Subgroups — C₂×C₂₃×#3233 #19208471

Order 5888.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 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, complemented
2	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3233	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×#485	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#482	C₂	1	2	yes	maximal
8	C₂×C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#533	—	2	4	no	—
14	C₂₃×#533	C₂²	1	8	yes	complemented
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#529	—	2	4	no	—
18	C₂₃×#529	C₂²	1	8	yes	—
19	C₂×C₂₃×#126	—	2	1	no	—
20	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#125	C₂²	1	2	yes	—
22	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#123	—	2	1	no	—
24	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
29	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₂₃	—	2	1	no	—
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#117	—	2	1	no	—
34	C₂×C₂₃×#117	C₂²	1	2	yes	—
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×#485	C₂²	1	16	yes	complemented
39	C₂₃×#482	C₂²	1	16	yes	—
40	C₂₃×#477	C₂²	1	8	yes	complemented
41	C₂₃×#476	C₂²	1	8	yes	complemented
42	C₂₃×#475	C₂²	1	8	yes	complemented
43	C₂₃×#474	C₂²	1	8	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	1	yes	characteristic
50	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#103	C₂²	1	2	yes	—
52	C₂×C₂₃×#103	C₂²	1	2	yes	—
53	C₂×C₂₃×#102	C₂²	1	2	yes	—
54	C₂×C₂₃×#102	C₂²	1	2	yes	—
55	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
56	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
57	C₂²×Q₈×C₂₃	—	2	1	no	—
58	C₂²×Q₈×C₂₃	—	2	1	no	—
59	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₂₃	—	2	1	no	—
61	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
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×C₄⋊Q₈	—	2	2	no	—
69	C₂₃×C₄⋊Q₈	—	2	2	no	—
70	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
71	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
72	C₂₃×#126	—	2	8	no	—
73	C₂₃×#125	—	2	2	no	—
74	C₂₃×#125	—	2	4	no	—
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	8	no	—
81	C₂₃×C₂²⋊Q₈	—	2	8	no	—
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
84	C₂₃×#121	—	2	2	no	—
85	C₂₃×#121	C₂³	1	4	yes	—
86	C₄×Q₈×C₂₃	—	2	8	no	—
87	C₄×Q₈×C₂₃	—	2	8	no	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
90	C₄×D₄×C₂₃	—	2	8	no	—
91	C₄×D₄×C₂₃	—	2	2	no	—
92	C₄×D₄×C₂₃	C₂³	1	4	yes	—
93	C₂₃×#117	—	2	4	no	—
94	C₂₃×#117	—	2	8	no	—
95	C₂₃×#117	C₂³	1	8	yes	—
96	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	2	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×C₂²⋊C₄×C₂₃	—	2	2	no	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
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×C₄²×C₂₃	—	2	1	no	—
115	C₂×C₄²×C₂₃	—	2	2	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	4	yes	—
127	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
128	C₂₃×#103	C₂³	1	8	yes	—
129	C₂₃×#103	C₂³	1	8	yes	—
130	C₂₃×#102	C₂³	1	8	yes	—
131	C₂₃×#102	C₂³	1	8	yes	—
132	C₂₃×C₂²⋊C₈	C₂³	1	8	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	—	2	4	no	—
139	C₂×Q₈×C₂₃	—	2	2	no	—
140	C₂×Q₈×C₂₃	—	2	2	no	—
141	C₂×Q₈×C₂₃	—	4	4	no	—
142	C₂×Q₈×C₂₃	—	4	4	no	—
143	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
144	C₂×D₄×C₂₃	—	2	4	no	—
145	C₂×D₄×C₂₃	—	2	1	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	4	4	no	—
149	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₂₃	—	2	4	no	—
151	C₂²×C₄×C₂₃	—	2	2	no	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	2	no	—
156	C₂²×C₄×C₂₃	—	2	2	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	4	2	no	—
160	C₂²×C₄×C₂₃	—	4	4	no	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
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	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
171	C₂×C₈×C₂₃	—	2	4	no	—
172	C₂×C₈×C₂₃	—	2	4	no	—
