Subgroups — C₂×C₂₃×#3273 #19208658

Order 5888.

3042 conjugacy classes of subgroups.

The subgroup lattice has 544 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	2	yes	maximal, complemented
2	C₂×C₂₃×#546	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3273	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#468	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#463	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#452	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#447	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₂₃	C₂²	1	2	yes	—
12	C₂³×D₄×C₂₃	C₂²	1	2	yes	complemented
13	C₂×C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
14	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	16	yes	complemented
15	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
17	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
18	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
19	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	complemented
22	C₂₃×#546	—	2	8	no	—
23	C₂₃×#546	C₂²	1	16	yes	complemented
24	C₂×C₂₃×#124	C₂²	1	2	yes	complemented
25	C₂×C₂₃×#123	—	2	2	no	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₂×C₂₃×#121	—	2	2	no	—
28	C₂×C₂₃×#121	C₂²	1	2	yes	—
29	C₂×C₂₃×#120	—	2	2	no	—
30	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
32	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₂₃×#495	C₂²	1	8	yes	complemented
34	C₂₃×#468	C₂²	1	16	yes	complemented
35	C₂₃×#463	C₂²	1	16	yes	complemented
36	C₂₃×#452	C₂²	1	16	yes	complemented
37	C₂₃×#447	C₂²	1	16	yes	complemented
38	C₄×D₈×C₂₃	C₂²	1	8	yes	complemented
39	C₂₃×#436	C₂²	1	8	yes	complemented
40	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, 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	2	yes	complemented
45	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
46	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
47	C₂⁵×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	—	2	4	no	—
49	C₂×#41×C₂₃	—	2	8	no	—
50	C₂×#41×C₂₃	—	4	4	no	—
51	C₂×#41×C₂₃	C₂³	1	8	yes	—
52	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
53	C₂²×D₄×C₂₃	—	2	8	no	—
54	C₂²×D₄×C₂₃	—	2	2	no	—
55	C₂²×D₄×C₂₃	—	2	2	no	—
56	C₂²×D₄×C₂₃	—	2	2	no	—
57	C₂²×D₄×C₂₃	—	2	4	no	—
58	C₂²×D₄×C₂₃	—	4	4	no	—
59	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
61	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
62	C₂²×D₄×C₂₃	C₂³	1	8	yes	complemented
63	C₂³×C₄×C₂₃	—	2	2	no	—
64	C₂³×C₄×C₂₃	—	2	2	no	—
65	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
66	C₂₃×AGL(1,ℤ/8)	—	2	64	no	—
67	C₂×Q₁₆×C₂₃	—	4	2	no	—
68	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	complemented
69	C₂×SD₁₆×C₂₃	—	2	8	no	—
70	C₂×SD₁₆×C₂₃	—	2	8	no	—
71	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
72	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
73	C₂×D₈×C₂₃	—	2	8	no	—
