Subgroups — C₂×C₂₃×#3251 #19208527

Order 5888.

2178 conjugacy classes of subgroups.

The subgroup lattice has 624 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, complemented
2	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3251	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#478	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#469	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#468	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#463	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	—	2	1	no	—
14	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#532	—	2	4	no	—
18	C₂₃×#532	C₂²	1	8	yes	complemented
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
21	C₂₃×#529	—	2	4	no	—
22	C₂₃×#529	C₂²	1	8	yes	complemented
23	C₂×C₂₃×#126	—	2	1	no	—
24	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#124	—	2	1	no	—
26	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#123	—	2	1	no	—
28	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#121	—	2	1	no	—
30	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#121	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₂₃	—	2	1	no	—
34	C₂×C₄×D₄×C₂₃	—	2	1	no	—
35	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
37	C₂×C₂₃×#117	—	2	1	no	—
38	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#478	C₂²	1	16	yes	complemented
40	C₂₃×#475	C₂²	1	8	yes	complemented
41	C₂₃×#474	C₂²	1	8	yes	complemented
42	C₂₃×#469	C₂²	1	16	yes	complemented
43	C₂₃×#468	C₂²	1	16	yes	complemented
44	C₂₃×#463	C₂²	1	16	yes	complemented
45	C₂₃×#460	C₂²	1	8	yes	complemented
46	C₂₃×#459	C₂²	1	8	yes	complemented
47	C₂×C₂₃×#107	C₂²	1	2	yes	—
48	C₂×C₂₃×#106	C₂²	1	2	yes	—
49	C₂₃×#423	C₂²	1	8	yes	complemented
50	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
52	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×#103	C₂²	1	2	yes	—
54	C₂×C₂₃×#102	C₂²	1	2	yes	—
55	C₂×C₂₃×#102	C₂²	1	2	yes	—
56	C₂×C₂₃×#102	C₂²	1	2	yes	—
57	C₂×#41×C₂₃	—	2	8	no	—
58	C₂×#41×C₂₃	—	4	8	no	—
59	C₂²×Q₈×C₂₃	—	2	1	no	—
60	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₂₃	—	2	2	no	—
62	C₂²×D₄×C₂₃	—	2	1	no	—
63	C₂²×D₄×C₂₃	—	2	1	no	—
64	C₂²×D₄×C₂₃	—	2	1	no	—
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
67	C₂³×C₄×C₂₃	—	2	2	no	—
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
70	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×SD₁₆×C₂₃	—	4	4	no	—
73	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
74	C₂×D₈×C₂₃	—	4	4	no	—
75	C₂×D₈×C₂₃	C₂³	1	8	yes	—
76	C₂×M₁₆×C₂₃	—	4	4	no	—
77	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
78	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
79	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
80	C₂₃×C₄⋊Q₈	—	2	2	no	—
81	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
82	C₂₃×#127	—	2	2	no	—
83	C₂₃×#127	C₂³	1	4	yes	—
84	C₂₃×#126	—	2	8	no	—
85	C₂₃×#125	—	2	2	no	—
86	C₂₃×#125	C₂³	1	4	yes	—
87	C₂₃×#124	—	2	8	no	—
88	C₂₃×#124	—	2	2	no	—
89	C₂₃×#124	C₂³	1	4	yes	—
90	C₂₃×#123	—	2	8	no	—
91	C₂₃×C₂²⋊Q₈	—	2	2	no	—
92	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
