Subgroups — C₂₃×#11132 #19158758

Order 5888.

864 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2911	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#421	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2896	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#410	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1740	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
15	C₂₃×#428	—	2	2	no	—
16	C₂₃×#428	—	2	2	no	—
17	C₂₃×#427	—	2	2	no	—
18	C₂₃×#421	—	2	2	no	—
19	C₂₃×#421	—	2	2	no	—
20	C₂×C₂₃×#104	—	2	2	no	—
21	C₂₃×#419	C₄	1	2	yes	complemented
22	C₂₃×#418	C₄	1	2	yes	complemented
23	C₂₃×#417	—	2	2	no	—
24	C₂₃×#416	—	2	2	no	—
25	C₂₃×#410	—	2	2	no	—
26	C₂₃×#410	—	2	2	no	—
27	C₂×C₂₃×#99	—	2	2	no	—
28	C₂₃×#344	—	2	2	no	—
29	C₂₃×#342	—	2	2	no	—
30	C₂₃×H(3,ℤ/4)	—	2	2	no	—
31	C₂×#41×C₂₃	—	2	1	no	—
32	C₂×#41×C₂₃	—	2	1	no	—
33	C₂×#41×C₂₃	—	2	1	no	—
34	C₂×#41×C₂₃	—	2	1	no	—
35	C₂×#41×C₂₃	—	2	1	no	—
36	C₂×#41×C₂₃	—	2	1	no	—
37	C₂×#41×C₂₃	—	2	1	no	—
38	C₂×#41×C₂₃	—	2	1	no	—
39	C₂×#41×C₂₃	—	2	1	no	—
40	C₂×#41×C₂₃	—	2	1	no	—
41	C₂×#41×C₂₃	D₄	1	2	yes	—
42	C₂×#41×C₂₃	Q₈	1	2	yes	—
43	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₂₃	—	2	1	no	—
45	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	—	2	1	no	—
47	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₂₃	—	2	1	no	—
49	C₂×M₁₆×C₂₃	—	2	1	no	—
50	C₂×M₁₆×C₂₃	—	2	1	no	—
51	C₂×M₁₆×C₂₃	—	2	1	no	—
52	C₂×M₁₆×C₂₃	—	2	1	no	—
53	C₂×M₁₆×C₂₃	—	2	1	no	—
54	C₂×M₁₆×C₂₃	—	2	1	no	—
55	C₂×M₁₆×C₂₃	—	2	1	no	—
56	C₂×M₁₆×C₂₃	—	2	1	no	—
57	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
59	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
60	C₂²×C₈×C₂₃	—	2	1	no	—
61	C₂²×C₈×C₂₃	D₄	1	2	yes	—
62	C₂₃×#117	—	2	1	no	—
63	C₂₃×#117	—	2	1	no	—
64	C₂₃×#117	—	2	1	no	—
65	C₂₃×#117	—	2	1	no	—
66	C₂₃×#117	—	2	1	no	—
67	C₂₃×#117	—	2	1	no	—
68	C₂₃×#117	—	2	1	no	—
69	C₂₃×#117	—	2	1	no	—
70	C₂₃×#117	—	2	1	no	—
71	C₂₃×#117	—	2	1	no	—
72	C₂₃×#117	—	2	1	no	—
73	C₂₃×#117	—	2	1	no	—
74	C₂₃×#117	—	2	1	no	—
75	C₂₃×#117	C₂×C₄	1	2	yes	—
76	C₂₃×#117	D₄	1	2	yes	—
77	C₂₃×#117	D₄	1	2	yes	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	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×C₂²⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
86	C₂×C₄²×C₂₃	—	2	1	no	—
87	C₂×C₄²×C₂₃	—	2	1	no	—
88	C₂×C₄²×C₂₃	—	2	1	no	—
89	C₂×C₄²×C₂₃	—	2	1	no	—
90	C₂₃×#107	—	2	4	no	—
91	C₂₃×#106	—	2	4	no	—
92	C₂₃×#104	—	2	8	no	—
93	C₂₃×#103	—	2	4	no	—
94	C₂₃×#102	—	2	4	no	—
95	C₂₃×#99	—	2	8	no	—
96	C₂⁴×C₂₃	—	2	1	no	—
97	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
98	#41×C₂₃	—	2	2	no	—
99	#41×C₂₃	—	2	2	no	—
100	#41×C₂₃	—	2	2	no	—
101	#41×C₂₃	—	2	2	no	—
102	#41×C₂₃	—	2	2	no	—
103	#41×C₂₃	—	2	2	no	—
104	#41×C₂₃	—	2	2	no	—
105	#41×C₂₃	—	2	2	no	—
106	#41×C₂₃	—	4	1	no	—
107	#41×C₂₃	—	4	1	no	—
108	#41×C₂₃	—	4	1	no	—
109	#41×C₂₃	—	4	1	no	—
110	#41×C₂₃	—	4	1	no	—
111	#41×C₂₃	—	4	1	no	—
112	#41×C₂₃	—	4	1	no	—
113	#41×C₂₃	—	4	1	no	—
114	C₂×Q₈×C₂₃	—	2	1	no	—
115	C₂×Q₈×C₂₃	—	2	1	no	—
116	C₂×Q₈×C₂₃	—	2	1	no	—
117	C₂×Q₈×C₂₃	—	2	1	no	—
118	C₂×Q₈×C₂₃	—	2	1	no	—
119	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
120	C₂×D₄×C₂₃	—	2	2	no	—
121	C₂×D₄×C₂₃	—	2	2	no	—
122	C₂×D₄×C₂₃	—	2	1	no	—
123	C₂×D₄×C₂₃	—	2	1	no	—
124	C₂×D₄×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	2	1	no	—
