Subgroups — C₂₃×#26078 #19173704

Order 5888.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#428	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2441	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2440	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2084	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1943	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1853	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1809	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1753	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#501	—	2	4	no	—
23	C₂₃×#428	—	2	4	no	—
24	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#399	—	2	1	no	—
38	C₂₃×#398	—	2	1	no	—
39	C₂₃×#396	—	2	1	no	—
40	C₂₃×#395	—	2	1	no	—
41	C₂₃×#389	—	2	1	no	—
42	C₂₃×#389	C₂²	1	1	yes	characteristic
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#389	C₂²	1	1	yes	characteristic
45	C₂₃×#389	C₂²	1	1	yes	characteristic
46	C₂₃×#387	—	2	1	no	—
47	C₂₃×#385	—	2	1	no	—
48	C₂₃×#384	C₂²	1	1	yes	characteristic
49	C₂₃×#384	C₂²	1	1	yes	characteristic
50	C₂₃×#380	C₂²	1	1	yes	characteristic
51	C₂₃×#380	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	—	2	1	no	—
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₂₃	—	2	2	no	—
65	C₂×M₁₆×C₂₃	—	2	2	no	—
66	C₂×M₁₆×C₂₃	Q₈	1	4	yes	complemented
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂₃×C₄⋊Q₈	—	2	2	no	—
72	C₂₃×C₄⋊Q₈	—	4	1	no	—
73	C₂₃×#125	—	2	2	no	—
74	C₂₃×#125	—	4	1	no	—
75	C₂₃×C₂²⋊Q₈	—	4	4	no	—
76	C₂₃×#117	—	2	2	no	—
77	C₂₃×#117	—	4	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂₃×#107	—	2	2	no	—
109	C₂₃×#107	—	2	2	no	—
110	C₂₃×#107	—	2	2	no	—
111	C₂₃×#107	D₄	1	4	yes	—
112	C₂₃×#106	—	2	2	no	—
113	C₂₃×#106	—	2	2	no	—
114	C₂₃×#106	—	2	2	no	—
115	C₂₃×#106	—	2	2	no	—
116	C₂₃×#106	—	4	1	no	—
117	C₂₃×#106	—	4	1	no	—
118	C₂₃×C₂²⋊C₈	Q₈	1	4	yes	complemented
119	C₂₃×#97	—	2	2	no	—
120	C₄×C₈×C₂₃	—	2	2	no	—
121	C₂₃×#95	—	2	1	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	D₄	1	1	yes	characteristic
128	C₂₃×#95	D₄	1	1	yes	characteristic
129	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Q₈×C₂₃	—	4	2	no	—
131	C₂×Q₈×C₂₃	—	4	2	no	—
132	C₂×Q₈×C₂₃	—	8	1	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	1	no	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	4	2	no	—
146	C₂²×C₄×C₂₃	—	4	2	no	—
147	C₂²×C₄×C₂₃	—	8	1	no	—
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
154	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
155	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
156	M₁₆×C₂₃	—	2	8	no	—
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
164	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
165	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
166	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
167	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
168	C₂×C₈×C₂₃	#41	1	2	yes	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	4	2	no	—
184	C₄⋊C₄×C₂₃	—	4	2	no	—
185	C₄⋊C₄×C₂₃	—	4	2	no	—
186	C₄⋊C₄×C₂₃	—	4	2	no	—
187	C₄⋊C₄×C₂₃	—	4	2	no	—
188	C₄⋊C₄×C₂₃	—	4	2	no	—
189	C₄⋊C₄×C₂₃	—	4	2	no	—
190	C₄⋊C₄×C₂₃	—	4	2	no	—
191	C₄⋊C₄×C₂₃	—	4	2	no	—
192	C₄⋊C₄×C₂₃	—	8	1	no	—
193	C₄⋊C₄×C₂₃	—	8	1	no	—
194	C₄⋊C₄×C₂₃	—	8	1	no	—
195	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
