Subgroups — C₂×C₂₃×#3223 #19208421

Order 5888.

2022 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#536	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3223	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#464	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#462	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#452	C₂	1	2	yes	maximal
11	C₂×C₂₃×#451	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂₃×#541	—	2	4	no	—
20	C₂₃×#541	C₂²	1	8	yes	complemented
21	C₂₃×#536	—	2	4	no	—
22	C₂₃×#536	C₂²	1	8	yes	—
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#126	—	2	2	no	—
27	C₂×C₂₃×#125	—	2	2	no	—
28	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#123	—	2	2	no	—
32	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
33	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
34	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
35	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
36	C₂₃×#497	C₂²	1	8	yes	complemented
37	C₂₃×#494	C₂²	1	8	yes	complemented
38	C₂₃×#491	C₂²	1	8	yes	complemented
39	C₂₃×#488	C₂²	1	8	yes	complemented
40	C₂₃×#464	C₂²	1	16	yes	complemented
41	C₂₃×#462	C₂²	1	16	yes	complemented
42	C₂₃×#452	C₂²	1	16	yes	—
43	C₂₃×#451	C₂²	1	16	yes	complemented
44	C₂₃×#433	C₂²	1	8	yes	complemented
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
46	C₂×C₂₃×#103	C₂²	1	2	yes	—
47	C₂×C₂₃×#103	C₂²	1	2	yes	—
48	C₂×C₂₃×#103	C₂²	1	2	yes	—
49	C₂×C₂₃×#102	C₂²	1	2	yes	—
50	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
51	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
52	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
53	C₂×#41×C₂₃	—	2	2	no	—
54	C₂×#41×C₂₃	—	4	4	no	—
55	C₂×#41×C₂₃	C₂³	1	4	yes	—
56	C₂²×Q₈×C₂₃	—	2	1	no	—
57	C₂²×Q₈×C₂₃	—	2	2	no	—
58	C₂²×Q₈×C₂₃	—	4	4	no	—
59	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
60	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
64	C₂²×D₄×C₂₃	—	2	1	no	—
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆×C₂₃	—	4	4	no	—
70	C₂×Q₁₆×C₂₃	—	4	4	no	—
71	C₂×Q₁₆×C₂₃	—	4	4	no	—
72	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
73	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
74	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₂₃	—	4	4	no	—
76	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
77	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
78	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
79	C₂₃×C₄⋊Q₈	—	2	2	no	—
80	C₂₃×C₄⋊Q₈	—	2	2	no	—
81	C₂₃×C₄⋊Q₈	—	2	2	no	—
82	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
83	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
84	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
85	C₂₃×#126	—	2	16	no	—
86	C₂₃×#125	—	2	16	no	—
87	C₂₃×#125	—	2	2	no	—
88	C₂₃×#125	C₂³	1	4	yes	—
89	C₂₃×#124	—	2	2	no	—
90	C₂₃×#124	—	2	2	no	—
91	C₂₃×#124	C₂³	1	4	yes	—
92	C₂₃×#124	C₂³	1	4	yes	—
93	C₂₃×#123	—	2	16	no	—
94	C₂₃×C₂²⋊Q₈	—	2	4	no	—
95	C₂₃×C₂²⋊Q₈	—	2	16	no	—
96	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
97	C₄×Q₈×C₂₃	—	2	4	no	—
98	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
99	C₂₃×#117	D₄	1	4	yes	—
100	C₂₃×#117	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
113	C₂×C₄²×C₂₃	—	2	2	no	—
114	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
117	C₂₃×#103	C₂³	1	8	yes	—
