Subgroups — C₂₃×#12748 #19160374

Order 5888.

934 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#576	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#359	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1816	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×H(5,𝔽₂)	—	2	1	no	—
9	C₂₃×#576	—	2	2	no	—
10	C₂₃×#576	—	2	2	no	—
11	C₂₃×#576	—	2	2	no	—
12	C₂₃×#576	C₄	1	4	yes	complemented
13	C₂×C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
14	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
15	C₂×C₂₃×#135	—	2	1	no	—
16	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₂₃	—	2	1	no	—
18	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
21	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#417	—	2	1	no	—
25	C₂₃×#361	—	2	2	no	—
26	C₂₃×#359	—	2	2	no	—
27	C₂₃×#358	—	2	1	no	—
28	C₂₃×#357	—	2	2	no	—
29	C₂₃×#357	—	2	1	no	—
30	C₂₃×#350	—	2	2	no	—
31	C₂₃×#343	—	2	1	no	—
32	C₂₃×H(5,𝔽₂)	—	2	4	no	—
33	C₂₃×H(5,𝔽₂)	—	2	4	no	—
34	C₂₃×H(5,𝔽₂)	—	4	2	no	—
35	C₂₃×H(5,𝔽₂)	—	4	2	no	—
36	C₂×#41×C₂₃	—	2	2	no	—
37	C₂×#41×C₂₃	—	2	1	no	—
38	C₂×#41×C₂₃	—	4	1	no	—
39	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
40	C₂²×D₄×C₂₃	—	2	2	no	—
41	C₂²×D₄×C₂₃	—	2	1	no	—
42	C₂²×D₄×C₂₃	—	4	2	no	—
43	C₂²×D₄×C₂₃	—	4	1	no	—
44	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
45	C₂₃×AGL(1,ℤ/8)	—	2	8	no	—
46	C₂₃×AGL(1,ℤ/8)	—	2	8	no	—
47	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
48	C₂₃×#135	—	2	4	no	—
49	C₂₃×#135	—	2	4	no	—
50	C₂₃×#135	—	2	1	no	—
51	C₂₃×#135	—	2	1	no	—
52	C₂₃×#135	C₂×C₄	1	2	yes	—
53	C₂₃×#135	D₄	1	2	yes	—
54	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₂₃	—	2	2	no	—
56	C₂×SD₁₆×C₂₃	—	2	1	no	—
57	C₂×D₈×C₂₃	—	2	2	no	—
58	C₂×D₈×C₂₃	—	2	4	no	—
59	C₂×D₈×C₂₃	—	2	4	no	—
60	C₂×D₈×C₂₃	—	2	1	no	—
61	C₂×D₈×C₂₃	—	2	1	no	—
62	C₂×D₈×C₂₃	—	2	1	no	—
63	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
64	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
65	C₂×D₈×C₂₃	D₄	1	2	yes	—
66	C₂×D₈×C₂₃	D₄	1	2	yes	—
67	C₂₃×#131	—	2	1	no	—
68	C₂₃×#131	—	2	1	no	—
69	C₂₃×#131	C₂×C₄	1	2	yes	—
70	C₂₃×#131	D₄	1	2	yes	—
71	C₂×M₁₆×C₂₃	—	2	1	no	—
72	C₂×M₁₆×C₂₃	—	2	1	no	—
73	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂₃×M₃₂	—	2	1	no	—
78	C₂₃×M₃₂	—	2	1	no	—
79	C₂×C₁₆×C₂₃	—	2	1	no	—
80	C₂×C₁₆×C₂₃	—	2	1	no	—
81	C₂×C₁₆×C₂₃	—	2	1	no	—
82	C₂×C₁₆×C₂₃	—	2	1	no	—
83	C₂₃×#108	—	2	1	no	—
84	C₂₃×#108	—	2	1	no	—
85	C₂₃×#107	—	2	1	no	—
86	C₂₃×#107	—	2	1	no	—
87	C₂₃×#103	—	4	1	no	—
88	C₂₃×#102	—	4	1	no	—
89	C₂⁴×C₂₃	—	4	2	no	—
90	C₂⁴×C₂₃	—	4	1	no	—
91	#41×C₂₃	—	2	1	no	—
92	#41×C₂₃	—	2	1	no	—
93	#41×C₂₃	—	2	8	no	—
94	#41×C₂₃	—	2	2	no	—
95	#41×C₂₃	—	2	2	no	—
96	#41×C₂₃	—	4	4	no	—
97	#41×C₂₃	—	4	4	no	—
98	#41×C₂₃	—	4	4	no	—
99	#41×C₂₃	—	4	1	no	—
100	#41×C₂₃	—	4	1	no	—
101	#41×C₂₃	C₂²⋊C₄	1	2	yes	—
