Subgroups — C₂×C₂₃×#1764 #19168044

Order 5888.

1558 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#104	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#416	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1764	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#339	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#131	—	2	4	no	—
11	C₂×C₂₃×#131	—	2	2	no	—
12	C₂×C₂₃×#131	C₄	1	8	yes	complemented
13	C₂×C₂₃×#131	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₂₃	—	2	1	no	—
15	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₂₃	—	2	1	no	—
18	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#104	C₄	1	8	yes	complemented
23	C₂×C₂₃×#104	C₂²	1	4	yes	—
24	C₂₃×#416	C₄	1	4	yes	complemented
25	C₂₃×#416	C₂²	1	4	yes	—
26	C₂×C₂₃×#103	C₄	1	2	yes	complemented
27	C₂×C₂₃×#102	C₄	1	2	yes	complemented
28	C₂₃×#405	—	2	2	no	—
29	C₂₃×#405	C₂²	1	4	yes	complemented
30	C₂×C₂₃×#97	—	2	1	no	—
31	C₂×C₄×C₈×C₂₃	—	2	4	no	—
32	C₂×C₄×C₈×C₂₃	—	2	1	no	—
33	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	complemented
34	C₂₃×#341	C₂²	1	4	yes	complemented
35	C₂₃×#339	C₂²	1	4	yes	complemented
36	C₂×#41×C₂₃	—	2	2	no	—
37	C₂×#41×C₂₃	—	2	1	no	—
38	C₂×#41×C₂₃	—	2	1	no	—
39	C₂×#41×C₂₃	—	2	2	no	—
40	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
41	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
42	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×#41×C₂₃	C₂³	1	2	yes	—
44	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₂₃	—	2	1	no	—
46	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	—	2	1	no	—
48	C₂³×C₄×C₂₃	—	2	1	no	—
49	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#131	—	2	16	no	—
51	C₂₃×#131	—	2	8	no	—
52	C₂₃×#131	C₂×C₄	1	16	yes	complemented
53	C₂×M₁₆×C₂₃	—	2	2	no	—
54	C₂×M₁₆×C₂₃	—	2	1	no	—
55	C₂×M₁₆×C₂₃	—	2	1	no	—
56	C₂×M₁₆×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	—	2	8	no	—
58	C₂×M₁₆×C₂₃	—	2	4	no	—
59	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
60	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
61	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂²×C₈×C₂₃	—	2	1	no	—
64	C₂²×C₈×C₂₃	—	2	2	no	—
65	C₂²×C₈×C₂₃	—	2	1	no	—
66	C₂²×C₈×C₂₃	—	2	2	no	—
67	C₂²×C₈×C₂₃	—	2	8	no	—
68	C₂²×C₈×C₂₃	—	2	4	no	—
69	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
70	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
71	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
72	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	D₄	1	4	yes	—
74	C₂²×C₈×C₂₃	D₄	1	2	yes	—
75	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
78	C₂₃×#117	—	2	1	no	—
79	C₂₃×#117	—	2	1	no	—
80	C₂₃×#117	C₂×C₄	1	2	yes	—
81	C₂₃×#117	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄²×C₂₃	—	2	2	no	—
85	C₂×C₄²×C₂₃	—	2	1	no	—
86	C₂×C₄²×C₂₃	—	2	1	no	—
87	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
88	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
89	C₂₃×#104	C₂×C₄	1	16	yes	complemented
90	C₂₃×#103	C₂×C₄	1	8	yes	complemented
91	C₂₃×#102	C₂×C₄	1	8	yes	complemented
92	C₂₃×#97	—	2	4	no	—
93	C₄×C₈×C₂₃	—	2	8	no	—
94	C₄×C₈×C₂₃	—	2	4	no	—
95	C₂⁴×C₂₃	—	2	1	no	—
96	C₂⁴×C₂₃	#41	1	1	yes	characteristic
97	#41×C₂₃	—	2	4	no	—
98	#41×C₂₃	—	2	4	no	—
99	#41×C₂₃	—	2	4	no	—
100	#41×C₂₃	—	2	4	no	—
101	#41×C₂₃	—	2	4	no	—
102	#41×C₂₃	—	2	2	no	—
103	#41×C₂₃	—	2	2	no	—
104	#41×C₂₃	C₄²	1	8	yes	—
105	#41×C₂₃	C₂²×C₄	1	4	yes	—
106	#41×C₂₃	C₂²×C₄	1	4	yes	—
107	C₂×Q₈×C₂₃	—	2	1	no	—
108	C₂×Q₈×C₂₃	—	2	2	no	—
109	C₂×Q₈×C₂₃	C₄²	1	4	yes	—
