Subgroups — C₂×C₂₃×#3232 #19208466

Order 5888.

1822 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×#529	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3232	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#499	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#488	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#481	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#480	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×#479	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#478	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#529	—	2	4	no	—
14	C₂₃×#529	C₂²	1	8	yes	—
15	C₂×C₂₃×#126	—	2	1	no	—
16	C₂×C₂₃×#125	—	2	1	no	—
17	C₂×C₂₃×#125	—	2	1	no	—
18	C₂×C₂₃×#125	—	2	4	no	—
19	C₂×C₂₃×#125	C₂²	1	2	yes	—
20	C₂×C₂₃×#125	C₂²	1	2	yes	—
21	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#125	C₂²	1	8	yes	complemented
23	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#123	—	2	1	no	—
25	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₂₃	—	2	1	no	—
29	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#117	—	2	1	no	—
31	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#499	C₂²	1	16	yes	complemented
37	C₂₃×#488	C₂²	1	16	yes	complemented
38	C₂₃×#484	C₂²	1	8	yes	complemented
39	C₂₃×#483	C₂²	1	8	yes	complemented
40	C₂₃×#481	C₂²	1	8	yes	—
41	C₂₃×#480	C₂²	1	8	yes	—
42	C₂₃×#479	C₂²	1	16	yes	complemented
43	C₂₃×#478	C₂²	1	16	yes	complemented
44	C₂₃×#431	C₂²	1	8	yes	complemented
45	C₂×C₂₃×#107	C₂²	1	2	yes	—
46	C₂×C₂₃×#107	C₂²	1	2	yes	—
47	C₂×C₂₃×#106	C₂²	1	2	yes	—
48	C₂×C₂₃×#106	C₂²	1	2	yes	—
49	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
50	C₂×C₂₃×#103	C₂²	1	2	yes	—
51	C₂×C₂₃×#103	C₂²	1	2	yes	—
52	C₂×C₂₃×#102	C₂²	1	2	yes	—
53	C₂×C₂₃×#102	C₂²	1	2	yes	—
54	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
57	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₂₃	—	2	1	no	—
59	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	—	2	1	no	—
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
68	C₂₃×#126	—	2	8	no	—
69	C₂₃×#125	—	2	4	no	—
70	C₂₃×#125	—	2	4	no	—
71	C₂₃×#125	—	2	8	no	—
72	C₂₃×#125	—	2	8	no	—
73	C₂₃×#125	—	2	2	no	—
74	C₂₃×#125	C₂³	1	8	yes	—
75	C₂₃×#125	C₂³	1	8	yes	—
76	C₂₃×#125	C₂³	1	4	yes	—
77	C₂₃×#124	—	2	2	no	—
78	C₂₃×#124	C₂³	1	4	yes	—
79	C₂₃×#123	—	2	8	no	—
80	C₂₃×C₂²⋊Q₈	—	2	2	no	—
81	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
82	C₂₃×#121	—	2	2	no	—
83	C₂₃×#121	C₂³	1	4	yes	—
84	C₄×Q₈×C₂₃	—	2	2	no	—
85	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
86	C₄×D₄×C₂₃	—	2	8	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	C₂³	1	4	yes	—
89	C₂₃×#117	—	2	8	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₄²×C₂₃	—	2	1	no	—
116	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₂₃	D₄	1	4	yes	—
119	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
122	C₂₃×#107	C₂³	1	8	yes	—
123	C₂₃×#107	C₂³	1	8	yes	—
124	C₂₃×#106	C₂³	1	8	yes	—
125	C₂₃×#106	C₂³	1	8	yes	—
126	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
