Subgroups — C₂×C₂₃×#2993 #19207886

Order 5888.

1926 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2993	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#436	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#435	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#433	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#424	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#529	C₄	1	8	yes	—
17	C₂₃×#529	C₂²	1	8	yes	—
18	C₂×C₂₃×#126	C₄	1	2	yes	—
19	C₂×C₂₃×#125	C₄	1	1	yes	characteristic
20	C₂×C₂₃×#124	C₄	1	1	yes	characteristic
21	C₂×C₂₃×#123	C₄	1	2	yes	—
22	C₂×C₂₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
23	C₂×C₂₃×#121	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₂₃×#117	C₂²	1	2	yes	—
28	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#446	C₂²	1	8	yes	complemented
30	C₈×Q₈×C₂₃	C₂²	1	8	yes	complemented
31	C₂₃×#436	C₂²	1	8	yes	complemented
32	C₂₃×#436	C₂²	1	16	yes	complemented
33	C₂₃×#435	C₂²	1	16	yes	complemented
34	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
35	C₂₃×#433	C₂²	1	16	yes	complemented
36	C₂₃×#431	C₂²	1	8	yes	complemented
37	C₂₃×#424	C₂²	1	16	yes	complemented
38	C₂₃×#423	C₂²	1	8	yes	complemented
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
42	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
44	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
46	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
49	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
50	C₂×C₂₃×#97	C₂²	1	2	yes	—
51	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
52	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
54	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
57	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
60	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₂₃	—	4	4	no	—
62	C₂×M₁₆×C₂₃	—	4	4	no	—
63	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
64	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
65	C₂²×C₈×C₂₃	—	4	4	no	—
66	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
69	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
73	C₂₃×#126	C₂×C₄	1	16	yes	—
74	C₂₃×#125	C₂×C₄	1	8	yes	—
75	C₂₃×#124	C₂×C₄	1	8	yes	—
76	C₂₃×#123	C₂×C₄	1	16	yes	—
77	C₂₃×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
78	C₂₃×#121	C₂×C₄	1	8	yes	—
79	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
80	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
81	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
82	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
83	C₄×D₄×C₂₃	C₂³	1	4	yes	—
84	C₄×D₄×C₂₃	C₂³	1	8	yes	—
85	C₂₃×#117	C₂×C₄	1	8	yes	—
86	C₂₃×#117	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
88	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
89	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
93	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
94	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
95	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
96	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
97	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
98	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
106	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
107	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
108	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
109	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
110	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
111	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
112	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
113	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
114	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
115	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
116	C₂₃×#97	C₂³	1	8	yes	—
117	C₄×C₈×C₂₃	C₂³	1	4	yes	—
118	C₄×C₈×C₂₃	C₂³	1	4	yes	—
119	C₄×C₈×C₂₃	C₂³	1	8	yes	—
120	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
121	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×Q₈×C₂₃	—	2	4	no	—
123	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	4	4	no	—
126	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
127	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
128	C₂²×C₄×C₂₃	—	2	4	no	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	2	no	—
133	C₂²×C₄×C₂₃	—	4	2	no	—
134	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
149	M₁₆×C₂₃	—	4	8	no	—
150	M₁₆×C₂₃	—	4	8	no	—
151	C₂×C₈×C₂₃	—	2	2	no	—
152	C₂×C₈×C₂₃	—	2	4	no	—
153	C₂×C₈×C₂₃	—	2	4	no	—
154	C₂×C₈×C₂₃	—	2	2	no	—
155	C₂×C₈×C₂₃	—	2	2	no	—
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	4	8	no	—
158	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
159	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
160	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
161	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
162	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
