Subgroups — C₂×C₂₃×#3217 #19208411

Order 5888.

2834 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3217	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#496	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#490	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#460	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#459	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#450	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#447	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₂₃	—	2	1	no	—
12	C₂³×D₄×C₂₃	—	2	1	no	—
13	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
15	C₂²×D₈×C₂₃	C₂²	1	4	yes	complemented
16	C₂×C₂₃×#127	—	2	1	no	—
17	C₂×C₂₃×#127	—	2	1	no	—
18	C₂×C₂₃×#127	—	2	4	no	—
19	C₂×C₂₃×#127	C₂²	1	2	yes	—
20	C₂×C₂₃×#127	C₂²	1	2	yes	—
21	C₂×C₂₃×#127	C₂²	1	8	yes	complemented
22	C₂₃×#529	—	2	4	no	—
23	C₂₃×#529	C₂²	1	8	yes	complemented
24	C₂×C₂₃×#126	—	2	1	no	—
25	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₂₃×#123	—	2	1	no	—
28	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄×D₄×C₂₃	—	2	1	no	—
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₂₃×#117	—	2	1	no	—
34	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#496	C₂²	1	16	yes	complemented
36	C₂₃×#490	C₂²	1	16	yes	complemented
37	C₂₃×#460	C₂²	1	16	yes	complemented
38	C₂₃×#459	C₂²	1	16	yes	complemented
39	C₂₃×#450	C₂²	1	16	yes	complemented
40	C₂₃×#447	C₂²	1	16	yes	complemented
41	C₂₃×#432	C₂²	1	8	yes	complemented
42	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
43	C₂×C₂₃×#102	C₂²	1	4	yes	complemented
44	C₂×C₂₃×#102	C₂²	1	4	yes	complemented
45	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
46	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
47	C₂⁵×C₂₃	—	2	2	no	—
48	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	—	2	2	no	—
51	C₂²×D₄×C₂₃	—	2	2	no	—
52	C₂²×D₄×C₂₃	—	2	2	no	—
53	C₂²×D₄×C₂₃	—	2	2	no	—
54	C₂²×D₄×C₂₃	—	2	8	no	—
55	C₂²×D₄×C₂₃	—	2	8	no	—
56	C₂²×D₄×C₂₃	—	2	4	no	—
57	C₂²×D₄×C₂₃	—	2	2	no	—
58	C₂²×D₄×C₂₃	—	4	8	no	—
59	C₂²×D₄×C₂₃	—	4	8	no	—
60	C₂²×D₄×C₂₃	—	4	8	no	—
61	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
63	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×SD₁₆×C₂₃	—	4	8	no	—
69	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	complemented
70	C₂×D₈×C₂₃	—	4	8	no	—
71	C₂×D₈×C₂₃	C₂³	1	16	yes	complemented
72	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
73	C₂₃×#127	—	2	4	no	—
74	C₂₃×#127	—	2	4	no	—
75	C₂₃×#127	—	2	8	no	—
76	C₂₃×#127	—	2	8	no	—
77	C₂₃×#127	C₂³	1	8	yes	—
78	C₂₃×#127	C₂³	1	8	yes	—
79	C₂₃×#126	—	2	8	no	—
80	C₂₃×#125	—	2	2	no	—
81	C₂₃×#125	C₂³	1	4	yes	complemented
82	C₂₃×#124	—	2	2	no	—
83	C₂₃×#124	C₂³	1	4	yes	complemented
84	C₂₃×#123	—	2	8	no	—
85	C₂₃×C₂²⋊Q₈	—	2	2	no	—
86	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
87	C₂₃×#121	—	2	2	no	—
88	C₂₃×#121	C₂³	1	4	yes	—
89	C₄×Q₈×C₂₃	—	2	2	no	—
90	C₄×Q₈×C₂₃	C₂³	1	4	yes	complemented
91	C₄×D₄×C₂₃	—	2	8	no	—
92	C₄×D₄×C₂₃	—	2	2	no	—
93	C₄×D₄×C₂₃	C₂³	1	4	yes	complemented