118	C₂₃×#103	C₂³	1	8	yes	—
119	C₂₃×#103	C₂³	1	8	yes	—
120	C₂₃×#102	C₂³	1	8	yes	—
121	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
122	C₂₃×#97	C₂³	1	4	yes	—
123	C₄×C₈×C₂₃	C₂³	1	4	yes	—
124	C₂⁴×C₂₃	—	2	1	no	—
125	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
126	#41×C₂₃	—	4	8	no	—
127	#41×C₂₃	—	4	8	no	—
128	C₂×Q₈×C₂₃	—	2	2	no	—
129	C₂×Q₈×C₂₃	—	2	4	no	—
130	C₂×Q₈×C₂₃	—	2	1	no	—
131	C₂×Q₈×C₂₃	—	2	1	no	—
132	C₂×Q₈×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	4	no	—
134	C₂×Q₈×C₂₃	—	2	4	no	—
135	C₂×Q₈×C₂₃	—	2	4	no	—
136	C₂×Q₈×C₂₃	—	2	4	no	—
137	C₂×Q₈×C₂₃	—	2	4	no	—
138	C₂×Q₈×C₂₃	—	4	8	no	—
139	C₂×Q₈×C₂₃	—	4	4	no	—
140	C₂×Q₈×C₂₃	—	4	8	no	—
141	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
142	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
143	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
144	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
145	C₂×D₄×C₂₃	—	2	4	no	—
146	C₂×D₄×C₂₃	—	2	1	no	—
147	C₂×D₄×C₂₃	—	2	4	no	—
148	C₂×D₄×C₂₃	—	4	4	no	—
149	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₂₃	—	2	4	no	—
151	C₂²×C₄×C₂₃	—	2	4	no	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	2	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	—	4	4	no	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
166	Q₁₆×C₂₃	—	4	8	no	—
167	Q₁₆×C₂₃	—	4	8	no	—
168	Q₁₆×C₂₃	—	4	8	no	—
169	SD₁₆×C₂₃	—	4	8	no	—
170	C₂×C₈×C₂₃	—	2	4	no	—
171	C₂×C₈×C₂₃	—	2	4	no	—
172	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
173	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₂₃	—	2	8	no	—
175	C₄⋊C₄×C₂₃	—	2	8	no	—
176	C₄⋊C₄×C₂₃	—	2	8	no	—
177	C₄⋊C₄×C₂₃	—	2	8	no	—
178	C₄⋊C₄×C₂₃	—	2	8	no	—
179	C₄⋊C₄×C₂₃	—	2	8	no	—
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
182	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
183	C₂²⋊C₄×C₂₃	—	2	8	no	—
184	C₂²⋊C₄×C₂₃	—	2	8	no	—
185	C₂²⋊C₄×C₂₃	—	2	8	no	—
186	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
187	C₄²×C₂₃	—	2	8	no	—
188	C₄²×C₂₃	C₂×D₄	1	2	yes	—
189	C₄²×C₂₃	C₂×D₄	1	2	yes	—
190	C₄²×C₂₃	C₂⁴	1	2	yes	—
191	C₄²×C₂₃	C₂⁴	1	2	yes	—
192	C₂×#3223	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₂₃	—	2	2	no	—
194	C₂³×C₂₃	—	2	4	no	—
195	C₂³×C₂₃	—	2	2	no	—
196	C₂³×C₂₃	—	4	2	no	—
197	C₂³×C₂₃	—	4	4	no	—
198	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
199	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
200	Q₈×C₂₃	—	2	4	no	—
201	Q₈×C₂₃	—	2	4	no	—
202	Q₈×C₂₃	—	2	4	no	—
203	Q₈×C₂₃	—	2	4	no	—
204	Q₈×C₂₃	—	2	4	no	—
205	Q₈×C₂₃	—	4	8	no	—
206	Q₈×C₂₃	—	4	4	no	—
207	Q₈×C₂₃	—	4	2	no	—
208	Q₈×C₂₃	—	4	2	no	—
209	Q₈×C₂₃	—	4	2	no	—
210	D₄×C₂₃	—	2	4	no	—
211	D₄×C₂₃	—	4	8	no	—
212	D₄×C₂₃	—	4	2	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	4	no	—
215	C₂×C₄×C₂₃	—	2	4	no	—
216	C₂×C₄×C₂₃	—	2	4	no	—
217	C₂×C₄×C₂₃	—	2	4	no	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	4	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	4	8	no	—
224	C₂×C₄×C₂₃	—	4	8	no	—
225	C₂×C₄×C₂₃	—	4	2	no	—
226	C₂×C₄×C₂₃	—	4	2	no	—
227	C₂×C₄×C₂₃	—	4	2	no	—
228	C₂×C₄×C₂₃	—	4	4	no	—
229	C₂×C₄×C₂₃	—	4	4	no	—
230	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
232	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
235	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
236	C₈×C₂₃	—	2	4	no	—
237	C₈×C₂₃	—	2	4	no	—