173	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
174	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
175	C₄⋊C₄×C₂₃	—	2	4	no	—
176	C₄⋊C₄×C₂₃	—	2	4	no	—
177	C₄⋊C₄×C₂₃	—	2	4	no	—
178	C₄⋊C₄×C₂₃	—	2	8	no	—
179	C₄⋊C₄×C₂₃	—	2	8	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	2	no	—
184	C₄⋊C₄×C₂₃	—	2	2	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
189	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
190	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
191	C₂²⋊C₄×C₂₃	—	2	8	no	—
192	C₂²⋊C₄×C₂₃	—	2	4	no	—
193	C₂²⋊C₄×C₂₃	—	2	4	no	—
194	C₂²⋊C₄×C₂₃	—	2	4	no	—
195	C₂²⋊C₄×C₂₃	—	2	4	no	—
196	C₂²⋊C₄×C₂₃	—	2	4	no	—
197	C₄²×C₂₃	—	2	4	no	—
198	C₄²×C₂₃	—	2	8	no	—
199	C₄²×C₂₃	C₂×D₄	1	4	yes	—
200	C₄²×C₂₃	C₂×D₄	1	2	yes	—
201	C₄²×C₂₃	C₂×D₄	1	4	yes	—
202	C₄²×C₂₃	C₂×D₄	1	2	yes	—
203	C₄²×C₂₃	C₂⁴	1	2	yes	—
204	C₄²×C₂₃	C₂⁴	1	2	yes	—
205	C₂×#3233	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₂₃	—	2	2	no	—
207	C₂³×C₂₃	—	2	2	no	—
208	C₂³×C₂₃	—	2	2	no	—
209	C₂³×C₂₃	—	2	2	no	—
210	C₂³×C₂₃	—	4	2	no	—
211	C₂³×C₂₃	—	4	2	no	—
212	C₂³×C₂₃	—	4	2	no	—
213	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
214	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
215	Q₈×C₂₃	—	2	2	no	—
216	Q₈×C₂₃	—	2	2	no	—
217	Q₈×C₂₃	—	4	8	no	—
218	Q₈×C₂₃	—	4	2	no	—
219	Q₈×C₂₃	—	4	8	no	—
220	D₄×C₂₃	—	2	2	no	—
221	D₄×C₂₃	—	2	2	no	—
222	D₄×C₂₃	—	4	8	no	—
223	D₄×C₂₃	—	4	2	no	—
224	C₂×C₄×C₂₃	—	2	2	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	8	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	2	4	no	—
235	C₂×C₄×C₂₃	—	2	4	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	4	4	no	—
238	C₂×C₄×C₂₃	—	4	2	no	—
239	C₂×C₄×C₂₃	—	4	2	no	—
240	C₂×C₄×C₂₃	—	4	4	no	—
241	C₂×C₄×C₂₃	—	4	2	no	—
242	C₂×C₄×C₂₃	—	4	4	no	—
243	C₂×C₄×C₂₃	—	4	8	no	—
244	C₂×C₄×C₂₃	#135	1	4	yes	—
245	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
252	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
253	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
254	C₈×C₂₃	—	2	4	no	—
255	C₈×C₂₃	—	2	4	no	—
256	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
257	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
258	#3233	C₂×C₂₃	1	16	yes	direct product factor
259	C₂×#499	C₂×C₂₃	1	2	yes	complemented
260	C₂×#488	C₂×C₂₃	1	2	yes	complemented
261	C₂×#485	C₂×C₂₃	1	2	yes	complemented
262	C₂×#482	C₂×C₂₃	1	2	yes	—
263	C₂×#477	C₂×C₂₃	1	1	yes	characteristic, complemented
264	C₂×#475	C₂×C₂₃	1	1	yes	characteristic, complemented
265	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
266	C₂×#474	C₂×C₂₃	1	1	yes	characteristic, complemented
267	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
268	C₂²×C₂₃	—	2	2	no	—
269	C₂²×C₂₃	—	2	2	no	—
270	C₂²×C₂₃	—	2	2	no	—
271	C₂²×C₂₃	—	4	4	no	—
272	C₂²×C₂₃	—	4	2	no	—
273	C₂²×C₂₃	—	4	2	no	—
274	C₂²×C₂₃	—	4	2	no	—
275	C₂²×C₂₃	—	8	4	no	—
276	C₂²×C₂₃	#536	1	2	yes	—
277	C₂²×C₂₃	C₂×#135	1	2	yes	—
278	C₂²×C₂₃	#575	1	2	yes	—
279	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
280	C₄×C₂₃	—	2	2	no	—
281	C₄×C₂₃	—	2	2	no	—
282	C₄×C₂₃	—	2	2	no	—
283	C₄×C₂₃	—	2	2	no	—
284	C₄×C₂₃	—	4	2	no	—
285	C₄×C₂₃	—	4	2	no	—
286	C₄×C₂₃	—	4	4	no	—
287	C₄×C₂₃	—	4	2	no	—
288	C₄×C₂₃	—	4	4	no	—
289	C₄×C₂₃	#536	1	2	yes	—
290	C₄×C₂₃	C₂×#135	1	4	yes	—
291	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
292	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
293	#533	—	2	4	no	—
294	#533	C₂²×C₂₃	1	8	yes	complemented
295	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
297	#529	—	2	4	no	—
298	#529	C₂²×C₂₃	1	8	yes	—
299	C₂×#126	—	2	1	no	—
300	C₂×#125	C₂²×C₂₃	1	2	yes	—
301	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
303	C₂×#123	—	2	1	no	—
304	C₂×C₂²⋊Q₈	—	2	1	no	—
305	C₂×C₂²⋊Q₈	—	2	1	no	—
306	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×C₄×Q₈	—	2	1	no	—
309	C₂×C₄×Q₈	—	2	1	no	—
310	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×C₄×D₄	—	2	1	no	—
