Subgroups — C₂×C₂₃×#3366 #19208930

Order 5888.

2198 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3366	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#490	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#478	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#459	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#442	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	—	2	1	no	—
14	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#554	—	2	4	no	—
17	C₂₃×#554	C₂²	1	8	yes	complemented
18	C₂₃×#550	—	2	4	no	—
19	C₂₃×#550	C₂²	1	8	yes	complemented
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#127	C₂²	1	2	yes	—
23	C₂×C₂₃×#125	C₂²	1	2	yes	—
24	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
26	C₂×C₂₃×#121	—	2	1	no	—
27	C₂×C₂₃×#121	—	2	2	no	—
28	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂₃×#490	C₂²	1	16	yes	complemented
35	C₂₃×#486	C₂²	1	8	yes	complemented
36	C₂₃×#478	C₂²	1	16	yes	complemented
37	C₂₃×#476	C₂²	1	8	yes	complemented
38	C₂₃×#459	C₂²	1	16	yes	complemented
39	C₂₃×#459	C₂²	1	8	yes	complemented
40	C₂₃×#446	C₂²	1	8	yes	complemented
41	C₂₃×#442	C₂²	1	16	yes	complemented
42	C₄×D₈×C₂₃	C₂²	1	8	yes	complemented
43	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#106	C₂²	1	2	yes	—
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
47	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×#102	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×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×#41×C₂₃	—	2	8	no	—
56	C₂×#41×C₂₃	—	4	8	no	—
57	C₂²×Q₈×C₂₃	—	2	1	no	—
58	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
59	C₂²×D₄×C₂₃	—	2	2	no	—
60	C₂²×D₄×C₂₃	—	2	1	no	—
61	C₂²×D₄×C₂₃	—	2	1	no	—
62	C₂²×D₄×C₂₃	—	2	2	no	—
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	—	2	2	no	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂³×C₄×C₂₃	—	2	2	no	—
70	C₂×D₈×C₂₃	—	2	4	no	—
71	C₂×D₈×C₂₃	—	2	8	no	—
72	C₂×D₈×C₂₃	C₂³	1	4	yes	—
73	C₂×D₈×C₂₃	C₂³	1	8	yes	—
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₂₃×#127	—	2	4	no	—
80	C₂₃×#127	—	2	2	no	—
81	C₂₃×#127	C₂³	1	4	yes	—
82	C₂₃×#127	C₂³	1	8	yes	—
83	C₂₃×#125	—	2	4	no	—
84	C₂₃×#125	C₂³	1	8	yes	—
85	C₂₃×C₂²⋊Q₈	—	2	8	no	—
86	C₂₃×C₂²⋊Q₈	—	2	16	no	—
87	C₂₃×#121	—	2	8	no	—
88	C₂₃×#121	—	2	16	no	—
89	C₄×Q₈×C₂₃	D₄	1	4	yes	—
90	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
91	C₄×D₄×C₂₃	—	2	4	no	—
92	C₄×D₄×C₂₃	—	2	2	no	—
93	C₄×D₄×C₂₃	—	2	4	no	—
94	C₄×D₄×C₂₃	—	2	2	no	—
95	C₄×D₄×C₂₃	C₂³	1	4	yes	—
96	C₄×D₄×C₂₃	C₂³	1	8	yes	—
97	C₄×D₄×C₂₃	C₂³	1	4	yes	—
98	C₄×D₄×C₂₃	C₂³	1	8	yes	—
99	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
103	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	D₄	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	1	no	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
112	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
114	C₂₃×#107	C₂³	1	4	yes	—
115	C₂₃×#106	C₂³	1	8	yes	—
116	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
117	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
118	C₂₃×#102	C₂³	1	4	yes	—
119	C₂₃×#102	C₂³	1	4	yes	—
120	C₂₃×#102	C₂³	1	8	yes	—
121	C₂₃×#102	C₂³	1	8	yes	—
122	C₂₃×#102	C₂³	1	8	yes	—
123	C₂₃×#102	C₂³	1	8	yes	—
124	C₂₃×#97	C₂³	1	8	yes	—
125	C₄×C₈×C₂₃	C₂³	1	4	yes	—
126	C₂⁴×C₂₃	—	2	1	no	—
127	C₂⁴×C₂₃	—	2	2	no	—
128	C₂⁴×C₂₃	—	2	1	no	—
129	#41×C₂₃	—	4	32	no	—
130	C₂×Q₈×C₂₃	—	2	4	no	—
131	C₂×Q₈×C₂₃	—	2	4	no	—
132	C₂×Q₈×C₂₃	—	8	2	no	—
133	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	2	2	no	—