94	C₂₃×#117	—	2	8	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	D₄	1	4	yes	—
110	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
113	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
114	C₂₃×#102	C₂³	1	16	yes	complemented
115	C₂₃×#102	C₂³	1	16	yes	complemented
116	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
117	C₂₃×#97	C₂³	1	8	yes	complemented
118	C₂⁴×C₂₃	—	2	8	no	—
119	C₂⁴×C₂₃	—	2	1	no	—
120	C₂⁴×C₂₃	—	2	4	no	—
121	C₂⁴×C₂₃	—	4	8	no	—
122	C₂⁴×C₂₃	—	4	8	no	—
123	C₂⁴×C₂₃	—	4	8	no	—
124	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Q₈×C₂₃	—	2	1	no	—
126	C₂×Q₈×C₂₃	—	2	4	no	—
127	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
128	C₂×D₄×C₂₃	—	2	4	no	—
129	C₂×D₄×C₂₃	—	2	1	no	—
130	C₂×D₄×C₂₃	—	2	8	no	—
131	C₂×D₄×C₂₃	—	2	4	no	—
132	C₂×D₄×C₂₃	—	2	8	no	—
133	C₂×D₄×C₂₃	—	2	8	no	—
134	C₂×D₄×C₂₃	—	2	8	no	—
135	C₂×D₄×C₂₃	—	2	8	no	—
136	C₂×D₄×C₂₃	—	2	4	no	—
137	C₂×D₄×C₂₃	—	2	8	no	—
138	C₂×D₄×C₂₃	—	2	8	no	—
139	C₂×D₄×C₂₃	—	4	16	no	—
140	C₂×D₄×C₂₃	—	4	8	no	—
141	C₂×D₄×C₂₃	—	4	8	no	—
142	C₂×D₄×C₂₃	—	4	8	no	—
143	C₂×D₄×C₂₃	—	4	8	no	—
144	C₂×D₄×C₂₃	—	4	32	no	—
145	C₂×D₄×C₂₃	—	4	16	no	—
146	C₂×D₄×C₂₃	—	4	32	no	—
147	C₂×D₄×C₂₃	—	4	4	no	—
148	C₂×D₄×C₂₃	—	4	8	no	—
149	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
150	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₂₃	—	2	4	no	—
152	C₂²×C₄×C₂₃	—	2	4	no	—
153	C₂²×C₄×C₂₃	—	2	4	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	4	4	no	—
159	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
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	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
169	SD₁₆×C₂₃	—	4	16	no	—
170	D₈×C₂₃	—	4	16	no	—
171	C₂×C₈×C₂₃	—	2	8	no	—
172	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
173	C₄⋊C₄×C₂₃	—	2	4	no	—
174	C₄⋊C₄×C₂₃	—	2	4	no	—
175	C₄⋊C₄×C₂₃	—	2	4	no	—
176	C₄⋊C₄×C₂₃	—	2	4	no	—
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
180	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
181	C₂²⋊C₄×C₂₃	—	2	4	no	—
182	C₂²⋊C₄×C₂₃	—	2	4	no	—
183	C₂²⋊C₄×C₂₃	—	2	4	no	—
184	C₂²⋊C₄×C₂₃	—	2	4	no	—
185	C₂²⋊C₄×C₂₃	—	2	4	no	—
186	C₄²×C₂₃	—	2	4	no	—
187	C₄²×C₂₃	C₂×D₄	1	4	yes	—
188	C₄²×C₂₃	C₂×D₄	1	2	yes	—
189	C₄²×C₂₃	C₂×D₄	1	2	yes	—
190	C₄²×C₂₃	C₂×D₄	1	4	yes	—
191	C₄²×C₂₃	C₂⁴	1	2	yes	—
192	C₄²×C₂₃	C₂⁴	1	2	yes	—
193	C₂×#3217	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₂₃	—	2	2	no	—
195	C₂³×C₂₃	—	2	2	no	—
196	C₂³×C₂₃	—	2	2	no	—
197	C₂³×C₂₃	—	2	8	no	—
198	C₂³×C₂₃	—	4	16	no	—
199	C₂³×C₂₃	—	4	16	no	—
200	C₂³×C₂₃	—	4	8	no	—
201	C₂³×C₂₃	—	4	2	no	—
202	C₂³×C₂₃	—	4	2	no	—
203	C₂³×C₂₃	—	4	16	no	—
204	C₂³×C₂₃	—	4	8	no	—
205	C₂³×C₂₃	—	4	8	no	—
206	C₂³×C₂₃	—	4	2	no	—
207	C₂³×C₂₃	—	8	32	no	—
208	C₂³×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
209	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
210	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
211	Q₈×C₂₃	—	2	4	no	—
212	Q₈×C₂₃	—	4	2	no	—
213	D₄×C₂₃	—	2	4	no	—
214	D₄×C₂₃	—	2	8	no	—
215	D₄×C₂₃	—	2	8	no	—
216	D₄×C₂₃	—	4	16	no	—
217	D₄×C₂₃	—	4	16	no	—
