Subgroups — C₂₃×#17717 #19165343

Order 5888.

1252 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1553	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1549	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1545	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1540	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₂₃	—	2	1	no	—
9	C₂₃×#532	—	2	1	no	—
10	C₂₃×#532	—	2	1	no	—
11	C₂₃×#532	—	2	1	no	—
12	C₂₃×#532	—	2	1	no	—
13	C₂₃×#532	C₄	1	2	yes	—
14	C₂₃×#532	C₄	1	2	yes	—
15	C₂₃×#532	C₂²	1	2	yes	—
16	C₂₃×#532	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#124	—	2	1	no	—
20	C₂×C₂₃×#121	—	2	1	no	—
21	C₂×C₂₃×#121	C₄	1	2	yes	—
22	C₂×C₄×D₄×C₂₃	—	2	1	no	—
23	C₂×C₄×D₄×C₂₃	C₄	1	2	yes	—
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#432	—	2	1	no	—
26	C₂₃×#432	C₂²	1	2	yes	complemented
27	C₂₃×#431	—	2	1	no	—
28	C₂₃×#431	—	2	1	no	—
29	C₂₃×#431	—	2	1	no	—
30	C₂₃×#431	C₂²	1	2	yes	—
31	C₂₃×#431	C₂²	1	2	yes	complemented
32	C₂₃×#431	C₂²	1	2	yes	complemented
33	C₂₃×#423	C₂²	1	2	yes	—
34	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
35	C₂₃×#407	—	2	1	no	—
36	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
37	C₄×M₁₆×C₂₃	—	2	1	no	—
38	C₂×C₄×C₈×C₂₃	—	2	1	no	—
39	C₂₃×#332	C₂²	1	8	yes	complemented
40	C₂₃×#331	C₂²	1	8	yes	complemented
41	C₂×#41×C₂₃	—	2	2	no	—
42	C₂×#41×C₂₃	—	2	2	no	—
43	C₂×#41×C₂₃	—	2	2	no	—
44	C₂×#41×C₂₃	—	2	2	no	—
45	C₂×#41×C₂₃	—	4	8	no	—
46	C₂²×Q₈×C₂₃	—	2	1	no	—
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₂₃	—	2	2	no	—
52	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₂₃	—	2	2	no	—
56	C₂²×C₈×C₂₃	—	2	2	no	—
57	C₂₃×C₄⋊Q₈	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
60	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
61	C₂₃×#127	—	2	1	no	—
62	C₂₃×#127	—	2	1	no	—
63	C₂₃×#127	C₂×C₄	1	2	yes	—
64	C₂₃×#127	C₂³	1	2	yes	—
65	C₂₃×#124	—	2	2	no	—
66	C₂₃×#124	—	2	2	no	—
67	C₂₃×#124	—	2	2	no	—
68	C₂₃×#124	—	2	2	no	—
69	C₂₃×#121	—	2	2	no	—
70	C₂₃×#121	—	2	2	no	—
71	C₂₃×#121	—	2	2	no	—
72	C₂₃×#121	—	2	2	no	—
73	C₂₃×#121	—	2	2	no	—
74	C₂₃×#121	—	2	2	no	—
75	C₂₃×#121	C₂×C₄	1	4	yes	—
76	C₂₃×#121	C₂×C₄	1	4	yes	—
77	C₄×D₄×C₂₃	—	2	2	no	—
78	C₄×D₄×C₂₃	—	2	2	no	—
79	C₄×D₄×C₂₃	—	2	2	no	—
80	C₄×D₄×C₂₃	—	2	2	no	—
81	C₄×D₄×C₂₃	—	2	2	no	—
82	C₄×D₄×C₂₃	—	2	2	no	—
83	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
84	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
87	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₂²⋊C₄×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	C₂³	1	4	yes	—
104	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
105	C₂₃×C₂²⋊C₈	—	2	2	no	—
106	C₂₃×C₂²⋊C₈	—	2	2	no	—
107	C₂₃×C₂²⋊C₈	—	2	2	no	—
108	C₂₃×C₂²⋊C₈	—	2	2	no	—
109	C₂₃×#97	—	2	2	no	—
110	C₄×C₈×C₂₃	—	2	2	no	—
111	C₄×C₈×C₂₃	—	2	2	no	—
112	C₄×C₈×C₂₃	—	2	2	no	—
113	C₂⁴×C₂₃	—	2	2	no	—
114	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
115	#41×C₂₃	—	4	8	no	—
116	#41×C₂₃	—	4	8	no	—
117	#41×C₂₃	—	4	8	no	—
118	#41×C₂₃	—	4	8	no	—
119	C₂×Q₈×C₂₃	—	2	2	no	—
120	C₂×Q₈×C₂₃	—	2	2	no	—
121	C₂×Q₈×C₂₃	—	4	4	no	—
122	C₂×D₄×C₂₃	—	2	4	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	2	no	—
125	C₂×D₄×C₂₃	—	2	2	no	—
126	C₂×D₄×C₂₃	—	2	2	no	—
127	C₂×D₄×C₂₃	—	2	2	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	4	4	no	—
130	C₂×D₄×C₂₃	—	4	4	no	—
131	C₂×D₄×C₂₃	—	4	8	no	—
132	C₂×D₄×C₂₃	—	4	4	no	—
133	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	4	no	—
138	C₂²×C₄×C₂₃	—	2	4	no	—
139	C₂²×C₄×C₂₃	—	4	2	no	—
140	C₂²×C₄×C₂₃	—	4	8	no	—
