Subgroups — C₂₃×#22953 #19170579

Order 5888.

868 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1940	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1916	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1851	C₂	1	2	yes	maximal
9	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1753	C₂	1	2	yes	maximal
11	C₄×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₂₃	—	2	4	no	—
17	C₂₃×#423	—	2	4	no	—
18	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#391	C₄	1	1	yes	characteristic
30	C₂₃×#389	C₄	1	1	yes	characteristic
31	C₂₃×#387	C₄	1	2	yes	—
32	C₂₃×#385	C₄	1	2	yes	—
33	C₂₃×#380	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₂₃×#336	C₂²	1	1	yes	characteristic
38	C₂₃×#336	C₂²	1	2	yes	—
39	C₂₃×#336	C₂²	1	2	yes	—
40	C₂₃×#336	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	1	yes	characteristic
42	C₂₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₂₃	—	2	4	no	—
48	C₂×M₁₆×C₂₃	—	2	4	no	—
49	C₂×M₁₆×C₂₃	—	4	2	no	—
50	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×C₂²⋊Q₈	—	2	4	no	—
57	C₄×Q₈×C₂₃	—	2	2	no	—
58	C₄×Q₈×C₂₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₂₃	—	2	2	no	—
68	C₂×C₄²×C₂₃	—	2	2	no	—
69	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
73	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×C₄⋊C₈	—	2	4	no	—
76	C₂₃×C₄⋊C₈	—	2	4	no	—
77	C₂₃×C₄⋊C₈	—	2	2	no	—
78	C₂₃×C₄⋊C₈	—	4	2	no	—
79	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
80	C₂₃×C₂²⋊C₈	—	2	2	no	—
81	C₂₃×C₂²⋊C₈	Q₈	1	4	yes	complemented
82	C₂₃×#97	—	4	2	no	—
83	C₄×C₈×C₂₃	—	2	4	no	—
84	C₄×C₈×C₂₃	—	2	2	no	—
85	C₄×C₈×C₂₃	—	2	2	no	—
86	C₂₃×#95	C₂×C₄	1	2	yes	—
87	C₂₃×#95	C₂×C₄	1	2	yes	—
88	C₂₃×#95	C₂×C₄	1	2	yes	—
89	C₂₃×#95	C₂×C₄	1	2	yes	—
90	C₂₃×#95	C₂×C₄	1	2	yes	—
91	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×Q₈×C₂₃	—	2	2	no	—
93	C₂×Q₈×C₂₃	—	2	2	no	—
94	C₂×Q₈×C₂₃	M₁₆	1	4	yes	complemented
95	C₂²×C₄×C₂₃	—	2	1	no	—
96	C₂²×C₄×C₂₃	—	2	1	no	—
97	C₂²×C₄×C₂₃	—	2	1	no	—
98	C₂²×C₄×C₂₃	—	2	1	no	—
99	C₂²×C₄×C₂₃	—	2	2	no	—
100	C₂²×C₄×C₂₃	—	2	2	no	—
101	C₂²×C₄×C₂₃	—	2	2	no	—
102	C₂²×C₄×C₂₃	—	2	2	no	—
103	C₂²×C₄×C₂₃	—	2	2	no	—
104	C₂²×C₄×C₂₃	—	4	1	no	—
105	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
115	C₂²×C₄×C₂₃	#41	1	2	yes	—
116	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₂₃	—	4	8	no	—
118	C₂×C₈×C₂₃	—	2	1	no	—
119	C₂×C₈×C₂₃	—	2	2	no	—
120	C₂×C₈×C₂₃	—	2	1	no	—
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	1	no	—
123	C₂×C₈×C₂₃	—	2	1	no	—
124	C₂×C₈×C₂₃	—	2	2	no	—
125	C₂×C₈×C₂₃	—	2	1	no	—
126	C₂×C₈×C₂₃	—	2	2	no	—
127	C₂×C₈×C₂₃	—	2	1	no	—
128	C₂×C₈×C₂₃	—	2	1	no	—
129	C₂×C₈×C₂₃	—	2	1	no	—
130	C₂×C₈×C₂₃	—	2	1	no	—
131	C₂×C₈×C₂₃	—	2	2	no	—
132	C₂×C₈×C₂₃	—	2	1	no	—
133	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
134	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
135	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
136	C₄⋊C₄×C₂₃	—	2	4	no	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	—	2	2	no	—
139	C₄⋊C₄×C₂₃	—	2	4	no	—
140	C₄⋊C₄×C₂₃	—	2	4	no	—
141	C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₂²⋊C₄×C₂₃	—	2	4	no	—
143	C₂²⋊C₄×C₂₃	—	4	2	no	—
144	C₄²×C₂₃	—	2	2	no	—
145	C₄²×C₂₃	—	2	2	no	—
146	C₄²×C₂₃	—	2	1	no	—
147	C₄²×C₂₃	—	2	2	no	—
148	C₄²×C₂₃	—	2	1	no	—
149	C₄²×C₂₃	—	2	2	no	—
150	C₄²×C₂₃	—	2	2	no	—
151	C₄²×C₂₃	—	2	2	no	—
152	C₄²×C₂₃	C₂×D₄	1	2	yes	—
153	C₄²×C₂₃	#41	1	2	yes	—