136	C₂×D₄×C₂₃	—	2	1	no	—
137	C₂×D₄×C₂₃	—	2	8	no	—
138	C₂×D₄×C₂₃	—	2	4	no	—
139	C₂×D₄×C₂₃	—	2	4	no	—
140	C₂×D₄×C₂₃	—	2	8	no	—
141	C₂×D₄×C₂₃	—	2	2	no	—
142	C₂×D₄×C₂₃	—	2	4	no	—
143	C₂×D₄×C₂₃	—	2	2	no	—
144	C₂×D₄×C₂₃	—	2	2	no	—
145	C₂×D₄×C₂₃	—	2	4	no	—
146	C₂×D₄×C₂₃	—	4	8	no	—
147	C₂×D₄×C₂₃	—	4	4	no	—
148	C₂×D₄×C₂₃	—	8	4	no	—
149	C₂×D₄×C₂₃	—	8	2	no	—
150	C₂×D₄×C₂₃	#41	1	4	yes	complemented
151	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
152	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
153	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	2	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	2	8	no	—
159	C₂²×C₄×C₂₃	—	2	4	no	—
160	C₂²×C₄×C₂₃	—	2	8	no	—
161	C₂²×C₄×C₂₃	—	4	8	no	—
162	C₂²×C₄×C₂₃	—	4	4	no	—
163	C₂²×C₄×C₂₃	—	4	4	no	—
164	C₂²×C₄×C₂₃	—	8	4	no	—
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
170	D₈×C₂₃	—	2	8	no	—
171	D₈×C₂₃	—	2	16	no	—
172	C₂×C₈×C₂₃	—	2	2	no	—
173	C₂×C₈×C₂₃	—	2	4	no	—
174	C₂×C₈×C₂₃	—	2	2	no	—
175	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
176	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
177	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₂₃	—	2	8	no	—
179	C₄⋊C₄×C₂₃	—	2	4	no	—
180	C₄⋊C₄×C₂₃	—	2	8	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	2	no	—
184	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
185	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
186	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
188	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
189	C₂²⋊C₄×C₂₃	—	2	4	no	—
190	C₂²⋊C₄×C₂₃	—	2	8	no	—
191	C₂²⋊C₄×C₂₃	—	2	4	no	—
192	C₂²⋊C₄×C₂₃	—	2	8	no	—
193	C₄²×C₂₃	C₂×D₄	1	4	yes	—
194	C₄²×C₂₃	C₂×D₄	1	2	yes	—
195	C₄²×C₂₃	C₂⁴	1	4	yes	—
196	C₄²×C₂₃	C₂⁴	1	2	yes	—
197	C₂×#3366	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₂₃	—	2	4	no	—
199	C₂³×C₂₃	—	2	2	no	—
200	C₂³×C₂₃	—	2	4	no	—
201	C₂³×C₂₃	—	2	2	no	—
202	C₂³×C₂₃	—	4	2	no	—
203	C₂³×C₂₃	—	4	2	no	—
204	C₂³×C₂₃	—	4	4	no	—
205	C₂³×C₂₃	—	4	2	no	—
206	C₂³×C₂₃	—	4	4	no	—
207	C₂³×C₂₃	—	4	2	no	—
208	C₂³×C₂₃	—	4	2	no	—
209	C₂³×C₂₃	—	4	4	no	—
210	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
211	Q₈×C₂₃	—	2	8	no	—
212	Q₈×C₂₃	—	8	4	no	—
213	D₄×C₂₃	—	2	4	no	—
214	D₄×C₂₃	—	2	2	no	—
215	D₄×C₂₃	—	2	4	no	—
216	D₄×C₂₃	—	2	2	no	—
217	D₄×C₂₃	—	2	2	no	—
218	D₄×C₂₃	—	2	4	no	—
219	D₄×C₂₃	—	4	16	no	—
220	D₄×C₂₃	—	4	4	no	—
221	D₄×C₂₃	—	4	2	no	—
222	D₄×C₂₃	—	4	8	no	—
223	D₄×C₂₃	—	8	8	no	—
224	D₄×C₂₃	—	8	4	no	—
225	D₄×C₂₃	C₂×#41	1	4	yes	complemented
226	C₂×C₄×C₂₃	—	2	4	no	—
227	C₂×C₄×C₂₃	—	2	4	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	4	no	—
232	C₂×C₄×C₂₃	—	4	16	no	—
233	C₂×C₄×C₂₃	—	4	2	no	—
234	C₂×C₄×C₂₃	—	4	2	no	—
235	C₂×C₄×C₂₃	—	4	4	no	—
236	C₂×C₄×C₂₃	—	4	8	no	—
237	C₂×C₄×C₂₃	—	4	8	no	—
238	C₂×C₄×C₂₃	—	8	8	no	—
239	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
240	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
243	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
246	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
247	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
248	C₈×C₂₃	—	2	4	no	—
249	C₈×C₂₃	—	2	2	no	—
250	C₈×C₂₃	—	2	2	no	—
251	C₂×#554	C₂×C₂₃	1	1	yes	characteristic, complemented
252	C₂×#550	C₂×C₂₃	1	1	yes	characteristic, complemented
253	#3366	C₂×C₂₃	1	16	yes	direct product factor
254	C₂×#490	C₂×C₂₃	1	2	yes	complemented
255	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
256	C₂×#478	C₂×C₂₃	1	2	yes	complemented
257	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