93	C₂₃×#121	—	2	8	no	—
94	C₂₃×#121	—	2	2	no	—
95	C₂₃×#121	—	2	4	no	—
96	C₂₃×#121	C₂³	1	8	yes	—
97	C₂₃×#121	C₂³	1	4	yes	—
98	C₄×Q₈×C₂₃	—	2	2	no	—
99	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
100	C₄×D₄×C₂₃	—	2	8	no	—
101	C₄×D₄×C₂₃	—	2	8	no	—
102	C₄×D₄×C₂₃	—	2	2	no	—
103	C₄×D₄×C₂₃	—	2	4	no	—
104	C₄×D₄×C₂₃	C₂³	1	4	yes	—
105	C₄×D₄×C₂₃	C₂³	1	8	yes	—
106	C₂₃×#117	—	2	8	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
122	C₂×C₄²×C₂₃	—	2	1	no	—
123	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₂₃	D₄	1	4	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
129	C₂₃×#107	C₂³	1	8	yes	—
130	C₂₃×#106	C₂³	1	8	yes	—
131	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
132	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
133	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
134	C₂₃×#103	C₂³	1	8	yes	—
135	C₂₃×#102	C₂³	1	8	yes	—
136	C₂₃×#102	C₂³	1	8	yes	—
137	C₂₃×#102	C₂³	1	8	yes	—
138	C₂⁴×C₂₃	—	2	1	no	—
139	C₂⁴×C₂₃	—	2	2	no	—
140	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
141	#41×C₂₃	—	4	32	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	4	no	—
144	C₂×Q₈×C₂₃	—	2	2	no	—
145	C₂×Q₈×C₂₃	—	2	2	no	—
146	C₂×Q₈×C₂₃	—	4	4	no	—
147	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
148	C₂×D₄×C₂₃	—	2	4	no	—
149	C₂×D₄×C₂₃	—	2	8	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	2	2	no	—
152	C₂×D₄×C₂₃	—	2	4	no	—
153	C₂×D₄×C₂₃	—	2	4	no	—
154	C₂×D₄×C₂₃	—	2	2	no	—
155	C₂×D₄×C₂₃	—	2	4	no	—
156	C₂×D₄×C₂₃	—	2	2	no	—
157	C₂×D₄×C₂₃	—	2	4	no	—
158	C₂×D₄×C₂₃	—	4	4	no	—
159	C₂×D₄×C₂₃	—	4	4	no	—
160	C₂×D₄×C₂₃	—	4	8	no	—
161	C₂×D₄×C₂₃	—	4	4	no	—
162	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
163	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₂₃	—	2	4	no	—
165	C₂²×C₄×C₂₃	—	2	2	no	—
166	C₂²×C₄×C₂₃	—	2	2	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
170	C₂²×C₄×C₂₃	—	2	2	no	—
171	C₂²×C₄×C₂₃	—	2	4	no	—
172	C₂²×C₄×C₂₃	—	2	8	no	—
173	C₂²×C₄×C₂₃	—	4	2	no	—
174	C₂²×C₄×C₂₃	—	4	8	no	—
175	C₂²×C₄×C₂₃	—	4	4	no	—
176	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
178	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
186	SD₁₆×C₂₃	—	4	8	no	—
187	D₈×C₂₃	—	4	8	no	—
188	M₁₆×C₂₃	—	4	8	no	—
189	C₂×C₈×C₂₃	—	2	4	no	—
190	C₂×C₈×C₂₃	—	2	4	no	—
191	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
192	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	—	2	4	no	—
195	C₄⋊C₄×C₂₃	—	2	4	no	—
196	C₄⋊C₄×C₂₃	—	2	4	no	—
197	C₄⋊C₄×C₂₃	—	2	4	no	—
198	C₄⋊C₄×C₂₃	—	2	2	no	—
199	C₄⋊C₄×C₂₃	—	2	1	no	—
200	C₄⋊C₄×C₂₃	—	2	1	no	—
201	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
203	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	4	no	—
207	C₂²⋊C₄×C₂₃	—	2	8	no	—
208	C₂²⋊C₄×C₂₃	—	2	4	no	—
209	C₂²⋊C₄×C₂₃	—	2	8	no	—
210	C₂²⋊C₄×C₂₃	—	2	4	no	—
211	C₄²×C₂₃	—	2	4	no	—
212	C₄²×C₂₃	C₂×D₄	1	4	yes	—
213	C₄²×C₂₃	C₂×D₄	1	2	yes	—
214	C₄²×C₂₃	C₂×D₄	1	4	yes	—
215	C₄²×C₂₃	C₂×D₄	1	2	yes	—
216	C₄²×C₂₃	C₂⁴	1	2	yes	—
217	C₄²×C₂₃	C₂⁴	1	2	yes	—
218	C₂×#3251	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₂₃	—	2	2	no	—