74	C₂×D₈×C₂₃	—	4	2	no	—
75	C₂×D₈×C₂₃	D₄	1	8	yes	complemented
76	C₂×D₈×C₂₃	C₂³	1	8	yes	—
77	C₂×D₈×C₂₃	C₂³	1	4	yes	—
78	C₂×D₈×C₂₃	C₂³	1	4	yes	—
79	C₂×M₁₆×C₂₃	—	2	8	no	—
80	C₂×M₁₆×C₂₃	C₂³	1	8	yes	complemented
81	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
82	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₂₃×#124	—	2	4	no	—
85	C₂₃×#124	C₂³	1	8	yes	complemented
86	C₂₃×#123	—	2	16	no	—
87	C₂₃×C₂²⋊Q₈	—	2	4	no	—
88	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	complemented
89	C₂₃×#121	—	2	4	no	—
90	C₂₃×#121	—	2	16	no	—
91	C₂₃×#121	C₂³	1	8	yes	—
92	C₂₃×#120	—	2	16	no	—
93	C₄×D₄×C₂₃	—	2	4	no	—
94	C₄×D₄×C₂₃	D₄	1	4	yes	—
95	C₄×D₄×C₂₃	C₂³	1	4	yes	—
96	C₄×D₄×C₂₃	C₂³	1	8	yes	complemented
97	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
103	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
106	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂₃×#107	C₂³	1	4	yes	complemented
109	C₂₃×C₄⋊C₈	C₂³	1	4	yes	complemented
110	C₂₃×#103	C₂³	1	8	yes	complemented
111	C₂₃×#102	C₂³	1	8	yes	complemented
112	C₂₃×#102	C₂³	1	8	yes	complemented
113	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	complemented
114	C₄×C₈×C₂₃	C₂³	1	4	yes	complemented
115	C₂⁴×C₂₃	—	2	2	no	—
116	C₂⁴×C₂₃	—	2	2	no	—
117	C₂⁴×C₂₃	—	2	8	no	—
118	C₂⁴×C₂₃	—	4	8	no	—
119	C₂⁴×C₂₃	—	4	8	no	—
120	C₂⁴×C₂₃	—	8	4	no	—
121	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
122	#41×C₂₃	—	2	16	no	—
123	#41×C₂₃	—	4	8	no	—
124	#41×C₂₃	—	4	16	no	—
125	C₂×Q₈×C₂₃	—	2	2	no	—
126	C₂×Q₈×C₂₃	—	2	4	no	—
127	C₂×Q₈×C₂₃	—	2	4	no	—
128	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
129	C₂×D₄×C₂₃	—	2	16	no	—
130	C₂×D₄×C₂₃	—	2	8	no	—
131	C₂×D₄×C₂₃	—	2	8	no	—
132	C₂×D₄×C₂₃	—	2	2	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	4	no	—
138	C₂×D₄×C₂₃	—	4	8	no	—
139	C₂×D₄×C₂₃	—	4	16	no	—
140	C₂×D₄×C₂₃	—	4	2	no	—
141	C₂×D₄×C₂₃	—	4	8	no	—
142	C₂×D₄×C₂₃	—	4	8	no	—
143	C₂×D₄×C₂₃	—	8	4	no	—
144	C₂×D₄×C₂₃	C₂×D₄	1	8	yes	—
145	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
146	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
147	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₂₃	—	2	4	no	—
149	C₂²×C₄×C₂₃	—	2	4	no	—
150	C₂²×C₄×C₂₃	—	2	2	no	—
151	C₂²×C₄×C₂₃	—	2	2	no	—
152	C₂²×C₄×C₂₃	—	2	8	no	—
153	C₂²×C₄×C₂₃	—	2	8	no	—
154	C₂²×C₄×C₂₃	—	4	8	no	—
155	C₂²×C₄×C₂₃	—	8	4	no	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₂₃	—	4	4	no	—
163	SD₁₆×C₂₃	—	2	16	no	—
164	SD₁₆×C₂₃	—	2	16	no	—
165	D₈×C₂₃	—	2	16	no	—
166	D₈×C₂₃	—	4	4	no	—
167	D₈×C₂₃	C₂×D₄	1	16	yes	complemented
168	M₁₆×C₂₃	—	2	16	no	—
169	C₂×C₈×C₂₃	—	2	4	no	—
170	C₂×C₈×C₂₃	—	2	2	no	—
171	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
172	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
173	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
174	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
