Subgroups — C₂×C₂₃×#3180 #19208312

Order 5888.

2278 conjugacy classes of subgroups.

The subgroup lattice has 608 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×#3180	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#489	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#466	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#465	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#442	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#440	C₂	1	2	yes	maximal, complemented
12	C₂²×C₂₃×#97	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	C₂³×C₈×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×#501	C₂²	1	8	yes	complemented
40	C₂₃×#496	C₂²	1	8	yes	complemented
41	C₂₃×#490	C₂²	1	8	yes	complemented
42	C₂₃×#489	C₂²	1	8	yes	complemented
43	C₂₃×#466	C₂²	1	16	yes	complemented
44	C₂₃×#465	C₂²	1	16	yes	complemented
45	C₂₃×#442	C₂²	1	16	yes	complemented
46	C₂₃×#440	C₂²	1	16	yes	complemented
47	C₂×C₂₃×#107	C₂²	1	2	yes	—
48	C₂×C₂₃×#106	C₂²	1	2	yes	—
49	C₂×C₂₃×#103	C₂²	1	2	yes	—
50	C₂×C₂₃×#102	C₂²	1	2	yes	—
51	C₂×C₂₃×#102	C₂²	1	2	yes	—
52	C₂×C₂₃×#102	C₂²	1	2	yes	—
53	C₂×C₂₃×#97	C₂²	1	2	yes	—
54	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×#97	C₂²	1	8	yes	complemented
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	—	2	8	no	—
73	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
74	C₂×D₈×C₂₃	—	2	8	no	—
75	C₂×D₈×C₂₃	C₂³	1	8	yes	—
76	C₂²×C₈×C₂₃	—	2	8	no	—
77	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
78	C₂²×C₈×C₂₃	C₂³	1	8	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	8	yes	—
105	C₄×D₄×C₂₃	C₂³	1	4	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×#103	C₂³	1	8	yes	—
132	C₂₃×#102	C₂³	1	8	yes	—
133	C₂₃×#102	C₂³	1	8	yes	—
134	C₂₃×#102	C₂³	1	8	yes	—
135	C₂₃×#97	C₂³	1	8	yes	—
136	C₂₃×#97	C₂³	1	4	yes	—
137	C₂₃×#97	C₂³	1	4	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	4	no	—
145	C₂×Q₈×C₂₃	—	4	4	no	—
146	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
147	C₂×D₄×C₂₃	—	2	4	no	—
148	C₂×D₄×C₂₃	—	2	8	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	—	2	2	no	—
151	C₂×D₄×C₂₃	—	2	4	no	—
152	C₂×D₄×C₂₃	—	2	4	no	—
153	C₂×D₄×C₂₃	—	2	4	no	—
154	C₂×D₄×C₂₃	—	2	4	no	—
155	C₂×D₄×C₂₃	—	2	4	no	—
156	C₂×D₄×C₂₃	—	4	4	no	—
157	C₂×D₄×C₂₃	—	4	4	no	—
158	C₂×D₄×C₂₃	—	4	4	no	—
159	C₂×D₄×C₂₃	—	4	8	no	—
160	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
161	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₂₃	—	2	4	no	—
163	C₂²×C₄×C₂₃	—	2	4	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	2	no	—
168	C₂²×C₄×C₂₃	—	2	4	no	—
169	C₂²×C₄×C₂₃	—	2	8	no	—
170	C₂²×C₄×C₂₃	—	4	2	no	—
171	C₂²×C₄×C₂₃	—	4	4	no	—
172	C₂²×C₄×C₂₃	—	4	8	no	—
173	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
175	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	4	yes	—
181	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
183	SD₁₆×C₂₃	—	2	16	no	—
184	D₈×C₂₃	—	2	16	no	—
185	C₂×C₈×C₂₃	—	2	4	no	—
186	C₂×C₈×C₂₃	—	2	16	no	—
187	C₂×C₈×C₂₃	#41	1	8	yes	complemented
188	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
189	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
190	C₄⋊C₄×C₂₃	—	2	4	no	—
