂×#459	C₂×C₂₃	1	4	yes	complemented
247	C₂×#424	C₂×C₂₃	1	1	yes	characteristic, complemented
248	C₂×#418	C₂×C₂₃	1	1	yes	characteristic, complemented
249	C₂×#416	C₂×C₂₃	1	1	yes	characteristic, complemented
250	C₂²×C₂₃	—	2	8	no	—
251	C₂²×C₂₃	—	2	4	no	—
252	C₂²×C₂₃	—	2	8	no	—
253	C₂²×C₂₃	—	4	16	no	—
254	C₂²×C₂₃	—	4	8	no	—
255	C₂²×C₂₃	—	4	2	no	—
256	C₂²×C₂₃	—	4	4	no	—
257	C₂²×C₂₃	—	4	4	no	—
258	C₂²×C₂₃	—	4	8	no	—
259	C₂²×C₂₃	—	8	16	no	—
260	C₂²×C₂₃	—	8	16	no	—
261	C₂²×C₂₃	—	8	4	no	—
262	C₂²×C₂₃	—	8	8	no	—
263	C₂²×C₂₃	C₂×#121	1	4	yes	—
264	C₂²×C₂₃	C₂×#121	1	2	yes	—
265	C₂²×C₂₃	#576	1	4	yes	—
266	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
267	C₄×C₂₃	—	2	8	no	—
268	C₄×C₂₃	—	4	2	no	—
269	C₄×C₂₃	—	4	4	no	—
270	C₄×C₂₃	C₂×#121	1	4	yes	—
271	C₄×C₂₃	C₂×#121	1	2	yes	—
272	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
273	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
274	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
275	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
276	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×AGL(1,ℤ/8)	—	2	8	no	—
278	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	8	yes	complemented
279	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
280	C₂²×D₈	—	2	8	no	—
281	C₂²×D₈	C₂²×C₂₃	1	4	yes	—
282	C₂²×D₈	C₂²×C₂₃	1	8	yes	complemented
283	C₂²×D₈	C₂²×C₂₃	1	2	yes	complemented
284	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
285	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
286	#535	—	2	4	no	—
287	#535	C₂²×C₂₃	1	8	yes	complemented
288	C₂×#127	C₂²×C₂₃	1	2	yes	complemented
289	C₂×#124	C₂²×C₂₃	1	2	yes	complemented
290	C₂×#121	—	2	2	no	—
291	C₂×#120	—	2	2	no	—
292	#518	—	2	4	no	—
293	#518	C₂²×C₂₃	1	8	yes	complemented
294	C₂×C₄×D₄	—	2	2	no	—
295	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
296	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
298	C₂²×C₂²⋊C₄	—	2	2	no	—
299	#463	C₂²×C₂₃	1	32	yes	complemented
300	#459	C₂²×C₂₃	1	32	yes	complemented
301	#424	C₂²×C₂₃	1	8	yes	complemented
302	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
303	#418	C₂²×C₂₃	1	8	yes	complemented
304	#416	C₂²×C₂₃	1	8	yes	complemented
305	C₂×#103	C₂²×C₂₃	1	2	yes	complemented
306	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
307	C₂×#102	C₂²×C₂₃	1	4	yes	complemented
308	C₂×C₂₃	—	2	4	no	—
309	C₂×C₂₃	—	4	8	no	—
310	C₂×C₂₃	—	8	2	no	—
311	C₂×C₂₃	—	8	4	no	—
312	C₂×C₂₃	#3041	1	4	yes	direct product factor
313	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
314	C₂×C₂₃	C₂×#576	1	2	yes	—
315	C₂⁵	—	2	1	no	—
316	C₂⁵	—	2	1	no	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	—	2	4	no	—
319	C₂×#41	—	2	4	no	—
320	C₂×#41	C₂³×C₂₃	1	4	yes	—
321	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	—	2	1	no	—
324	C₂²×D₄	—	2	2	no	—
325	C₂²×D₄	—	2	4	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	—	2	2	no	—
328	C₂²×D₄	—	2	4	no	—
329	C₂²×D₄	—	2	4	no	—
330	C₂²×D₄	—	2	4	no	—