102	#41×C₂₃	SD₁₆	1	2	yes	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
105	C₂×D₄×C₂₃	—	2	1	no	—
106	C₂×D₄×C₂₃	—	2	8	no	—
107	C₂×D₄×C₂₃	—	2	2	no	—
108	C₂×D₄×C₂₃	—	2	1	no	—
109	C₂×D₄×C₂₃	—	2	2	no	—
110	C₂×D₄×C₂₃	—	2	2	no	—
111	C₂×D₄×C₂₃	—	4	2	no	—
112	C₂×D₄×C₂₃	—	4	4	no	—
113	C₂×D₄×C₂₃	—	4	8	no	—
114	C₂×D₄×C₂₃	—	4	8	no	—
115	C₂×D₄×C₂₃	—	4	4	no	—
116	C₂×D₄×C₂₃	—	4	2	no	—
117	C₂×D₄×C₂₃	—	4	1	no	—
118	C₂×D₄×C₂₃	—	4	4	no	—
119	C₂×D₄×C₂₃	—	4	1	no	—
120	C₂×D₄×C₂₃	—	4	1	no	—
121	C₂×D₄×C₂₃	—	4	1	no	—
122	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
123	C₂²×C₄×C₂₃	—	2	1	no	—
124	C₂²×C₄×C₂₃	—	4	2	no	—
125	C₂²×C₄×C₂₃	—	4	1	no	—
126	C₂²×C₄×C₂₃	—	4	1	no	—
127	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
128	Q₁₆×C₂₃	—	2	1	no	—
129	Q₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
130	SD₁₆×C₂₃	—	2	8	no	—
131	SD₁₆×C₂₃	—	2	2	no	—
132	SD₁₆×C₂₃	—	2	2	no	—
133	D₈×C₂₃	—	2	8	no	—
134	D₈×C₂₃	—	2	2	no	—
135	D₈×C₂₃	—	2	1	no	—
136	D₈×C₂₃	—	2	2	no	—
137	D₈×C₂₃	—	2	2	no	—
138	D₈×C₂₃	C₂²⋊C₄	1	2	yes	—
139	D₈×C₂₃	C₂²⋊C₄	1	4	yes	—
140	M₁₆×C₂₃	—	2	1	no	—
141	M₁₆×C₂₃	—	2	2	no	—
142	M₁₆×C₂₃	—	2	2	no	—
143	M₁₆×C₂₃	—	4	1	no	—
144	M₁₆×C₂₃	—	4	1	no	—
145	M₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
146	C₂×C₈×C₂₃	—	2	1	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	—	2	2	no	—
149	C₂×C₈×C₂₃	—	4	1	no	—
150	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₄⋊C₄×C₂₃	—	4	1	no	—
157	C₄⋊C₄×C₂₃	—	4	1	no	—
158	C₁₆×C₂₃	—	4	1	no	—
159	C₁₆×C₂₃	—	4	1	no	—
160	#12748	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₂₃	—	2	1	no	—
162	C₂³×C₂₃	—	4	8	no	—
163	C₂³×C₂₃	—	4	2	no	—
164	C₂³×C₂₃	—	4	1	no	—
165	C₂³×C₂₃	—	4	2	no	—
166	C₂³×C₂₃	—	4	2	no	—
167	C₂³×C₂₃	—	16	1	no	—
168	C₂³×C₂₃	—	16	1	no	—
169	C₂³×C₂₃	—	16	2	no	—
170	C₂³×C₂₃	—	16	2	no	—
171	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
172	Q₈×C₂₃	—	2	1	no	—
173	Q₈×C₂₃	—	2	2	no	—
174	Q₈×C₂₃	—	4	1	no	—
175	Q₈×C₂₃	#102	1	2	yes	—
176	D₄×C₂₃	—	2	2	no	—
177	D₄×C₂₃	—	2	2	no	—
178	D₄×C₂₃	—	2	2	no	—
179	D₄×C₂₃	—	2	4	no	—
180	D₄×C₂₃	—	2	1	no	—
181	D₄×C₂₃	—	4	8	no	—
182	D₄×C₂₃	—	4	8	no	—
183	D₄×C₂₃	—	4	2	no	—
184	D₄×C₂₃	—	4	2	no	—
185	D₄×C₂₃	—	4	2	no	—
186	D₄×C₂₃	—	4	2	no	—
187	D₄×C₂₃	—	4	1	no	—
188	D₄×C₂₃	—	4	2	no	—
189	D₄×C₂₃	#102	1	2	yes	—
190	C₂×C₄×C₂₃	—	2	1	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	4	8	no	—
194	C₂×C₄×C₂₃	—	4	1	no	—
195	C₂×C₄×C₂₃	—	4	2	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	8	1	no	—
198	C₂×C₄×C₂₃	—	16	1	no	—
199	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
201	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
203	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
204	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₈×C₂₃	—	2	2	no	—
206	C₈×C₂₃	—	8	1	no	—
207	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