110	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₂₃	—	2	4	no	—
112	C₂×D₄×C₂₃	—	2	2	no	—
113	C₂×D₄×C₂₃	—	2	4	no	—
114	C₂×D₄×C₂₃	—	2	2	no	—
115	C₂×D₄×C₂₃	—	2	2	no	—
116	C₂×D₄×C₂₃	—	2	1	no	—
117	C₂×D₄×C₂₃	C₄²	1	4	yes	—
118	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₂₃	—	2	8	no	—
120	C₂²×C₄×C₂₃	—	2	4	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	2	no	—
123	C₂²×C₄×C₂₃	—	2	4	no	—
124	C₂²×C₄×C₂₃	—	2	2	no	—
125	C₂²×C₄×C₂₃	—	2	4	no	—
126	C₂²×C₄×C₂₃	—	2	2	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	#41	1	2	yes	—
137	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
138	M₁₆×C₂₃	—	2	4	no	—
139	M₁₆×C₂₃	—	2	4	no	—
140	M₁₆×C₂₃	—	2	2	no	—
141	M₁₆×C₂₃	—	2	2	no	—
142	M₁₆×C₂₃	—	2	16	no	—
143	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
144	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
145	C₂×C₈×C₂₃	—	2	4	no	—
146	C₂×C₈×C₂₃	—	2	4	no	—
147	C₂×C₈×C₂₃	—	2	4	no	—
148	C₂×C₈×C₂₃	—	2	4	no	—
149	C₂×C₈×C₂₃	—	2	2	no	—
150	C₂×C₈×C₂₃	—	2	2	no	—
151	C₂×C₈×C₂₃	—	2	2	no	—
152	C₂×C₈×C₂₃	—	2	2	no	—
153	C₂×C₈×C₂₃	—	2	16	no	—
154	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
155	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
156	C₂×C₈×C₂₃	C₂²⋊C₄	1	8	yes	—
157	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
158	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
159	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
164	C₂²⋊C₄×C₂₃	—	2	2	no	—
165	C₂²⋊C₄×C₂₃	—	2	2	no	—
166	C₄²×C₂₃	—	2	4	no	—
167	C₄²×C₂₃	—	2	2	no	—
168	C₄²×C₂₃	—	2	2	no	—
169	C₄²×C₂₃	—	2	2	no	—
170	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
171	C₂×#1764	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₂₃	—	2	4	no	—
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	—	2	4	no	—
175	C₂³×C₂₃	—	4	4	no	—
176	C₂³×C₂₃	#117	1	2	yes	—
177	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
178	C₂³×C₂₃	C₄×D₄	1	2	yes	—
179	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
180	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
181	Q₈×C₂₃	—	2	2	no	—
182	Q₈×C₂₃	—	2	4	no	—
183	Q₈×C₂₃	C₂×C₄²	1	4	yes	—
184	D₄×C₂₃	—	2	4	no	—
185	D₄×C₂₃	—	2	4	no	—
186	D₄×C₂₃	—	2	8	no	—
187	D₄×C₂₃	—	2	2	no	—
188	D₄×C₂₃	—	2	4	no	—
189	D₄×C₂₃	C₂×C₄²	1	4	yes	—
190	C₂×C₄×C₂₃	—	2	8	no	—
191	C₂×C₄×C₂₃	—	2	16	no	—
192	C₂×C₄×C₂₃	—	2	4	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	4	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	2	8	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	4	4	no	—
200	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
201	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
202	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
203	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
204	C₂×C₄×C₂₃	#117	1	2	yes	—
205	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
208	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
211	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
218	C₈×C₂₃	—	2	4	no	—
219	C₈×C₂₃	—	2	4	no	—
220	C₈×C₂₃	—	2	4	no	—
221	C₈×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
222	C₈×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
223	C₂²×#131	C₂×C₂₃	1	1	yes	characteristic
224	C₂²×#104	C₂×C₂₃	1	1	yes	characteristic
225	C₂×#416	C₂×C₂₃	1	1	yes	characteristic
226	C₂×#405	C₂×C₂₃	1	1	yes	characteristic, complemented
227	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
228	#1764	C₂×C₂₃	1	8	yes	direct product factor
229	C₂×#341	C₂×C₂₃	1	1	yes	characteristic, complemented
