Subgroups — C₂×C₂₃×#3003 #19207980

Order 5888.

7114 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3003	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₂₃×#450	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#447	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#418	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁶×C₂₃	—	2	1	no	—
10	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic, complemented
11	C₂³×D₄×C₂₃	—	2	16	no	—
12	C₂³×D₄×C₂₃	—	2	2	no	—
13	C₂³×D₄×C₂₃	—	2	1	no	—
14	C₂³×D₄×C₂₃	—	2	4	no	—
15	C₂³×D₄×C₂₃	C₂²	1	4	yes	—
16	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
17	C₂³×D₄×C₂₃	C₂²	1	8	yes	complemented
18	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
20	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	16	yes	complemented
21	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
22	C₂²×D₈×C₂₃	C₂²	1	4	yes	complemented
23	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	complemented
24	C₂₃×#525	—	2	4	no	—
25	C₂₃×#525	C₂²	1	8	yes	complemented
26	C₂×C₂₃×#123	—	2	1	no	—
27	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
28	C₂×C₂₃×#121	C₂²	1	2	yes	complemented
29	C₂×C₂₃×#120	—	2	1	no	—
30	C₂×C₄×D₄×C₂₃	—	2	2	no	—
31	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#450	C₂²	1	32	yes	complemented
33	C₂₃×#447	C₂²	1	32	yes	complemented
34	C₂₃×#418	C₂²	1	16	yes	complemented
35	C₂×C₂₃×#102	C₂²	1	8	yes	complemented
36	C₂₃×#407	C₂²	1	8	yes	complemented
37	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
38	C₂⁵×C₂₃	—	2	16	no	—
39	C₂⁵×C₂₃	—	2	4	no	—
40	C₂⁵×C₂₃	—	2	2	no	—
41	C₂⁵×C₂₃	—	2	8	no	—
42	C₂⁵×C₂₃	—	4	16	no	—
43	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
44	C₂×#41×C₂₃	—	2	2	no	—
45	C₂×#41×C₂₃	—	2	8	no	—
46	C₂×#41×C₂₃	C₂³	1	4	yes	complemented
47	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic, complemented
48	C₂²×D₄×C₂₃	—	2	16	no	—
49	C₂²×D₄×C₂₃	—	2	32	no	—
50	C₂²×D₄×C₂₃	—	2	64	no	—
51	C₂²×D₄×C₂₃	—	2	32	no	—
52	C₂²×D₄×C₂₃	—	2	64	no	—
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	—	2	2	no	—
55	C₂²×D₄×C₂₃	—	2	8	no	—
56	C₂²×D₄×C₂₃	—	2	4	no	—
57	C₂²×D₄×C₂₃	—	2	4	no	—
58	C₂²×D₄×C₂₃	—	2	2	no	—
59	C₂²×D₄×C₂₃	—	2	8	no	—
60	C₂²×D₄×C₂₃	—	2	16	no	—
61	C₂²×D₄×C₂₃	—	2	8	no	—
62	C₂²×D₄×C₂₃	D₄	1	16	yes	complemented
63	C₂²×D₄×C₂₃	C₂³	1	16	yes	—
64	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
65	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
66	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic, complemented
67	C₂³×C₄×C₂₃	—	2	8	no	—
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂³×C₄×C₂₃	D₄	1	2	yes	—
70	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	D₄	1	4	yes	—
72	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
74	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×AGL(1,ℤ/8)	—	2	64	no	—
76	C₂×SD₁₆×C₂₃	—	2	16	no	—