312	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×#117	—	2	1	no	—
314	C₂×#117	C₂²×C₂₃	1	2	yes	—
315	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
316	#499	C₂²×C₂₃	1	16	yes	complemented
317	#488	C₂²×C₂₃	1	16	yes	complemented
318	#485	C₂²×C₂₃	1	16	yes	complemented
319	#482	C₂²×C₂₃	1	16	yes	—
320	#477	C₂²×C₂₃	1	8	yes	complemented
321	#476	C₂²×C₂₃	1	8	yes	complemented
322	#475	C₂²×C₂₃	1	8	yes	complemented
323	#474	C₂²×C₂₃	1	8	yes	complemented
324	#431	C₂²×C₂₃	1	8	yes	complemented
325	C₂×#107	C₂²×C₂₃	1	2	yes	—
326	C₂×#107	C₂²×C₂₃	1	2	yes	—
327	C₂×#106	C₂²×C₂₃	1	2	yes	—
328	C₂×#106	C₂²×C₂₃	1	2	yes	—
329	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×#103	C₂²×C₂₃	1	2	yes	—
332	C₂×#103	C₂²×C₂₃	1	2	yes	—
333	C₂×#102	C₂²×C₂₃	1	2	yes	—
334	C₂×#102	C₂²×C₂₃	1	2	yes	—
335	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
336	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
337	C₂×C₂₃	—	4	2	no	—
338	C₂×C₂₃	—	8	2	no	—
339	C₂×C₂₃	#3233	1	4	yes	direct product factor
340	C₂×C₂₃	C₂×#536	1	1	yes	characteristic
341	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
342	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
343	C₂²×Q₈	—	2	1	no	—
344	C₂²×Q₈	—	2	1	no	—
345	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
346	C₂²×D₄	—	2	1	no	—
347	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
348	C₂³×C₄	—	2	1	no	—
349	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
350	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
353	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
354	C₄⋊Q₈	—	2	2	no	—
355	C₄⋊Q₈	—	2	2	no	—
356	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
357	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
358	#126	—	2	8	no	—
359	#125	—	2	2	no	—
360	#125	—	2	4	no	—
361	#125	C₂³×C₂₃	1	4	yes	—
362	#125	C₂³×C₂₃	1	8	yes	—
363	#124	—	2	2	no	—
364	#124	C₂³×C₂₃	1	4	yes	—
365	#123	—	2	8	no	—
366	C₂²⋊Q₈	—	2	8	no	—
367	C₂²⋊Q₈	—	2	8	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
370	#121	—	2	2	no	—
371	#121	C₂³×C₂₃	1	4	yes	—
372	C₄×Q₈	—	2	8	no	—
373	C₄×Q₈	—	2	8	no	—
374	C₄×Q₈	—	2	2	no	—
375	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
376	C₄×D₄	—	2	8	no	—
377	C₄×D₄	—	2	2	no	—
378	C₄×D₄	C₂³×C₂₃	1	4	yes	—
379	#117	—	2	4	no	—
380	#117	—	2	8	no	—
381	#117	C₂³×C₂₃	1	8	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	2	no	—
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	—	2	2	no	—
395	C₂×C₂²⋊C₄	—	2	1	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₄²	—	2	1	no	—
401	C₂×C₄²	—	2	2	no	—
402	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
403	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
404	C₂×C₄²	D₄×C₂₃	1	4	yes	—
405	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
408	#107	C₂³×C₂₃	1	8	yes	—
409	#107	C₂³×C₂₃	1	8	yes	—
410	#106	C₂³×C₂₃	1	8	yes	—
411	#106	C₂³×C₂₃	1	8	yes	—
412	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
413	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
414	#103	C₂³×C₂₃	1	8	yes	—
415	#103	C₂³×C₂₃	1	8	yes	—
416	#102	C₂³×C₂₃	1	8	yes	—
417	#102	C₂³×C₂₃	1	8	yes	—
418	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
419	C₄×C₈	C₂³×C₂₃	1	8	yes	—
420	C₂₃	C₂×#3233	1	1	yes	characteristic, direct product factor, Sylow
421	C₂⁴	—	2	1	no	—
422	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	4	no	—
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	—	4	4	no	—
429	C₂×Q₈	—	4	4	no	—
430	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	4	4	no	—
436	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
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	4	yes	—
454	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
455	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
458	C₂×C₈	—	2	4	no	—
459	C₂×C₈	—	2	4	no	—
460	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