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	2	4	no	—
224	C₂³×C₂₃	—	4	2	no	—
225	C₂³×C₂₃	—	4	4	no	—
226	C₂³×C₂₃	—	4	4	no	—
227	C₂³×C₂₃	—	4	2	no	—
228	C₂³×C₂₃	—	4	4	no	—
229	C₂³×C₂₃	—	4	2	no	—
230	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
231	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₂₃	—	2	2	no	—
233	Q₈×C₂₃	—	2	2	no	—
234	Q₈×C₂₃	—	4	2	no	—
235	Q₈×C₂₃	—	4	8	no	—
236	D₄×C₂₃	—	2	2	no	—
237	D₄×C₂₃	—	2	4	no	—
238	D₄×C₂₃	—	2	2	no	—
239	D₄×C₂₃	—	2	4	no	—
240	D₄×C₂₃	—	4	8	no	—
241	D₄×C₂₃	—	4	8	no	—
242	D₄×C₂₃	—	4	16	no	—
243	D₄×C₂₃	—	4	2	no	—
244	D₄×C₂₃	—	4	4	no	—
245	D₄×C₂₃	—	4	8	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	4	no	—
251	C₂×C₄×C₂₃	—	2	8	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	2	no	—
257	C₂×C₄×C₂₃	—	2	4	no	—
258	C₂×C₄×C₂₃	—	4	4	no	—
259	C₂×C₄×C₂₃	—	4	16	no	—
260	C₂×C₄×C₂₃	—	4	2	no	—
261	C₂×C₄×C₂₃	—	4	2	no	—
262	C₂×C₄×C₂₃	—	4	4	no	—
263	C₂×C₄×C₂₃	—	4	2	no	—
264	C₂×C₄×C₂₃	—	4	8	no	—
265	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
266	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
273	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
274	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
275	C₈×C₂₃	—	2	4	no	—
276	C₈×C₂₃	—	2	4	no	—
277	C₂×#532	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₂×#529	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#3251	C₂×C₂₃	1	16	yes	direct product factor
280	C₂×#475	C₂×C₂₃	1	1	yes	characteristic, complemented
281	C₂×#478	C₂×C₂₃	1	2	yes	complemented
282	C₂×#474	C₂×C₂₃	1	1	yes	characteristic, complemented
283	C₂×#469	C₂×C₂₃	1	2	yes	complemented
284	C₂×#468	C₂×C₂₃	1	2	yes	complemented
285	C₂×#460	C₂×C₂₃	1	1	yes	characteristic, complemented
286	C₂×#463	C₂×C₂₃	1	2	yes	complemented
287	C₂×#459	C₂×C₂₃	1	1	yes	characteristic, complemented
288	C₂×#423	C₂×C₂₃	1	1	yes	characteristic, complemented
289	C₂²×C₂₃	—	2	2	no	—
290	C₂²×C₂₃	—	2	2	no	—
291	C₂²×C₂₃	—	2	2	no	—
292	C₂²×C₂₃	—	4	4	no	—
293	C₂²×C₂₃	—	4	2	no	—
294	C₂²×C₂₃	—	4	4	no	—
295	C₂²×C₂₃	—	4	4	no	—
296	C₂²×C₂₃	—	4	2	no	—
297	C₂²×C₂₃	—	4	4	no	—
298	C₂²×C₂₃	—	4	2	no	—
299	C₂²×C₂₃	—	8	8	no	—
300	C₂²×C₂₃	—	8	4	no	—
301	C₂²×C₂₃	#537	1	2	yes	—
302	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
303	C₂²×C₂₃	#575	1	2	yes	—
304	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
305	C₄×C₂₃	—	2	2	no	—
306	C₄×C₂₃	—	2	2	no	—
307	C₄×C₂₃	—	2	2	no	—
308	C₄×C₂₃	—	2	2	no	—
309	C₄×C₂₃	—	4	2	no	—
310	C₄×C₂₃	—	4	2	no	—
311	C₄×C₂₃	—	4	4	no	—
312	C₄×C₂₃	—	4	2	no	—
313	C₄×C₂₃	#537	1	2	yes	—
314	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
315	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
316	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
317	C₂²×#41	—	2	1	no	—
318	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
319	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
320	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
321	#532	—	2	4	no	—
322	#532	C₂²×C₂₃	1	8	yes	complemented
323	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
325	#529	—	2	4	no	—