238	C₂×#541	C₂×C₂₃	1	1	yes	characteristic, complemented
239	C₂×#536	C₂×C₂₃	1	1	yes	characteristic
240	#3223	C₂×C₂₃	1	16	yes	direct product factor
241	C₂×#497	C₂×C₂₃	1	1	yes	characteristic, complemented
242	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
243	C₂×#491	C₂×C₂₃	1	1	yes	characteristic, complemented
244	C₂×#488	C₂×C₂₃	1	1	yes	characteristic, complemented
245	C₂×#464	C₂×C₂₃	1	2	yes	complemented
246	C₂×#462	C₂×C₂₃	1	2	yes	complemented
247	C₂×#452	C₂×C₂₃	1	2	yes	—
248	C₂×#451	C₂×C₂₃	1	2	yes	complemented
249	C₂×#433	C₂×C₂₃	1	1	yes	characteristic, complemented
250	C₂²×C₂₃	—	2	2	no	—
251	C₂²×C₂₃	—	2	4	no	—
252	C₂²×C₂₃	—	4	4	no	—
253	C₂²×C₂₃	—	4	2	no	—
254	C₂²×C₂₃	—	4	4	no	—
255	C₂²×C₂₃	—	8	4	no	—
256	C₂²×C₂₃	#535	1	2	yes	—
257	C₂²×C₂₃	#578	1	4	yes	—
258	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
259	C₄×C₂₃	—	2	2	no	—
260	C₄×C₂₃	—	2	4	no	—
261	C₄×C₂₃	—	2	4	no	—
262	C₄×C₂₃	—	4	2	no	—
263	C₄×C₂₃	—	4	2	no	—
264	C₄×C₂₃	—	4	2	no	—
265	C₄×C₂₃	—	4	4	no	—
266	C₄×C₂₃	—	4	4	no	—
267	C₄×C₂₃	#535	1	2	yes	—
268	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
269	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
270	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
271	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
272	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
273	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
274	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
275	#541	—	2	4	no	—
276	#541	C₂²×C₂₃	1	8	yes	complemented
277	#536	—	2	4	no	—
278	#536	C₂²×C₂₃	1	8	yes	—
279	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
282	C₂×#126	—	2	2	no	—
283	C₂×#125	—	2	2	no	—
284	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
285	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
286	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
287	C₂×#123	—	2	2	no	—
288	C₂×C₂²⋊Q₈	—	2	2	no	—
289	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
290	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
291	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
292	#497	C₂²×C₂₃	1	8	yes	complemented
293	#494	C₂²×C₂₃	1	8	yes	complemented
294	#491	C₂²×C₂₃	1	8	yes	complemented
295	#488	C₂²×C₂₃	1	8	yes	complemented
296	#464	C₂²×C₂₃	1	16	yes	complemented
297	#462	C₂²×C₂₃	1	16	yes	complemented
298	#452	C₂²×C₂₃	1	16	yes	—
299	#451	C₂²×C₂₃	1	16	yes	complemented
300	#433	C₂²×C₂₃	1	8	yes	complemented
301	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
302	C₂×#103	C₂²×C₂₃	1	2	yes	—
303	C₂×#103	C₂²×C₂₃	1	2	yes	—
304	C₂×#103	C₂²×C₂₃	1	2	yes	—
305	C₂×#102	C₂²×C₂₃	1	2	yes	—
306	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
307	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
309	C₂×C₂₃	—	4	2	no	—
310	C₂×C₂₃	—	8	2	no	—
311	C₂×C₂₃	#3223	1	4	yes	direct product factor
312	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
313	C₂×C₂₃	C₂×#578	1	2	yes	—
314	C₂×#41	—	2	2	no	—
315	C₂×#41	—	4	4	no	—
316	C₂×#41	C₂³×C₂₃	1	4	yes	—
317	C₂²×Q₈	—	2	1	no	—
318	C₂²×Q₈	—	2	2	no	—
319	C₂²×Q₈	—	4	4	no	—
320	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
321	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
322	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
323	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