197	C₂²⋊C₄×C₂₃	—	2	2	no	—
198	C₂²⋊C₄×C₂₃	—	2	2	no	—
199	C₂²⋊C₄×C₂₃	—	4	2	no	—
200	C₂²⋊C₄×C₂₃	—	4	2	no	—
201	C₂²⋊C₄×C₂₃	—	8	1	no	—
202	C₄²×C₂₃	—	2	1	no	—
203	C₄²×C₂₃	—	2	1	no	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	—	2	1	no	—
206	C₄²×C₂₃	—	4	2	no	—
207	#26078	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂³×C₂₃	—	2	1	no	—
209	C₂³×C₂₃	—	2	1	no	—
210	C₂³×C₂₃	—	2	1	no	—
211	C₂³×C₂₃	—	2	1	no	—
212	C₂³×C₂₃	—	2	1	no	—
213	C₂³×C₂₃	—	2	1	no	—
214	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
215	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
216	Q₈×C₂₃	—	8	4	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	8	4	no	—
258	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
259	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
260	C₂×C₄×C₂₃	#137	1	2	yes	—
261	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
263	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
264	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
266	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
267	C₈×C₂₃	—	2	2	no	—
268	C₈×C₂₃	—	2	2	no	—
269	C₈×C₂₃	—	2	2	no	—
270	C₈×C₂₃	C₂²⋊Q₈	1	4	yes	complemented
271	C₂×#501	C₂×C₂₃	1	1	yes	characteristic, complemented
272	C₂×#428	C₂×C₂₃	1	1	yes	characteristic
273	#2441	C₂×C₂₃	1	1	yes	characteristic
274	#2440	C₂×C₂₃	1	1	yes	characteristic
275	#2084	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#2083	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#2058	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1943	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1942	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#1854	C₂×C₂₃	1	1	yes	characteristic
282	#1853	C₂×C₂₃	1	1	yes	characteristic
283	#1844	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1809	C₂×C₂₃	1	1	yes	characteristic
285	#1753	C₂×C₂₃	1	1	yes	characteristic
286	C₂²×C₂₃	—	2	2	no	—
287	C₂²×C₂₃	—	2	2	no	—
288	C₂²×C₂₃	—	2	2	no	—
289	C₂²×C₂₃	—	4	1	no	—
290	C₂²×C₂₃	—	4	1	no	—
291	C₂²×C₂₃	—	4	1	no	—
292	C₂²×C₂₃	—	4	1	no	—
293	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
294	C₂²×C₂₃	D₄²	1	1	yes	characteristic
295	C₂²×C₂₃	#547	1	1	yes	characteristic
296	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
298	C₂²×C₂₃	#577	1	1	yes	characteristic
299	C₂²×C₂₃	#577	1	1	yes	characteristic
300	C₄×C₂₃	—	2	2	no	—
301	C₄×C₂₃	—	2	2	no	—
302	C₄×C₂₃	—	2	2	no	—
303	C₄×C₂₃	—	4	1	no	—
304	C₄×C₂₃	—	4	2	no	—
305	C₄×C₂₃	—	4	2	no	—
306	C₄×C₂₃	—	4	1	no	—
307	C₄×C₂₃	—	8	1	no	—
308	C₄×C₂₃	—	8	1	no	—
309	C₄×C₂₃	—	8	1	no	—
310	C₄×C₂₃	—	8	1	no	—
311	C₄×C₂₃	—	8	1	no	—
312	C₄×C₂₃	—	8	1	no	—
313	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
314	C₄×C₂₃	#544	1	1	yes	characteristic
315	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
316	C₄×C₂₃	#554	1	1	yes	characteristic
317	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
319	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×C₂²⋊Q₈	—	2	1	no	—
321	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
322	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
323	#501	—	2	4	no	—
324	#428	—	2	4	no	—
325	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
333	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
338	#399	—	2	1	no	—
339	#398	—	2	1	no	—