141	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	M₁₆×C₂₃	—	4	4	no	—
160	C₂×C₈×C₂₃	—	2	4	no	—
161	C₂×C₈×C₂₃	—	2	4	no	—
162	C₂×C₈×C₂₃	—	4	4	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	—	2	2	no	—
166	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
167	C₂²⋊C₄×C₂₃	—	2	4	no	—
168	C₂²⋊C₄×C₂₃	—	2	4	no	—
169	C₂²⋊C₄×C₂₃	—	2	4	no	—
170	C₂²⋊C₄×C₂₃	—	2	4	no	—
171	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
172	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
187	#17717	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₂₃	—	2	2	no	—
189	C₂³×C₂₃	—	2	2	no	—
190	C₂³×C₂₃	—	2	4	no	—
191	C₂³×C₂₃	—	4	1	no	—
192	C₂³×C₂₃	—	4	4	no	—
193	C₂³×C₂₃	—	4	4	no	—
194	C₂³×C₂₃	—	4	4	no	—
195	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	Q₈×C₂₃	—	4	4	no	—
199	Q₈×C₂₃	—	4	4	no	—
200	D₄×C₂₃	—	4	4	no	—
201	D₄×C₂₃	—	4	4	no	—
202	D₄×C₂₃	—	4	8	no	—
203	D₄×C₂₃	—	4	8	no	—
204	D₄×C₂₃	—	4	4	no	—
205	D₄×C₂₃	—	4	4	no	—
206	D₄×C₂₃	—	4	4	no	—
207	D₄×C₂₃	—	4	4	no	—
208	C₂×C₄×C₂₃	—	2	2	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	2	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	4	no	—
222	C₂×C₄×C₂₃	—	4	1	no	—
223	C₂×C₄×C₂₃	—	4	1	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	8	no	—
226	C₂×C₄×C₂₃	—	4	4	no	—
227	C₂×C₄×C₂₃	—	4	8	no	—
228	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₂₃	#135	1	4	yes	—
238	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
242	C₈×C₂₃	—	4	4	no	—
243	C₈×C₂₃	—	4	4	no	—
244	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
245	#2920	C₂×C₂₃	1	1	yes	characteristic, complemented
246	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
247	#1553	C₂×C₂₃	1	4	yes	complemented
248	#1549	C₂×C₂₃	1	2	yes	complemented
249	#1545	C₂×C₂₃	1	2	yes	complemented
250	#1540	C₂×C₂₃	1	4	yes	complemented
251	C₂²×C₂₃	—	2	1	no	—
252	C₂²×C₂₃	—	2	1	no	—
253	C₂²×C₂₃	—	2	1	no	—
254	C₂²×C₂₃	—	2	1	no	—
255	C₂²×C₂₃	—	2	1	no	—
256	C₂²×C₂₃	—	2	1	no	—
257	C₂²×C₂₃	—	2	1	no	—
258	C₂²×C₂₃	—	4	4	no	—
259	C₂²×C₂₃	—	4	4	no	—
260	C₂²×C₂₃	—	4	4	no	—
261	C₂²×C₂₃	—	4	4	no	—
262	C₂²×C₂₃	—	8	8	no	—
263	C₂²×C₂₃	#416	1	2	yes	—
264	C₂²×C₂₃	#418	1	2	yes	—
265	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
266	C₄×C₂₃	—	2	1	no	—
267	C₄×C₂₃	—	2	1	no	—
268	C₄×C₂₃	—	2	1	no	—
269	C₄×C₂₃	—	2	1	no	—
270	C₄×C₂₃	—	2	1	no	—
271	C₄×C₂₃	—	2	1	no	—
272	C₄×C₂₃	—	2	1	no	—
273	C₄×C₂₃	—	4	4	no	—
274	C₄×C₂₃	#413	1	2	yes	—
275	C₄×C₂₃	#416	1	2	yes	—
276	C₄×C₂₃	#418	1	2	yes	—
277	C₄×C₂₃	C₂×#135	1	2	yes	—
278	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
279	C₂²×#41	—	2	1	no	—
280	#532	—	2	1	no	—
281	#532	—	2	1	no	—
282	#532	—	2	1	no	—
283	#532	—	2	1	no	—
284	#532	C₄×C₂₃	1	2	yes	—
285	#532	C₄×C₂₃	1	2	yes	—
286	#532	C₂²×C₂₃	1	2	yes	—
287	#532	C₂²×C₂₃	1	2	yes	—
288	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
289	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
290	C₂×#124	—	2	1	no	—
291	C₂×#121	—	2	1	no	—
292	C₂×#121	C₄×C₂₃	1	2	yes	—
293	C₂×C₄×D₄	—	2	1	no	—
294	C₂×C₄×D₄	C₄×C₂₃	1	2	yes	—
295	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
296	#432	—	2	1	no	—
297	#432	C₂²×C₂₃	1	2	yes	complemented
298	#431	—	2	1	no	—
299	#431	—	2	1	no	—
300	#431	—	2	1	no	—
301	#431	C₂²×C₂₃	1	2	yes	—
302	#431	C₂²×C₂₃	1	2	yes	complemented
303	#431	C₂²×C₂₃	1	2	yes	complemented
304	#423	C₂²×C₂₃	1	2	yes	—
305	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
306	#407	—	2	1	no	—
307	C₂×C₂²⋊C₈	—	2	1	no	—
308	C₄×M₁₆	—	2	1	no	—
309	C₂×C₄×C₈	—	2	1	no	—
310	#332	C₂²×C₂₃	1	8	yes	complemented