127	C₂₃×#103	C₂³	1	8	yes	—
128	C₂₃×#103	C₂³	1	8	yes	—
129	C₂₃×#102	C₂³	1	8	yes	—
130	C₂₃×#102	C₂³	1	8	yes	—
131	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
132	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
133	C₄×C₈×C₂₃	C₂³	1	8	yes	—
134	C₂⁴×C₂₃	—	2	1	no	—
135	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	4	no	—
138	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
139	C₂×D₄×C₂₃	—	2	4	no	—
140	C₂×D₄×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	4	no	—
142	C₂×D₄×C₂₃	—	4	4	no	—
143	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₂₃	—	2	4	no	—
145	C₂²×C₄×C₂₃	—	2	4	no	—
146	C₂²×C₄×C₂₃	—	2	4	no	—
147	C₂²×C₄×C₂₃	—	2	4	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	2	no	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	4	no	—
154	C₂²×C₄×C₂₃	—	4	2	no	—
155	C₂²×C₄×C₂₃	—	4	4	no	—
156	C₂²×C₄×C₂₃	—	4	4	no	—
157	C₂²×C₄×C₂₃	—	4	4	no	—
158	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
168	C₂×C₈×C₂₃	—	2	4	no	—
169	C₂×C₈×C₂₃	—	2	4	no	—
170	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
171	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
172	C₄⋊C₄×C₂₃	—	2	8	no	—
173	C₄⋊C₄×C₂₃	—	2	8	no	—
174	C₄⋊C₄×C₂₃	—	2	8	no	—
175	C₄⋊C₄×C₂₃	—	2	8	no	—
176	C₄⋊C₄×C₂₃	—	2	4	no	—
177	C₄⋊C₄×C₂₃	—	2	4	no	—
178	C₄⋊C₄×C₂₃	—	2	2	no	—
179	C₄⋊C₄×C₂₃	—	2	2	no	—
180	C₄⋊C₄×C₂₃	—	2	4	no	—
181	C₄⋊C₄×C₂₃	—	2	4	no	—
182	C₄⋊C₄×C₂₃	—	2	4	no	—
183	C₄⋊C₄×C₂₃	—	2	4	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
189	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
190	C₂²⋊C₄×C₂₃	—	2	4	no	—
191	C₂²⋊C₄×C₂₃	—	2	4	no	—
192	C₂²⋊C₄×C₂₃	—	2	4	no	—
193	C₂²⋊C₄×C₂₃	—	2	4	no	—
194	C₂²⋊C₄×C₂₃	—	2	4	no	—
195	C₄²×C₂₃	—	2	4	no	—
196	C₄²×C₂₃	C₂×D₄	1	4	yes	—
197	C₄²×C₂₃	C₂×D₄	1	2	yes	—
198	C₄²×C₂₃	C₂×D₄	1	2	yes	—
199	C₄²×C₂₃	C₂×D₄	1	4	yes	—
200	C₄²×C₂₃	C₂⁴	1	2	yes	—
201	C₄²×C₂₃	C₂⁴	1	2	yes	—
202	C₂×#3232	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂³×C₂₃	—	2	2	no	—
204	C₂³×C₂₃	—	2	2	no	—
205	C₂³×C₂₃	—	2	2	no	—
206	C₂³×C₂₃	—	4	2	no	—
207	C₂³×C₂₃	—	4	2	no	—
208	C₂³×C₂₃	—	4	2	no	—
209	C₂³×C₂₃	#135	1	4	yes	—
210	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
211	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₂₃	—	2	4	no	—
213	Q₈×C₂₃	—	4	2	no	—
214	D₄×C₂₃	—	2	4	no	—
215	D₄×C₂₃	—	4	8	no	—
216	D₄×C₂₃	—	4	2	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	2	8	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	4	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	2	4	no	—
230	C₂×C₄×C₂₃	—	4	4	no	—
231	C₂×C₄×C₂₃	—	4	8	no	—
232	C₂×C₄×C₂₃	—	4	8	no	—
233	C₂×C₄×C₂₃	—	4	2	no	—
234	C₂×C₄×C₂₃	—	4	8	no	—
235	C₂×C₄×C₂₃	—	4	4	no	—
236	C₂×C₄×C₂₃	—	4	2	no	—
237	C₂×C₄×C₂₃	—	4	2	no	—
238	C₂×C₄×C₂₃	—	4	4	no	—
239	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
246	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
247	C₈×C₂₃	—	2	4	no	—
248	C₈×C₂₃	—	2	4	no	—
249	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