258	C₂×#459	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₂×#459	C₂×C₂₃	1	2	yes	complemented
260	C₂×#446	C₂×C₂₃	1	1	yes	characteristic, complemented
261	C₂×#442	C₂×C₂₃	1	2	yes	complemented
262	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
263	C₂²×C₂₃	—	2	4	no	—
264	C₂²×C₂₃	—	4	8	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	—	4	2	no	—
267	C₂²×C₂₃	—	4	4	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	—	4	4	no	—
270	C₂²×C₂₃	—	4	2	no	—
271	C₂²×C₂₃	—	4	2	no	—
272	C₂²×C₂₃	—	4	4	no	—
273	C₂²×C₂₃	—	8	8	no	—
274	C₂²×C₂₃	—	8	4	no	—
275	C₂²×C₂₃	#548	1	2	yes	—
276	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
277	C₂²×C₂₃	#576	1	2	yes	—
278	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
279	C₄×C₂₃	—	2	4	no	—
280	C₄×C₂₃	—	2	4	no	—
281	C₄×C₂₃	—	2	2	no	—
282	C₄×C₂₃	—	4	2	no	—
283	C₄×C₂₃	—	4	2	no	—
284	C₄×C₂₃	—	4	4	no	—
285	C₄×C₂₃	#548	1	2	yes	—
286	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
287	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
288	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
289	C₂²×#41	—	2	1	no	—
290	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
291	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
292	#554	—	2	4	no	—
293	#554	C₂²×C₂₃	1	8	yes	complemented
294	#550	—	2	4	no	—
295	#550	C₂²×C₂₃	1	8	yes	complemented
296	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
298	C₂×#127	C₂²×C₂₃	1	2	yes	—
299	C₂×#125	C₂²×C₂₃	1	2	yes	—
300	C₂×C₂²⋊Q₈	—	2	1	no	—
301	C₂×C₂²⋊Q₈	—	2	2	no	—
302	C₂×#121	—	2	1	no	—
303	C₂×#121	—	2	2	no	—
304	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
306	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
309	C₂²×C₄⋊C₄	—	2	1	no	—
310	#490	C₂²×C₂₃	1	16	yes	complemented
311	#486	C₂²×C₂₃	1	8	yes	complemented
312	#478	C₂²×C₂₃	1	16	yes	complemented
313	#476	C₂²×C₂₃	1	8	yes	complemented
314	#459	C₂²×C₂₃	1	16	yes	complemented
315	#459	C₂²×C₂₃	1	8	yes	complemented
316	#446	C₂²×C₂₃	1	8	yes	complemented
317	#442	C₂²×C₂₃	1	16	yes	complemented
318	C₄×D₈	C₂²×C₂₃	1	8	yes	complemented
319	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×#106	C₂²×C₂₃	1	2	yes	—
321	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
322	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×#102	C₂²×C₂₃	1	2	yes	—
324	C₂×#102	C₂²×C₂₃	1	2	yes	—
325	C₂×#102	C₂²×C₂₃	1	2	yes	—
326	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#102	C₂²×C₂₃	1	2	yes	—
328	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#97	C₂²×C₂₃	1	2	yes	—
330	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×C₂₃	—	4	4	no	—
332	C₂×C₂₃	—	8	4	no	—
333	C₂×C₂₃	—	8	2	no	—
334	C₂×C₂₃	#3366	1	4	yes	direct product factor
335	C₂×C₂₃	C₂×#548	1	1	yes	characteristic
336	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
337	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
338	C₂×#41	—	2	8	no	—
339	C₂×#41	—	4	8	no	—
340	C₂²×Q₈	—	2	1	no	—
341	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
342	C₂²×D₄	—	2	2	no	—
343	C₂²×D₄	—	2	1	no	—
344	C₂²×D₄	—	2	1	no	—
345	C₂²×D₄	—	2	2	no	—
346	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
347	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
348	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	—	2	2	no	—
351	C₂³×C₄	—	2	1	no	—
352	C₂³×C₄	—	2	2	no	—
353	C₂×D₈	—	2	4	no	—
354	C₂×D₈	—	2	8	no	—
355	C₂×D₈	C₂³×C₂₃	1	4	yes	—
356	C₂×D₈	C₂³×C₂₃	1	8	yes	—
357	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
358	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