311	#331	C₂²×C₂₃	1	8	yes	complemented
312	C₂×C₂₃	—	2	2	no	—
313	C₂×C₂₃	—	2	2	no	—
314	C₂×C₂₃	—	4	1	no	—
315	C₂×C₂₃	—	8	4	no	—
316	C₂×C₂₃	C₂×#413	1	1	yes	characteristic
317	C₂×C₂₃	C₂×#416	1	1	yes	characteristic
318	C₂×C₂₃	C₂×#418	1	1	yes	characteristic
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	2	2	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	4	8	no	—
324	C₂²×Q₈	—	2	1	no	—
325	C₂²×D₄	—	2	2	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
329	C₂³×C₄	—	2	2	no	—
330	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
331	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
332	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
333	C₂×M₁₆	—	2	2	no	—
334	C₂²×C₈	—	2	2	no	—
335	C₄⋊Q₈	—	2	1	no	—
336	C₄⋊Q₈	—	2	1	no	—
337	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
338	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
339	#127	—	2	1	no	—
340	#127	—	2	1	no	—
341	#127	C₂×C₄×C₂₃	1	2	yes	—
342	#127	C₂³×C₂₃	1	2	yes	—
343	#124	—	2	2	no	—
344	#124	—	2	2	no	—
345	#124	—	2	2	no	—
346	#124	—	2	2	no	—
347	#121	—	2	2	no	—
348	#121	—	2	2	no	—
349	#121	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	C₂×C₄×C₂₃	1	4	yes	—
354	#121	C₂×C₄×C₂₃	1	4	yes	—
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
362	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₂²⋊C₄	—	2	2	no	—
367	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
368	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
369	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
370	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
371	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
372	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
373	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
374	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
375	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
379	C₄⋊C₈	—	2	2	no	—
380	C₄⋊C₈	—	2	2	no	—
381	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
382	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
383	C₂²⋊C₈	—	2	2	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	C₂²⋊C₈	—	2	2	no	—
387	#97	—	2	2	no	—
388	C₄×C₈	—	2	2	no	—
389	C₄×C₈	—	2	2	no	—
390	C₄×C₈	—	2	2	no	—
391	C₂₃	#17717	1	1	yes	characteristic, direct product factor, Sylow
392	C₂⁴	—	2	2	no	—
393	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
394	#41	—	4	8	no	—
395	#41	—	4	8	no	—
396	#41	—	4	8	no	—
397	#41	—	4	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	4	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	4	4	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	8	no	—
411	C₂×D₄	—	4	4	no	—
412	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	4	8	no	—
420	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
421	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
422	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
423	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
424	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
425	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
427	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
428	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
429	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
438	M₁₆	—	4	4	no	—
439	C₂×C₈	—	2	4	no	—
440	C₂×C₈	—	2	4	no	—
441	C₂×C₈	—	4	4	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
451	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
452	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
453	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