77	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	complemented
78	C₂×D₈×C₂₃	—	2	16	no	—
79	C₂×D₈×C₂₃	C₂³	1	16	yes	complemented
80	C₂×M₁₆×C₂₃	—	2	8	no	—
81	C₂×M₁₆×C₂₃	C₂³	1	8	yes	complemented
82	C₂²×C₈×C₂₃	C₂³	1	4	yes	complemented
83	C₂₃×#123	—	2	8	no	—
84	C₂₃×C₂²⋊Q₈	—	2	4	no	—
85	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	complemented
86	C₂₃×#121	—	2	4	no	—
87	C₂₃×#121	C₂³	1	8	yes	complemented
88	C₂₃×#120	—	2	8	no	—
89	C₄×D₄×C₂₃	—	2	16	no	—
90	C₂₃×#117	—	2	2	no	—
91	C₂₃×#117	C₂³	1	4	yes	complemented
92	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	complemented
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
97	C₂×C₄²×C₂₃	—	2	1	no	—
98	C₂₃×#102	C₂³	1	32	yes	complemented
99	C₂₃×C₂²⋊C₈	C₂³	1	16	yes	complemented
100	C₂⁴×C₂₃	—	2	4	no	—
101	C₂⁴×C₂₃	—	2	8	no	—
102	C₂⁴×C₂₃	—	2	32	no	—
103	C₂⁴×C₂₃	—	2	64	no	—
104	C₂⁴×C₂₃	—	2	16	no	—
105	C₂⁴×C₂₃	—	2	16	no	—
106	C₂⁴×C₂₃	—	2	8	no	—
107	C₂⁴×C₂₃	—	2	4	no	—
108	C₂⁴×C₂₃	—	2	1	no	—
109	C₂⁴×C₂₃	—	4	128	no	—
110	C₂⁴×C₂₃	—	4	16	no	—
111	C₂⁴×C₂₃	—	4	16	no	—
112	C₂⁴×C₂₃	—	4	64	no	—
113	C₂⁴×C₂₃	—	4	16	no	—
114	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
115	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
117	#41×C₂₃	—	2	16	no	—
118	#41×C₂₃	—	4	8	no	—
119	C₂×Q₈×C₂₃	—	2	1	no	—
120	C₂×Q₈×C₂₃	—	2	4	no	—
121	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	complemented
122	C₂×D₄×C₂₃	—	2	32	no	—
123	C₂×D₄×C₂₃	—	2	64	no	—
124	C₂×D₄×C₂₃	—	2	64	no	—
125	C₂×D₄×C₂₃	—	2	128	no	—
126	C₂×D₄×C₂₃	—	2	64	no	—
127	C₂×D₄×C₂₃	—	2	8	no	—
128	C₂×D₄×C₂₃	—	2	4	no	—
129	C₂×D₄×C₂₃	—	2	16	no	—
130	C₂×D₄×C₂₃	—	2	32	no	—
131	C₂×D₄×C₂₃	—	2	16	no	—
132	C₂×D₄×C₂₃	—	2	8	no	—
133	C₂×D₄×C₂₃	—	2	4	no	—
134	C₂×D₄×C₂₃	—	2	8	no	—
135	C₂×D₄×C₂₃	—	2	16	no	—
136	C₂×D₄×C₂₃	—	2	4	no	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	4	4	no	—
139	C₂×D₄×C₂₃	—	4	8	no	—
140	C₂×D₄×C₂₃	C₂×D₄	1	16	yes	—
141	C₂×D₄×C₂₃	C₂×D₄	1	32	yes	complemented
142	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
143	C₂×D₄×C₂₃	C₂⁴	1	2	yes	complemented
144	C₂²×C₄×C₂₃	—	2	4	no	—
145	C₂²×C₄×C₂₃	—	2	8	no	—
146	C₂²×C₄×C₂₃	—	2	16	no	—
147	C₂²×C₄×C₂₃	—	2	8	no	—
148	C₂²×C₄×C₂₃	—	2	16	no	—
149	C₂²×C₄×C₂₃	—	2	2	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	4	no	—
152	C₂²×C₄×C₂₃	—	4	4	no	—
153	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
155	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	2	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	8	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
163	SD₁₆×C₂₃	—	2	32	no	—
164	D₈×C₂₃	—	2	32	no	—
165	M₁₆×C₂₃	—	2	16	no	—
166	C₂×C₈×C₂₃	—	2	8	no	—
167	C₂×C₈×C₂₃	C₂⁴	1	8	yes	complemented
168	C₄⋊C₄×C₂₃	—	2	8	no	—
169	C₄⋊C₄×C₂₃	—	2	2	no	—
170	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	complemented
171	C₂²⋊C₄×C₂₃	—	2	4	no	—
172	C₂²⋊C₄×C₂₃	—	2	8	no	—
173	C₂²⋊C₄×C₂₃	—	2	8	no	—
174	C₄²×C₂₃	—	2	4	no	—