359	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
360	C₄⋊Q₈	—	2	2	no	—
361	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
362	#127	—	2	4	no	—
363	#127	—	2	2	no	—
364	#127	C₂³×C₂₃	1	4	yes	—
365	#127	C₂³×C₂₃	1	8	yes	—
366	#125	—	2	4	no	—
367	#125	C₂³×C₂₃	1	8	yes	—
368	C₂²⋊Q₈	—	2	8	no	—
369	C₂²⋊Q₈	—	2	16	no	—
370	#121	—	2	8	no	—
371	#121	—	2	16	no	—
372	C₄×Q₈	D₄×C₂₃	1	4	yes	—
373	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	4	no	—
377	C₄×D₄	—	2	2	no	—
378	C₄×D₄	C₂³×C₂₃	1	4	yes	—
379	C₄×D₄	C₂³×C₂₃	1	8	yes	—
380	C₄×D₄	C₂³×C₂₃	1	4	yes	—
381	C₄×D₄	C₂³×C₂₃	1	8	yes	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	—	2	8	no	—
386	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
387	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
391	C₂×C₂²⋊C₄	—	2	1	no	—
392	C₂×C₂²⋊C₄	—	2	2	no	—
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	2	no	—
395	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
396	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
397	#107	C₂³×C₂₃	1	4	yes	—
398	#106	C₂³×C₂₃	1	8	yes	—
399	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
400	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
401	#102	C₂³×C₂₃	1	4	yes	—
402	#102	C₂³×C₂₃	1	4	yes	—
403	#102	C₂³×C₂₃	1	8	yes	—
404	#102	C₂³×C₂₃	1	8	yes	—
405	#102	C₂³×C₂₃	1	8	yes	—
406	#102	C₂³×C₂₃	1	8	yes	—
407	#97	C₂³×C₂₃	1	8	yes	—
408	C₄×C₈	C₂³×C₂₃	1	4	yes	—
409	C₂₃	C₂×#3366	1	1	yes	characteristic, direct product factor, Sylow
410	C₂⁴	—	2	1	no	—
411	C₂⁴	—	2	2	no	—
412	C₂⁴	—	2	1	no	—
413	#41	—	4	32	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	8	2	no	—
417	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	8	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	4	8	no	—
431	C₂×D₄	—	4	4	no	—
432	C₂×D₄	—	8	4	no	—
433	C₂×D₄	—	8	2	no	—
434	C₂×D₄	#41×C₂₃	1	4	yes	complemented
435	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
436	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
437	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	8	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	8	no	—
445	C₂²×C₄	—	4	8	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	8	4	no	—
449	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
454	D₈	—	2	8	no	—
455	D₈	—	2	16	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	4	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
460	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
461	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
469	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
470	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
471	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
472	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	8	no	—
477	C₄²	C₂×D₄×C₂₃	1	4	yes	—
478	C₄²	C₂×D₄×C₂₃	1	2	yes	—
479	C₄²	C₂⁴×C₂₃	1	2	yes	—
480	C₄²	C₂⁴×C₂₃	1	4	yes	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	—	4	2	no	—
487	C₂³	—	4	4	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	4	4	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	4	2	no	—
492	C₂³	—	4	4	no	—
493	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
494	Q₈	—	2	8	no	—
495	Q₈	—	8	4	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	2	no	—
498	D₄	—	2	4	no	—
499	D₄	—	2	2	no	—
500	D₄	—	2	2	no	—
501	D₄	—	2	4	no	—
502	D₄	—	4	16	no	—
503	D₄	—	4	4	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	8	no	—
506	D₄	—	8	8	no	—
507	D₄	—	8	4	no	—
508	D₄	C₂×#41×C₂₃	1	4	yes	complemented
