Subgroups — C₂₃×#17486 #19165112

Order 5888.

1026 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1493	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1487	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1481	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₂₃	—	2	1	no	—
8	C₂₃×#533	—	2	1	no	—
9	C₂₃×#533	—	2	1	no	—
10	C₂₃×#533	—	2	1	no	—
11	C₂₃×#533	—	2	1	no	—
12	C₂₃×#533	C₄	1	2	yes	—
13	C₂₃×#533	C₄	1	2	yes	—
14	C₂₃×#533	C₂²	1	2	yes	—
15	C₂₃×#533	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
18	C₂×C₂₃×#125	—	2	1	no	—
19	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₂₃×C₂²⋊Q₈	C₄	1	2	yes	—
21	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
22	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂₃×#117	—	2	1	no	—
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#431	—	2	1	no	—
26	C₂₃×#431	—	2	1	no	—
27	C₂₃×#431	C₂²	1	2	yes	—
28	C₂₃×#431	C₂²	1	2	yes	—
29	C₂₃×#423	—	2	1	no	—
30	C₂₃×#423	—	2	1	no	—
31	C₂₃×#423	C₂²	1	2	yes	—
32	C₂₃×#423	C₂²	1	2	yes	—
33	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
34	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
35	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
37	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
38	C₂₃×Q₈⋊C₈	C₂²	1	16	yes	complemented
39	C₂²×Q₈×C₂₃	—	2	1	no	—
40	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
44	C₂×M₁₆×C₂₃	—	2	2	no	—
45	C₂×M₁₆×C₂₃	—	2	2	no	—
46	C₂×M₁₆×C₂₃	—	4	4	no	—
47	C₂²×C₈×C₂₃	—	2	2	no	—
48	C₂²×C₈×C₂₃	—	2	2	no	—
49	C₂₃×C₄⋊Q₈	—	2	1	no	—
50	C₂₃×C₄⋊Q₈	—	2	1	no	—
51	C₂₃×C₄⋊Q₈	—	2	1	no	—
52	C₂₃×C₄⋊Q₈	—	2	1	no	—
53	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
54	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
55	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
56	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
57	C₂₃×#125	—	2	2	no	—
58	C₂₃×#125	—	2	2	no	—
59	C₂₃×#125	—	2	2	no	—
60	C₂₃×#125	—	2	2	no	—
61	C₂₃×C₂²⋊Q₈	—	2	2	no	—
62	C₂₃×C₂²⋊Q₈	—	2	2	no	—
63	C₂₃×C₂²⋊Q₈	—	2	2	no	—
64	C₂₃×C₂²⋊Q₈	—	2	2	no	—
65	C₂₃×C₂²⋊Q₈	—	2	2	no	—
66	C₂₃×C₂²⋊Q₈	—	2	2	no	—
67	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
68	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
69	C₄×Q₈×C₂₃	—	2	2	no	—
70	C₄×Q₈×C₂₃	—	2	2	no	—
71	C₄×Q₈×C₂₃	—	2	2	no	—
72	C₄×Q₈×C₂₃	—	2	2	no	—
73	C₄×Q₈×C₂₃	—	2	2	no	—
74	C₄×Q₈×C₂₃	—	2	2	no	—
75	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
76	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
77	C₂₃×#117	—	2	2	no	—
78	C₂₃×#117	—	2	2	no	—
79	C₂₃×#117	—	2	2	no	—
80	C₂₃×#117	—	2	2	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	2	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
87	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₄²×C₂₃	—	2	2	no	—
89	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂₃×C₄⋊C₈	—	2	2	no	—
102	C₂₃×C₄⋊C₈	—	2	2	no	—
103	C₂₃×C₄⋊C₈	—	2	2	no	—
104	C₂₃×C₄⋊C₈	—	2	2	no	—
105	C₂₃×C₄⋊C₈	—	2	2	no	—
106	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
107	C₂₃×C₂²⋊C₈	—	2	2	no	—
108	C₂₃×C₂²⋊C₈	—	2	2	no	—
109	C₄×C₈×C₂₃	—	2	2	no	—
110	C₄×C₈×C₂₃	C₂³	1	4	yes	—
111	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×Q₈×C₂₃	—	2	2	no	—
113	C₂×Q₈×C₂₃	—	2	2	no	—
114	C₂×Q₈×C₂₃	—	2	2	no	—
115	C₂×Q₈×C₂₃	—	4	4	no	—
116	C₂×Q₈×C₂₃	—	4	4	no	—
117	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
118	C₂²×C₄×C₂₃	—	2	2	no	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	4	2	no	—
121	C₂²×C₄×C₂₃	—	4	2	no	—
122	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
132	C₂²×C₄×C₂₃	Q₁₆	1	4	yes	—
133	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	M₁₆×C₂₃	—	4	4	no	—
143	M₁₆×C₂₃	—	4	4	no	—
144	C₂×C₈×C₂₃	—	2	4	no	—