191	C₄⋊C₄×C₂₃	—	2	4	no	—
192	C₄⋊C₄×C₂₃	—	2	4	no	—
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	—	2	4	no	—
195	C₄⋊C₄×C₂₃	—	2	2	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
199	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
200	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
201	C₂²⋊C₄×C₂₃	—	2	4	no	—
202	C₂²⋊C₄×C₂₃	—	2	4	no	—
203	C₂²⋊C₄×C₂₃	—	2	4	no	—
204	C₂²⋊C₄×C₂₃	—	2	8	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	8	no	—
207	C₂²⋊C₄×C₂₃	—	2	4	no	—
208	C₄²×C₂₃	—	2	4	no	—
209	C₄²×C₂₃	C₂×D₄	1	4	yes	—
210	C₄²×C₂₃	C₂×D₄	1	2	yes	—
211	C₄²×C₂₃	C₂×D₄	1	4	yes	—
212	C₄²×C₂₃	C₂×D₄	1	2	yes	—
213	C₄²×C₂₃	C₂⁴	1	2	yes	—
214	C₄²×C₂₃	C₂⁴	1	2	yes	—
215	C₂×#3180	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂³×C₂₃	—	2	2	no	—
217	C₂³×C₂₃	—	2	2	no	—
218	C₂³×C₂₃	—	2	2	no	—
219	C₂³×C₂₃	—	2	4	no	—
220	C₂³×C₂₃	—	4	2	no	—
221	C₂³×C₂₃	—	4	4	no	—
222	C₂³×C₂₃	—	4	4	no	—
223	C₂³×C₂₃	—	4	2	no	—
224	C₂³×C₂₃	—	4	2	no	—
225	C₂³×C₂₃	—	4	4	no	—
226	C₂³×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
227	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
228	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
229	Q₈×C₂₃	—	2	4	no	—
230	Q₈×C₂₃	—	4	8	no	—
231	Q₈×C₂₃	—	4	2	no	—
232	D₄×C₂₃	—	2	4	no	—
233	D₄×C₂₃	—	2	4	no	—
234	D₄×C₂₃	—	2	4	no	—
235	D₄×C₂₃	—	4	8	no	—
236	D₄×C₂₃	—	4	8	no	—
237	D₄×C₂₃	—	4	8	no	—
238	D₄×C₂₃	—	4	16	no	—
239	D₄×C₂₃	—	4	2	no	—
240	D₄×C₂₃	—	4	4	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	8	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	4	no	—
248	C₂×C₄×C₂₃	—	2	4	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	4	4	no	—
254	C₂×C₄×C₂₃	—	4	8	no	—
255	C₂×C₄×C₂₃	—	4	16	no	—
256	C₂×C₄×C₂₃	—	4	2	no	—
257	C₂×C₄×C₂₃	—	4	4	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	2	no	—
260	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
267	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
268	C₈×C₂₃	—	2	4	no	—
269	C₈×C₂₃	C₂×#41	1	8	yes	complemented
270	C₂×#532	C₂×C₂₃	1	1	yes	characteristic, complemented
271	C₂×#529	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#3180	C₂×C₂₃	1	16	yes	direct product factor
273	C₂×#501	C₂×C₂₃	1	1	yes	characteristic, complemented
274	C₂×#496	C₂×C₂₃	1	1	yes	characteristic, complemented
275	C₂×#490	C₂×C₂₃	1	1	yes	characteristic, complemented
276	C₂×#489	C₂×C₂₃	1	1	yes	characteristic, complemented
277	C₂×#466	C₂×C₂₃	1	2	yes	complemented
278	C₂×#465	C₂×C₂₃	1	2	yes	complemented
279	C₂×#442	C₂×C₂₃	1	2	yes	complemented
280	C₂×#440	C₂×C₂₃	1	2	yes	complemented
281	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
282	C₂²×C₂₃	—	2	8	no	—
283	C₂²×C₂₃	—	2	2	no	—
284	C₂²×C₂₃	—	2	2	no	—
285	C₂²×C₂₃	—	4	2	no	—
286	C₂²×C₂₃	—	4	4	no	—
287	C₂²×C₂₃	—	4	4	no	—
288	C₂²×C₂₃	—	4	2	no	—
289	C₂²×C₂₃	—	4	2	no	—
290	C₂²×C₂₃	—	4	4	no	—
291	C₂²×C₂₃	—	8	8	no	—
292	C₂²×C₂₃	—	8	4	no	—
293	C₂²×C₂₃	#532	1	2	yes	—
294	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
295	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
296	C₂²×C₂₃	#575	1	2	yes	—
297	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
298	C₄×C₂₃	—	2	2	no	—
299	C₄×C₂₃	—	2	2	no	—
300	C₄×C₂₃	—	2	2	no	—