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	4	16	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	8	8	no	—
522	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
523	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
524	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
525	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
526	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
528	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
529	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
530	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
531	C₈	—	2	2	no	—
532	C₈	—	2	4	no	—
533	C₈	—	2	2	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	8	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	8	8	no	—
545	C₂²	—	8	4	no	—
546	C₂²	C₂₃×#548	1	2	yes	—
547	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
548	C₂²	C₂₃×#576	1	2	yes	—
549	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
550	C₄	—	2	4	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	4	no	—
556	C₄	C₂₃×#548	1	2	yes	—
557	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
558	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
559	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
560	C₂	—	4	4	no	—
561	C₂	—	8	4	no	—
562	C₂	—	8	2	no	—
563	C₂	C₂₃×#3366	1	4	yes	direct product factor
564	C₂	C₂×C₂₃×#548	1	1	yes	characteristic
565	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
566	C₂	C₂×C₂₃×#576	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 ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#554	C₂	semidirect
2. C₂×C₂₃×#550	C₂	semidirect
3. C₂₃×#3366	C₂	direct (N × H)
4. C₂×C₂₃×#490	C₂	semidirect
5. C₂×C₂₃×#486	C₂	semidirect
6. C₂×C₂₃×#478	C₂	semidirect
7. C₂×C₂₃×#476	C₂	semidirect
8. C₂×C₂₃×#459	C₂	semidirect
9. C₂×C₂₃×#459	C₂	semidirect
10. C₂×C₂₃×#446	C₂	semidirect
11. C₂×C₂₃×#442	C₂	semidirect
12. C₂×C₄×D₈×C₂₃	C₂	semidirect
17. C₂₃×#554	C₂²	semidirect
19. C₂₃×#550	C₂²	semidirect
34. C₂₃×#490	C₂²	semidirect
35. C₂₃×#486	C₂²	semidirect
36. C₂₃×#478	C₂²	semidirect
37. C₂₃×#476	C₂²	semidirect
38. C₂₃×#459	C₂²	semidirect
39. C₂₃×#459	C₂²	semidirect
40. C₂₃×#446	C₂²	semidirect
41. C₂₃×#442	C₂²	semidirect
42. C₄×D₈×C₂₃	C₂²	semidirect
150. C₂×D₄×C₂₃	#41	semidirect
197. C₂×#3366	C₂₃	direct (N × H)
225. D₄×C₂₃	C₂×#41	semidirect
251. C₂×#554	C₂×C₂₃	semidirect
252. C₂×#550	C₂×C₂₃	semidirect
253. #3366	C₂×C₂₃	direct (N × H)
254. C₂×#490	C₂×C₂₃	semidirect
255. C₂×#486	C₂×C₂₃	semidirect
256. C₂×#478	C₂×C₂₃	semidirect
257. C₂×#476	C₂×C₂₃	semidirect
258. C₂×#459	C₂×C₂₃	semidirect
259. C₂×#459	C₂×C₂₃	semidirect
260. C₂×#446	C₂×C₂₃	semidirect
261. C₂×#442	C₂×C₂₃	semidirect
262. C₂×C₄×D₈	C₂×C₂₃	semidirect
293. #554	C₂²×C₂₃	semidirect
295. #550	C₂²×C₂₃	semidirect
310. #490	C₂²×C₂₃	semidirect
311. #486	C₂²×C₂₃	semidirect
312. #478	C₂²×C₂₃	semidirect
313. #476	C₂²×C₂₃	semidirect
314. #459	C₂²×C₂₃	semidirect
315. #459	C₂²×C₂₃	semidirect
316. #446	C₂²×C₂₃	semidirect
317. #442	C₂²×C₂₃	semidirect
318. C₄×D₈	C₂²×C₂₃	semidirect
334. C₂×C₂₃	#3366	direct (N × H)
409. C₂₃	C₂×#3366	direct (N × H)
434. C₂×D₄	#41×C₂₃	semidirect
508. D₄	C₂×#41×C₂₃	semidirect
563. C₂	C₂₃×#3366	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×#554	2
C₂×C₂₃×#550	2
C₂₃×#3366	2
C₂×C₂₃×#490	2
C₂×C₂₃×#486	2
C₂×C₂₃×#478	2
C₂×C₂₃×#476	2
C₂×C₂₃×#459	2
C₂×C₂₃×#446	2
C₂×C₂₃×#442	2
C₂×C₄×D₈×C₂₃	2
C₂×#3366	23

Subgroups — C2×C23×#3366 #19208930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3366 #19208930