175	C₄⋊C₄×C₂₃	—	2	8	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
179	C₂²⋊C₄×C₂₃	—	2	8	no	—
180	C₂²⋊C₄×C₂₃	—	2	8	no	—
181	C₂²⋊C₄×C₂₃	—	2	8	no	—
182	C₂²⋊C₄×C₂₃	—	2	8	no	—
183	C₂²⋊C₄×C₂₃	—	2	8	no	—
184	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
185	C₄²×C₂₃	C₂×D₄	1	2	yes	—
186	C₄²×C₂₃	C₂⁴	1	2	yes	—
187	C₂×#3273	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₂₃	—	2	4	no	—
189	C₂³×C₂₃	—	2	8	no	—
190	C₂³×C₂₃	—	2	4	no	—
191	C₂³×C₂₃	—	2	4	no	—
192	C₂³×C₂₃	—	2	4	no	—
193	C₂³×C₂₃	—	2	4	no	—
194	C₂³×C₂₃	—	4	16	no	—
195	C₂³×C₂₃	—	4	16	no	—
196	C₂³×C₂₃	—	4	4	no	—
197	C₂³×C₂₃	—	4	4	no	—
198	C₂³×C₂₃	—	4	4	no	—
199	C₂³×C₂₃	—	4	4	no	—
200	C₂³×C₂₃	—	4	4	no	—
201	C₂³×C₂₃	—	8	8	no	—
202	C₂³×C₂₃	—	8	32	no	—
203	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
204	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₂₃	—	2	4	no	—
206	Q₈×C₂₃	—	2	4	no	—
207	Q₈×C₂₃	—	4	4	no	—
208	D₄×C₂₃	—	2	8	no	—
209	D₄×C₂₃	—	2	4	no	—
210	D₄×C₂₃	—	2	4	no	—
211	D₄×C₂₃	—	2	4	no	—
212	D₄×C₂₃	—	4	4	no	—
213	D₄×C₂₃	—	4	16	no	—
214	D₄×C₂₃	—	4	16	no	—
215	D₄×C₂₃	—	4	4	no	—
216	D₄×C₂₃	—	8	8	no	—
217	D₄×C₂₃	C₂²×D₄	1	8	yes	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	8	no	—
222	C₂×C₄×C₂₃	—	2	4	no	—
223	C₂×C₄×C₂₃	—	2	4	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	4	no	—
226	C₂×C₄×C₂₃	—	4	16	no	—
227	C₂×C₄×C₂₃	—	4	4	no	—
228	C₂×C₄×C₂₃	—	4	4	no	—
229	C₂×C₄×C₂₃	—	8	8	no	—
230	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
236	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
237	C₈×C₂₃	—	2	4	no	—
238	C₈×C₂₃	—	2	2	no	—
239	C₈×C₂₃	C₂²×D₄	1	4	yes	—
240	C₂²×AGL(1,ℤ/8)	C₂×C₂₃	1	2	yes	complemented
241	C₂×#546	C₂×C₂₃	1	2	yes	complemented
242	#3273	C₂×C₂₃	1	16	yes	direct product factor
243	C₂×#495	C₂×C₂₃	1	1	yes	characteristic, complemented
244	C₂×#468	C₂×C₂₃	1	2	yes	complemented
245	C₂×#463	C₂×C₂₃	1	2	yes	complemented
246	C₂×#452	C₂×C₂₃	1	2	yes	complemented
247	C₂×#447	C₂×C₂₃	1	2	yes	complemented
248	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
249	C₂×#436	C₂×C₂₃	1	1	yes	characteristic, complemented
250	C₂²×C₂₃	—	2	4	no	—
251	C₂²×C₂₃	—	2	8	no	—
252	C₂²×C₂₃	—	2	4	no	—
253	C₂²×C₂₃	—	2	4	no	—
254	C₂²×C₂₃	—	4	8	no	—
255	C₂²×C₂₃	—	4	16	no	—
256	C₂²×C₂₃	—	4	4	no	—
257	C₂²×C₂₃	—	4	4	no	—
258	C₂²×C₂₃	—	4	4	no	—
259	C₂²×C₂₃	—	4	4	no	—
260	C₂²×C₂₃	—	4	4	no	—
261	C₂²×C₂₃	—	8	32	no	—
262	C₂²×C₂₃	—	8	8	no	—
263	C₂²×C₂₃	D₄²	1	2	yes	—
264	C₂²×C₂₃	#575	1	2	yes	—
265	C₂²×C₂₃	#576	1	2	yes	—
266	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
267	C₄×C₂₃	—	2	4	no	—
268	C₄×C₂₃	—	2	2	no	—
269	C₄×C₂₃	—	2	2	no	—
270	C₄×C₂₃	—	4	4	no	—
271	C₄×C₂₃	—	4	4	no	—
272	C₄×C₂₃	D₄²	1	2	yes	—
273	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
274	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
275	C₂²×#41	C₂²×C₂₃	1	2	yes	—
276	C₂³×D₄	C₂²×C₂₃	1	2	yes	complemented
277	C₂×AGL(1,ℤ/8)	—	2	16	no	—