175	C₂×#3003	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₂₃	—	2	32	no	—
177	C₂³×C₂₃	—	2	8	no	—
178	C₂³×C₂₃	—	2	4	no	—
179	C₂³×C₂₃	—	2	4	no	—
180	C₂³×C₂₃	—	2	4	no	—
181	C₂³×C₂₃	—	2	16	no	—
182	C₂³×C₂₃	—	2	64	no	—
183	C₂³×C₂₃	—	2	16	no	—
184	C₂³×C₂₃	—	2	32	no	—
185	C₂³×C₂₃	—	2	8	no	—
186	C₂³×C₂₃	—	2	2	no	—
187	C₂³×C₂₃	—	4	256	no	—
188	C₂³×C₂₃	—	4	64	no	—
189	C₂³×C₂₃	—	4	128	no	—
190	C₂³×C₂₃	—	4	32	no	—
191	C₂³×C₂₃	—	4	32	no	—
192	C₂³×C₂₃	—	4	16	no	—
193	C₂³×C₂₃	—	4	32	no	—
194	C₂³×C₂₃	—	4	2	no	—
195	C₂³×C₂₃	—	4	4	no	—
196	C₂³×C₂₃	#120	1	4	yes	—
197	C₂³×C₂₃	AGL(1,ℤ/8)	1	8	yes	complemented
198	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
199	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
200	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₂₃	—	2	4	no	—
202	Q₈×C₂₃	—	4	2	no	—
203	D₄×C₂₃	—	2	64	no	—
204	D₄×C₂₃	—	2	16	no	—
205	D₄×C₂₃	—	2	32	no	—
206	D₄×C₂₃	—	2	8	no	—
207	D₄×C₂₃	—	2	4	no	—
208	D₄×C₂₃	—	4	8	no	—
209	D₄×C₂₃	—	4	16	no	—
210	D₄×C₂₃	—	4	2	no	—
211	D₄×C₂₃	C₂²×D₄	1	16	yes	—
212	C₂×C₄×C₂₃	—	2	8	no	—
213	C₂×C₄×C₂₃	—	2	4	no	—
214	C₂×C₄×C₂₃	—	2	16	no	—
215	C₂×C₄×C₂₃	—	2	8	no	—
216	C₂×C₄×C₂₃	—	2	8	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	#120	1	4	yes	—
223	C₂×C₄×C₂₃	#120	1	8	yes	—
224	C₂×C₄×C₂₃	C₂²×D₄	1	4	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	4	yes	—
228	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
229	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
230	C₈×C₂₃	—	2	8	no	—
231	C₂⁴×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂²×AGL(1,ℤ/8)	C₂×C₂₃	1	2	yes	complemented
233	C₂×#525	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#3003	C₂×C₂₃	1	16	yes	direct product factor
235	C₂×#450	C₂×C₂₃	1	4	yes	complemented
236	C₂×#447	C₂×C₂₃	1	4	yes	complemented
237	C₂×#418	C₂×C₂₃	1	2	yes	complemented
238	C₂×#407	C₂×C₂₃	1	1	yes	characteristic, complemented
239	C₂²×C₂₃	—	2	32	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	2	8	no	—
242	C₂²×C₂₃	—	2	16	no	—
243	C₂²×C₂₃	—	2	4	no	—
244	C₂²×C₂₃	—	2	4	no	—
245	C₂²×C₂₃	—	2	16	no	—
246	C₂²×C₂₃	—	4	64	no	—
247	C₂²×C₂₃	—	4	128	no	—
248	C₂²×C₂₃	—	4	32	no	—
249	C₂²×C₂₃	—	4	16	no	—
250	C₂²×C₂₃	—	4	2	no	—
251	C₂²×C₂₃	—	4	4	no	—
252	C₂²×C₂₃	—	8	4	no	—
253	C₂²×C₂₃	C₂×#120	1	4	yes	—
254	C₂²×C₂₃	C₂×#120	1	2	yes	—
255	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
256	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	8	yes	complemented
257	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
258	C₄×C₂₃	—	2	4	no	—
259	C₄×C₂₃	—	4	2	no	—
260	C₄×C₂₃	—	4	4	no	—
261	C₄×C₂₃	C₂×#120	1	4	yes	—
262	C₄×C₂₃	C₂×#120	1	2	yes	—
263	C₄×C₂₃	C₂³×D₄	1	2	yes	—
264	C₂⁶	—	2	1	no	—
265	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic, complemented
266	C₂³×D₄	—	2	16	no	—
267	C₂³×D₄	—	2	2	no	—
268	C₂³×D₄	—	2	1	no	—
269	C₂³×D₄	—	2	4	no	—
270	C₂³×D₄	C₂²×C₂₃	1	4	yes	—