250	C₂²×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
251	#3232	C₂×C₂₃	1	16	yes	direct product factor
252	C₂×#499	C₂×C₂₃	1	2	yes	complemented
253	C₂×#488	C₂×C₂₃	1	2	yes	complemented
254	C₂×#484	C₂×C₂₃	1	1	yes	characteristic, complemented
255	C₂×#481	C₂×C₂₃	1	1	yes	characteristic
256	C₂×#483	C₂×C₂₃	1	1	yes	characteristic, complemented
257	C₂×#480	C₂×C₂₃	1	1	yes	characteristic
258	C₂×#479	C₂×C₂₃	1	2	yes	complemented
259	C₂×#478	C₂×C₂₃	1	2	yes	complemented
260	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
261	C₂²×C₂₃	—	2	2	no	—
262	C₂²×C₂₃	—	2	8	no	—
263	C₂²×C₂₃	—	2	2	no	—
264	C₂²×C₂₃	—	4	2	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	—	4	2	no	—
267	C₂²×C₂₃	—	8	4	no	—
268	C₂²×C₂₃	#536	1	2	yes	—
269	C₂²×C₂₃	C₂×#135	1	4	yes	—
270	C₂²×C₂₃	C₂×#135	1	2	yes	—
271	C₂²×C₂₃	#575	1	2	yes	—
272	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
273	C₄×C₂₃	—	2	2	no	—
274	C₄×C₂₃	—	2	2	no	—
275	C₄×C₂₃	—	2	2	no	—
276	C₄×C₂₃	—	2	2	no	—
277	C₄×C₂₃	—	2	2	no	—
278	C₄×C₂₃	—	4	2	no	—
279	C₄×C₂₃	—	4	4	no	—
280	C₄×C₂₃	—	4	2	no	—
281	C₄×C₂₃	—	4	2	no	—
282	C₄×C₂₃	—	4	4	no	—
283	C₄×C₂₃	#536	1	2	yes	—
284	C₄×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
285	#529	—	2	4	no	—
286	#529	C₂²×C₂₃	1	8	yes	—
287	C₂×#126	—	2	1	no	—
288	C₂×#125	—	2	1	no	—
289	C₂×#125	—	2	1	no	—
290	C₂×#125	—	2	4	no	—
291	C₂×#125	C₂²×C₂₃	1	2	yes	—
292	C₂×#125	C₂²×C₂₃	1	2	yes	—
293	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
294	C₂×#125	C₂²×C₂₃	1	8	yes	complemented
295	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×#123	—	2	1	no	—
297	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
298	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
299	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
300	C₂×C₄×D₄	—	2	1	no	—
301	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×#117	—	2	1	no	—
303	C₂²×C₄⋊C₄	—	2	1	no	—
304	C₂²×C₄⋊C₄	—	2	1	no	—
305	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
306	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
307	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
308	#499	C₂²×C₂₃	1	16	yes	complemented
309	#488	C₂²×C₂₃	1	16	yes	complemented
310	#484	C₂²×C₂₃	1	8	yes	complemented
311	#483	C₂²×C₂₃	1	8	yes	complemented
312	#481	C₂²×C₂₃	1	8	yes	—
313	#480	C₂²×C₂₃	1	8	yes	—
314	#479	C₂²×C₂₃	1	16	yes	complemented
315	#478	C₂²×C₂₃	1	16	yes	complemented
316	#431	C₂²×C₂₃	1	8	yes	complemented
317	C₂×#107	C₂²×C₂₃	1	2	yes	—
318	C₂×#107	C₂²×C₂₃	1	2	yes	—
319	C₂×#106	C₂²×C₂₃	1	2	yes	—
320	C₂×#106	C₂²×C₂₃	1	2	yes	—
321	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
322	C₂×#103	C₂²×C₂₃	1	2	yes	—
323	C₂×#103	C₂²×C₂₃	1	2	yes	—
324	C₂×#102	C₂²×C₂₃	1	2	yes	—
325	C₂×#102	C₂²×C₂₃	1	2	yes	—
326	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
329	C₂×C₂₃	—	2	4	no	—
330	C₂×C₂₃	—	8	2	no	—
331	C₂×C₂₃	#3232	1	4	yes	direct product factor
332	C₂×C₂₃	C₂×#536	1	1	yes	characteristic
333	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