340	#396	—	2	1	no	—
341	#395	—	2	1	no	—
342	#389	—	2	1	no	—
343	#389	C₂²×C₂₃	1	1	yes	characteristic
344	#389	C₂²×C₂₃	1	1	yes	characteristic
345	#389	C₂²×C₂₃	1	1	yes	characteristic
346	#389	C₂²×C₂₃	1	1	yes	characteristic
347	#387	—	2	1	no	—
348	#385	—	2	1	no	—
349	#384	C₂²×C₂₃	1	1	yes	characteristic
350	#384	C₂²×C₂₃	1	1	yes	characteristic
351	#380	C₂²×C₂₃	1	1	yes	characteristic
352	#380	C₂²×C₂₃	1	1	yes	characteristic
353	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
354	#340	C₂²×C₂₃	1	1	yes	characteristic
355	#340	C₂²×C₂₃	1	1	yes	characteristic
356	#340	C₂²×C₂₃	1	1	yes	characteristic
357	#340	C₂²×C₂₃	1	1	yes	characteristic
358	#340	C₂²×C₂₃	1	1	yes	characteristic
359	#340	C₂²×C₂₃	1	1	yes	characteristic
360	#336	C₂²×C₂₃	1	1	yes	characteristic
361	#336	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×C₂₃	—	4	2	no	—
363	C₂×C₂₃	#2440	1	1	yes	characteristic
364	C₂×C₂₃	#3163	1	1	yes	characteristic
365	C₂×C₂₃	#3165	1	1	yes	characteristic
366	C₂×C₂₃	#3272	1	1	yes	characteristic
367	C₂×C₂₃	#3278	1	1	yes	characteristic
368	C₂×C₂₃	#3344	1	1	yes	characteristic
369	C₂×C₂₃	#3345	1	1	yes	characteristic
370	C₂²×Q₈	—	2	1	no	—
371	C₂³×C₄	—	2	1	no	—
372	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
373	C₂×M₁₆	—	2	2	no	—
374	C₂×M₁₆	—	2	2	no	—
375	C₂×M₁₆	Q₈×C₂₃	1	4	yes	complemented
376	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₄⋊Q₈	—	2	2	no	—
381	C₄⋊Q₈	—	4	1	no	—
382	#125	—	2	2	no	—
383	#125	—	4	1	no	—
384	C₂²⋊Q₈	—	4	4	no	—
385	#117	—	2	2	no	—
386	#117	—	4	1	no	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	4	1	no	—
401	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
409	C₂×C₂²⋊C₄	—	2	1	no	—
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
413	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
414	C₂×C₄²	—	2	1	no	—
415	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
417	#107	—	2	2	no	—
418	#107	—	2	2	no	—
419	#107	—	2	2	no	—
420	#107	D₄×C₂₃	1	4	yes	—
421	#106	—	2	2	no	—
422	#106	—	2	2	no	—
423	#106	—	2	2	no	—
424	#106	—	2	2	no	—
425	#106	—	4	1	no	—
426	#106	—	4	1	no	—
427	C₂²⋊C₈	Q₈×C₂₃	1	4	yes	complemented
428	#97	—	2	2	no	—
429	C₄×C₈	—	2	2	no	—
430	#95	—	2	1	no	—
431	#95	—	2	1	no	—
432	#95	—	2	1	no	—
433	#95	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	D₄×C₂₃	1	1	yes	characteristic
437	#95	D₄×C₂₃	1	1	yes	characteristic
438	C₂₃	#26078	1	1	yes	characteristic, direct product factor, Sylow
439	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	4	2	no	—
442	C₂×Q₈	—	8	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	4	2	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	8	1	no	—
458	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
463	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
464	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
466	M₁₆	—	2	8	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
474	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
475	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
476	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
477	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
478	C₂×C₈	#41×C₂₃	1	2	yes	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	4	2	no	—
494	C₄⋊C₄	—	4	2	no	—
495	C₄⋊C₄	—	4	2	no	—
496	C₄⋊C₄	—	4	2	no	—
497	C₄⋊C₄	—	4	2	no	—