218	D₄×C₂₃	—	4	16	no	—
219	D₄×C₂₃	—	4	16	no	—
220	D₄×C₂₃	—	4	8	no	—
221	D₄×C₂₃	—	4	2	no	—
222	D₄×C₂₃	—	4	8	no	—
223	D₄×C₂₃	—	4	16	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	8	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	4	no	—
230	C₂×C₄×C₂₃	—	2	8	no	—
231	C₂×C₄×C₂₃	—	2	4	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	2	2	no	—
235	C₂×C₄×C₂₃	—	4	2	no	—
236	C₂×C₄×C₂₃	—	4	2	no	—
237	C₂×C₄×C₂₃	—	4	2	no	—
238	C₂×C₄×C₂₃	—	4	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	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	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
247	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
248	C₈×C₂₃	—	2	8	no	—
249	C₂²×#127	C₂×C₂₃	1	1	yes	characteristic, complemented
250	C₂×#529	C₂×C₂₃	1	1	yes	characteristic, complemented
251	#3217	C₂×C₂₃	1	16	yes	direct product factor
252	C₂×#496	C₂×C₂₃	1	2	yes	complemented
253	C₂×#490	C₂×C₂₃	1	2	yes	complemented
254	C₂×#460	C₂×C₂₃	1	2	yes	complemented
255	C₂×#459	C₂×C₂₃	1	2	yes	complemented
256	C₂×#450	C₂×C₂₃	1	2	yes	complemented
257	C₂×#447	C₂×C₂₃	1	2	yes	complemented
258	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₂²×C₂₃	—	2	2	no	—
260	C₂²×C₂₃	—	2	8	no	—
261	C₂²×C₂₃	—	2	2	no	—
262	C₂²×C₂₃	—	4	8	no	—
263	C₂²×C₂₃	—	4	2	no	—
264	C₂²×C₂₃	—	4	2	no	—
265	C₂²×C₂₃	—	4	8	no	—
266	C₂²×C₂₃	—	4	8	no	—
267	C₂²×C₂₃	—	4	2	no	—
268	C₂²×C₂₃	—	8	32	no	—
269	C₂²×C₂₃	—	8	4	no	—
270	C₂²×C₂₃	—	8	16	no	—
271	C₂²×C₂₃	#535	1	2	yes	—
272	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
273	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
274	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
275	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
276	C₄×C₂₃	—	2	2	no	—
277	C₄×C₂₃	—	2	2	no	—
278	C₄×C₂₃	—	2	2	no	—
279	C₄×C₂₃	—	2	2	no	—
280	C₄×C₂₃	—	4	2	no	—
281	C₄×C₂₃	—	4	2	no	—
282	C₄×C₂₃	—	4	2	no	—
283	C₄×C₂₃	#535	1	2	yes	—
284	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
285	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
286	C₂³×D₄	—	2	1	no	—
287	C₂³×D₄	—	2	1	no	—
288	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
289	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
290	C₂²×D₈	C₂²×C₂₃	1	4	yes	complemented
291	C₂×#127	—	2	1	no	—
292	C₂×#127	—	2	1	no	—
293	C₂×#127	—	2	4	no	—
294	C₂×#127	C₂²×C₂₃	1	2	yes	—
295	C₂×#127	C₂²×C₂₃	1	2	yes	—
296	C₂×#127	C₂²×C₂₃	1	8	yes	complemented
297	#529	—	2	4	no	—
298	#529	C₂²×C₂₃	1	8	yes	complemented
299	C₂×#126	—	2	1	no	—
300	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic, complemented
301	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
302	C₂×#123	—	2	1	no	—
303	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
306	C₂×C₄×D₄	—	2	1	no	—
307	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
308	C₂×#117	—	2	1	no	—
309	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
310	#496	C₂²×C₂₃	1	16	yes	complemented
311	#490	C₂²×C₂₃	1	16	yes	complemented
312	#460	C₂²×C₂₃	1	16	yes	complemented
313	#459	C₂²×C₂₃	1	16	yes	complemented