209	C₈×C₂₃	#120	1	1	yes	characteristic
210	C₈×C₂₃	#120	1	1	yes	characteristic
211	C₂×#576	C₂×C₂₃	1	1	yes	characteristic
212	C₂×#361	C₂×C₂₃	1	1	yes	characteristic, complemented
213	C₂×#359	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#2139	C₂×C₂₃	1	1	yes	characteristic, complemented
215	C₂×#357	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂×#350	C₂×C₂₃	1	1	yes	characteristic, complemented
217	#1816	C₂×C₂₃	1	1	yes	characteristic, complemented
218	C₂²×C₂₃	—	2	1	no	—
219	C₂²×C₂₃	—	2	1	no	—
220	C₂²×C₂₃	—	2	2	no	—
221	C₂²×C₂₃	—	4	1	no	—
222	C₂²×C₂₃	—	4	1	no	—
223	C₂²×C₂₃	—	4	4	no	—
224	C₂²×C₂₃	—	4	2	no	—
225	C₂²×C₂₃	—	8	2	no	—
226	C₂²×C₂₃	—	8	1	no	—
227	C₂²×C₂₃	—	8	2	no	—
228	C₂²×C₂₃	—	8	1	no	—
229	C₂²×C₂₃	—	16	2	no	—
230	C₂²×C₂₃	—	16	2	no	—
231	C₂²×C₂₃	—	16	8	no	—
232	C₂²×C₂₃	#418	1	1	yes	characteristic
233	C₂²×C₂₃	#447	1	1	yes	characteristic
234	C₂²×C₂₃	#450	1	1	yes	characteristic
235	C₄×C₂₃	—	2	2	no	—
236	C₄×C₂₃	—	4	2	no	—
237	C₄×C₂₃	—	16	1	no	—
238	C₄×C₂₃	#379	1	1	yes	characteristic
239	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
240	C₄×C₂₃	#447	1	1	yes	characteristic
241	C₄×C₂₃	#450	1	1	yes	characteristic
242	C₂×H(5,𝔽₂)	—	2	1	no	—
243	#576	—	2	2	no	—
244	#576	—	2	2	no	—
245	#576	—	2	2	no	—
246	#576	C₄×C₂₃	1	4	yes	complemented
247	C₂×AGL(1,ℤ/8)	—	2	2	no	—
248	C₂×AGL(1,ℤ/8)	—	2	1	no	—
249	C₂×#135	—	2	1	no	—
250	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
251	C₂²×D₈	—	2	1	no	—
252	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
254	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
255	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
256	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic, complemented
257	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic, complemented
258	#417	—	2	1	no	—
259	#361	—	2	2	no	—
260	#359	—	2	2	no	—
261	#358	—	2	1	no	—
262	#357	—	2	2	no	—
263	#357	—	2	1	no	—
264	#350	—	2	2	no	—
265	#343	—	2	1	no	—
266	C₂×C₂₃	—	2	1	no	—
267	C₂×C₂₃	—	2	1	no	—
268	C₂×C₂₃	—	4	2	no	—
269	C₂×C₂₃	—	8	4	no	—
270	C₂×C₂₃	—	8	2	no	—
271	C₂×C₂₃	#1786	1	1	yes	characteristic
272	C₂×C₂₃	#2192	1	1	yes	characteristic
273	C₂×C₂₃	#2194	1	1	yes	characteristic
274	H(5,𝔽₂)	—	2	4	no	—
275	H(5,𝔽₂)	—	2	4	no	—
276	H(5,𝔽₂)	—	4	2	no	—
277	H(5,𝔽₂)	—	4	2	no	—
278	C₂×#41	—	2	2	no	—
279	C₂×#41	—	2	1	no	—
280	C₂×#41	—	4	1	no	—
281	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	—	4	2	no	—
285	C₂²×D₄	—	4	1	no	—
286	AGL(1,ℤ/8)	—	2	4	no	—
287	AGL(1,ℤ/8)	—	2	8	no	—
288	AGL(1,ℤ/8)	—	2	8	no	—
289	AGL(1,ℤ/8)	—	2	4	no	—
290	#135	—	2	4	no	—
291	#135	—	2	4	no	—
292	#135	—	2	1	no	—
293	#135	—	2	1	no	—
294	#135	C₂×C₄×C₂₃	1	2	yes	—
295	#135	D₄×C₂₃	1	2	yes	—
296	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
297	C₂×SD₁₆	—	2	2	no	—
298	C₂×SD₁₆	—	2	1	no	—