126	C₂×D₄×C₂₃	—	2	1	no	—
127	C₂×D₄×C₂₃	—	4	1	no	—
128	C₂×D₄×C₂₃	—	4	1	no	—
129	C₂×D₄×C₂₃	—	4	1	no	—
130	C₂×D₄×C₂₃	—	4	1	no	—
131	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	2	2	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	4	1	no	—
137	C₂²×C₄×C₂₃	—	4	1	no	—
138	C₂²×C₄×C₂₃	—	4	1	no	—
139	C₂²×C₄×C₂₃	—	4	1	no	—
140	C₂²×C₄×C₂₃	—	4	1	no	—
141	C₂²×C₄×C₂₃	—	4	1	no	—
142	C₂²×C₄×C₂₃	—	4	1	no	—
143	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
157	M₁₆×C₂₃	—	2	2	no	—
158	M₁₆×C₂₃	—	2	2	no	—
159	M₁₆×C₂₃	—	2	2	no	—
160	M₁₆×C₂₃	—	2	2	no	—
161	M₁₆×C₂₃	—	2	2	no	—
162	M₁₆×C₂₃	—	2	2	no	—
163	M₁₆×C₂₃	—	4	1	no	—
164	M₁₆×C₂₃	—	4	1	no	—
165	C₂×C₈×C₂₃	—	2	2	no	—
166	C₂×C₈×C₂₃	—	2	2	no	—
167	C₂×C₈×C₂₃	—	2	2	no	—
168	C₂×C₈×C₂₃	—	4	1	no	—
169	C₂×C₈×C₂₃	—	4	1	no	—
170	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
171	C₄⋊C₄×C₂₃	—	2	2	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	2	no	—
174	C₄⋊C₄×C₂₃	—	4	1	no	—
175	C₄⋊C₄×C₂₃	—	4	1	no	—
176	C₄⋊C₄×C₂₃	—	4	1	no	—
177	C₄⋊C₄×C₂₃	—	4	1	no	—
178	C₄⋊C₄×C₂₃	—	4	1	no	—
179	C₂²⋊C₄×C₂₃	—	2	2	no	—
180	C₂²⋊C₄×C₂₃	—	2	2	no	—
181	C₂²⋊C₄×C₂₃	—	2	2	no	—
182	C₂²⋊C₄×C₂₃	—	2	2	no	—
183	C₂²⋊C₄×C₂₃	—	2	2	no	—
184	C₂²⋊C₄×C₂₃	—	4	1	no	—
185	C₂²⋊C₄×C₂₃	—	4	1	no	—
186	C₂²⋊C₄×C₂₃	—	4	1	no	—
187	C₄²×C₂₃	—	2	2	no	—
188	C₄²×C₂₃	—	2	2	no	—
189	C₄²×C₂₃	—	2	2	no	—
190	C₄²×C₂₃	—	4	1	no	—
191	C₄²×C₂₃	—	4	1	no	—
192	C₄²×C₂₃	—	4	1	no	—
193	C₄²×C₂₃	—	4	1	no	—
194	C₄²×C₂₃	—	4	1	no	—
195	#11132	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₂₃	—	2	2	no	—
197	C₂³×C₂₃	—	4	1	no	—
198	C₂³×C₂₃	—	4	1	no	—
199	C₂³×C₂₃	—	8	1	no	—
200	C₂³×C₂₃	—	8	2	no	—
201	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
202	C₂³×C₂₃	#120	1	1	yes	characteristic
203	C₂³×C₂₃	#121	1	1	yes	characteristic
204	C₂³×C₂₃	#121	1	1	yes	characteristic
205	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
206	C₂³×C₂₃	#123	1	1	yes	characteristic
207	C₂³×C₂₃	#127	1	1	yes	characteristic
208	Q₈×C₂₃	—	2	2	no	—
209	Q₈×C₂₃	—	2	2	no	—
210	Q₈×C₂₃	—	4	1	no	—
211	Q₈×C₂₃	—	4	1	no	—
212	D₄×C₂₃	—	2	2	no	—
213	D₄×C₂₃	—	2	2	no	—
214	D₄×C₂₃	—	4	2	no	—
215	D₄×C₂₃	—	4	2	no	—
216	D₄×C₂₃	—	4	2	no	—
217	D₄×C₂₃	—	4	2	no	—
218	D₄×C₂₃	—	4	1	no	—
219	D₄×C₂₃	—	4	1	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	4	1	no	—
222	C₂×C₄×C₂₃	—	4	1	no	—
223	C₂×C₄×C₂₃	—	4	2	no	—
224	C₂×C₄×C₂₃	—	4	2	no	—
225	C₂×C₄×C₂₃	—	4	2	no	—
226	C₂×C₄×C₂₃	—	4	2	no	—
227	C₂×C₄×C₂₃	—	4	2	no	—
228	C₂×C₄×C₂₃	—	4	2	no	—
229	C₂×C₄×C₂₃	—	4	2	no	—
230	C₂×C₄×C₂₃	—	4	2	no	—
231	C₂×C₄×C₂₃	—	8	2	no	—
232	C₂×C₄×C₂₃	—	8	2	no	—
233	C₂×C₄×C₂₃	—	8	2	no	—
234	C₂×C₄×C₂₃	—	8	2	no	—
235	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
238	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
239	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
240	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
241	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
242	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
243	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
244	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
245	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
246	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
247	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
248	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