154	#22953	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₂₃	—	2	1	no	—
156	C₂³×C₂₃	—	2	1	no	—
157	C₂³×C₂₃	—	2	1	no	—
158	C₂³×C₂₃	—	2	1	no	—
159	C₂³×C₂₃	—	2	1	no	—
160	C₂³×C₂₃	—	2	1	no	—
161	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
162	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
163	Q₈×C₂₃	—	2	8	no	—
164	C₂×C₄×C₂₃	—	2	1	no	—
165	C₂×C₄×C₂₃	—	2	1	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	1	no	—
168	C₂×C₄×C₂₃	—	2	1	no	—
169	C₂×C₄×C₂₃	—	2	2	no	—
170	C₂×C₄×C₂₃	—	2	1	no	—
171	C₂×C₄×C₂₃	—	2	1	no	—
172	C₂×C₄×C₂₃	—	2	2	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	2	2	no	—
176	C₂×C₄×C₂₃	—	2	1	no	—
177	C₂×C₄×C₂₃	—	2	2	no	—
178	C₂×C₄×C₂₃	—	2	2	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	1	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	2	no	—
184	C₂×C₄×C₂₃	—	2	2	no	—
185	C₂×C₄×C₂₃	—	2	1	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	4	4	no	—
189	C₂×C₄×C₂₃	—	4	4	no	—
190	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
193	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
194	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
195	C₂×C₄×C₂₃	#131	1	2	yes	—
196	C₂×C₄×C₂₃	#131	1	2	yes	—
197	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
199	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
202	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
203	C₈×C₂₃	—	2	4	no	—
204	C₈×C₂₃	—	2	2	no	—
205	C₈×C₂₃	—	2	2	no	—
206	C₈×C₂₃	—	4	1	no	—
207	C₈×C₂₃	—	4	2	no	—
208	C₈×C₂₃	—	4	1	no	—
209	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
210	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
211	C₄×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
212	#1984	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#1940	C₂×C₂₃	1	2	yes	complemented
214	#1939	C₂×C₂₃	1	1	yes	characteristic, complemented
215	#1916	C₂×C₂₃	1	2	yes	complemented
216	#1851	C₂×C₂₃	1	2	yes	—
217	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#1753	C₂×C₂₃	1	2	yes	—
219	C₄×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic
220	C₂²×C₂₃	—	2	2	no	—
221	C₂²×C₂₃	—	2	2	no	—
222	C₂²×C₂₃	—	2	2	no	—
223	C₂²×C₂₃	—	4	1	no	—
224	C₂²×C₂₃	—	4	1	no	—
225	C₂²×C₂₃	—	4	1	no	—
226	C₂²×C₂₃	—	4	1	no	—
227	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
229	C₂²×C₂₃	#520	1	1	yes	characteristic
230	C₂²×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
231	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
232	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
233	C₂²×C₂₃	#568	1	1	yes	characteristic
234	C₄×C₂₃	—	2	2	no	—
235	C₄×C₂₃	—	2	4	no	—
236	C₄×C₂₃	—	2	2	no	—
237	C₄×C₂₃	—	2	2	no	—
238	C₄×C₂₃	—	4	1	no	—
239	C₄×C₂₃	—	4	2	no	—
240	C₄×C₂₃	—	4	2	no	—
241	C₄×C₂₃	—	4	2	no	—
242	C₄×C₂₃	#435	1	4	yes	—
243	C₄×C₂₃	#546	1	1	yes	characteristic
244	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
245	C₄×C₂₃	#552	1	1	yes	characteristic
246	C₄×C₂₃	#554	1	1	yes	characteristic
247	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
248	C₂×C₂²⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
250	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
251	C₈×Q₈	—	2	4	no	—
252	#423	—	2	4	no	—
253	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
254	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
255	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