334	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
335	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
336	C₂²×D₄	—	2	1	no	—
337	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
338	C₂³×C₄	—	2	1	no	—
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	—	2	1	no	—
341	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
342	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
345	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
346	#126	—	2	8	no	—
347	#125	—	2	4	no	—
348	#125	—	2	4	no	—
349	#125	—	2	8	no	—
350	#125	—	2	8	no	—
351	#125	—	2	2	no	—
352	#125	C₂³×C₂₃	1	8	yes	—
353	#125	C₂³×C₂₃	1	8	yes	—
354	#125	C₂³×C₂₃	1	4	yes	—
355	#124	—	2	2	no	—
356	#124	C₂³×C₂₃	1	4	yes	—
357	#123	—	2	8	no	—
358	C₂²⋊Q₈	—	2	2	no	—
359	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
360	#121	—	2	2	no	—
361	#121	C₂³×C₂₃	1	4	yes	—
362	C₄×Q₈	—	2	2	no	—
363	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
364	C₄×D₄	—	2	8	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	C₂³×C₂₃	1	4	yes	—
367	#117	—	2	8	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	—	2	2	no	—
371	C₂×C₄⋊C₄	—	2	2	no	—
372	C₂×C₄⋊C₄	—	2	8	no	—
373	C₂×C₄⋊C₄	—	2	8	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
386	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₂²⋊C₄	—	2	1	no	—
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
395	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
396	C₂×C₄²	D₄×C₂₃	1	4	yes	—
397	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
400	#107	C₂³×C₂₃	1	8	yes	—
401	#107	C₂³×C₂₃	1	8	yes	—
402	#106	C₂³×C₂₃	1	8	yes	—
403	#106	C₂³×C₂₃	1	8	yes	—
404	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
405	#103	C₂³×C₂₃	1	8	yes	—
406	#103	C₂³×C₂₃	1	8	yes	—
407	#102	C₂³×C₂₃	1	8	yes	—
408	#102	C₂³×C₂₃	1	8	yes	—
409	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
410	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
411	C₄×C₈	C₂³×C₂₃	1	8	yes	—
412	C₂₃	C₂×#3232	1	1	yes	characteristic, direct product factor, Sylow
413	C₂⁴	—	2	1	no	—
414	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	4	no	—
437	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
440	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
444	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
447	C₂×C₈	—	2	4	no	—
448	C₂×C₈	—	2	4	no	—
449	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
450	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
451	C₄⋊C₄	—	2	8	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
466	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
467	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
468	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₄²	—	2	4	no	—
475	C₄²	C₂×D₄×C₂₃	1	4	yes	—
476	C₄²	C₂×D₄×C₂₃	1	2	yes	—
477	C₄²	C₂×D₄×C₂₃	1	2	yes	—
478	C₄²	C₂×D₄×C₂₃	1	4	yes	—
479	C₄²	C₂⁴×C₂₃	1	2	yes	—
480	C₄²	C₂⁴×C₂₃	1	2	yes	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	—	4	2	no	—
487	C₂³	C₂₃×#135	1	4	yes	—
488	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