326	#529	C₂²×C₂₃	1	8	yes	complemented
327	C₂×#126	—	2	1	no	—
328	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#124	—	2	1	no	—
330	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×#123	—	2	1	no	—
332	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
333	C₂×#121	—	2	1	no	—
334	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×#121	C₂²×C₂₃	1	2	yes	—
336	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×C₄×D₄	—	2	1	no	—
338	C₂×C₄×D₄	—	2	1	no	—
339	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
340	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×#117	—	2	1	no	—
342	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
343	#478	C₂²×C₂₃	1	16	yes	complemented
344	#475	C₂²×C₂₃	1	8	yes	complemented
345	#474	C₂²×C₂₃	1	8	yes	complemented
346	#469	C₂²×C₂₃	1	16	yes	complemented
347	#468	C₂²×C₂₃	1	16	yes	complemented
348	#463	C₂²×C₂₃	1	16	yes	complemented
349	#460	C₂²×C₂₃	1	8	yes	complemented
350	#459	C₂²×C₂₃	1	8	yes	complemented
351	C₂×#107	C₂²×C₂₃	1	2	yes	—
352	C₂×#106	C₂²×C₂₃	1	2	yes	—
353	#423	C₂²×C₂₃	1	8	yes	complemented
354	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
355	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×#103	C₂²×C₂₃	1	2	yes	—
358	C₂×#102	C₂²×C₂₃	1	2	yes	—
359	C₂×#102	C₂²×C₂₃	1	2	yes	—
360	C₂×#102	C₂²×C₂₃	1	2	yes	—
361	C₂×C₂₃	—	4	2	no	—
362	C₂×C₂₃	—	8	4	no	—
363	C₂×C₂₃	—	8	2	no	—
364	C₂×C₂₃	#3251	1	4	yes	direct product factor
365	C₂×C₂₃	C₂×#537	1	1	yes	characteristic
366	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
367	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
368	C₂×#41	—	2	8	no	—
369	C₂×#41	—	4	8	no	—
370	C₂²×Q₈	—	2	1	no	—
371	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
372	C₂²×D₄	—	2	1	no	—
373	C₂²×D₄	—	2	2	no	—
374	C₂²×D₄	—	2	1	no	—
375	C₂²×D₄	—	2	1	no	—
376	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
377	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
378	C₂³×C₄	—	2	1	no	—
379	C₂³×C₄	—	2	2	no	—
380	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
381	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂×SD₁₆	—	4	4	no	—
384	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
385	C₂×D₈	—	4	4	no	—
386	C₂×D₈	C₂³×C₂₃	1	8	yes	—
387	C₂×M₁₆	—	4	4	no	—
388	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
389	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
390	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
391	C₄⋊Q₈	—	2	2	no	—
392	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
393	#127	—	2	2	no	—
394	#127	C₂³×C₂₃	1	4	yes	—
395	#126	—	2	8	no	—
396	#125	—	2	2	no	—
397	#125	C₂³×C₂₃	1	4	yes	—
398	#124	—	2	8	no	—
399	#124	—	2	2	no	—
400	#124	C₂³×C₂₃	1	4	yes	—
401	#123	—	2	8	no	—
402	C₂²⋊Q₈	—	2	2	no	—
403	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
404	#121	—	2	8	no	—
405	#121	—	2	2	no	—
406	#121	—	2	4	no	—
407	#121	C₂³×C₂₃	1	4	yes	—
408	#121	C₂³×C₂₃	1	8	yes	—
409	C₄×Q₈	—	2	2	no	—
410	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
411	C₄×D₄	—	2	8	no	—
412	C₄×D₄	—	2	8	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	C₂³×C₂₃	1	4	yes	—