230	C₂×#339	C₂×C₂₃	1	1	yes	characteristic, complemented
231	C₂²×C₂₃	—	2	2	no	—
232	C₂²×C₂₃	—	2	2	no	—
233	C₂²×C₂₃	—	2	4	no	—
234	C₂²×C₂₃	—	2	4	no	—
235	C₂²×C₂₃	—	2	4	no	—
236	C₂²×C₂₃	—	4	4	no	—
237	C₂²×C₂₃	—	4	8	no	—
238	C₂²×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
239	C₂²×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
240	C₂²×C₂₃	#443	1	4	yes	—
241	C₂²×C₂₃	C₂×#117	1	1	yes	characteristic
242	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
243	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
244	C₄×C₂₃	—	2	8	no	—
245	C₄×C₂₃	—	2	4	no	—
246	C₄×C₂₃	—	4	4	no	—
247	C₄×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
248	C₄×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
249	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
250	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
251	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
252	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
253	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
254	C₂×#131	—	2	4	no	—
255	C₂×#131	—	2	2	no	—
256	C₂×#131	C₄×C₂₃	1	8	yes	complemented
257	C₂×#131	C₂²×C₂₃	1	4	yes	—
258	C₂²×M₁₆	—	2	1	no	—
259	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
260	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
261	C₂³×C₈	—	2	1	no	—
262	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
263	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
265	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×#104	C₄×C₂₃	1	8	yes	complemented
267	C₂×#104	C₂²×C₂₃	1	4	yes	—
268	#416	C₄×C₂₃	1	4	yes	complemented
269	#416	C₂²×C₂₃	1	4	yes	—
270	C₂×#103	C₄×C₂₃	1	2	yes	complemented
271	C₂×#102	C₄×C₂₃	1	2	yes	complemented
272	#405	—	2	2	no	—
273	#405	C₂²×C₂₃	1	4	yes	complemented
274	C₂×#97	—	2	1	no	—
275	C₂×C₄×C₈	—	2	4	no	—
276	C₂×C₄×C₈	—	2	1	no	—
277	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	complemented
278	#341	C₂²×C₂₃	1	4	yes	complemented
279	#339	C₂²×C₂₃	1	4	yes	complemented
280	C₂×C₂₃	—	2	1	no	—
281	C₂×C₂₃	—	2	1	no	—
282	C₂×C₂₃	—	2	2	no	—
283	C₂×C₂₃	—	4	4	no	—
284	C₂×C₂₃	#1764	1	4	yes	direct product factor
285	C₂×C₂₃	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₂₃	C₂×#443	1	2	yes	—
287	C₂×#41	—	2	2	no	—
288	C₂×#41	—	2	1	no	—
289	C₂×#41	—	2	1	no	—
290	C₂×#41	—	2	2	no	—
291	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
292	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
293	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
294	C₂×#41	C₂³×C₂₃	1	2	yes	—
295	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
298	C₂³×C₄	—	2	1	no	—
299	C₂³×C₄	—	2	1	no	—
300	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
301	#131	—	2	16	no	—
302	#131	—	2	8	no	—
303	#131	C₂×C₄×C₂₃	1	16	yes	complemented
304	C₂×M₁₆	—	2	2	no	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂×M₁₆	—	2	1	no	—
307	C₂×M₁₆	—	2	2	no	—
308	C₂×M₁₆	—	2	8	no	—
309	C₂×M₁₆	—	2	4	no	—
310	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
311	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
312	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
313	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
314	C₂²×C₈	—	2	1	no	—
315	C₂²×C₈	—	2	2	no	—
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	—	2	2	no	—
318	C₂²×C₈	—	2	8	no	—
319	C₂²×C₈	—	2	4	no	—
320	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
321	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
322	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