145	C₂×C₈×C₂₃	—	2	4	no	—
146	C₂×C₈×C₂₃	—	4	4	no	—
147	C₂×C₈×C₂₃	—	4	4	no	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	—	2	2	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	4	no	—
155	C₄⋊C₄×C₂₃	—	2	4	no	—
156	C₄⋊C₄×C₂₃	—	2	4	no	—
157	C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	2	4	no	—
159	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
160	C₂²⋊C₄×C₂₃	—	2	4	no	—
161	C₂²⋊C₄×C₂₃	—	2	4	no	—
162	C₄²×C₂₃	—	2	4	no	—
163	C₄²×C₂₃	—	2	4	no	—
164	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
165	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
166	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
167	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
168	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
169	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
170	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
172	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
180	#17486	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂³×C₂₃	—	4	1	no	—
182	C₂³×C₂₃	—	4	1	no	—
183	C₂³×C₂₃	#103	1	4	yes	—
184	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	Q₈×C₂₃	—	2	8	no	—
188	Q₈×C₂₃	—	4	4	no	—
189	Q₈×C₂₃	—	4	4	no	—
190	Q₈×C₂₃	—	4	4	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	4	1	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	4	1	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	4	1	no	—
204	C₂×C₄×C₂₃	—	4	1	no	—
205	C₂×C₄×C₂₃	—	4	4	no	—
206	C₂×C₄×C₂₃	—	4	4	no	—
207	C₂×C₄×C₂₃	#103	1	4	yes	—
208	C₂×C₄×C₂₃	#103	1	4	yes	—
209	C₂×C₄×C₂₃	#103	1	4	yes	—
210	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
220	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
221	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
222	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
223	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
224	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
225	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
226	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	C₈×C₂₃	—	4	4	no	—
229	C₈×C₂₃	—	4	4	no	—
230	C₂×#533	C₂×C₂₃	1	1	yes	characteristic
231	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
232	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
233	#1493	C₂×C₂₃	1	4	yes	complemented
234	#1487	C₂×C₂₃	1	4	yes	complemented
235	#1481	C₂×C₂₃	1	4	yes	complemented
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	1	no	—
238	C₂²×C₂₃	—	2	1	no	—
239	C₂²×C₂₃	—	2	1	no	—
240	C₂²×C₂₃	—	2	1	no	—
241	C₂²×C₂₃	—	2	1	no	—
242	C₂²×C₂₃	—	4	4	no	—
243	C₂²×C₂₃	C₂×#103	1	2	yes	—
244	C₂²×C₂₃	C₂×#103	1	2	yes	—
245	C₂²×C₂₃	C₂×#103	1	2	yes	—
246	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
247	C₄×C₂₃	—	2	1	no	—
248	C₄×C₂₃	—	2	1	no	—
249	C₄×C₂₃	—	2	1	no	—
250	C₄×C₂₃	—	2	1	no	—
251	C₄×C₂₃	—	2	1	no	—
252	C₄×C₂₃	—	2	1	no	—
253	C₄×C₂₃	—	4	4	no	—
254	C₄×C₂₃	—	4	4	no	—
255	C₄×C₂₃	C₂×#103	1	2	yes	—
256	C₄×C₂₃	C₂×#103	1	2	yes	—
257	C₄×C₂₃	C₂×#103	1	2	yes	—
258	C₄×C₂₃	#421	1	2	yes	—
259	C₄×C₂₃	#568	1	2	yes	—
260	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
261	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
262	C₂²×M₁₆	—	2	1	no	—
263	#533	—	2	1	no	—
264	#533	—	2	1	no	—
265	#533	—	2	1	no	—
266	#533	—	2	1	no	—
267	#533	C₄×C₂₃	1	2	yes	—
268	#533	C₄×C₂₃	1	2	yes	—
269	#533	C₂²×C₂₃	1	2	yes	—
270	#533	C₂²×C₂₃	1	2	yes	—
271	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
272	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
273	C₂×#125	—	2	1	no	—
274	C₂×C₂²⋊Q₈	—	2	1	no	—
275	C₂×C₂²⋊Q₈	C₄×C₂₃	1	2	yes	—
276	C₂×C₄×Q₈	—	2	1	no	—
277	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
278	C₂×#117	—	2	1	no	—