271	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
272	C₂³×D₄	C₂²×C₂₃	1	8	yes	complemented
273	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
274	C₂×AGL(1,ℤ/8)	—	2	16	no	—
275	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	16	yes	complemented
276	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
277	C₂²×D₈	C₂²×C₂₃	1	4	yes	complemented
278	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	complemented
279	#525	—	2	4	no	—
280	#525	C₂²×C₂₃	1	8	yes	complemented
281	C₂×#123	—	2	1	no	—
282	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
283	C₂×#121	C₂²×C₂₃	1	2	yes	complemented
284	C₂×#120	—	2	1	no	—
285	C₂×C₄×D₄	—	2	2	no	—
286	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic, complemented
287	#450	C₂²×C₂₃	1	32	yes	complemented
288	#447	C₂²×C₂₃	1	32	yes	complemented
289	#418	C₂²×C₂₃	1	16	yes	complemented
290	C₂×#102	C₂²×C₂₃	1	8	yes	complemented
291	#407	C₂²×C₂₃	1	8	yes	complemented
292	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	complemented
293	C₂×C₂₃	—	2	4	no	—
294	C₂×C₂₃	—	2	8	no	—
295	C₂×C₂₃	—	4	16	no	—
296	C₂×C₂₃	—	8	2	no	—
297	C₂×C₂₃	#3003	1	4	yes	direct product factor
298	C₂×C₂₃	C₂²×#120	1	1	yes	characteristic
299	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
300	C₂⁵	—	2	16	no	—
301	C₂⁵	—	2	4	no	—
302	C₂⁵	—	2	2	no	—
303	C₂⁵	—	2	8	no	—
304	C₂⁵	—	4	16	no	—
305	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
306	C₂×#41	—	2	2	no	—
307	C₂×#41	—	2	8	no	—
308	C₂×#41	C₂³×C₂₃	1	4	yes	complemented
309	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic, complemented
310	C₂²×D₄	—	2	16	no	—
311	C₂²×D₄	—	2	32	no	—
312	C₂²×D₄	—	2	64	no	—
313	C₂²×D₄	—	2	32	no	—
314	C₂²×D₄	—	2	64	no	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	—	2	4	no	—
318	C₂²×D₄	—	2	8	no	—
319	C₂²×D₄	—	2	2	no	—
320	C₂²×D₄	—	2	4	no	—
321	C₂²×D₄	—	2	8	no	—
322	C₂²×D₄	—	2	16	no	—
323	C₂²×D₄	—	2	8	no	—
324	C₂²×D₄	D₄×C₂₃	1	16	yes	complemented
325	C₂²×D₄	C₂³×C₂₃	1	16	yes	—
326	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
327	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
328	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic, complemented
329	C₂³×C₄	—	2	8	no	—
330	C₂³×C₄	—	2	1	no	—
331	C₂³×C₄	D₄×C₂₃	1	2	yes	—
332	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
333	C₂³×C₄	D₄×C₂₃	1	4	yes	—
334	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
335	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
336	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
337	AGL(1,ℤ/8)	—	2	64	no	—
338	C₂×SD₁₆	—	2	16	no	—
339	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	complemented
340	C₂×D₈	—	2	16	no	—
341	C₂×D₈	C₂³×C₂₃	1	16	yes	complemented
342	C₂×M₁₆	—	2	8	no	—
343	C₂×M₁₆	C₂³×C₂₃	1	8	yes	complemented
344	C₂²×C₈	C₂³×C₂₃	1	4	yes	complemented
345	#123	—	2	8	no	—
346	C₂²⋊Q₈	—	2	4	no	—
347	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	complemented