301	C₄×C₂₃	—	2	2	no	—
302	C₄×C₂₃	—	2	2	no	—
303	C₄×C₂₃	—	4	2	no	—
304	C₄×C₂₃	—	4	4	no	—
305	C₄×C₂₃	—	4	2	no	—
306	C₄×C₂₃	—	4	2	no	—
307	C₄×C₂₃	#532	1	2	yes	—
308	C₄×C₂₃	C₂²×#41	1	2	yes	—
309	C₂²×#41	—	2	1	no	—
310	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
311	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
312	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
313	#532	—	2	4	no	—
314	#532	C₂²×C₂₃	1	8	yes	complemented
315	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
317	#529	—	2	4	no	—
318	#529	C₂²×C₂₃	1	8	yes	complemented
319	C₂×#126	—	2	1	no	—
320	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#124	—	2	1	no	—
322	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×#123	—	2	1	no	—
324	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×#121	—	2	1	no	—
326	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#121	C₂²×C₂₃	1	2	yes	—
328	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×C₄×D₄	—	2	1	no	—
330	C₂×C₄×D₄	—	2	1	no	—
331	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
333	C₂×#117	—	2	1	no	—
334	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
335	#501	C₂²×C₂₃	1	8	yes	complemented
336	#496	C₂²×C₂₃	1	8	yes	complemented
337	#490	C₂²×C₂₃	1	8	yes	complemented
338	#489	C₂²×C₂₃	1	8	yes	complemented
339	#466	C₂²×C₂₃	1	16	yes	complemented
340	#465	C₂²×C₂₃	1	16	yes	complemented
341	#442	C₂²×C₂₃	1	16	yes	complemented
342	#440	C₂²×C₂₃	1	16	yes	complemented
343	C₂×#107	C₂²×C₂₃	1	2	yes	—
344	C₂×#106	C₂²×C₂₃	1	2	yes	—
345	C₂×#103	C₂²×C₂₃	1	2	yes	—
346	C₂×#102	C₂²×C₂₃	1	2	yes	—
347	C₂×#102	C₂²×C₂₃	1	2	yes	—
348	C₂×#102	C₂²×C₂₃	1	2	yes	—
349	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×#97	C₂²×C₂₃	1	2	yes	—
352	C₂×#97	C₂²×C₂₃	1	8	yes	complemented
353	C₂×C₂₃	—	2	4	no	—
354	C₂×C₂₃	—	8	4	no	—
355	C₂×C₂₃	—	8	2	no	—
356	C₂×C₂₃	#3180	1	4	yes	direct product factor
357	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
358	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
359	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
360	C₂×#41	—	2	8	no	—
361	C₂×#41	—	4	8	no	—
362	C₂²×Q₈	—	2	1	no	—
363	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
364	C₂²×D₄	—	2	1	no	—
365	C₂²×D₄	—	2	2	no	—
366	C₂²×D₄	—	2	1	no	—
367	C₂²×D₄	—	2	1	no	—
368	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
369	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
370	C₂³×C₄	—	2	1	no	—
371	C₂³×C₄	—	2	2	no	—
372	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
373	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
374	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
375	C₂×SD₁₆	—	2	8	no	—
376	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
377	C₂×D₈	—	2	8	no	—
378	C₂×D₈	C₂³×C₂₃	1	8	yes	—
379	C₂²×C₈	—	2	8	no	—
380	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
381	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
382	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
383	C₄⋊Q₈	—	2	2	no	—
384	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
385	#127	—	2	2	no	—
386	#127	C₂³×C₂₃	1	4	yes	—
387	#126	—	2	8	no	—
388	#125	—	2	2	no	—
389	#125	C₂³×C₂₃	1	4	yes	—