279	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
280	#431	—	2	1	no	—
281	#431	—	2	1	no	—
282	#431	C₂²×C₂₃	1	2	yes	—
283	#431	C₂²×C₂₃	1	2	yes	—
284	#423	—	2	1	no	—
285	#423	—	2	1	no	—
286	#423	C₂²×C₂₃	1	2	yes	—
287	#423	C₂²×C₂₃	1	2	yes	—
288	C₂×C₄⋊C₈	—	2	1	no	—
289	C₂×C₄⋊C₈	—	2	1	no	—
290	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
291	C₂×C₂²⋊C₈	—	2	1	no	—
292	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
293	Q₈⋊C₈	C₂²×C₂₃	1	16	yes	complemented
294	C₂×C₂₃	—	2	2	no	—
295	C₂×C₂₃	—	2	2	no	—
296	C₂×C₂₃	—	2	2	no	—
297	C₂×C₂₃	#2879	1	1	yes	characteristic
298	C₂×C₂₃	C₂²×#103	1	1	yes	characteristic
299	C₂×C₂₃	C₂×#421	1	1	yes	characteristic
300	C₂²×Q₈	—	2	1	no	—
301	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
302	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
303	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
304	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
305	C₂×M₁₆	—	2	2	no	—
306	C₂×M₁₆	—	2	2	no	—
307	C₂×M₁₆	—	4	4	no	—
308	C₂²×C₈	—	2	2	no	—
309	C₂²×C₈	—	2	2	no	—
310	C₄⋊Q₈	—	2	1	no	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	—	2	1	no	—
313	C₄⋊Q₈	—	2	1	no	—
314	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
315	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
316	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
317	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
318	#125	—	2	2	no	—
319	#125	—	2	2	no	—
320	#125	—	2	2	no	—
321	#125	—	2	2	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	—	2	2	no	—
324	C₂²⋊Q₈	—	2	2	no	—
325	C₂²⋊Q₈	—	2	2	no	—
326	C₂²⋊Q₈	—	2	2	no	—
327	C₂²⋊Q₈	—	2	2	no	—
328	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
329	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	—	2	2	no	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	—	2	2	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
337	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
338	#117	—	2	2	no	—
339	#117	—	2	2	no	—
340	#117	—	2	2	no	—
341	#117	—	2	2	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	2	no	—
347	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₄²	—	2	2	no	—
350	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
351	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
352	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
353	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
354	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
355	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
356	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
357	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
358	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
359	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
362	C₄⋊C₈	—	2	2	no	—
363	C₄⋊C₈	—	2	2	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	2	no	—
366	C₄⋊C₈	—	2	2	no	—
367	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
368	C₂²⋊C₈	—	2	2	no	—
369	C₂²⋊C₈	—	2	2	no	—
370	C₄×C₈	—	2	2	no	—
371	C₄×C₈	C₂³×C₂₃	1	4	yes	—
372	C₂₃	#17486	1	1	yes	characteristic, direct product factor, Sylow
373	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	4	4	no	—
378	C₂×Q₈	—	4	4	no	—
379	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
385	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
386	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
387	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
388	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
389	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
390	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
394	C₂²×C₄	Q₁₆×C₂₃	1	4	yes	—
395	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