348	#121	—	2	4	no	—
349	#121	C₂³×C₂₃	1	8	yes	complemented
350	#120	—	2	8	no	—
351	C₄×D₄	—	2	16	no	—
352	#117	—	2	2	no	—
353	#117	C₂³×C₂₃	1	4	yes	complemented
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	complemented
356	C₂×C₂²⋊C₄	—	2	1	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₄²	—	2	1	no	—
360	#102	C₂³×C₂₃	1	32	yes	complemented
361	C₂²⋊C₈	C₂³×C₂₃	1	16	yes	complemented
362	C₂₃	C₂×#3003	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	2	4	no	—
364	C₂⁴	—	2	8	no	—
365	C₂⁴	—	2	32	no	—
366	C₂⁴	—	2	64	no	—
367	C₂⁴	—	2	16	no	—
368	C₂⁴	—	2	16	no	—
369	C₂⁴	—	2	4	no	—
370	C₂⁴	—	2	8	no	—
371	C₂⁴	—	2	1	no	—
372	C₂⁴	—	4	128	no	—
373	C₂⁴	—	4	16	no	—
374	C₂⁴	—	4	16	no	—
375	C₂⁴	—	4	64	no	—
376	C₂⁴	—	4	16	no	—
377	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
378	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
379	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
380	#41	—	2	16	no	—
381	#41	—	4	8	no	—
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	complemented
385	C₂×D₄	—	2	32	no	—
386	C₂×D₄	—	2	64	no	—
387	C₂×D₄	—	2	64	no	—
388	C₂×D₄	—	2	128	no	—
389	C₂×D₄	—	2	64	no	—
390	C₂×D₄	—	2	8	no	—
391	C₂×D₄	—	2	4	no	—
392	C₂×D₄	—	2	16	no	—
393	C₂×D₄	—	2	32	no	—
394	C₂×D₄	—	2	16	no	—
395	C₂×D₄	—	2	8	no	—
396	C₂×D₄	—	2	16	no	—
397	C₂×D₄	—	2	8	no	—
398	C₂×D₄	—	2	4	no	—
399	C₂×D₄	—	2	4	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	4	4	no	—
402	C₂×D₄	—	4	8	no	—
403	C₂×D₄	C₂×D₄×C₂₃	1	16	yes	—
404	C₂×D₄	C₂×D₄×C₂₃	1	32	yes	complemented
405	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
406	C₂×D₄	C₂⁴×C₂₃	1	2	yes	complemented
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	16	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	2	16	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	4	4	no	—
416	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
417	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
418	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
419	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
420	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
422	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
423	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
424	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
425	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
426	SD₁₆	—	2	32	no	—
427	D₈	—	2	32	no	—
428	M₁₆	—	2	16	no	—
429	C₂×C₈	—	2	8	no	—
430	C₂×C₈	C₂⁴×C₂₃	1	8	yes	complemented
431	C₄⋊C₄	—	2	8	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	complemented
434	C₂²⋊C₄	—	2	4	no	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₄²	—	2	4	no	—
438	C₂³	—	2	32	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	16	no	—
442	C₂³	—	2	8	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	64	no	—
445	C₂³	—	2	8	no	—
446	C₂³	—	2	16	no	—
447	C₂³	—	2	32	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	4	256	no	—