255	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
256	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
257	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
258	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
259	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
260	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
261	C₂×C₄×C₂₃	#126	1	1	yes	characteristic
262	C₂×C₄×C₂₃	#126	1	1	yes	characteristic
263	C₈×C₂₃	—	2	4	no	—
264	C₈×C₂₃	—	4	2	no	—
265	#2911	C₂×C₂₃	1	1	yes	characteristic, complemented
266	C₂×#421	C₂×C₂₃	1	1	yes	characteristic
267	#2896	C₂×C₂₃	1	1	yes	characteristic
268	C₂×#410	C₂×C₂₃	1	1	yes	characteristic
269	C₂×#344	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1740	C₂×C₂₃	1	1	yes	characteristic, complemented
271	C₂×H(3,ℤ/4)	C₂×C₂₃	1	1	yes	characteristic, complemented
272	C₂²×C₂₃	—	4	1	no	—
273	C₂²×C₂₃	—	4	1	no	—
274	C₂²×C₂₃	—	4	1	no	—
275	C₂²×C₂₃	—	4	1	no	—
276	C₂²×C₂₃	—	4	1	no	—
277	C₂²×C₂₃	—	4	1	no	—
278	C₂²×C₂₃	—	4	1	no	—
279	C₂²×C₂₃	—	4	2	no	—
280	C₂²×C₂₃	—	8	2	no	—
281	C₂²×C₂₃	—	8	2	no	—
282	C₂²×C₂₃	—	8	2	no	—
283	C₂²×C₂₃	#385	1	1	yes	characteristic
284	C₂²×C₂₃	#388	1	1	yes	characteristic
285	C₂²×C₂₃	#390	1	1	yes	characteristic
286	C₂²×C₂₃	#392	1	1	yes	characteristic
287	C₂²×C₂₃	#394	1	1	yes	characteristic
288	C₂²×C₂₃	#396	1	1	yes	characteristic
289	C₂²×C₂₃	#399	1	1	yes	characteristic
290	C₄×C₂₃	—	4	2	no	—
291	C₄×C₂₃	—	8	2	no	—
292	C₄×C₂₃	—	8	2	no	—
293	C₄×C₂₃	—	8	2	no	—
294	C₄×C₂₃	—	8	2	no	—
295	C₄×C₂₃	#380	1	1	yes	characteristic
296	C₄×C₂₃	#385	1	1	yes	characteristic
297	C₄×C₂₃	#386	1	1	yes	characteristic
298	C₄×C₂₃	#388	1	1	yes	characteristic
299	C₄×C₂₃	#393	1	1	yes	characteristic
300	C₄×C₂₃	#394	1	1	yes	characteristic
301	C₄×C₂₃	#396	1	1	yes	characteristic
302	C₄×C₂₃	#397	1	1	yes	characteristic
303	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
304	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
305	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
306	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
309	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
310	#428	—	2	2	no	—
311	#428	—	2	2	no	—
312	#427	—	2	2	no	—
313	#421	—	2	2	no	—
314	#421	—	2	2	no	—
315	C₂×#104	—	2	2	no	—
316	#419	C₄×C₂₃	1	2	yes	complemented
317	#418	C₄×C₂₃	1	2	yes	complemented
318	#417	—	2	2	no	—
319	#416	—	2	2	no	—
320	#410	—	2	2	no	—
321	#410	—	2	2	no	—
322	C₂×#99	—	2	2	no	—
323	#344	—	2	2	no	—
324	#342	—	2	2	no	—
325	H(3,ℤ/4)	—	2	2	no	—
326	C₂×C₂₃	—	2	1	no	—
327	C₂×C₂₃	—	2	1	no	—
328	C₂×C₂₃	—	2	1	no	—
329	C₂×C₂₃	—	2	1	no	—
330	C₂×C₂₃	—	2	1	no	—
331	C₂×C₂₃	—	2	1	no	—
332	C₂×C₂₃	—	2	1	no	—
333	C₂×C₂₃	—	8	2	no	—
334	C₂×C₂₃	#1439	1	1	yes	characteristic
335	C₂×#41	—	2	1	no	—
336	C₂×#41	—	2	1	no	—
337	C₂×#41	—	2	1	no	—
338	C₂×#41	—	2	1	no	—
339	C₂×#41	—	2	1	no	—
340	C₂×#41	—	2	1	no	—
341	C₂×#41	—	2	1	no	—
342	C₂×#41	—	2	1	no	—
343	C₂×#41	—	2	1	no	—
344	C₂×#41	—	2	1	no	—
345	C₂×#41	D₄×C₂₃	1	2	yes	—
346	C₂×#41	Q₈×C₂₃	1	2	yes	—
347	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
348	C₂²×D₄	—	2	1	no	—
349	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
350	C₂³×C₄	—	2	1	no	—
351	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	1	no	—
355	C₂×M₁₆	—	2	1	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	—	2	1	no	—