278	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	16	yes	complemented
279	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
280	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
281	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
282	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
283	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
284	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
285	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	complemented
286	#546	—	2	8	no	—
287	#546	C₂²×C₂₃	1	16	yes	complemented
288	C₂×#124	C₂²×C₂₃	1	2	yes	complemented
289	C₂×#123	—	2	2	no	—
290	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
291	C₂×#121	—	2	2	no	—
292	C₂×#121	C₂²×C₂₃	1	2	yes	—
293	C₂×#120	—	2	2	no	—
294	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
296	C₂²×C₂²⋊C₄	—	2	2	no	—
297	#495	C₂²×C₂₃	1	8	yes	complemented
298	#468	C₂²×C₂₃	1	16	yes	complemented
299	#463	C₂²×C₂₃	1	16	yes	complemented
300	#452	C₂²×C₂₃	1	16	yes	complemented
301	#447	C₂²×C₂₃	1	16	yes	complemented
302	C₄×D₈	C₂²×C₂₃	1	8	yes	complemented
303	#436	C₂²×C₂₃	1	8	yes	complemented
304	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic, complemented
305	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
306	C₂×#103	C₂²×C₂₃	1	2	yes	complemented
307	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
308	C₂×#102	C₂²×C₂₃	1	2	yes	complemented
309	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
310	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
311	C₂×C₂₃	—	4	4	no	—
312	C₂×C₂₃	—	4	8	no	—
313	C₂×C₂₃	—	8	4	no	—
314	C₂×C₂₃	#3273	1	4	yes	direct product factor
315	C₂×C₂₃	C₂×D₄²	1	1	yes	characteristic
316	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
317	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
318	C₂⁵	—	2	2	no	—
319	C₂×#41	—	2	4	no	—
320	C₂×#41	—	2	8	no	—
321	C₂×#41	—	4	4	no	—
322	C₂×#41	C₂³×C₂₃	1	8	yes	—
323	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
324	C₂²×D₄	—	2	8	no	—
325	C₂²×D₄	—	2	2	no	—
326	C₂²×D₄	—	2	2	no	—
327	C₂²×D₄	—	2	2	no	—
328	C₂²×D₄	—	2	4	no	—
329	C₂²×D₄	—	4	4	no	—
330	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
331	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
332	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
333	C₂²×D₄	C₂³×C₂₃	1	8	yes	complemented
334	C₂³×C₄	—	2	2	no	—
335	C₂³×C₄	—	2	2	no	—
336	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
337	AGL(1,ℤ/8)	—	2	64	no	—
338	C₂×Q₁₆	—	4	2	no	—
339	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	complemented
340	C₂×SD₁₆	—	2	8	no	—
341	C₂×SD₁₆	—	2	8	no	—
342	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
343	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
344	C₂×D₈	—	2	8	no	—
345	C₂×D₈	—	4	2	no	—
346	C₂×D₈	D₄×C₂₃	1	8	yes	complemented
347	C₂×D₈	C₂³×C₂₃	1	8	yes	—
348	C₂×D₈	C₂³×C₂₃	1	4	yes	—