323	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
324	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
325	C₂²×C₈	D₄×C₂₃	1	4	yes	—
326	C₂²×C₈	D₄×C₂₃	1	2	yes	—
327	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
328	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
329	#117	—	2	1	no	—
330	#117	—	2	1	no	—
331	#117	C₂×C₄×C₂₃	1	2	yes	—
332	#117	C₂³×C₂₃	1	2	yes	—
333	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
334	C₂×C₂²⋊C₄	—	2	1	no	—
335	C₂×C₄²	—	2	1	no	—
336	C₂×C₄²	—	2	2	no	—
337	C₂×C₄²	—	2	1	no	—
338	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
339	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
340	#104	C₂×C₄×C₂₃	1	16	yes	complemented
341	#103	C₂×C₄×C₂₃	1	8	yes	complemented
342	#102	C₂×C₄×C₂₃	1	8	yes	complemented
343	#97	—	2	4	no	—
344	C₄×C₈	—	2	8	no	—
345	C₄×C₈	—	2	4	no	—
346	C₂₃	C₂×#1764	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	—	2	1	no	—
348	C₂⁴	#41×C₂₃	1	1	yes	characteristic
349	#41	—	2	4	no	—
350	#41	—	2	4	no	—
351	#41	—	2	4	no	—
352	#41	—	2	4	no	—
353	#41	—	2	4	no	—
354	#41	—	2	2	no	—
355	#41	—	2	2	no	—
356	#41	C₄²×C₂₃	1	8	yes	—
357	#41	C₂²×C₄×C₂₃	1	4	yes	—
358	#41	C₂²×C₄×C₂₃	1	4	yes	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	C₄²×C₂₃	1	4	yes	—
362	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
363	C₂×D₄	—	2	4	no	—
364	C₂×D₄	—	2	4	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	C₄²×C₂₃	1	4	yes	—
370	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
381	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
382	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
383	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
387	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
388	C₂²×C₄	#41×C₂₃	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
390	M₁₆	—	2	4	no	—
391	M₁₆	—	2	4	no	—
392	M₁₆	—	2	2	no	—
393	M₁₆	—	2	2	no	—
394	M₁₆	—	2	16	no	—
395	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
396	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
397	C₂×C₈	—	2	4	no	—
398	C₂×C₈	—	2	4	no	—
399	C₂×C₈	—	2	4	no	—
400	C₂×C₈	—	2	4	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	16	no	—
406	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
407	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
408	C₂×C₈	C₂²⋊C₄×C₂₃	1	8	yes	—
409	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
410	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
411	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
412	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
413	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	4	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	4	4	no	—
427	C₂³	C₂₃×#117	1	2	yes	—
428	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
429	C₂³	C₄×D₄×C₂₃	1	2	yes	—
430	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
431	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
432	Q₈	—	2	4	no	—
433	Q₈	—	2	2	no	—
434	Q₈	C₂×C₄²×C₂₃	1	4	yes	—
435	D₄	—	2	8	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	2	4	no	—
439	D₄	—	2	2	no	—
440	D₄	C₂×C₄²×C₂₃	1	4	yes	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	16	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
452	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
453	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
455	C₂×C₄	C₂₃×#117	1	2	yes	—
456	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
457	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