498	C₄⋊C₄	—	4	2	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	—	4	2	no	—
501	C₄⋊C₄	—	4	2	no	—
502	C₄⋊C₄	—	8	1	no	—
503	C₄⋊C₄	—	8	1	no	—
504	C₄⋊C₄	—	8	1	no	—
505	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
506	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	4	2	no	—
510	C₂²⋊C₄	—	4	2	no	—
511	C₂²⋊C₄	—	8	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	4	2	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
524	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
525	Q₈	—	8	4	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	8	4	no	—
567	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
568	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
569	C₂×C₄	C₂₃×#137	1	2	yes	—
570	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
571	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
572	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
573	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
574	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
575	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
576	C₈	—	2	2	no	—
577	C₈	—	2	2	no	—
578	C₈	—	2	2	no	—
579	C₈	C₂₃×C₂²⋊Q₈	1	4	yes	complemented
580	C₂²	—	2	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
588	C₂²	D₄²×C₂₃	1	1	yes	characteristic
589	C₂²	C₂₃×#547	1	1	yes	characteristic
590	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
591	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
592	C₂²	C₂₃×#577	1	1	yes	characteristic
593	C₂²	C₂₃×#577	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	2	no	—
600	C₄	—	4	1	no	—
601	C₄	—	8	1	no	—
602	C₄	—	8	1	no	—
603	C₄	—	8	1	no	—
604	C₄	—	8	1	no	—
605	C₄	—	8	1	no	—
606	C₄	—	8	1	no	—
607	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
608	C₄	C₂₃×#544	1	1	yes	characteristic
609	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
610	C₄	C₂₃×#554	1	1	yes	characteristic
611	C₂	—	4	2	no	—
612	C₂	C₂₃×#2440	1	1	yes	characteristic
613	C₂	C₂₃×#3163	1	1	yes	characteristic
614	C₂	C₂₃×#3165	1	1	yes	characteristic
615	C₂	C₂₃×#3272	1	1	yes	characteristic
616	C₂	C₂₃×#3278	1	1	yes	characteristic
617	C₂	C₂₃×#3344	1	1	yes	characteristic
618	C₂	C₂₃×#3345	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#501	C₂	semidirect
5. C₂₃×#2084	C₂	semidirect
6. C₂₃×#2083	C₂	semidirect
7. C₂₃×#2059	C₂	semidirect
8. C₂₃×#2058	C₂	semidirect
9. C₂₃×#1943	C₂	semidirect
10. C₂₃×#1942	C₂	semidirect
13. C₂₃×#1844	C₂	semidirect
66. C₂×M₁₆×C₂₃	Q₈	semidirect
118. C₂₃×C₂²⋊C₈	Q₈	semidirect
207. #26078	C₂₃	direct (N × H)
270. C₈×C₂₃	C₂²⋊Q₈	semidirect
271. C₂×#501	C₂×C₂₃	semidirect
275. #2084	C₂×C₂₃	semidirect
276. #2083	C₂×C₂₃	semidirect
277. #2059	C₂×C₂₃	semidirect
278. #2058	C₂×C₂₃	semidirect
279. #1943	C₂×C₂₃	semidirect
280. #1942	C₂×C₂₃	semidirect
283. #1844	C₂×C₂₃	semidirect
375. C₂×M₁₆	Q₈×C₂₃	semidirect
427. C₂²⋊C₈	Q₈×C₂₃	semidirect
438. C₂₃	#26078	direct (N × H)
579. C₈	C₂₃×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#501	2
C₂×C₂₃×#428	2
C₂₃×#2441	2
C₂₃×#2440	2
C₂₃×#2084	2
C₂₃×#2083	2
C₂₃×#2059	2
C₂₃×#2058	2
C₂₃×#1943	2
C₂₃×#1942	2
C₂₃×#1854	2
C₂₃×#1853	2
C₂₃×#1844	2
C₂₃×#1809	2
C₂₃×#1753	2
#26078	23

Subgroups — C23×#26078 #19173704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26078 #19173704