416	C₄×D₄	C₂³×C₂₃	1	8	yes	—
417	#117	—	2	8	no	—
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	2	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₂²⋊C₄	—	2	2	no	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₄²	—	2	1	no	—
434	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
435	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
436	C₂×C₄²	D₄×C₂₃	1	4	yes	—
437	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
440	#107	C₂³×C₂₃	1	8	yes	—
441	#106	C₂³×C₂₃	1	8	yes	—
442	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
443	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
444	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
445	#103	C₂³×C₂₃	1	8	yes	—
446	#102	C₂³×C₂₃	1	8	yes	—
447	#102	C₂³×C₂₃	1	8	yes	—
448	#102	C₂³×C₂₃	1	8	yes	—
449	C₂₃	C₂×#3251	1	1	yes	characteristic, direct product factor, Sylow
450	C₂⁴	—	2	1	no	—
451	C₂⁴	—	2	2	no	—
452	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
453	#41	—	4	32	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	4	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	4	4	no	—
459	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	8	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	4	4	no	—
471	C₂×D₄	—	4	8	no	—
472	C₂×D₄	—	4	4	no	—
473	C₂×D₄	—	4	4	no	—
474	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
475	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	8	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	8	no	—
487	C₂²×C₄	—	4	4	no	—
488	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
490	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
494	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
495	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
498	SD₁₆	—	4	8	no	—
499	D₈	—	4	8	no	—
500	M₁₆	—	4	8	no	—
501	C₂×C₈	—	2	4	no	—
502	C₂×C₈	—	2	4	no	—
503	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
504	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
514	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
515	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	8	no	—
522	C₂²⋊C₄	—	2	4	no	—
523	C₄²	—	2	4	no	—
524	C₄²	C₂×D₄×C₂₃	1	4	yes	—
525	C₄²	C₂×D₄×C₂₃	1	2	yes	—
526	C₄²	C₂×D₄×C₂₃	1	4	yes	—
527	C₄²	C₂×D₄×C₂₃	1	2	yes	—
528	C₄²	C₂⁴×C₂₃	1	2	yes	—
529	C₄²	C₂⁴×C₂₃	1	2	yes	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	4	4	no	—
536	C₂³	—	4	2	no	—
537	C₂³	—	4	4	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	4	2	no	—
540	C₂³	—	4	4	no	—
541	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
542	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
543	Q₈	—	2	2	no	—
544	Q₈	—	2	2	no	—
545	Q₈	—	4	8	no	—
546	Q₈	—	4	2	no	—
547	D₄	—	2	2	no	—
548	D₄	—	2	4	no	—
549	D₄	—	2	2	no	—
550	D₄	—	2	4	no	—
551	D₄	—	4	8	no	—
552	D₄	—	4	16	no	—
553	D₄	—	4	8	no	—
554	D₄	—	4	8	no	—
555	D₄	—	4	2	no	—
556	D₄	—	4	4	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	8	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	16	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	2	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	8	no	—