260	C₂×#97	C₂²×C₂₃	1	2	yes	—
261	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
262	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
263	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
264	#391	C₄×C₂₃	1	1	yes	characteristic
265	#389	C₄×C₂₃	1	1	yes	characteristic
266	#387	C₄×C₂₃	1	2	yes	—
267	#385	C₄×C₂₃	1	2	yes	—
268	#380	C₄×C₂₃	1	1	yes	characteristic
269	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
270	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
271	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
272	#336	C₂²×C₂₃	1	2	yes	—
273	#336	C₂²×C₂₃	1	2	yes	—
274	#336	C₂²×C₂₃	1	1	yes	characteristic
275	#336	C₂²×C₂₃	1	2	yes	—
276	#336	C₂²×C₂₃	1	1	yes	characteristic
277	#336	C₂²×C₂₃	1	2	yes	—
278	C₂×C₂₃	—	4	2	no	—
279	C₂×C₂₃	#2360	1	1	yes	characteristic
280	C₂×C₂₃	C₂×#435	1	1	yes	characteristic
281	C₂×C₂₃	#2933	1	1	yes	characteristic
282	C₂×C₂₃	#2960	1	1	yes	characteristic
283	C₂×C₂₃	Q₈×M₁₆	1	1	yes	characteristic
284	C₂×C₂₃	#2991	1	1	yes	characteristic
285	C₂×C₂₃	#2993	1	1	yes	characteristic
286	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
287	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
288	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
289	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
290	C₂×M₁₆	—	2	4	no	—
291	C₂×M₁₆	—	2	4	no	—
292	C₂×M₁₆	—	4	2	no	—
293	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
294	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
297	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
299	C₂²⋊Q₈	—	2	4	no	—
300	C₄×Q₈	—	2	2	no	—
301	C₄×Q₈	—	2	2	no	—
302	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
304	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
305	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
309	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	2	no	—
312	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
313	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
315	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
316	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
317	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
318	C₄⋊C₈	—	2	4	no	—
319	C₄⋊C₈	—	2	4	no	—
320	C₄⋊C₈	—	2	2	no	—
321	C₄⋊C₈	—	4	2	no	—
322	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
323	C₂²⋊C₈	—	2	2	no	—
324	C₂²⋊C₈	Q₈×C₂₃	1	4	yes	complemented
325	#97	—	4	2	no	—
326	C₄×C₈	—	2	4	no	—
327	C₄×C₈	—	2	2	no	—
328	C₄×C₈	—	2	2	no	—
329	#95	C₂×C₄×C₂₃	1	2	yes	—
330	#95	C₂×C₄×C₂₃	1	2	yes	—
331	#95	C₂×C₄×C₂₃	1	2	yes	—
332	#95	C₂×C₄×C₂₃	1	2	yes	—
333	#95	C₂×C₄×C₂₃	1	2	yes	—
334	C₂₃	#22953	1	1	yes	characteristic, direct product factor, Sylow
335	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	M₁₆×C₂₃	1	4	yes	complemented
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	4	1	no	—
349	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
350	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
352	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
353	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
354	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
355	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
356	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
357	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
358	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
359	C₂²×C₄	#41×C₂₃	1	2	yes	—