349	C₂×D₈	C₂³×C₂₃	1	4	yes	—
350	C₂×M₁₆	—	2	8	no	—
351	C₂×M₁₆	C₂³×C₂₃	1	8	yes	complemented
352	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
353	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
354	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
355	#124	—	2	4	no	—
356	#124	C₂³×C₂₃	1	8	yes	complemented
357	#123	—	2	16	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	complemented
360	#121	—	2	4	no	—
361	#121	—	2	16	no	—
362	#121	C₂³×C₂₃	1	8	yes	—
363	#120	—	2	16	no	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	D₄×C₂₃	1	4	yes	—
366	C₄×D₄	C₂³×C₂₃	1	4	yes	—
367	C₄×D₄	C₂³×C₂₃	1	8	yes	complemented
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	2	16	no	—
377	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
378	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
379	#107	C₂³×C₂₃	1	4	yes	complemented
380	C₄⋊C₈	C₂³×C₂₃	1	4	yes	complemented
381	#103	C₂³×C₂₃	1	8	yes	complemented
382	#102	C₂³×C₂₃	1	8	yes	complemented
383	#102	C₂³×C₂₃	1	8	yes	complemented
384	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	complemented
385	C₄×C₈	C₂³×C₂₃	1	4	yes	complemented
386	C₂₃	C₂×#3273	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	—	2	2	no	—
388	C₂⁴	—	2	2	no	—
389	C₂⁴	—	2	8	no	—
390	C₂⁴	—	4	8	no	—
391	C₂⁴	—	4	8	no	—
392	C₂⁴	—	8	4	no	—
393	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
394	#41	—	2	16	no	—
395	#41	—	4	8	no	—
396	#41	—	4	16	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
401	C₂×D₄	—	2	16	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	8	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	4	8	no	—
411	C₂×D₄	—	4	16	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	—	4	8	no	—
414	C₂×D₄	—	4	8	no	—
415	C₂×D₄	—	8	4	no	—
416	C₂×D₄	C₂×D₄×C₂₃	1	8	yes	—
417	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
418	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
419	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	4	8	no	—
427	C₂²×C₄	—	8	4	no	—
428	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
431	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
432	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
434	Q₁₆	—	4	4	no	—
435	SD₁₆	—	2	16	no	—
436	SD₁₆	—	2	16	no	—
437	D₈	—	2	16	no	—
438	D₈	—	4	4	no	—
439	D₈	C₂×D₄×C₂₃	1	16	yes	complemented
440	M₁₆	—	2	16	no	—
441	C₂×C₈	—	2	4	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
444	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
445	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
446	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
450	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	8	no	—
456	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
457	C₄²	C₂×D₄×C₂₃	1	2	yes	—
458	C₄²	C₂⁴×C₂₃	1	2	yes	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	4	no	—
463	C₂³	—	2	8	no	—
464	C₂³	—	2	4	no	—
465	C₂³	—	4	16	no	—