314	#450	C₂²×C₂₃	1	16	yes	complemented
315	#447	C₂²×C₂₃	1	16	yes	complemented
316	#432	C₂²×C₂₃	1	8	yes	complemented
317	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
318	C₂×#102	C₂²×C₂₃	1	4	yes	complemented
319	C₂×#102	C₂²×C₂₃	1	4	yes	complemented
320	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
321	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
322	C₂×C₂₃	—	2	4	no	—
323	C₂×C₂₃	—	8	2	no	—
324	C₂×C₂₃	—	8	8	no	—
325	C₂×C₂₃	#3217	1	4	yes	direct product factor
326	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
327	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
329	C₂⁵	—	2	2	no	—
330	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	2	no	—
333	C₂²×D₄	—	2	4	no	—
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	—	2	2	no	—
337	C₂²×D₄	—	2	2	no	—
338	C₂²×D₄	—	2	8	no	—
339	C₂²×D₄	—	2	8	no	—
340	C₂²×D₄	—	4	8	no	—
341	C₂²×D₄	—	4	8	no	—
342	C₂²×D₄	—	4	8	no	—
343	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
344	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
345	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
346	C₂³×C₄	—	2	1	no	—
347	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
348	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
350	C₂×SD₁₆	—	4	8	no	—
351	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	complemented
352	C₂×D₈	—	4	8	no	—
353	C₂×D₈	C₂³×C₂₃	1	16	yes	complemented
354	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
355	#127	—	2	4	no	—
356	#127	—	2	4	no	—
357	#127	—	2	8	no	—
358	#127	—	2	8	no	—
359	#127	C₂³×C₂₃	1	8	yes	—
360	#127	C₂³×C₂₃	1	8	yes	—
361	#126	—	2	8	no	—
362	#125	—	2	2	no	—
363	#125	C₂³×C₂₃	1	4	yes	complemented
364	#124	—	2	2	no	—
365	#124	C₂³×C₂₃	1	4	yes	complemented
366	#123	—	2	8	no	—
367	C₂²⋊Q₈	—	2	2	no	—
368	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
369	#121	—	2	2	no	—
370	#121	C₂³×C₂₃	1	4	yes	—
371	C₄×Q₈	—	2	2	no	—
372	C₄×Q₈	C₂³×C₂₃	1	4	yes	complemented
373	C₄×D₄	—	2	8	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	C₂³×C₂₃	1	4	yes	complemented
376	#117	—	2	8	no	—
377	C₂×C₄⋊C₄	—	2	1	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	C₂³×C₂₃	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
390	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
391	C₂×C₄²	D₄×C₂₃	1	4	yes	—
392	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
395	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
396	#102	C₂³×C₂₃	1	16	yes	complemented
397	#102	C₂³×C₂₃	1	16	yes	complemented
398	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
399	#97	C₂³×C₂₃	1	8	yes	complemented
400	C₂₃	C₂×#3217	1	1	yes	characteristic, direct product factor, Sylow
401	C₂⁴	—	2	1	no	—
402	C₂⁴	—	2	8	no	—
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	4	8	no	—
405	C₂⁴	—	4	8	no	—
406	C₂⁴	—	4	8	no	—
407	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	4	16	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	8	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	4	32	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂×D₄	—	4	32	no	—