450	C₂³	—	4	64	no	—
451	C₂³	—	4	128	no	—
452	C₂³	—	4	32	no	—
453	C₂³	—	4	32	no	—
454	C₂³	—	4	16	no	—
455	C₂³	—	4	32	no	—
456	C₂³	—	4	4	no	—
457	C₂³	—	4	2	no	—
458	C₂³	C₂₃×#120	1	4	yes	—
459	C₂³	C₂₃×AGL(1,ℤ/8)	1	8	yes	complemented
460	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
461	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
462	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
463	Q₈	—	2	4	no	—
464	Q₈	—	4	2	no	—
465	D₄	—	2	64	no	—
466	D₄	—	2	8	no	—
467	D₄	—	2	4	no	—
468	D₄	—	2	16	no	—
469	D₄	—	2	32	no	—
470	D₄	—	4	16	no	—
471	D₄	—	4	8	no	—
472	D₄	—	4	2	no	—
473	D₄	C₂²×D₄×C₂₃	1	16	yes	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	16	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	4	4	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	8	no	—
484	C₂×C₄	C₂₃×#120	1	4	yes	—
485	C₂×C₄	C₂₃×#120	1	8	yes	—
486	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
487	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
488	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
489	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
490	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
491	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
492	C₈	—	2	8	no	—
493	C₂²	—	2	32	no	—
494	C₂²	—	2	4	no	—
495	C₂²	—	2	8	no	—
496	C₂²	—	2	16	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	2	4	no	—
499	C₂²	—	2	16	no	—
500	C₂²	—	4	128	no	—
501	C₂²	—	4	64	no	—
502	C₂²	—	4	32	no	—
503	C₂²	—	4	16	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	4	no	—
506	C₂²	—	8	4	no	—
507	C₂²	C₂×C₂₃×#120	1	4	yes	—
508	C₂²	C₂×C₂₃×#120	1	2	yes	—
509	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
510	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	8	yes	complemented
511	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
512	C₄	—	2	4	no	—
513	C₄	—	4	2	no	—
514	C₄	—	4	4	no	—
515	C₄	C₂×C₂₃×#120	1	4	yes	—
516	C₄	C₂×C₂₃×#120	1	2	yes	—
517	C₄	C₂³×D₄×C₂₃	1	2	yes	—
518	C₂	—	2	8	no	—
519	C₂	—	2	4	no	—
520	C₂	—	4	16	no	—
521	C₂	—	8	2	no	—
522	C₂	C₂₃×#3003	1	4	yes	direct product factor
523	C₂	C₂²×C₂₃×#120	1	1	yes	characteristic
524	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	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 ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₄×C₂₃	C₂	semidirect
2. C₂²×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₂₃×#525	C₂	semidirect
4. C₂₃×#3003	C₂	direct (N × H)
5. C₂×C₂₃×#450	C₂	semidirect
6. C₂×C₂₃×#447	C₂	semidirect
7. C₂×C₂₃×#418	C₂	semidirect
8. C₂×C₂₃×#407	C₂	semidirect
10. C₂²×#41×C₂₃	C₂²	semidirect
17. C₂³×D₄×C₂₃	C₂²	semidirect
20. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
21. C₂²×SD₁₆×C₂₃	C₂²	semidirect
22. C₂²×D₈×C₂₃	C₂²	semidirect
23. C₂²×M₁₆×C₂₃	C₂²	semidirect
25. C₂₃×#525	C₂²	semidirect
27. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