576	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
577	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
582	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
583	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
584	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
585	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
586	C₈	—	2	4	no	—
587	C₈	—	2	4	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	4	4	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	8	8	no	—
599	C₂²	—	8	4	no	—
600	C₂²	C₂₃×#537	1	2	yes	—
601	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
602	C₂²	C₂₃×#575	1	2	yes	—
603	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	C₂₃×#537	1	2	yes	—
613	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
614	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
615	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
616	C₂	—	4	2	no	—
617	C₂	—	8	4	no	—
618	C₂	—	8	2	no	—
619	C₂	C₂₃×#3251	1	4	yes	direct product factor
620	C₂	C₂×C₂₃×#537	1	1	yes	characteristic
621	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
622	C₂	C₂×C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
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×#532	C₂	semidirect
2. C₂×C₂₃×#529	C₂	semidirect
3. C₂₃×#3251	C₂	direct (N × H)
4. C₂×C₂₃×#475	C₂	semidirect
5. C₂×C₂₃×#478	C₂	semidirect
6. C₂×C₂₃×#474	C₂	semidirect
7. C₂×C₂₃×#469	C₂	semidirect
8. C₂×C₂₃×#468	C₂	semidirect
9. C₂×C₂₃×#460	C₂	semidirect
10. C₂×C₂₃×#463	C₂	semidirect
11. C₂×C₂₃×#459	C₂	semidirect
12. C₂×C₂₃×#423	C₂	semidirect
18. C₂₃×#532	C₂²	semidirect
22. C₂₃×#529	C₂²	semidirect
39. C₂₃×#478	C₂²	semidirect
40. C₂₃×#475	C₂²	semidirect
41. C₂₃×#474	C₂²	semidirect
42. C₂₃×#469	C₂²	semidirect
43. C₂₃×#468	C₂²	semidirect
44. C₂₃×#463	C₂²	semidirect
45. C₂₃×#460	C₂²	semidirect
46. C₂₃×#459	C₂²	semidirect
49. C₂₃×#423	C₂²	semidirect
218. C₂×#3251	C₂₃	direct (N × H)
277. C₂×#532	C₂×C₂₃	semidirect
278. C₂×#529	C₂×C₂₃	semidirect
279. #3251	C₂×C₂₃	direct (N × H)
280. C₂×#475	C₂×C₂₃	semidirect
281. C₂×#478	C₂×C₂₃	semidirect
282. C₂×#474	C₂×C₂₃	semidirect
283. C₂×#469	C₂×C₂₃	semidirect
284. C₂×#468	C₂×C₂₃	semidirect
285. C₂×#460	C₂×C₂₃	semidirect
286. C₂×#463	C₂×C₂₃	semidirect
287. C₂×#459	C₂×C₂₃	semidirect
288. C₂×#423	C₂×C₂₃	semidirect
322. #532	C₂²×C₂₃	semidirect
326. #529	C₂²×C₂₃	semidirect
343. #478	C₂²×C₂₃	semidirect
344. #475	C₂²×C₂₃	semidirect
345. #474	C₂²×C₂₃	semidirect
346. #469	C₂²×C₂₃	semidirect
347. #468	C₂²×C₂₃	semidirect
348. #463	C₂²×C₂₃	semidirect
349. #460	C₂²×C₂₃	semidirect
350. #459	C₂²×C₂₃	semidirect
353. #423	C₂²×C₂₃	semidirect
364. C₂×C₂₃	#3251	direct (N × H)
449. C₂₃	C₂×#3251	direct (N × H)
619. C₂	C₂₃×#3251	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
C₂×C₂₃×#529	2
C₂₃×#3251	2
C₂×C₂₃×#475	2
C₂×C₂₃×#478	2
C₂×C₂₃×#474	2
C₂×C₂₃×#469	2
C₂×C₂₃×#468	2
C₂×C₂₃×#460	2
C₂×C₂₃×#463	2
C₂×C₂₃×#459	2
C₂×C₂₃×#423	2
C₂×#3251	23

Subgroups — C2×C23×#3251 #19208527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3251 #19208527