331	C₂²×D₄	D₄×C₂₃	1	8	yes	complemented
332	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
333	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
334	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
335	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
336	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
337	C₂³×C₄	—	2	1	no	—
338	C₂³×C₄	—	2	4	no	—
339	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
340	AGL(1,ℤ/8)	—	2	32	no	—
341	C₂×SD₁₆	—	2	8	no	—
342	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
343	C₂×D₈	—	2	16	no	—
344	C₂×D₈	—	2	8	no	—
345	C₂×D₈	—	2	32	no	—
346	C₂×D₈	C₂³×C₂₃	1	16	yes	—
347	C₂×D₈	C₂³×C₂₃	1	8	yes	complemented
348	C₂×M₁₆	—	2	4	no	—
349	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
350	C₂²×C₈	—	2	4	no	—
351	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
352	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
353	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
354	#127	—	2	4	no	—
355	#127	C₂³×C₂₃	1	8	yes	complemented
356	#124	—	2	4	no	—
357	#124	C₂³×C₂₃	1	8	yes	complemented
358	#121	—	2	16	no	—
359	#120	—	2	16	no	—
360	C₄×D₄	—	2	16	no	—
361	C₄×D₄	—	2	8	no	—
362	C₄×D₄	C₂³×C₂₃	1	16	yes	complemented
363	#117	—	2	2	no	—
364	#117	D₄×C₂₃	1	4	yes	—
365	#117	C₂³×C₂₃	1	4	yes	—
366	#117	C₂³×C₂₃	1	4	yes	—
367	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
369	C₂×C₂²⋊C₄	—	2	1	no	—
370	C₂×C₂²⋊C₄	—	2	4	no	—
371	C₂×C₂²⋊C₄	—	2	4	no	—
372	C₂×C₂²⋊C₄	—	2	16	no	—
373	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
374	C₂×C₄²	—	2	1	no	—
375	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
376	C₄⋊C₈	C₂³×C₂₃	1	16	yes	complemented
377	#103	C₂³×C₂₃	1	8	yes	complemented
378	#102	C₂³×C₂₃	1	8	yes	complemented
379	#102	C₂³×C₂₃	1	16	yes	complemented
380	C₂₃	C₂×#3041	1	1	yes	characteristic, direct product factor, Sylow
381	C₂⁴	—	2	2	no	—
382	C₂⁴	—	2	8	no	—
383	C₂⁴	—	2	4	no	—
384	C₂⁴	—	2	2	no	—
385	C₂⁴	—	2	1	no	—
386	C₂⁴	—	2	4	no	—
387	C₂⁴	—	4	4	no	—
388	C₂⁴	—	8	4	no	—
389	C₂⁴	—	8	4	no	—
390	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
391	#41	—	2	8	no	—
392	#41	—	2	8	no	—
393	#41	—	4	8	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
398	C₂×D₄	—	2	4	no	—
399	C₂×D₄	—	2	8	no	—
400	C₂×D₄	—	2	16	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	8	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	16	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	4	4	no	—
416	C₂×D₄	—	4	4	no	—
417	C₂×D₄	—	4	8	no	—
418	C₂×D₄	—	8	8	no	—
419	C₂×D₄	C₂×D₄×C₂₃	1	8	yes	—
420	C₂×D₄	C₂×D₄×C₂₃	1	16	yes	complemented
421	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
422	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
423	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	16	no	—
428	C₂²×C₄	—	4	4	no	—
429	C₂²×C₄	—	8	8	no	—
430	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
431	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