28. C₂×C₂₃×#121	C₂²	semidirect
31. C₂×C₂₃×#117	C₂²	semidirect
32. C₂₃×#450	C₂²	semidirect
33. C₂₃×#447	C₂²	semidirect
34. C₂₃×#418	C₂²	semidirect
35. C₂×C₂₃×#102	C₂²	semidirect
36. C₂₃×#407	C₂²	semidirect
37. C₂×C₂₃×C₂²⋊C₈	C₂²	semidirect
46. C₂×#41×C₂₃	C₂³	semidirect
47. C₂²×Q₈×C₂₃	C₂³	semidirect
62. C₂²×D₄×C₂₃	D₄	semidirect
66. C₂²×D₄×C₂₃	C₂³	semidirect
77. C₂×SD₁₆×C₂₃	C₂³	semidirect
79. C₂×D₈×C₂₃	C₂³	semidirect
81. C₂×M₁₆×C₂₃	C₂³	semidirect
82. C₂²×C₈×C₂₃	C₂³	semidirect
85. C₂₃×C₂²⋊Q₈	C₂³	semidirect
87. C₂₃×#121	C₂³	semidirect
91. C₂₃×#117	C₂³	semidirect
93. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
98. C₂₃×#102	C₂³	semidirect
99. C₂₃×C₂²⋊C₈	C₂³	semidirect
121. C₂×Q₈×C₂₃	C₂⁴	semidirect
141. C₂×D₄×C₂₃	C₂×D₄	semidirect
143. C₂×D₄×C₂₃	C₂⁴	semidirect
167. C₂×C₈×C₂₃	C₂⁴	semidirect
170. C₄⋊C₄×C₂₃	C₂⁴	semidirect
175. C₂×#3003	C₂₃	direct (N × H)
197. C₂³×C₂₃	AGL(1,ℤ/8)	semidirect
231. C₂⁴×D₄	C₂×C₂₃	semidirect
232. C₂²×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
233. C₂×#525	C₂×C₂₃	semidirect
234. #3003	C₂×C₂₃	direct (N × H)
235. C₂×#450	C₂×C₂₃	semidirect
236. C₂×#447	C₂×C₂₃	semidirect
237. C₂×#418	C₂×C₂₃	semidirect
238. C₂×#407	C₂×C₂₃	semidirect
256. C₂²×C₂₃	C₂×AGL(1,ℤ/8)	semidirect
265. C₂²×#41	C₂²×C₂₃	semidirect
272. C₂³×D₄	C₂²×C₂₃	semidirect
275. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
276. C₂²×SD₁₆	C₂²×C₂₃	semidirect
277. C₂²×D₈	C₂²×C₂₃	semidirect
278. C₂²×M₁₆	C₂²×C₂₃	semidirect
280. #525	C₂²×C₂₃	semidirect
282. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
283. C₂×#121	C₂²×C₂₃	semidirect
286. C₂×#117	C₂²×C₂₃	semidirect
287. #450	C₂²×C₂₃	semidirect
288. #447	C₂²×C₂₃	semidirect
289. #418	C₂²×C₂₃	semidirect
290. C₂×#102	C₂²×C₂₃	semidirect
291. #407	C₂²×C₂₃	semidirect
292. C₂×C₂²⋊C₈	C₂²×C₂₃	semidirect
297. C₂×C₂₃	#3003	direct (N × H)
308. C₂×#41	C₂³×C₂₃	semidirect
309. C₂²×Q₈	C₂³×C₂₃	semidirect
324. C₂²×D₄	D₄×C₂₃	semidirect
328. C₂²×D₄	C₂³×C₂₃	semidirect
339. C₂×SD₁₆	C₂³×C₂₃	semidirect
341. C₂×D₈	C₂³×C₂₃	semidirect
343. C₂×M₁₆	C₂³×C₂₃	semidirect
344. C₂²×C₈	C₂³×C₂₃	semidirect
347. C₂²⋊Q₈	C₂³×C₂₃	semidirect
349. #121	C₂³×C₂₃	semidirect
353. #117	C₂³×C₂₃	semidirect
355. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
360. #102	C₂³×C₂₃	semidirect
361. C₂²⋊C₈	C₂³×C₂₃	semidirect
362. C₂₃	C₂×#3003	direct (N × H)
384. C₂×Q₈	C₂⁴×C₂₃	semidirect
404. C₂×D₄	C₂×D₄×C₂₃	semidirect
406. C₂×D₄	C₂⁴×C₂₃	semidirect
430. C₂×C₈	C₂⁴×C₂₃	semidirect
433. C₄⋊C₄	C₂⁴×C₂₃	semidirect
459. C₂³	C₂₃×AGL(1,ℤ/8)	semidirect
510. C₂²	C₂×C₂₃×AGL(1,ℤ/8)	semidirect
522. C₂	C₂₃×#3003	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₂₃	2
C₂²×C₂₃×AGL(1,ℤ/8)	2
C₂×C₂₃×#525	2
C₂₃×#3003	2
C₂×C₂₃×#450	2
C₂×C₂₃×#447	2
C₂×C₂₃×#418	2
C₂×C₂₃×#407	2
C₂×#3003	23

Subgroups — C2×C23×#3003 #19207980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3003 #19207980