458	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
459	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
460	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
461	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
462	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
463	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
464	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
465	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
466	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
467	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
468	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
469	C₈	—	2	4	no	—
470	C₈	—	2	4	no	—
471	C₈	—	2	4	no	—
472	C₈	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
473	C₈	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	8	no	—
481	C₂²	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
482	C₂²	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
483	C₂²	C₂₃×#443	1	4	yes	—
484	C₂²	C₂×C₂₃×#117	1	1	yes	characteristic
485	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
486	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
487	C₄	—	2	8	no	—
488	C₄	—	2	4	no	—
489	C₄	—	4	4	no	—
490	C₄	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
491	C₄	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
492	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
493	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
494	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
495	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
496	C₂	—	2	1	no	—
497	C₂	—	2	2	no	—
498	C₂	—	2	1	no	—
499	C₂	—	4	4	no	—
500	C₂	C₂₃×#1764	1	4	yes	direct product factor
501	C₂	C₂×C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
502	C₂	C₂×C₂₃×#443	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₃	D₄
commutator	C₄	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₄×D₄
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₂₃×#405	C₂	semidirect
5. C₂²×C₄×C₈×C₂₃	C₂	semidirect
6. C₂₃×#1764	C₂	direct (N × H)
7. C₂×C₂₃×#341	C₂	semidirect
8. C₂×C₂₃×#339	C₂	semidirect
12. C₂×C₂₃×#131	C₄	semidirect
22. C₂×C₂₃×#104	C₄	semidirect
24. C₂₃×#416	C₄	semidirect
26. C₂×C₂₃×#103	C₄	semidirect
27. C₂×C₂₃×#102	C₄	semidirect
29. C₂₃×#405	C₂²	semidirect
33. C₂×C₄×C₈×C₂₃	C₂²	semidirect
34. C₂₃×#341	C₂²	semidirect
35. C₂₃×#339	C₂²	semidirect
52. C₂₃×#131	C₂×C₄	semidirect
89. C₂₃×#104	C₂×C₄	semidirect
90. C₂₃×#103	C₂×C₄	semidirect
91. C₂₃×#102	C₂×C₄	semidirect
171. C₂×#1764	C₂₃	direct (N × H)
226. C₂×#405	C₂×C₂₃	semidirect
227. C₂²×C₄×C₈	C₂×C₂₃	semidirect
228. #1764	C₂×C₂₃	direct (N × H)
229. C₂×#341	C₂×C₂₃	semidirect
230. C₂×#339	C₂×C₂₃	semidirect
256. C₂×#131	C₄×C₂₃	semidirect
266. C₂×#104	C₄×C₂₃	semidirect
268. #416	C₄×C₂₃	semidirect
270. C₂×#103	C₄×C₂₃	semidirect
271. C₂×#102	C₄×C₂₃	semidirect
273. #405	C₂²×C₂₃	semidirect
277. C₂×C₄×C₈	C₂²×C₂₃	semidirect
278. #341	C₂²×C₂₃	semidirect
279. #339	C₂²×C₂₃	semidirect
284. C₂×C₂₃	#1764	direct (N × H)
303. #131	C₂×C₄×C₂₃	semidirect
340. #104	C₂×C₄×C₂₃	semidirect
341. #103	C₂×C₄×C₂₃	semidirect
342. #102	C₂×C₄×C₂₃	semidirect
346. C₂₃	C₂×#1764	direct (N × H)
500. C₂	C₂₃×#1764	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×#131	2
C₂²×C₂₃×#104	2
C₂×C₂₃×#416	2
C₂×C₂₃×#405	2
C₂²×C₄×C₈×C₂₃	2
C₂₃×#1764	2
C₂×C₂₃×#341	2
C₂×C₂₃×#339	2
C₂×#1764	23

Subgroups — C2×C23×#1764 #19168044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1764 #19168044