429	C₂×D₄	—	4	16	no	—
430	C₂×D₄	—	4	4	no	—
431	C₂×D₄	—	4	8	no	—
432	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
433	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	4	4	no	—
442	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
444	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
448	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
449	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
452	SD₁₆	—	4	16	no	—
453	D₈	—	4	16	no	—
454	C₂×C₈	—	2	8	no	—
455	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
463	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₄²	—	2	4	no	—
470	C₄²	C₂×D₄×C₂₃	1	4	yes	—
471	C₄²	C₂×D₄×C₂₃	1	2	yes	—
472	C₄²	C₂×D₄×C₂₃	1	2	yes	—
473	C₄²	C₂×D₄×C₂₃	1	4	yes	—
474	C₄²	C₂⁴×C₂₃	1	2	yes	—
475	C₄²	C₂⁴×C₂₃	1	2	yes	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	8	no	—
480	C₂³	—	4	16	no	—
481	C₂³	—	4	16	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	4	16	no	—
484	C₂³	—	4	8	no	—
485	C₂³	—	4	8	no	—
486	C₂³	—	4	2	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	8	32	no	—
490	C₂³	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
491	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
492	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
493	Q₈	—	2	4	no	—
494	Q₈	—	4	2	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	8	no	—
497	D₄	—	2	8	no	—
498	D₄	—	4	16	no	—
499	D₄	—	4	16	no	—
500	D₄	—	4	16	no	—
501	D₄	—	4	16	no	—
502	D₄	—	4	8	no	—
503	D₄	—	4	16	no	—
504	D₄	—	4	8	no	—
505	D₄	—	4	2	no	—
506	C₂×C₄	—	2	8	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
522	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
523	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
524	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
525	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
526	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
527	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
528	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
529	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
530	C₈	—	2	8	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	4	8	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	8	no	—
538	C₂²	—	4	8	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	8	32	no	—
541	C₂²	—	8	16	no	—
542	C₂²	—	8	4	no	—
543	C₂²	C₂₃×#535	1	2	yes	—
544	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
545	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
546	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
547	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	2	no	—
551	C₄	—	2	2	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	C₂₃×#535	1	2	yes	—
556	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
557	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
558	C₂	—	2	4	no	—
559	C₂	—	8	8	no	—
560	C₂	—	8	2	no	—
561	C₂	C₂₃×#3217	1	4	yes	direct product factor
562	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
563	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