299	C₂×D₈	—	2	2	no	—
300	C₂×D₈	—	2	4	no	—
301	C₂×D₈	—	2	4	no	—
302	C₂×D₈	—	2	1	no	—
303	C₂×D₈	—	2	1	no	—
304	C₂×D₈	—	2	1	no	—
305	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
306	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
307	C₂×D₈	D₄×C₂₃	1	2	yes	—
308	C₂×D₈	D₄×C₂₃	1	2	yes	—
309	#131	—	2	1	no	—
310	#131	—	2	1	no	—
311	#131	C₂×C₄×C₂₃	1	2	yes	—
312	#131	D₄×C₂₃	1	2	yes	—
313	C₂×M₁₆	—	2	1	no	—
314	C₂×M₁₆	—	2	1	no	—
315	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
318	C₂×C₄⋊C₄	—	2	1	no	—
319	M₃₂	—	2	1	no	—
320	M₃₂	—	2	1	no	—
321	C₂×C₁₆	—	2	1	no	—
322	C₂×C₁₆	—	2	1	no	—
323	C₂×C₁₆	—	2	1	no	—
324	C₂×C₁₆	—	2	1	no	—
325	#108	—	2	1	no	—
326	#108	—	2	1	no	—
327	#107	—	2	1	no	—
328	#107	—	2	1	no	—
329	#103	—	4	1	no	—
330	#102	—	4	1	no	—
331	C₂₃	#12748	1	1	yes	characteristic, direct product factor, Sylow
332	C₂⁴	—	4	2	no	—
333	C₂⁴	—	4	1	no	—
334	#41	—	2	1	no	—
335	#41	—	2	1	no	—
336	#41	—	2	8	no	—
337	#41	—	2	2	no	—
338	#41	—	2	2	no	—
339	#41	—	4	4	no	—
340	#41	—	4	4	no	—
341	#41	—	4	4	no	—
342	#41	—	4	1	no	—
343	#41	—	4	1	no	—
344	#41	C₂²⋊C₄×C₂₃	1	2	yes	—
345	#41	SD₁₆×C₂₃	1	2	yes	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	8	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	4	2	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	8	no	—
357	C₂×D₄	—	4	8	no	—
358	C₂×D₄	—	4	4	no	—
359	C₂×D₄	—	4	2	no	—
360	C₂×D₄	—	4	1	no	—
361	C₂×D₄	—	4	4	no	—
362	C₂×D₄	—	4	1	no	—
363	C₂×D₄	—	4	1	no	—
364	C₂×D₄	—	4	1	no	—
365	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	1	no	—
369	C₂²×C₄	—	4	1	no	—
370	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
371	Q₁₆	—	2	1	no	—
372	Q₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
373	SD₁₆	—	2	8	no	—
374	SD₁₆	—	2	2	no	—
375	SD₁₆	—	2	2	no	—
376	D₈	—	2	8	no	—
377	D₈	—	2	2	no	—
378	D₈	—	2	1	no	—
379	D₈	—	2	2	no	—
380	D₈	—	2	2	no	—
381	D₈	C₂²⋊C₄×C₂₃	1	2	yes	—
382	D₈	C₂²⋊C₄×C₂₃	1	4	yes	—
383	M₁₆	—	2	1	no	—
384	M₁₆	—	2	2	no	—
385	M₁₆	—	2	2	no	—
386	M₁₆	—	4	1	no	—
387	M₁₆	—	4	1	no	—
388	M₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	4	1	no	—
393	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
394	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₄⋊C₄	—	4	1	no	—
400	C₄⋊C₄	—	4	1	no	—
401	C₁₆	—	4	1	no	—
402	C₁₆	—	4	1	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	4	8	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	1	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	16	1	no	—
410	C₂³	—	16	1	no	—
411	C₂³	—	16	2	no	—
412	C₂³	—	16	2	no	—
413	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
414	Q₈	—	2	1	no	—
415	Q₈	—	2	2	no	—
416	Q₈	—	4	1	no	—
417	Q₈	C₂₃×#102	1	2	yes	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	—	2	2	no	—
421	D₄	—	2	4	no	—
422	D₄	—	2	1	no	—
423	D₄	—	4	8	no	—
424	D₄	—	4	8	no	—