360	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
361	M₁₆	—	4	8	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
378	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
379	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₂²⋊C₄	—	2	4	no	—
387	C₂²⋊C₄	—	4	2	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	C₂×D₄×C₂₃	1	2	yes	—
397	C₄²	#41×C₂₃	1	2	yes	—
398	C₂³	—	2	1	no	—
399	C₂³	—	2	1	no	—
400	C₂³	—	2	1	no	—
401	C₂³	—	2	1	no	—
402	C₂³	—	2	1	no	—
403	C₂³	—	2	1	no	—
404	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
405	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
406	Q₈	—	2	8	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	—	4	4	no	—
433	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
434	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
435	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
436	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
437	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
438	C₂×C₄	C₂₃×#131	1	2	yes	—
439	C₂×C₄	C₂₃×#131	1	2	yes	—
440	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
441	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
444	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
445	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
446	C₈	—	2	2	no	—
447	C₈	—	2	2	no	—
448	C₈	—	2	4	no	—
449	C₈	—	4	1	no	—
450	C₈	—	4	2	no	—
451	C₈	—	4	1	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	2	2	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	4	1	no	—
458	C₂²	—	4	1	no	—
459	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
460	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
461	C₂²	C₂₃×#520	1	1	yes	characteristic
462	C₂²	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
463	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
464	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
465	C₂²	C₂₃×#568	1	1	yes	characteristic
466	C₄	—	2	4	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	4	1	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	C₂₃×#435	1	4	yes	—
475	C₄	C₂₃×#546	1	1	yes	characteristic
476	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
477	C₄	C₂₃×#552	1	1	yes	characteristic
478	C₄	C₂₃×#554	1	1	yes	characteristic
479	C₂	—	4	2	no	—
480	C₂	C₂₃×#2360	1	1	yes	characteristic
481	C₂	C₂×C₂₃×#435	1	1	yes	characteristic
482	C₂	C₂₃×#2933	1	1	yes	characteristic
483	C₂	C₂₃×#2960	1	1	yes	characteristic
484	C₂	Q₈×M₁₆×C₂₃	1	1	yes	characteristic
485	C₂	C₂₃×#2991	1	1	yes	characteristic
486	C₂	C₂₃×#2993	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#1984	C₂	semidirect
5. C₂₃×#1940	C₂	semidirect
6. C₂₃×#1939	C₂	semidirect
7. C₂₃×#1916	C₂	semidirect
9. C₂₃×#1770	C₂	semidirect
81. C₂₃×C₂²⋊C₈	Q₈	semidirect
94. C₂×Q₈×C₂₃	M₁₆	semidirect
154. #22953	C₂₃	direct (N × H)
209. C₂×C₈×Q₈	C₂×C₂₃	semidirect
212. #1984	C₂×C₂₃	semidirect
213. #1940	C₂×C₂₃	semidirect
214. #1939	C₂×C₂₃	semidirect
215. #1916	C₂×C₂₃	semidirect
217. #1770	C₂×C₂₃	semidirect
324. C₂²⋊C₈	Q₈×C₂₃	semidirect
334. C₂₃	#22953	direct (N × H)
338. C₂×Q₈	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₂₃	2
C₂×C₂₃×#423	2
C₄×C₂₃×C₂²⋊Q₈	2
C₂₃×#1984	2
C₂₃×#1940	2
C₂₃×#1939	2
C₂₃×#1916	2
C₂₃×#1851	2
C₂₃×#1770	2
C₂₃×#1753	2
C₄×C₂₃×C₂²⋊C₈	2
#22953	23

Subgroups — C23×#22953 #19170579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22953 #19170579