359	C₂×M₁₆	—	2	1	no	—
360	C₂×M₁₆	—	2	1	no	—
361	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
362	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
363	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
364	C₂²×C₈	—	2	1	no	—
365	C₂²×C₈	D₄×C₂₃	1	2	yes	—
366	#117	—	2	1	no	—
367	#117	—	2	1	no	—
368	#117	—	2	1	no	—
369	#117	—	2	1	no	—
370	#117	—	2	1	no	—
371	#117	—	2	1	no	—
372	#117	—	2	1	no	—
373	#117	—	2	1	no	—
374	#117	—	2	1	no	—
375	#117	—	2	1	no	—
376	#117	—	2	1	no	—
377	#117	—	2	1	no	—
378	#117	—	2	1	no	—
379	#117	C₂×C₄×C₂₃	1	2	yes	—
380	#117	D₄×C₂₃	1	2	yes	—
381	#117	D₄×C₂₃	1	2	yes	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	—	2	1	no	—
394	#107	—	2	4	no	—
395	#106	—	2	4	no	—
396	#104	—	2	8	no	—
397	#103	—	2	4	no	—
398	#102	—	2	4	no	—
399	#99	—	2	8	no	—
400	C₂₃	#11132	1	1	yes	characteristic, direct product factor, Sylow
401	C₂⁴	—	2	1	no	—
402	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
403	#41	—	2	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	2	no	—
407	#41	—	2	2	no	—
408	#41	—	2	2	no	—
409	#41	—	2	2	no	—
410	#41	—	2	2	no	—
411	#41	—	4	1	no	—
412	#41	—	4	1	no	—
413	#41	—	4	1	no	—
414	#41	—	4	1	no	—
415	#41	—	4	1	no	—
416	#41	—	4	1	no	—
417	#41	—	4	1	no	—
418	#41	—	4	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	4	1	no	—
433	C₂×D₄	—	4	1	no	—
434	C₂×D₄	—	4	1	no	—
435	C₂×D₄	—	4	1	no	—
436	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	4	1	no	—
442	C₂²×C₄	—	4	1	no	—
443	C₂²×C₄	—	4	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	—	4	1	no	—
448	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
457	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
458	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
459	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
460	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
461	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
462	M₁₆	—	2	2	no	—
463	M₁₆	—	2	2	no	—
464	M₁₆	—	2	2	no	—
465	M₁₆	—	2	2	no	—
466	M₁₆	—	2	2	no	—
467	M₁₆	—	2	2	no	—
468	M₁₆	—	4	1	no	—
469	M₁₆	—	4	1	no	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	4	1	no	—
474	C₂×C₈	—	4	1	no	—
475	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	4	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₄²	—	2	2	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	4	1	no	—
496	C₄²	—	4	1	no	—
497	C₄²	—	4	1	no	—
498	C₄²	—	4	1	no	—
499	C₄²	—	4	1	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	4	1	no	—
503	C₂³	—	8	1	no	—
504	C₂³	—	8	2	no	—
505	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
506	C₂³	C₂₃×#120	1	1	yes	characteristic
507	C₂³	C₂₃×#121	1	1	yes	characteristic
508	C₂³	C₂₃×#121	1	1	yes	characteristic
509	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
510	C₂³	C₂₃×#123	1	1	yes	characteristic
511	C₂³	C₂₃×#127	1	1	yes	characteristic
512	Q₈	—	2	2	no	—
513	Q₈	—	2	2	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	D₄	—	2	2	no	—
517	D₄	—	2	2	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	2	no	—
521	D₄	—	4	2	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	8	2	no	—
536	C₂×C₄	—	8	2	no	—
537	C₂×C₄	—	8	2	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
540	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
541	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
