Subgroups — C₂₃×#22487 #19170113

Order 5888.

1258 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1892	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1872	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1824	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#340	C₂	1	2	yes	maximal, complemented
12	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#554	—	2	4	no	—
14	C₂₃×#554	C₄	1	8	yes	complemented
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#125	—	2	1	no	—
17	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
18	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	—	2	1	no	—
21	C₂×C₄×D₄×C₂₃	—	2	1	no	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₈×D₄×C₂₃	—	2	4	no	—
26	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#103	C₂²	1	2	yes	—
30	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#389	—	2	1	no	—
33	C₂₃×#385	—	2	2	no	—
34	C₂₃×#384	C₂²	1	1	yes	characteristic
35	C₂₃×#380	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#340	C₂²	1	2	yes	—
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#340	C₂²	1	1	yes	characteristic
41	C₂₃×#340	C₂²	1	2	yes	—
42	C₂₃×#340	C₂²	1	2	yes	—
43	C₂₃×#340	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₂₃	—	2	1	no	—
45	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	—	2	2	no	—
48	C₂³×C₄×C₂₃	—	2	2	no	—
49	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₂₃	—	2	4	no	—
51	C₂²×C₈×C₂₃	—	2	4	no	—
52	C₂²×C₈×C₂₃	—	4	2	no	—
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×C₄⋊Q₈	—	2	2	no	—
57	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
58	C₂₃×#125	—	2	8	no	—
59	C₂₃×C₂²⋊Q₈	—	2	4	no	—
60	C₂₃×C₂²⋊Q₈	—	2	16	no	—
61	C₂₃×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
62	C₄×Q₈×C₂₃	—	2	2	no	—
63	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
64	C₄×D₄×C₂₃	—	2	8	no	—
65	C₄×D₄×C₂₃	—	4	4	no	—
66	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
67	C₄×D₄×C₂₃	D₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
69	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
79	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
88	C₂×C₄²×C₂₃	—	2	1	no	—
89	C₂×C₄²×C₂₃	—	2	1	no	—
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂₃×C₄⋊C₈	—	2	2	no	—
92	C₂₃×#103	—	2	2	no	—
93	C₂₃×#103	—	2	2	no	—
94	C₂₃×#103	—	4	2	no	—
95	C₂₃×C₂²⋊C₈	—	2	4	no	—
96	C₄×C₈×C₂₃	—	2	2	no	—
97	C₂₃×#95	—	2	1	no	—
98	C₂₃×#95	—	2	2	no	—
99	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₂₃	—	2	4	no	—
101	C₂×Q₈×C₂₃	—	2	1	no	—
102	C₂×Q₈×C₂₃	—	2	1	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×Q₈×C₂₃	—	2	1	no	—
106	C₂×Q₈×C₂₃	—	8	1	no	—
107	C₂×Q₈×C₂₃	—	8	1	no	—
108	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
109	C₂×D₄×C₂₃	C₂²⋊C₄	1	4	yes	—
110	C₂×D₄×C₂₃	SD₁₆	1	4	yes	—
111	C₂×D₄×C₂₃	Q₁₆	1	4	yes	—
112	C₂²×C₄×C₂₃	—	2	2	no	—
113	C₂²×C₄×C₂₃	—	2	2	no	—
114	C₂²×C₄×C₂₃	—	2	2	no	—
115	C₂²×C₄×C₂₃	—	2	2	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	1	no	—
118	C₂²×C₄×C₂₃	—	2	1	no	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	2	8	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	2	no	—
123	C₂²×C₄×C₂₃	—	4	4	no	—
124	C₂²×C₄×C₂₃	—	4	4	no	—
125	C₂²×C₄×C₂₃	—	8	2	no	—
126	C₂²×C₄×C₂₃	—	8	2	no	—
127	C₂²×C₄×C₂₃	—	8	2	no	—
128	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
129	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
134	C₂×C₈×C₂₃	—	2	1	no	—
135	C₂×C₈×C₂₃	—	2	1	no	—
136	C₂×C₈×C₂₃	—	2	2	no	—
137	C₂×C₈×C₂₃	—	2	1	no	—
138	C₂×C₈×C₂₃	—	2	1	no	—
139	C₂×C₈×C₂₃	—	2	2	no	—
140	C₂×C₈×C₂₃	—	2	2	no	—
141	C₂×C₈×C₂₃	—	2	1	no	—
142	C₂×C₈×C₂₃	—	4	8	no	—
143	C₂×C₈×C₂₃	#41	1	2	yes	—
144	C₄⋊C₄×C₂₃	—	2	8	no	—
145	C₄⋊C₄×C₂₃	—	2	4	no	—