489	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
490	Q₈	—	2	4	no	—
491	Q₈	—	4	2	no	—
492	D₄	—	2	4	no	—
493	D₄	—	4	8	no	—
494	D₄	—	4	2	no	—
495	C₂×C₄	—	2	8	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	4	4	no	—
509	C₂×C₄	—	4	8	no	—
510	C₂×C₄	—	4	8	no	—
511	C₂×C₄	—	4	8	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
519	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
520	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
521	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
523	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
524	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
525	C₈	—	2	4	no	—
526	C₈	—	2	4	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	8	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	8	4	no	—
534	C₂²	C₂₃×#536	1	2	yes	—
535	C₂²	C₂×C₂₃×#135	1	4	yes	—
536	C₂²	C₂×C₂₃×#135	1	2	yes	—
537	C₂²	C₂₃×#575	1	2	yes	—
538	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
539	C₄	—	2	2	no	—
540	C₄	—	2	2	no	—
541	C₄	—	2	2	no	—
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	4	2	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	4	no	—
549	C₄	C₂₃×#536	1	2	yes	—
550	C₄	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
551	C₂	—	2	4	no	—
552	C₂	—	8	2	no	—
553	C₂	C₂₃×#3232	1	4	yes	direct product factor
554	C₂	C₂×C₂₃×#536	1	1	yes	characteristic
555	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
556	C₂	C₂×C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×#125	C₂	semidirect
3. C₂₃×#3232	C₂	direct (N × H)
4. C₂×C₂₃×#499	C₂	semidirect
5. C₂×C₂₃×#488	C₂	semidirect
6. C₂×C₂₃×#484	C₂	semidirect
8. C₂×C₂₃×#483	C₂	semidirect
10. C₂×C₂₃×#479	C₂	semidirect
11. C₂×C₂₃×#478	C₂	semidirect
12. C₂×C₂₃×#431	C₂	semidirect
22. C₂×C₂₃×#125	C₂²	semidirect
36. C₂₃×#499	C₂²	semidirect
37. C₂₃×#488	C₂²	semidirect
38. C₂₃×#484	C₂²	semidirect
39. C₂₃×#483	C₂²	semidirect
42. C₂₃×#479	C₂²	semidirect
43. C₂₃×#478	C₂²	semidirect
44. C₂₃×#431	C₂²	semidirect
202. C₂×#3232	C₂₃	direct (N × H)
250. C₂²×#125	C₂×C₂₃	semidirect
251. #3232	C₂×C₂₃	direct (N × H)
252. C₂×#499	C₂×C₂₃	semidirect
253. C₂×#488	C₂×C₂₃	semidirect
254. C₂×#484	C₂×C₂₃	semidirect
256. C₂×#483	C₂×C₂₃	semidirect
258. C₂×#479	C₂×C₂₃	semidirect
259. C₂×#478	C₂×C₂₃	semidirect
260. C₂×#431	C₂×C₂₃	semidirect
294. C₂×#125	C₂²×C₂₃	semidirect
308. #499	C₂²×C₂₃	semidirect
309. #488	C₂²×C₂₃	semidirect
310. #484	C₂²×C₂₃	semidirect
311. #483	C₂²×C₂₃	semidirect
314. #479	C₂²×C₂₃	semidirect
315. #478	C₂²×C₂₃	semidirect
316. #431	C₂²×C₂₃	semidirect
331. C₂×C₂₃	#3232	direct (N × H)
412. C₂₃	C₂×#3232	direct (N × H)
553. C₂	C₂₃×#3232	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×#529	2
C₂²×C₂₃×#125	2
C₂₃×#3232	2
C₂×C₂₃×#499	2
C₂×C₂₃×#488	2
C₂×C₂₃×#484	2
C₂×C₂₃×#481	2
C₂×C₂₃×#483	2
C₂×C₂₃×#480	2
C₂×C₂₃×#479	2
C₂×C₂₃×#478	2
C₂×C₂₃×#431	2
C₂×#3232	23

Subgroups — C2×C23×#3232 #19208466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3232 #19208466