542	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
543	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
544	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
545	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
546	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
547	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
548	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
549	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
550	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
551	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
552	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
553	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
554	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
555	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
556	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
557	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
558	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
559	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
560	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
561	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
562	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
563	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
564	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
565	C₂×C₄	C₂₃×#126	1	1	yes	characteristic
566	C₂×C₄	C₂₃×#126	1	1	yes	characteristic
567	C₈	—	2	4	no	—
568	C₈	—	4	2	no	—
569	C₂²	—	4	1	no	—
570	C₂²	—	4	1	no	—
571	C₂²	—	4	1	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	8	2	no	—
578	C₂²	—	8	2	no	—
579	C₂²	—	8	2	no	—
580	C₂²	C₂₃×#385	1	1	yes	characteristic
581	C₂²	C₂₃×#388	1	1	yes	characteristic
582	C₂²	C₂₃×#390	1	1	yes	characteristic
583	C₂²	C₂₃×#392	1	1	yes	characteristic
584	C₂²	C₂₃×#394	1	1	yes	characteristic
585	C₂²	C₂₃×#396	1	1	yes	characteristic
586	C₂²	C₂₃×#399	1	1	yes	characteristic
587	C₄	—	4	2	no	—
588	C₄	—	8	2	no	—
589	C₄	—	8	2	no	—
590	C₄	—	8	2	no	—
591	C₄	—	8	2	no	—
592	C₄	C₂₃×#380	1	1	yes	characteristic
593	C₄	C₂₃×#385	1	1	yes	characteristic
594	C₄	C₂₃×#386	1	1	yes	characteristic
595	C₄	C₂₃×#388	1	1	yes	characteristic
596	C₄	C₂₃×#393	1	1	yes	characteristic
597	C₄	C₂₃×#394	1	1	yes	characteristic
598	C₄	C₂₃×#396	1	1	yes	characteristic
599	C₄	C₂₃×#397	1	1	yes	characteristic
600	C₂	—	2	1	no	—
601	C₂	—	2	1	no	—
602	C₂	—	2	1	no	—
603	C₂	—	2	1	no	—
604	C₂	—	2	1	no	—
605	C₂	—	2	1	no	—
606	C₂	—	2	1	no	—
607	C₂	—	8	2	no	—
608	C₂	C₂₃×#1439	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#397
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂×C₂₃	#1439
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2911	C₂	semidirect
5. C₂×C₂₃×#344	C₂	semidirect
6. C₂₃×#1740	C₂	semidirect
7. C₂×C₂₃×H(3,ℤ/4)	C₂	semidirect
21. C₂₃×#419	C₄	semidirect
22. C₂₃×#418	C₄	semidirect
195. #11132	C₂₃	direct (N × H)
265. #2911	C₂×C₂₃	semidirect
269. C₂×#344	C₂×C₂₃	semidirect
270. #1740	C₂×C₂₃	semidirect
271. C₂×H(3,ℤ/4)	C₂×C₂₃	semidirect
316. #419	C₄×C₂₃	semidirect
317. #418	C₄×C₂₃	semidirect
400. C₂₃	#11132	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2911	2
C₂×C₂₃×#421	2
C₂₃×#2896	2
C₂×C₂₃×#410	2
C₂×C₂₃×#344	2
C₂₃×#1740	2
C₂×C₂₃×H(3,ℤ/4)	2
#11132	23

Subgroups — C23×#11132 #19158758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11132 #19158758