564	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	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. C₂²×C₂₃×#127	C₂	semidirect
2. C₂×C₂₃×#529	C₂	semidirect
3. C₂₃×#3217	C₂	direct (N × H)
4. C₂×C₂₃×#496	C₂	semidirect
5. C₂×C₂₃×#490	C₂	semidirect
6. C₂×C₂₃×#460	C₂	semidirect
7. C₂×C₂₃×#459	C₂	semidirect
8. C₂×C₂₃×#450	C₂	semidirect
9. C₂×C₂₃×#447	C₂	semidirect
10. C₂×C₂₃×#432	C₂	semidirect
14. C₂²×SD₁₆×C₂₃	C₂²	semidirect
15. C₂²×D₈×C₂₃	C₂²	semidirect
21. C₂×C₂₃×#127	C₂²	semidirect
23. C₂₃×#529	C₂²	semidirect
25. C₂×C₂₃×#125	C₂²	semidirect
26. C₂×C₂₃×#124	C₂²	semidirect
30. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
32. C₂×C₄×D₄×C₂₃	C₂²	semidirect
35. C₂₃×#496	C₂²	semidirect
36. C₂₃×#490	C₂²	semidirect
37. C₂₃×#460	C₂²	semidirect
38. C₂₃×#459	C₂²	semidirect
39. C₂₃×#450	C₂²	semidirect
40. C₂₃×#447	C₂²	semidirect
41. C₂₃×#432	C₂²	semidirect
42. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
43. C₂×C₂₃×#102	C₂²	semidirect
44. C₂×C₂₃×#102	C₂²	semidirect
46. C₂×C₂₃×#97	C₂²	semidirect
69. C₂×SD₁₆×C₂₃	C₂³	semidirect
71. C₂×D₈×C₂₃	C₂³	semidirect
81. C₂₃×#125	C₂³	semidirect
83. C₂₃×#124	C₂³	semidirect
90. C₄×Q₈×C₂₃	C₂³	semidirect
93. C₄×D₄×C₂₃	C₂³	semidirect
113. C₂₃×C₄⋊C₈	C₂³	semidirect
114. C₂₃×#102	C₂³	semidirect
115. C₂₃×#102	C₂³	semidirect
117. C₂₃×#97	C₂³	semidirect
193. C₂×#3217	C₂₃	direct (N × H)
249. C₂²×#127	C₂×C₂₃	semidirect
250. C₂×#529	C₂×C₂₃	semidirect
251. #3217	C₂×C₂₃	direct (N × H)
252. C₂×#496	C₂×C₂₃	semidirect
253. C₂×#490	C₂×C₂₃	semidirect
254. C₂×#460	C₂×C₂₃	semidirect
255. C₂×#459	C₂×C₂₃	semidirect
256. C₂×#450	C₂×C₂₃	semidirect
257. C₂×#447	C₂×C₂₃	semidirect
258. C₂×#432	C₂×C₂₃	semidirect
289. C₂²×SD₁₆	C₂²×C₂₃	semidirect
290. C₂²×D₈	C₂²×C₂₃	semidirect
296. C₂×#127	C₂²×C₂₃	semidirect
298. #529	C₂²×C₂₃	semidirect
300. C₂×#125	C₂²×C₂₃	semidirect
301. C₂×#124	C₂²×C₂₃	semidirect
305. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
307. C₂×C₄×D₄	C₂²×C₂₃	semidirect
310. #496	C₂²×C₂₃	semidirect
311. #490	C₂²×C₂₃	semidirect
312. #460	C₂²×C₂₃	semidirect
313. #459	C₂²×C₂₃	semidirect
314. #450	C₂²×C₂₃	semidirect
315. #447	C₂²×C₂₃	semidirect
316. #432	C₂²×C₂₃	semidirect
317. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
318. C₂×#102	C₂²×C₂₃	semidirect
319. C₂×#102	C₂²×C₂₃	semidirect
321. C₂×#97	C₂²×C₂₃	semidirect
325. C₂×C₂₃	#3217	direct (N × H)
351. C₂×SD₁₆	C₂³×C₂₃	semidirect
353. C₂×D₈	C₂³×C₂₃	semidirect
363. #125	C₂³×C₂₃	semidirect
365. #124	C₂³×C₂₃	semidirect
372. C₄×Q₈	C₂³×C₂₃	semidirect
375. C₄×D₄	C₂³×C₂₃	semidirect
395. C₄⋊C₈	C₂³×C₂₃	semidirect
396. #102	C₂³×C₂₃	semidirect
397. #102	C₂³×C₂₃	semidirect
399. #97	C₂³×C₂₃	semidirect
400. C₂₃	C₂×#3217	direct (N × H)
561. C₂	C₂₃×#3217	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₂₃×#127	2
C₂×C₂₃×#529	2
C₂₃×#3217	2
C₂×C₂₃×#496	2
C₂×C₂₃×#490	2
C₂×C₂₃×#460	2
C₂×C₂₃×#459	2
C₂×C₂₃×#450	2
C₂×C₂₃×#447	2
C₂×C₂₃×#432	2
C₂×#3217	23

Subgroups — C2×C23×#3217 #19208411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3217 #19208411