163	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
164	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
165	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
166	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
167	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
168	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
170	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
171	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
173	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
174	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
175	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
176	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
177	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
178	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
179	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
180	C₄²×C₂₃	C₂⁴	1	2	yes	—
181	C₄²×C₂₃	C₂⁴	1	2	yes	—
182	C₄²×C₂₃	C₂⁴	1	4	yes	—
183	C₂×#2993	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₂₃	—	2	4	no	—
185	C₂³×C₂₃	—	2	2	no	—
186	C₂³×C₂₃	—	2	2	no	—
187	C₂³×C₂₃	—	2	4	no	—
188	C₂³×C₂₃	—	2	2	no	—
189	C₂³×C₂₃	—	2	4	no	—
190	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
191	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
192	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
193	Q₈×C₂₃	—	2	8	no	—
194	D₄×C₂₃	—	2	8	no	—
195	D₄×C₂₃	—	4	8	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	2	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	8	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	8	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	2	no	—
209	C₂×C₄×C₂₃	—	2	4	no	—
210	C₂×C₄×C₂₃	—	4	4	no	—
211	C₂×C₄×C₂₃	#131	1	4	yes	—
212	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
215	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
216	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
217	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
218	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
219	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
221	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
222	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
223	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
224	C₈×C₂₃	—	2	2	no	—
225	C₈×C₂₃	—	2	4	no	—
226	C₈×C₂₃	—	2	4	no	—
227	C₈×C₂₃	—	2	2	no	—
228	C₈×C₂₃	—	2	2	no	—
229	C₈×C₂₃	—	2	2	no	—
230	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
231	#2993	C₂×C₂₃	1	16	yes	direct product factor
232	C₂×#446	C₂×C₂₃	1	1	yes	characteristic, complemented
233	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
234	C₂×#436	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂×#436	C₂×C₂₃	1	2	yes	complemented
236	C₂×#435	C₂×C₂₃	1	2	yes	complemented
237	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
238	C₂×#433	C₂×C₂₃	1	2	yes	complemented
239	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
240	C₂×#424	C₂×C₂₃	1	2	yes	complemented
241	C₂×#423	C₂×C₂₃	1	1	yes	characteristic, complemented
242	C₂²×C₂₃	—	2	4	no	—
243	C₂²×C₂₃	—	2	2	no	—
244	C₂²×C₂₃	—	2	2	no	—
245	C₂²×C₂₃	—	2	4	no	—
246	C₂²×C₂₃	—	2	2	no	—
247	C₂²×C₂₃	—	2	4	no	—
248	C₂²×C₂₃	—	4	4	no	—
249	C₂²×C₂₃	—	4	8	no	—
250	C₂²×C₂₃	#520	1	2	yes	—
251	C₂²×C₂₃	C₂×#131	1	2	yes	—
252	C₂²×C₂₃	#568	1	2	yes	—
253	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
254	C₄×C₂₃	—	2	4	no	—
255	C₄×C₂₃	—	2	2	no	—
256	C₄×C₂₃	—	2	2	no	—
257	C₄×C₂₃	—	2	4	no	—
258	C₄×C₂₃	—	2	2	no	—
259	C₄×C₂₃	—	2	2	no	—
260	C₄×C₂₃	—	2	4	no	—
261	C₄×C₂₃	#520	1	2	yes	—
262	C₄×C₂₃	C₂×#131	1	4	yes	—
263	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
264	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
265	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
266	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
267	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
268	#529	C₄×C₂₃	1	8	yes	—
269	#529	C₂²×C₂₃	1	8	yes	—
270	C₂×#126	C₄×C₂₃	1	2	yes	—
271	C₂×#125	C₄×C₂₃	1	1	yes	characteristic
272	C₂×#124	C₄×C₂₃	1	1	yes	characteristic
273	C₂×#123	C₄×C₂₃	1	2	yes	—
274	C₂×C₂²⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
275	C₂×#121	C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
278	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
279	C₂×#117	C₂²×C₂₃	1	2	yes	—
280	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
281	#446	C₂²×C₂₃	1	8	yes	complemented
282	C₈×Q₈	C₂²×C₂₃	1	8	yes	complemented
283	#436	C₂²×C₂₃	1	8	yes	complemented
284	#436	C₂²×C₂₃	1	16	yes	complemented
285	#435	C₂²×C₂₃	1	16	yes	complemented
286	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
287	#433	C₂²×C₂₃	1	16	yes	complemented
288	#431	C₂²×C₂₃	1	8	yes	complemented
289	#424	C₂²×C₂₃	1	16	yes	complemented
290	#423	C₂²×C₂₃	1	8	yes	complemented
291	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
292	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