466	C₂³	—	4	16	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	4	4	no	—
470	C₂³	—	4	4	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	8	8	no	—
473	C₂³	—	8	32	no	—
474	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
475	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
476	Q₈	—	2	4	no	—
477	Q₈	—	2	4	no	—
478	Q₈	—	4	4	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	8	no	—
483	D₄	—	4	4	no	—
484	D₄	—	4	16	no	—
485	D₄	—	4	16	no	—
486	D₄	—	4	4	no	—
487	D₄	—	8	8	no	—
488	D₄	C₂²×D₄×C₂₃	1	8	yes	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	4	16	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	8	8	no	—
501	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
504	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
506	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
507	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
508	C₈	—	2	4	no	—
509	C₈	—	2	2	no	—
510	C₈	C₂²×D₄×C₂₃	1	4	yes	—
511	C₂²	—	2	4	no	—
512	C₂²	—	2	8	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	2	4	no	—
515	C₂²	—	4	8	no	—
516	C₂²	—	4	16	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	4	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	8	32	no	—
523	C₂²	—	8	8	no	—
524	C₂²	D₄²×C₂₃	1	2	yes	—
525	C₂²	C₂₃×#575	1	2	yes	—
526	C₂²	C₂₃×#576	1	2	yes	—
527	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
528	C₄	—	2	4	no	—
529	C₄	—	2	2	no	—
530	C₄	—	2	2	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	4	no	—
533	C₄	D₄²×C₂₃	1	2	yes	—
534	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
535	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
536	C₂	—	4	4	no	—
537	C₂	—	4	8	no	—
538	C₂	—	8	4	no	—
539	C₂	C₂₃×#3273	1	4	yes	direct product factor
540	C₂	C₂×D₄²×C₂₃	1	1	yes	characteristic
541	C₂	C₂×C₂₃×#575	1	1	yes	characteristic
542	C₂	C₂×C₂₃×#576	1	1	yes	characteristic

	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×C₄×D₄×C₂₃ ◁ 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. C₂×C₂₃×#546	C₂	semidirect
3. C₂₃×#3273	C₂	direct (N × H)
4. C₂×C₂₃×#495	C₂	semidirect
5. C₂×C₂₃×#468	C₂	semidirect
6. C₂×C₂₃×#463	C₂	semidirect
7. C₂×C₂₃×#452	C₂	semidirect
8. C₂×C₂₃×#447	C₂	semidirect
9. C₂×C₄×D₈×C₂₃	C₂	semidirect
10. C₂×C₂₃×#436	C₂	semidirect
12. C₂³×D₄×C₂₃	C₂²	semidirect
14. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
15. C₂²×Q₁₆×C₂₃	C₂²	semidirect
16. C₂²×SD₁₆×C₂₃	C₂²	semidirect
17. C₂²×SD₁₆×C₂₃	C₂²	semidirect
21. C₂²×M₁₆×C₂₃	C₂²	semidirect
23. C₂₃×#546	C₂²	semidirect
24. C₂×C₂₃×#124	C₂²	semidirect
26. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
31. C₂×C₄×D₄×C₂₃	C₂²	semidirect
33. C₂₃×#495	C₂²	semidirect
34. C₂₃×#468	C₂²	semidirect
35. C₂₃×#463	C₂²	semidirect
36. C₂₃×#452	C₂²	semidirect
37. C₂₃×#447	C₂²	semidirect
38. C₄×D₈×C₂₃	C₂²	semidirect
39. C₂₃×#436	C₂²	semidirect
40. C₂×C₂₃×#107	C₂²	semidirect
41. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
42. C₂×C₂₃×#103	C₂²	semidirect
43. C₂×C₂₃×#102	C₂²	semidirect
44. C₂×C₂₃×#102	C₂²	semidirect
45. C₂×C₂₃×C₂²⋊C₈	C₂²	semidirect
46. C₂×C₄×C₈×C₂₃	C₂²	semidirect
62. C₂²×D₄×C₂₃	C₂³	semidirect
68. C₂×Q₁₆×C₂₃	C₂³	semidirect
71. C₂×SD₁₆×C₂₃	C₂³	semidirect
72. C₂×SD₁₆×C₂₃	C₂³	semidirect
75. C₂×D₈×C₂₃	D₄	semidirect
80. C₂×M₁₆×C₂₃	C₂³	semidirect
85. C₂₃×#124	C₂³	semidirect
88. C₂₃×C₂²⋊Q₈	C₂³	semidirect
96. C₄×D₄×C₂₃	C₂³	semidirect
108. C₂₃×#107	C₂³	semidirect
109. C₂₃×C₄⋊C₈	C₂³	semidirect
110. C₂₃×#103	C₂³	semidirect
111. C₂₃×#102	C₂³	semidirect
112. C₂₃×#102	C₂³	semidirect
113. C₂₃×C₂²⋊C₈	C₂³	semidirect
114. C₄×C₈×C₂₃	C₂³	semidirect
167. D₈×C₂₃	C₂×D₄	semidirect
187. C₂×#3273	C₂₃	direct (N × H)
240. C₂²×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
241. C₂×#546	C₂×C₂₃	semidirect
242. #3273	C₂×C₂₃	direct (N × H)
243. C₂×#495	C₂×C₂₃	semidirect
244. C₂×#468	C₂×C₂₃	semidirect
245. C₂×#463	C₂×C₂₃	semidirect
246. C₂×#452	C₂×C₂₃	semidirect
247. C₂×#447	C₂×C₂₃	semidirect
248. C₂×C₄×D₈	C₂×C₂₃	semidirect
249. C₂×#436	C₂×C₂₃	semidirect
276. C₂³×D₄	C₂²×C₂₃	semidirect
278. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
279. C₂²×Q₁₆	C₂²×C₂₃	semidirect
280. C₂²×SD₁₆	C₂²×C₂₃	semidirect
281. C₂²×SD₁₆	C₂²×C₂₃	semidirect
285. C₂²×M₁₆	C₂²×C₂₃	semidirect
287. #546	C₂²×C₂₃	semidirect
288. C₂×#124	C₂²×C₂₃	semidirect
290. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
295. C₂×C₄×D₄	C₂²×C₂₃	semidirect
297. #495	C₂²×C₂₃	semidirect
298. #468	C₂²×C₂₃	semidirect
299. #463	C₂²×C₂₃	semidirect
300. #452	C₂²×C₂₃	semidirect
301. #447	C₂²×C₂₃	semidirect
302. C₄×D₈	C₂²×C₂₃	semidirect
303. #436	C₂²×C₂₃	semidirect
304. C₂×#107	C₂²×C₂₃	semidirect
305. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
306. C₂×#103	C₂²×C₂₃	semidirect
307. C₂×#102	C₂²×C₂₃	semidirect
308. C₂×#102	C₂²×C₂₃	semidirect
309. C₂×C₂²⋊C₈	C₂²×C₂₃	semidirect
310. C₂×C₄×C₈	C₂²×C₂₃	semidirect
314. C₂×C₂₃	#3273	direct (N × H)
333. C₂²×D₄	C₂³×C₂₃	semidirect
339. C₂×Q₁₆	C₂³×C₂₃	semidirect
342. C₂×SD₁₆	C₂³×C₂₃	semidirect
343. C₂×SD₁₆	C₂³×C₂₃	semidirect
346. C₂×D₈	D₄×C₂₃	semidirect
351. C₂×M₁₆	C₂³×C₂₃	semidirect
356. #124	C₂³×C₂₃	semidirect
359. C₂²⋊Q₈	C₂³×C₂₃	semidirect
367. C₄×D₄	C₂³×C₂₃	semidirect
379. #107	C₂³×C₂₃	semidirect
380. C₄⋊C₈	C₂³×C₂₃	semidirect
381. #103	C₂³×C₂₃	semidirect
382. #102	C₂³×C₂₃	semidirect
383. #102	C₂³×C₂₃	semidirect
384. C₂²⋊C₈	C₂³×C₂₃	semidirect
385. C₄×C₈	C₂³×C₂₃	semidirect
386. C₂₃	C₂×#3273	direct (N × H)
439. D₈	C₂×D₄×C₂₃	semidirect
539. C₂	C₂₃×#3273	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
C₂×C₂₃×#546	2
C₂₃×#3273	2
C₂×C₂₃×#495	2
C₂×C₂₃×#468	2
C₂×C₂₃×#463	2
C₂×C₂₃×#452	2
C₂×C₂₃×#447	2
C₂×C₄×D₈×C₂₃	2
C₂×C₂₃×#436	2
C₂×#3273	23

Subgroups — C2×C23×#3273 #19208658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3273 #19208658