390	#124	—	2	8	no	—
391	#124	—	2	2	no	—
392	#124	C₂³×C₂₃	1	4	yes	—
393	#123	—	2	8	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
396	#121	—	2	8	no	—
397	#121	—	2	2	no	—
398	#121	—	2	4	no	—
399	#121	C₂³×C₂₃	1	4	yes	—
400	#121	C₂³×C₂₃	1	8	yes	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
403	C₄×D₄	—	2	8	no	—
404	C₄×D₄	—	2	8	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	4	no	—
407	C₄×D₄	C₂³×C₂₃	1	4	yes	—
408	C₄×D₄	C₂³×C₂₃	1	8	yes	—
409	#117	—	2	8	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₂²⋊C₄	—	2	1	no	—
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	—	2	2	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	2	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₄²	—	2	1	no	—
426	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
427	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
428	C₂×C₄²	D₄×C₂₃	1	4	yes	—
429	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
432	#107	C₂³×C₂₃	1	8	yes	—
433	#106	C₂³×C₂₃	1	8	yes	—
434	#103	C₂³×C₂₃	1	8	yes	—
435	#102	C₂³×C₂₃	1	8	yes	—
436	#102	C₂³×C₂₃	1	8	yes	—
437	#102	C₂³×C₂₃	1	8	yes	—
438	#97	C₂³×C₂₃	1	8	yes	—
439	#97	C₂³×C₂₃	1	4	yes	—
440	#97	C₂³×C₂₃	1	4	yes	—
441	C₂₃	C₂×#3180	1	1	yes	characteristic, direct product factor, Sylow
442	C₂⁴	—	2	1	no	—
443	C₂⁴	—	2	2	no	—
444	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
445	#41	—	4	32	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	4	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	4	4	no	—
450	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	4	4	no	—
461	C₂×D₄	—	4	4	no	—
462	C₂×D₄	—	4	8	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
465	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	8	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	4	8	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
484	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
485	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
487	SD₁₆	—	2	16	no	—
488	D₈	—	2	16	no	—
489	C₂×C₈	—	2	4	no	—
490	C₂×C₈	—	2	16	no	—
491	C₂×C₈	#41×C₂₃	1	8	yes	complemented
492	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
493	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
503	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
504	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	8	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	8	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	C₂×D₄×C₂₃	1	4	yes	—
514	C₄²	C₂×D₄×C₂₃	1	2	yes	—
515	C₄²	C₂×D₄×C₂₃	1	4	yes	—
516	C₄²	C₂×D₄×C₂₃	1	2	yes	—
517	C₄²	C₂⁴×C₂₃	1	2	yes	—
518	C₄²	C₂⁴×C₂₃	1	2	yes	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	4	no	—
523	C₂³	—	4	4	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	2	no	—
526	C₂³	—	4	4	no	—
527	C₂³	—	4	2	no	—
528	C₂³	—	4	4	no	—
529	C₂³	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
530	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
531	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
532	Q₈	—	2	4	no	—
533	Q₈	—	4	8	no	—
534	Q₈	—	4	2	no	—
535	D₄	—	2	4	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	4	no	—
538	D₄	—	4	8	no	—
539	D₄	—	4	8	no	—
540	D₄	—	4	16	no	—