293	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
296	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
297	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
298	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
299	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
300	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
301	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
302	C₂×#97	C₂²×C₂₃	1	2	yes	—
303	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
306	C₂×C₂₃	—	4	2	no	—
307	C₂×C₂₃	—	4	4	no	—
308	C₂×C₂₃	#2993	1	4	yes	direct product factor
309	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
310	C₂×C₂₃	C₂²×#131	1	1	yes	characteristic
311	C₂×C₂₃	C₂×#568	1	1	yes	characteristic
312	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
313	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
314	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
315	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
316	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
317	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
318	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
319	C₂×M₁₆	—	4	4	no	—
320	C₂×M₁₆	—	4	4	no	—
321	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
322	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
323	C₂²×C₈	—	4	4	no	—
324	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
326	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
327	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
328	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
331	#126	C₂×C₄×C₂₃	1	16	yes	—
332	#125	C₂×C₄×C₂₃	1	8	yes	—
333	#124	C₂×C₄×C₂₃	1	8	yes	—
334	#123	C₂×C₄×C₂₃	1	16	yes	—
335	C₂²⋊Q₈	C₂×C₄×C₂₃	1	8	yes	—
336	#121	C₂×C₄×C₂₃	1	8	yes	—
337	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
338	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
339	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
340	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
341	C₄×D₄	C₂³×C₂₃	1	4	yes	—
342	C₄×D₄	C₂³×C₂₃	1	8	yes	—
343	#117	C₂×C₄×C₂₃	1	8	yes	—
344	#117	C₂³×C₂₃	1	8	yes	—
345	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
346	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
347	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
348	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
349	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
351	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
352	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
353	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
354	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
355	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
356	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
357	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
358	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
359	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
361	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
362	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
363	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
364	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
365	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
366	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
367	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
368	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
369	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
370	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
371	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
372	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
373	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
374	#97	C₂³×C₂₃	1	8	yes	—
375	C₄×C₈	C₂³×C₂₃	1	4	yes	—
376	C₄×C₈	C₂³×C₂₃	1	4	yes	—
377	C₄×C₈	C₂³×C₂₃	1	8	yes	—
378	C₂₃	C₂×#2993	1	1	yes	characteristic, direct product factor, Sylow
379	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
380	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	4	4	no	—
385	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
386	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
395	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
399	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
400	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
401	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
402	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
403	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
404	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
408	M₁₆	—	4	8	no	—
409	M₁₆	—	4	8	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	4	no	—
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	4	8	no	—
417	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
418	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
419	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