404	M₁₆	—	4	4	no	—
405	M₁₆	—	4	4	no	—
406	C₂×C₈	—	2	4	no	—
407	C₂×C₈	—	2	4	no	—
408	C₂×C₈	—	4	4	no	—
409	C₂×C₈	—	4	4	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₄²	—	2	4	no	—
425	C₄²	—	2	4	no	—
426	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
427	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
428	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
429	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
430	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
431	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
432	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
433	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
434	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
435	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
440	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
442	C₂³	—	4	1	no	—
443	C₂³	—	4	1	no	—
444	C₂³	C₂₃×#103	1	4	yes	—
445	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
446	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
447	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
448	Q₈	—	2	8	no	—
449	Q₈	—	4	4	no	—
450	Q₈	—	4	4	no	—
451	Q₈	—	4	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	4	no	—
468	C₂×C₄	C₂₃×#103	1	4	yes	—
469	C₂×C₄	C₂₃×#103	1	4	yes	—
470	C₂×C₄	C₂₃×#103	1	4	yes	—
471	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
472	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
473	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
474	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
475	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
476	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
477	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
478	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
480	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
481	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
482	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
483	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
484	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
485	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
486	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
488	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
489	C₈	—	4	4	no	—
490	C₈	—	4	4	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	4	4	no	—
498	C₂²	C₂×C₂₃×#103	1	2	yes	—
499	C₂²	C₂×C₂₃×#103	1	2	yes	—
500	C₂²	C₂×C₂₃×#103	1	2	yes	—
501	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
502	C₄	—	2	1	no	—
503	C₄	—	2	1	no	—
504	C₄	—	2	1	no	—
505	C₄	—	2	1	no	—
506	C₄	—	2	1	no	—
507	C₄	—	2	1	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	4	no	—
510	C₄	C₂×C₂₃×#103	1	2	yes	—
511	C₄	C₂×C₂₃×#103	1	2	yes	—
512	C₄	C₂×C₂₃×#103	1	2	yes	—
513	C₄	C₂₃×#421	1	2	yes	—
514	C₄	C₂₃×#568	1	2	yes	—
515	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
516	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
517	C₂	—	2	2	no	—
518	C₂	—	2	2	no	—
519	C₂	—	2	2	no	—
520	C₂	C₂₃×#2879	1	1	yes	characteristic
521	C₂	C₂²×C₂₃×#103	1	1	yes	characteristic
522	C₂	C₂×C₂₃×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1493	C₂	semidirect
5. C₂₃×#1487	C₂	semidirect
6. C₂₃×#1481	C₂	semidirect
38. C₂₃×Q₈⋊C₈	C₂²	semidirect
180. #17486	C₂₃	direct (N × H)
233. #1493	C₂×C₂₃	semidirect
234. #1487	C₂×C₂₃	semidirect
235. #1481	C₂×C₂₃	semidirect
293. Q₈⋊C₈	C₂²×C₂₃	semidirect
372. C₂₃	#17486	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×#533	2
C₂×C₂₃×#431	2
C₂×C₂₃×#423	2
C₂₃×#1493	2
C₂₃×#1487	2
C₂₃×#1481	2
#17486	23

Subgroups — C23×#17486 #19165112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17486 #19165112