146	C₄⋊C₄×C₂₃	—	2	4	no	—
147	C₄⋊C₄×C₂₃	—	2	8	no	—
148	C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₄⋊C₄×C₂₃	—	2	2	no	—
150	C₄⋊C₄×C₂₃	—	2	1	no	—
151	C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	4	4	no	—
156	C₄⋊C₄×C₂₃	—	4	2	no	—
157	C₄⋊C₄×C₂₃	—	8	1	no	—
158	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
159	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
160	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
161	C₄⋊C₄×C₂₃	#41	1	2	yes	—
162	C₂²⋊C₄×C₂₃	—	2	8	no	—
163	C₂²⋊C₄×C₂₃	—	4	4	no	—
164	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
165	C₄²×C₂₃	—	2	4	no	—
166	C₄²×C₂₃	—	4	2	no	—
167	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
168	C₄²×C₂₃	C₂×D₄	1	2	yes	—
169	#22487	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₂₃	—	2	2	no	—
171	C₂³×C₂₃	—	2	2	no	—
172	C₂³×C₂₃	—	2	2	no	—
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
175	C₂³×C₂₃	C₂×SD₁₆	1	4	yes	—
176	C₂³×C₂₃	C₂×Q₁₆	1	4	yes	—
177	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₂₃	—	2	2	no	—
179	Q₈×C₂₃	—	2	2	no	—
180	Q₈×C₂₃	—	4	1	no	—
181	Q₈×C₂₃	—	4	1	no	—
182	Q₈×C₂₃	—	8	4	no	—
183	D₄×C₂₃	#103	1	16	yes	complemented
184	C₂×C₄×C₂₃	—	2	2	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	1	no	—
187	C₂×C₄×C₂₃	—	2	1	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	4	no	—
191	C₂×C₄×C₂₃	—	2	4	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	4	1	no	—
197	C₂×C₄×C₂₃	—	4	1	no	—
198	C₂×C₄×C₂₃	—	4	1	no	—
199	C₂×C₄×C₂₃	—	4	1	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	1	no	—
206	C₂×C₄×C₂₃	—	4	1	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	4	2	no	—
209	C₂×C₄×C₂₃	—	4	1	no	—
210	C₂×C₄×C₂₃	—	8	8	no	—
211	C₂×C₄×C₂₃	—	8	8	no	—
212	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
214	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
215	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
216	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
217	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
223	C₈×C₂₃	—	2	2	no	—
224	C₈×C₂₃	—	4	2	no	—
225	C₈×C₂₃	—	4	1	no	—
226	C₂×#554	C₂×C₂₃	1	1	yes	characteristic
227	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
228	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
229	#1990	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#1987	C₂×C₂₃	1	1	yes	characteristic, complemented
231	#1892	C₂×C₂₃	1	2	yes	complemented
232	#1872	C₂×C₂₃	1	2	yes	complemented
233	#1824	C₂×C₂₃	1	2	yes	complemented
234	#1803	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₄×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂×#340	C₂×C₂₃	1	2	yes	complemented
237	C₂²×C₂₃	—	2	4	no	—
238	C₂²×C₂₃	—	2	4	no	—
239	C₂²×C₂₃	—	2	4	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	2	4	no	—
242	C₂²×C₂₃	—	2	4	no	—
243	C₂²×C₂₃	C₂×#103	1	8	yes	—
244	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
245	C₂²×C₂₃	#546	1	1	yes	characteristic
246	C₂²×C₂₃	#547	1	1	yes	characteristic
247	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
248	C₂²×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
249	C₂²×C₂₃	#576	1	1	yes	characteristic
250	C₂²×C₂₃	#577	1	1	yes	characteristic
251	C₄×C₂₃	—	2	4	no	—
252	C₄×C₂₃	—	2	2	no	—
253	C₄×C₂₃	—	4	2	no	—
254	C₄×C₂₃	—	4	2	no	—
255	C₄×C₂₃	—	4	4	no	—
256	C₄×C₂₃	—	4	2	no	—
257	C₄×C₂₃	—	8	1	no	—
258	C₄×C₂₃	—	8	2	no	—
259	C₄×C₂₃	C₂×#103	1	4	yes	—
260	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
261	C₄×C₂₃	#520	1	1	yes	characteristic
262	C₄×C₂₃	#546	1	1	yes	characteristic
263	C₄×C₂₃	#547	1	1	yes	characteristic
264	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
265	#554	—	2	4	no	—
266	#554	C₄×C₂₃	1	8	yes	complemented