541	D₄	—	4	8	no	—
542	D₄	—	4	2	no	—
543	D₄	—	4	4	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	8	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	16	no	—
558	C₂×C₄	—	4	8	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	2	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
567	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
569	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
570	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
571	C₈	—	2	4	no	—
572	C₈	C₂×#41×C₂₃	1	8	yes	complemented
573	C₂²	—	2	8	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	8	8	no	—
583	C₂²	—	8	4	no	—
584	C₂²	C₂₃×#532	1	2	yes	—
585	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
586	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
587	C₂²	C₂₃×#575	1	2	yes	—
588	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	4	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	2	no	—
598	C₄	C₂₃×#532	1	2	yes	—
599	C₄	C₂²×#41×C₂₃	1	2	yes	—
600	C₂	—	2	4	no	—
601	C₂	—	8	4	no	—
602	C₂	—	8	2	no	—
603	C₂	C₂₃×#3180	1	4	yes	direct product factor
604	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
605	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
606	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×#3180	C₂	direct (N × H)
4. C₂×C₂₃×#501	C₂	semidirect
5. C₂×C₂₃×#496	C₂	semidirect
6. C₂×C₂₃×#490	C₂	semidirect
7. C₂×C₂₃×#489	C₂	semidirect
8. C₂×C₂₃×#466	C₂	semidirect
9. C₂×C₂₃×#465	C₂	semidirect
10. C₂×C₂₃×#442	C₂	semidirect
11. C₂×C₂₃×#440	C₂	semidirect
12. C₂²×C₂₃×#97	C₂	semidirect
18. C₂₃×#532	C₂²	semidirect
22. C₂₃×#529	C₂²	semidirect
39. C₂₃×#501	C₂²	semidirect
40. C₂₃×#496	C₂²	semidirect
41. C₂₃×#490	C₂²	semidirect
42. C₂₃×#489	C₂²	semidirect
43. C₂₃×#466	C₂²	semidirect
44. C₂₃×#465	C₂²	semidirect
45. C₂₃×#442	C₂²	semidirect
46. C₂₃×#440	C₂²	semidirect
56. C₂×C₂₃×#97	C₂²	semidirect
187. C₂×C₈×C₂₃	#41	semidirect
215. C₂×#3180	C₂₃	direct (N × H)
269. C₈×C₂₃	C₂×#41	semidirect
270. C₂×#532	C₂×C₂₃	semidirect
271. C₂×#529	C₂×C₂₃	semidirect
272. #3180	C₂×C₂₃	direct (N × H)
273. C₂×#501	C₂×C₂₃	semidirect
274. C₂×#496	C₂×C₂₃	semidirect
275. C₂×#490	C₂×C₂₃	semidirect
276. C₂×#489	C₂×C₂₃	semidirect
277. C₂×#466	C₂×C₂₃	semidirect
278. C₂×#465	C₂×C₂₃	semidirect
279. C₂×#442	C₂×C₂₃	semidirect
280. C₂×#440	C₂×C₂₃	semidirect
281. C₂²×#97	C₂×C₂₃	semidirect
314. #532	C₂²×C₂₃	semidirect
318. #529	C₂²×C₂₃	semidirect
335. #501	C₂²×C₂₃	semidirect
336. #496	C₂²×C₂₃	semidirect
337. #490	C₂²×C₂₃	semidirect
338. #489	C₂²×C₂₃	semidirect
339. #466	C₂²×C₂₃	semidirect
340. #465	C₂²×C₂₃	semidirect
341. #442	C₂²×C₂₃	semidirect
342. #440	C₂²×C₂₃	semidirect
352. C₂×#97	C₂²×C₂₃	semidirect
356. C₂×C₂₃	#3180	direct (N × H)
441. C₂₃	C₂×#3180	direct (N × H)
491. C₂×C₈	#41×C₂₃	semidirect
572. C₈	C₂×#41×C₂₃	semidirect
603. C₂	C₂₃×#3180	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×#3180	2
C₂×C₂₃×#501	2
C₂×C₂₃×#496	2
C₂×C₂₃×#490	2
C₂×C₂₃×#489	2
C₂×C₂₃×#466	2
C₂×C₂₃×#465	2
C₂×C₂₃×#442	2
C₂×C₂₃×#440	2
C₂²×C₂₃×#97	2
C₂×#3180	23

Subgroups — C2×C23×#3180 #19208312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3180 #19208312