461	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	8	no	—
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
475	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
476	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
477	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
478	C₂²⋊C₄	—	2	8	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₄²	—	2	4	no	—
485	C₄²	—	2	8	no	—
486	C₄²	C₂×D₄×C₂₃	1	4	yes	—
487	C₄²	C₂×D₄×C₂₃	1	2	yes	—
488	C₄²	C₂×D₄×C₂₃	1	4	yes	—
489	C₄²	C₂×D₄×C₂₃	1	2	yes	—
490	C₄²	C₂⁴×C₂₃	1	2	yes	—
491	C₄²	C₂⁴×C₂₃	1	2	yes	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	2	no	—
499	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
500	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
501	Q₈	—	2	2	no	—
502	Q₈	—	2	2	no	—
503	Q₈	—	4	8	no	—
504	Q₈	—	4	8	no	—
505	Q₈	—	4	2	no	—
506	D₄	—	2	2	no	—
507	D₄	—	2	2	no	—
508	D₄	—	4	8	no	—
509	D₄	—	4	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	4	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	8	no	—
530	C₂×C₄	C₂₃×#135	1	4	yes	—
531	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
532	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
534	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
535	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
536	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
537	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
538	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
539	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
540	C₈	—	2	4	no	—
541	C₈	—	2	4	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	8	4	no	—
550	C₂²	C₂₃×#536	1	2	yes	—
551	C₂²	C₂×C₂₃×#135	1	2	yes	—
552	C₂²	C₂₃×#575	1	2	yes	—
553	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	4	2	no	—
559	C₄	—	4	2	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	4	no	—
563	C₄	C₂₃×#536	1	2	yes	—
564	C₄	C₂×C₂₃×#135	1	4	yes	—
565	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
566	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
567	C₂	—	4	2	no	—
568	C₂	—	8	2	no	—
569	C₂	C₂₃×#3233	1	4	yes	direct product factor
570	C₂	C₂×C₂₃×#536	1	1	yes	characteristic
571	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
572	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
1. C₂×C₂₃×#533	C₂	semidirect
3. C₂₃×#3233	C₂	direct (N × H)
4. C₂×C₂₃×#499	C₂	semidirect
5. C₂×C₂₃×#488	C₂	semidirect
6. C₂×C₂₃×#485	C₂	semidirect
8. C₂×C₂₃×#477	C₂	semidirect
9. C₂×C₂₃×#475	C₂	semidirect
10. C₂×C₂₃×#476	C₂	semidirect
11. C₂×C₂₃×#474	C₂	semidirect
12. C₂×C₂₃×#431	C₂	semidirect
14. C₂₃×#533	C₂²	semidirect
36. C₂₃×#499	C₂²	semidirect
37. C₂₃×#488	C₂²	semidirect
38. C₂₃×#485	C₂²	semidirect
40. C₂₃×#477	C₂²	semidirect
41. C₂₃×#476	C₂²	semidirect
42. C₂₃×#475	C₂²	semidirect
43. C₂₃×#474	C₂²	semidirect
44. C₂₃×#431	C₂²	semidirect
205. C₂×#3233	C₂₃	direct (N × H)
256. C₂×#533	C₂×C₂₃	semidirect
258. #3233	C₂×C₂₃	direct (N × H)
259. C₂×#499	C₂×C₂₃	semidirect
260. C₂×#488	C₂×C₂₃	semidirect
261. C₂×#485	C₂×C₂₃	semidirect
263. C₂×#477	C₂×C₂₃	semidirect
264. C₂×#475	C₂×C₂₃	semidirect
265. C₂×#476	C₂×C₂₃	semidirect
266. C₂×#474	C₂×C₂₃	semidirect
267. C₂×#431	C₂×C₂₃	semidirect
294. #533	C₂²×C₂₃	semidirect
316. #499	C₂²×C₂₃	semidirect
317. #488	C₂²×C₂₃	semidirect
318. #485	C₂²×C₂₃	semidirect
320. #477	C₂²×C₂₃	semidirect
321. #476	C₂²×C₂₃	semidirect
322. #475	C₂²×C₂₃	semidirect
323. #474	C₂²×C₂₃	semidirect
324. #431	C₂²×C₂₃	semidirect
339. C₂×C₂₃	#3233	direct (N × H)
420. C₂₃	C₂×#3233	direct (N × H)
569. C₂	C₂₃×#3233	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×#529	2
C₂₃×#3233	2
C₂×C₂₃×#499	2
C₂×C₂₃×#488	2
C₂×C₂₃×#485	2
C₂×C₂₃×#482	2
C₂×C₂₃×#477	2
C₂×C₂₃×#475	2
C₂×C₂₃×#476	2
C₂×C₂₃×#474	2
C₂×C₂₃×#431	2
C₂×#3233	23

Subgroups — C2×C23×#3233 #19208471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3233 #19208471