433	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
434	C₂²×C₄	#41×C₂₃	1	4	yes	—
435	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
437	SD₁₆	—	2	16	no	—
438	D₈	—	2	32	no	—
439	D₈	—	2	16	no	—
440	M₁₆	—	2	8	no	—
441	C₂×C₈	—	2	4	no	—
442	C₂×C₈	—	2	8	no	—
443	C₂×C₈	—	2	4	no	—
444	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
445	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
448	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	16	no	—
451	C₂²⋊C₄	—	2	16	no	—
452	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
453	C₄²	—	2	4	no	—
454	C₄²	C₂×D₄×C₂₃	1	4	yes	—
455	C₄²	C₂⁴×C₂₃	1	4	yes	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	8	no	—
459	C₂³	—	2	16	no	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	4	8	no	—
463	C₂³	—	4	4	no	—
464	C₂³	—	4	8	no	—
465	C₂³	—	4	2	no	—
466	C₂³	—	4	8	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	8	16	no	—
469	C₂³	—	8	8	no	—
470	C₂³	—	8	8	no	—
471	C₂³	—	8	16	no	—
472	C₂³	C₂₃×#121	1	4	yes	—
473	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
474	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
475	Q₈	—	2	2	no	—
476	Q₈	—	2	2	no	—
477	Q₈	—	4	2	no	—
478	D₄	—	2	4	no	—
479	D₄	—	2	2	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	2	no	—
483	D₄	—	2	8	no	—
484	D₄	—	2	16	no	—
485	D₄	—	4	8	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	4	no	—
488	D₄	—	4	2	no	—
489	D₄	—	8	16	no	—
490	D₄	C₂²×D₄×C₂₃	1	8	yes	—
491	C₂×C₄	—	2	8	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	4	4	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	8	no	—
497	C₂×C₄	—	8	16	no	—
498	C₂×C₄	C₂₃×#121	1	4	yes	—
499	C₂×C₄	C₂₃×#121	1	8	yes	—
500	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
501	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
503	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
504	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
505	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
506	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
507	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
508	C₈	—	2	4	no	—
509	C₈	—	2	4	no	—
510	C₂²	—	2	8	no	—
511	C₂²	—	2	4	no	—
512	C₂²	—	2	8	no	—
513	C₂²	—	4	16	no	—
514	C₂²	—	4	8	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	8	no	—
519	C₂²	—	8	16	no	—
520	C₂²	—	8	16	no	—
521	C₂²	—	8	8	no	—
522	C₂²	—	8	4	no	—
523	C₂²	C₂×C₂₃×#121	1	4	yes	—
524	C₂²	C₂×C₂₃×#121	1	2	yes	—
525	C₂²	C₂₃×#576	1	4	yes	—
526	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
527	C₄	—	2	8	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	4	no	—
530	C₄	C₂×C₂₃×#121	1	4	yes	—
531	C₄	C₂×C₂₃×#121	1	2	yes	—
532	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
533	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
534	C₂	—	2	4	no	—
535	C₂	—	4	8	no	—
536	C₂	—	8	4	no	—
537	C₂	—	8	2	no	—
538	C₂	C₂₃×#3041	1	4	yes	direct product factor
539	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