267	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
268	C₂×#125	—	2	1	no	—
269	C₂×C₂²⋊Q₈	—	2	2	no	—
270	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
271	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×C₄×D₄	—	2	1	no	—
273	C₂×C₄×D₄	—	2	1	no	—
274	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
275	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
276	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
277	C₈×D₄	—	2	4	no	—
278	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
279	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×#103	C₂²×C₂₃	1	2	yes	—
282	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
283	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
284	#389	—	2	1	no	—
285	#385	—	2	2	no	—
286	#384	C₂²×C₂₃	1	1	yes	characteristic
287	#380	C₂²×C₂₃	1	2	yes	—
288	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
289	#340	C₂²×C₂₃	1	2	yes	—
290	#340	C₂²×C₂₃	1	1	yes	characteristic
291	#340	C₂²×C₂₃	1	2	yes	—
292	#340	C₂²×C₂₃	1	2	yes	—
293	#340	C₂²×C₂₃	1	2	yes	—
294	#340	C₂²×C₂₃	1	2	yes	—
295	#340	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×C₂₃	—	2	8	no	—
297	C₂×C₂₃	#2339	1	1	yes	characteristic
298	C₂×C₂₃	C₂²×#103	1	1	yes	characteristic
299	C₂×C₂₃	#2897	1	1	yes	characteristic
300	C₂×C₂₃	#3295	1	1	yes	characteristic
301	C₂×C₂₃	#3299	1	1	yes	characteristic
302	C₂×C₂₃	#3335	1	1	yes	characteristic
303	C₂×C₂₃	#3338	1	1	yes	characteristic
304	C₂²×Q₈	—	2	1	no	—
305	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
306	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
307	C₂³×C₄	—	2	2	no	—
308	C₂³×C₄	—	2	2	no	—
309	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
310	C₂²×C₈	—	2	4	no	—
311	C₂²×C₈	—	2	4	no	—
312	C₂²×C₈	—	4	2	no	—
313	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
314	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
316	C₄⋊Q₈	—	2	2	no	—
317	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
318	#125	—	2	8	no	—
319	C₂²⋊Q₈	—	2	4	no	—
320	C₂²⋊Q₈	—	2	16	no	—
321	C₂²⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
322	C₄×Q₈	—	2	2	no	—
323	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
324	C₄×D₄	—	2	8	no	—
325	C₄×D₄	—	4	4	no	—
326	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
327	C₄×D₄	D₄×C₂₃	1	4	yes	—
328	C₂×C₄⋊C₄	—	2	2	no	—
329	C₂×C₄⋊C₄	—	2	4	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	4	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	4	2	no	—
338	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	complemented
339	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
351	C₄⋊C₈	—	2	2	no	—
352	#103	—	2	2	no	—
353	#103	—	2	2	no	—
354	#103	—	4	2	no	—
355	C₂²⋊C₈	—	2	4	no	—
356	C₄×C₈	—	2	2	no	—
357	#95	—	2	1	no	—
358	#95	—	2	2	no	—
359	C₂₃	#22487	1	1	yes	characteristic, direct product factor, Sylow
360	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	8	1	no	—
368	C₂×Q₈	—	8	1	no	—
369	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
370	C₂×D₄	C₂²⋊C₄×C₂₃	1	4	yes	—
371	C₂×D₄	SD₁₆×C₂₃	1	4	yes	—
372	C₂×D₄	Q₁₆×C₂₃	1	4	yes	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	8	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	8	2	no	—
387	C₂²×C₄	—	8	2	no	—
388	C₂²×C₄	—	8	2	no	—
389	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
390	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
393	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	4	8	no	—
404	C₂×C₈	#41×C₂₃	1	2	yes	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	4	4	no	—
417	C₄⋊C₄	—	4	2	no	—
418	C₄⋊C₄	—	8	1	no	—
419	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
420	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