420	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
421	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
422	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
423	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
424	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
425	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
426	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
427	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
428	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
429	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
430	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
431	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
432	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
433	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
434	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
435	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
436	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
437	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
438	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
439	C₄²	C₂⁴×C₂₃	1	2	yes	—
440	C₄²	C₂⁴×C₂₃	1	4	yes	—
441	C₄²	C₂⁴×C₂₃	1	2	yes	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	2	no	—
448	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
449	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
450	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
451	Q₈	—	2	8	no	—
452	D₄	—	2	8	no	—
453	D₄	—	4	8	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	C₂₃×#131	1	4	yes	—
470	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
471	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
472	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
473	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
474	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
475	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
476	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
477	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
478	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
480	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
481	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
482	C₈	—	2	2	no	—
483	C₈	—	2	4	no	—
484	C₈	—	2	2	no	—
485	C₈	—	2	4	no	—
486	C₈	—	2	2	no	—
487	C₈	—	2	2	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	8	no	—
496	C₂²	C₂₃×#520	1	2	yes	—
497	C₂²	C₂×C₂₃×#131	1	2	yes	—
498	C₂²	C₂₃×#568	1	2	yes	—
499	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
500	C₄	—	2	4	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	4	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	4	no	—
507	C₄	C₂₃×#520	1	2	yes	—
508	C₄	C₂×C₂₃×#131	1	4	yes	—
509	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
510	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
511	C₂	—	4	2	no	—
512	C₂	—	4	4	no	—
513	C₂	C₂₃×#2993	1	4	yes	direct product factor
514	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
515	C₂	C₂²×C₂₃×#131	1	1	yes	characteristic
516	C₂	C₂×C₂₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂×C₂₃×#121	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2993	C₂	direct (N × H)
3. C₂×C₂₃×#446	C₂	semidirect
4. C₂×C₈×Q₈×C₂₃	C₂	semidirect
5. C₂×C₂₃×#436	C₂	semidirect
6. C₂×C₂₃×#436	C₂	semidirect
7. C₂×C₂₃×#435	C₂	semidirect
8. C₂×C₈×D₄×C₂₃	C₂	semidirect
9. C₂×C₂₃×#433	C₂	semidirect
10. C₂×C₂₃×#431	C₂	semidirect
11. C₂×C₂₃×#424	C₂	semidirect
12. C₂×C₂₃×#423	C₂	semidirect
29. C₂₃×#446	C₂²	semidirect
30. C₈×Q₈×C₂₃	C₂²	semidirect
31. C₂₃×#436	C₂²	semidirect
32. C₂₃×#436	C₂²	semidirect
33. C₂₃×#435	C₂²	semidirect
34. C₈×D₄×C₂₃	C₂²	semidirect
35. C₂₃×#433	C₂²	semidirect
36. C₂₃×#431	C₂²	semidirect
37. C₂₃×#424	C₂²	semidirect
38. C₂₃×#423	C₂²	semidirect
183. C₂×#2993	C₂₃	direct (N × H)
231. #2993	C₂×C₂₃	direct (N × H)
232. C₂×#446	C₂×C₂₃	semidirect
233. C₂×C₈×Q₈	C₂×C₂₃	semidirect
234. C₂×#436	C₂×C₂₃	semidirect
235. C₂×#436	C₂×C₂₃	semidirect
236. C₂×#435	C₂×C₂₃	semidirect
237. C₂×C₈×D₄	C₂×C₂₃	semidirect
238. C₂×#433	C₂×C₂₃	semidirect
239. C₂×#431	C₂×C₂₃	semidirect
240. C₂×#424	C₂×C₂₃	semidirect
241. C₂×#423	C₂×C₂₃	semidirect
281. #446	C₂²×C₂₃	semidirect
282. C₈×Q₈	C₂²×C₂₃	semidirect
283. #436	C₂²×C₂₃	semidirect
284. #436	C₂²×C₂₃	semidirect
285. #435	C₂²×C₂₃	semidirect
286. C₈×D₄	C₂²×C₂₃	semidirect
287. #433	C₂²×C₂₃	semidirect
288. #431	C₂²×C₂₃	semidirect
289. #424	C₂²×C₂₃	semidirect
290. #423	C₂²×C₂₃	semidirect
308. C₂×C₂₃	#2993	direct (N × H)
378. C₂₃	C₂×#2993	direct (N × H)
513. C₂	C₂₃×#2993	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×#529	2
C₂₃×#2993	2
C₂×C₂₃×#446	2
C₂×C₈×Q₈×C₂₃	2
C₂×C₂₃×#436	2
C₂×C₂₃×#435	2
C₂×C₈×D₄×C₂₃	2
C₂×C₂₃×#433	2
C₂×C₂₃×#431	2
C₂×C₂₃×#424	2
C₂×C₂₃×#423	2
C₂×#2993	23

Subgroups — C2×C23×#2993 #19207886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2993 #19207886