540	C₂	C₂×C₂₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
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×AGL(1,ℤ/8)	C₂	semidirect
2. C₂³×D₈×C₂₃	C₂	semidirect
3. C₂×C₂₃×#535	C₂	semidirect
4. C₂×C₂₃×#518	C₂	semidirect
5. C₂₃×#3041	C₂	direct (N × H)
6. C₂×C₂₃×#463	C₂	semidirect
7. C₂×C₂₃×#459	C₂	semidirect
8. C₂×C₂₃×#424	C₂	semidirect
9. C₂×C₂₃×#418	C₂	semidirect
10. C₂×C₂₃×#416	C₂	semidirect
15. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂²×SD₁₆×C₂₃	C₂²	semidirect
19. C₂²×D₈×C₂₃	C₂²	semidirect
20. C₂²×D₈×C₂₃	C₂²	semidirect
24. C₂₃×#535	C₂²	semidirect
25. C₂×C₂₃×#127	C₂²	semidirect
26. C₂×C₂₃×#124	C₂²	semidirect
30. C₂₃×#518	C₂²	semidirect
32. C₂×C₄×D₄×C₂₃	C₂²	semidirect
36. C₂₃×#463	C₂²	semidirect
37. C₂₃×#459	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
43. C₂×C₂₃×#102	C₂²	semidirect
44. C₂×C₂₃×#102	C₂²	semidirect
61. C₂²×D₄×C₂₃	D₄	semidirect
72. C₂×SD₁₆×C₂₃	C₂³	semidirect
77. C₂×D₈×C₂₃	C₂³	semidirect
85. C₂₃×#127	C₂³	semidirect
87. C₂₃×#124	C₂³	semidirect
92. C₄×D₄×C₂₃	C₂³	semidirect
106. C₂₃×C₄⋊C₈	C₂³	semidirect
107. C₂₃×#103	C₂³	semidirect
108. C₂₃×#102	C₂³	semidirect
109. C₂₃×#102	C₂³	semidirect
149. C₂×D₄×C₂₃	C₂×D₄	semidirect
185. C₂×#3041	C₂₃	direct (N × H)
240. C₂²×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
241. C₂³×D₈	C₂×C₂₃	semidirect
242. C₂×#535	C₂×C₂₃	semidirect
243. C₂×#518	C₂×C₂₃	semidirect
244. #3041	C₂×C₂₃	direct (N × H)
245. C₂×#463	C₂×C₂₃	semidirect
246. C₂×#459	C₂×C₂₃	semidirect
247. C₂×#424	C₂×C₂₃	semidirect
248. C₂×#418	C₂×C₂₃	semidirect
249. C₂×#416	C₂×C₂₃	semidirect
278. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
279. C₂²×SD₁₆	C₂²×C₂₃	semidirect
282. C₂²×D₈	C₂²×C₂₃	semidirect
283. C₂²×D₈	C₂²×C₂₃	semidirect
287. #535	C₂²×C₂₃	semidirect
288. C₂×#127	C₂²×C₂₃	semidirect
289. C₂×#124	C₂²×C₂₃	semidirect
293. #518	C₂²×C₂₃	semidirect
295. C₂×C₄×D₄	C₂²×C₂₃	semidirect
299. #463	C₂²×C₂₃	semidirect
300. #459	C₂²×C₂₃	semidirect
301. #424	C₂²×C₂₃	semidirect
302. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
303. #418	C₂²×C₂₃	semidirect
304. #416	C₂²×C₂₃	semidirect
305. C₂×#103	C₂²×C₂₃	semidirect
306. C₂×#102	C₂²×C₂₃	semidirect
307. C₂×#102	C₂²×C₂₃	semidirect
312. C₂×C₂₃	#3041	direct (N × H)
331. C₂²×D₄	D₄×C₂₃	semidirect
342. C₂×SD₁₆	C₂³×C₂₃	semidirect
347. C₂×D₈	C₂³×C₂₃	semidirect
355. #127	C₂³×C₂₃	semidirect
357. #124	C₂³×C₂₃	semidirect
362. C₄×D₄	C₂³×C₂₃	semidirect
376. C₄⋊C₈	C₂³×C₂₃	semidirect
377. #103	C₂³×C₂₃	semidirect
378. #102	C₂³×C₂₃	semidirect
379. #102	C₂³×C₂₃	semidirect
380. C₂₃	C₂×#3041	direct (N × H)
420. C₂×D₄	C₂×D₄×C₂₃	semidirect
538. C₂	C₂₃×#3041	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×AGL(1,ℤ/8)	2
C₂³×D₈×C₂₃	2
C₂×C₂₃×#535	2
C₂×C₂₃×#518	2
C₂₃×#3041	2
C₂×C₂₃×#463	2
C₂×C₂₃×#459	2
C₂×C₂₃×#424	2
C₂×C₂₃×#418	2
C₂×C₂₃×#416	2
C₂×#3041	23

Subgroups — C2×C23×#3041 #19208042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3041 #19208042