421	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
422	C₄⋊C₄	#41×C₂₃	1	2	yes	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
426	C₄²	—	2	4	no	—
427	C₄²	—	4	2	no	—
428	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
429	C₄²	C₂×D₄×C₂₃	1	2	yes	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
435	C₂³	C₂×SD₁₆×C₂₃	1	4	yes	—
436	C₂³	C₂×Q₁₆×C₂₃	1	4	yes	—
437	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
438	Q₈	—	2	2	no	—
439	Q₈	—	2	2	no	—
440	Q₈	—	4	1	no	—
441	Q₈	—	4	1	no	—
442	Q₈	—	8	4	no	—
443	D₄	C₂₃×#103	1	16	yes	complemented
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	8	8	no	—
471	C₂×C₄	—	8	8	no	—
472	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
473	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
474	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
475	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
476	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
477	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
481	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
482	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
483	C₈	—	2	2	no	—
484	C₈	—	4	1	no	—
485	C₈	—	4	2	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	4	no	—
492	C₂²	C₂×C₂₃×#103	1	8	yes	—
493	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
494	C₂²	C₂₃×#546	1	1	yes	characteristic
495	C₂²	C₂₃×#547	1	1	yes	characteristic
496	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
497	C₂²	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
498	C₂²	C₂₃×#576	1	1	yes	characteristic
499	C₂²	C₂₃×#577	1	1	yes	characteristic
500	C₄	—	2	4	no	—
501	C₄	—	2	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	4	no	—
506	C₄	—	8	2	no	—
507	C₄	—	8	1	no	—
508	C₄	C₂×C₂₃×#103	1	4	yes	—
509	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
510	C₄	C₂₃×#520	1	1	yes	characteristic
511	C₄	C₂₃×#546	1	1	yes	characteristic
512	C₄	C₂₃×#547	1	1	yes	characteristic
513	C₂	—	2	8	no	—
514	C₂	C₂₃×#2339	1	1	yes	characteristic
515	C₂	C₂²×C₂₃×#103	1	1	yes	characteristic
516	C₂	C₂₃×#2897	1	1	yes	characteristic
517	C₂	C₂₃×#3295	1	1	yes	characteristic
518	C₂	C₂₃×#3299	1	1	yes	characteristic
519	C₂	C₂₃×#3335	1	1	yes	characteristic
520	C₂	C₂₃×#3338	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
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
4. C₂₃×#1990	C₂	semidirect
5. C₂₃×#1987	C₂	semidirect
6. C₂₃×#1892	C₂	semidirect
7. C₂₃×#1872	C₂	semidirect
8. C₂₃×#1824	C₂	semidirect
9. C₂₃×#1803	C₂	semidirect
10. C₄×C₂₃×#103	C₂	semidirect
11. C₂×C₂₃×#340	C₂	semidirect
14. C₂₃×#554	C₄	semidirect
57. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
61. C₂₃×C₂²⋊Q₈	C₂×C₄	semidirect
63. C₄×Q₈×C₂₃	C₂×C₄	semidirect
78. C₂×C₄⋊C₄×C₂₃	C₂×C₄	semidirect
169. #22487	C₂₃	direct (N × H)
183. D₄×C₂₃	#103	semidirect
229. #1990	C₂×C₂₃	semidirect
230. #1987	C₂×C₂₃	semidirect
231. #1892	C₂×C₂₃	semidirect
232. #1872	C₂×C₂₃	semidirect
233. #1824	C₂×C₂₃	semidirect
234. #1803	C₂×C₂₃	semidirect
235. C₄×#103	C₂×C₂₃	semidirect
236. C₂×#340	C₂×C₂₃	semidirect
266. #554	C₄×C₂₃	semidirect
317. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
321. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
323. C₄×Q₈	C₂×C₄×C₂₃	semidirect
338. C₂×C₄⋊C₄	C₂×C₄×C₂₃	semidirect
359. C₂₃	#22487	direct (N × H)
443. D₄	C₂₃×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#554	2
C₂×C₈×D₄×C₂₃	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1990	2
C₂₃×#1987	2
C₂₃×#1892	2
C₂₃×#1872	2
C₂₃×#1824	2
C₂₃×#1803	2
C₄×C₂₃×#103	2
C₂×C₂₃×#340	2
#22487	23

Subgroups — C23×#22487 #19170113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22487 #19170113