454	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
455	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
456	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
457	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
458	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
462	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
464	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
465	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	4	1	no	—
470	C₂³	—	4	4	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	4	4	no	—
473	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
474	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
475	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
476	Q₈	—	4	4	no	—
477	Q₈	—	4	4	no	—
478	D₄	—	4	4	no	—
479	D₄	—	4	4	no	—
480	D₄	—	4	8	no	—
481	D₄	—	4	8	no	—
482	D₄	—	4	4	no	—
483	D₄	—	4	4	no	—
484	D₄	—	4	4	no	—
485	D₄	—	4	4	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
507	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
508	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
509	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
510	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
511	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
512	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
513	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
514	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
515	C₂×C₄	C₂₃×#135	1	4	yes	—
516	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
517	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
518	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
520	C₈	—	4	4	no	—
521	C₈	—	4	4	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	8	8	no	—
534	C₂²	C₂₃×#416	1	2	yes	—
535	C₂²	C₂₃×#418	1	2	yes	—
536	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
537	C₄	—	2	1	no	—
538	C₄	—	2	1	no	—
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	4	4	no	—
545	C₄	C₂₃×#413	1	2	yes	—
546	C₄	C₂₃×#416	1	2	yes	—
547	C₄	C₂₃×#418	1	2	yes	—
548	C₄	C₂×C₂₃×#135	1	2	yes	—
549	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
550	C₂	—	2	2	no	—
551	C₂	—	2	2	no	—
552	C₂	—	4	1	no	—
553	C₂	—	8	4	no	—
554	C₂	C₂×C₂₃×#413	1	1	yes	characteristic
555	C₂	C₂×C₂₃×#416	1	1	yes	characteristic
556	C₂	C₂×C₂₃×#418	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
prime radical	C₂×C₂₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
2. C₂₃×#2920	C₂	semidirect
3. C₂×C₂₃×#431	C₂	semidirect
4. C₂₃×#1553	C₂	semidirect
5. C₂₃×#1549	C₂	semidirect
6. C₂₃×#1545	C₂	semidirect
7. C₂₃×#1540	C₂	semidirect
26. C₂₃×#432	C₂²	semidirect
31. C₂₃×#431	C₂²	semidirect
32. C₂₃×#431	C₂²	semidirect
34. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
39. C₂₃×#332	C₂²	semidirect
40. C₂₃×#331	C₂²	semidirect
187. #17717	C₂₃	direct (N × H)
245. #2920	C₂×C₂₃	semidirect
246. C₂×#431	C₂×C₂₃	semidirect
247. #1553	C₂×C₂₃	semidirect
248. #1549	C₂×C₂₃	semidirect
249. #1545	C₂×C₂₃	semidirect
250. #1540	C₂×C₂₃	semidirect
297. #432	C₂²×C₂₃	semidirect
302. #431	C₂²×C₂₃	semidirect
303. #431	C₂²×C₂₃	semidirect
305. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
310. #332	C₂²×C₂₃	semidirect
311. #331	C₂²×C₂₃	semidirect
391. C₂₃	#17717	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×#532	2
C₂₃×#2920	2
C₂×C₂₃×#431	2
C₂₃×#1553	2
C₂₃×#1549	2
C₂₃×#1545	2
C₂₃×#1540	2
#17717	23

Subgroups — C23×#17717 #19165343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17717 #19165343