425	D₄	—	4	2	no	—
426	D₄	—	4	2	no	—
427	D₄	—	4	2	no	—
428	D₄	—	4	2	no	—
429	D₄	—	4	1	no	—
430	D₄	—	4	2	no	—
431	D₄	C₂₃×#102	1	2	yes	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	8	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	8	1	no	—
440	C₂×C₄	—	16	1	no	—
441	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
442	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
443	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
444	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
445	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
446	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
447	C₈	—	2	2	no	—
448	C₈	—	8	1	no	—
449	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
450	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
451	C₈	C₂₃×#120	1	1	yes	characteristic
452	C₈	C₂₃×#120	1	1	yes	characteristic
453	C₂²	—	2	1	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	4	1	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	1	no	—
462	C₂²	—	8	2	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	16	2	no	—
465	C₂²	—	16	2	no	—
466	C₂²	—	16	8	no	—
467	C₂²	C₂₃×#418	1	1	yes	characteristic
468	C₂²	C₂₃×#447	1	1	yes	characteristic
469	C₂²	C₂₃×#450	1	1	yes	characteristic
470	C₄	—	2	2	no	—
471	C₄	—	4	2	no	—
472	C₄	—	16	1	no	—
473	C₄	C₂₃×#379	1	1	yes	characteristic
474	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
475	C₄	C₂₃×#447	1	1	yes	characteristic
476	C₄	C₂₃×#450	1	1	yes	characteristic
477	C₂	—	2	1	no	—
478	C₂	—	2	1	no	—
479	C₂	—	4	2	no	—
480	C₂	—	8	4	no	—
481	C₂	—	8	2	no	—
482	C₂	C₂₃×#1786	1	1	yes	characteristic
483	C₂	C₂₃×#2192	1	1	yes	characteristic
484	C₂	C₂₃×#2194	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#447
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×C₂₃×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#361	C₂	semidirect
3. C₂×C₂₃×#359	C₂	semidirect
4. C₂₃×#2139	C₂	semidirect
5. C₂×C₂₃×#357	C₂	semidirect
6. C₂×C₂₃×#350	C₂	semidirect
7. C₂₃×#1816	C₂	semidirect
12. C₂₃×#576	C₄	semidirect
20. C₂×C₂₃×M₃₂	C₂²	semidirect
21. C₂²×C₁₆×C₂₃	C₂²	semidirect
22. C₂×C₂₃×#108	C₂²	semidirect
23. C₂×C₂₃×#107	C₂²	semidirect
160. #12748	C₂₃	direct (N × H)
212. C₂×#361	C₂×C₂₃	semidirect
213. C₂×#359	C₂×C₂₃	semidirect
214. #2139	C₂×C₂₃	semidirect
215. C₂×#357	C₂×C₂₃	semidirect
216. C₂×#350	C₂×C₂₃	semidirect
217. #1816	C₂×C₂₃	semidirect
246. #576	C₄×C₂₃	semidirect
254. C₂×M₃₂	C₂²×C₂₃	semidirect
255. C₂²×C₁₆	C₂²×C₂₃	semidirect
256. C₂×#108	C₂²×C₂₃	semidirect
257. C₂×#107	C₂²×C₂₃	semidirect
331. C₂₃	#12748	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#576	2
C₂×C₂₃×#361	2
C₂×C₂₃×#359	2
C₂₃×#2139	2
C₂×C₂₃×#357	2
C₂×C₂₃×#350	2
C₂₃×#1816	2
#12748	23

Subgroups — C23×#12748 #19160374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12748 #19160374