324	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
325	C₂²×D₄	—	2	1	no	—
326	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
327	C₂³×C₄	—	2	1	no	—
328	C₂³×C₄	—	2	1	no	—
329	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
330	C₂×Q₁₆	—	4	4	no	—
331	C₂×Q₁₆	—	4	4	no	—
332	C₂×Q₁₆	—	4	4	no	—
333	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
334	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
335	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
336	C₂×SD₁₆	—	4	4	no	—
337	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
338	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
339	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
340	C₄⋊Q₈	—	2	2	no	—
341	C₄⋊Q₈	—	2	2	no	—
342	C₄⋊Q₈	—	2	2	no	—
343	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
344	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
345	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
346	#126	—	2	16	no	—
347	#125	—	2	16	no	—
348	#125	—	2	2	no	—
349	#125	C₂³×C₂₃	1	4	yes	—
350	#124	—	2	2	no	—
351	#124	—	2	2	no	—
352	#124	C₂³×C₂₃	1	4	yes	—
353	#124	C₂³×C₂₃	1	4	yes	—
354	#123	—	2	16	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	—	2	16	no	—
357	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
358	C₄×Q₈	—	2	4	no	—
359	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
360	#117	D₄×C₂₃	1	4	yes	—
361	#117	C₂³×C₂₃	1	4	yes	—
362	C₂×C₄⋊C₄	—	2	2	no	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	2	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
374	C₂×C₄²	—	2	2	no	—
375	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
377	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
378	#103	C₂³×C₂₃	1	8	yes	—
379	#103	C₂³×C₂₃	1	8	yes	—
380	#103	C₂³×C₂₃	1	8	yes	—
381	#102	C₂³×C₂₃	1	8	yes	—
382	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
383	#97	C₂³×C₂₃	1	4	yes	—
384	C₄×C₈	C₂³×C₂₃	1	4	yes	—
385	C₂₃	C₂×#3223	1	1	yes	characteristic, direct product factor, Sylow
386	C₂⁴	—	2	1	no	—
387	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
388	#41	—	4	8	no	—
389	#41	—	4	8	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	4	8	no	—
401	C₂×Q₈	—	4	4	no	—
402	C₂×Q₈	—	4	8	no	—
403	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
404	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
405	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
406	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	—	4	4	no	—
422	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
423	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
424	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
426	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
427	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
428	Q₁₆	—	4	8	no	—
429	Q₁₆	—	4	8	no	—
430	Q₁₆	—	4	8	no	—
431	SD₁₆	—	4	8	no	—
432	C₂×C₈	—	2	4	no	—
433	C₂×C₈	—	2	4	no	—
434	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
435	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	—	2	8	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	8	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
444	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
445	C₂²⋊C₄	—	2	8	no	—
446	C₂²⋊C₄	—	2	8	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
449	C₄²	—	2	8	no	—
450	C₄²	C₂×D₄×C₂₃	1	2	yes	—
451	C₄²	C₂×D₄×C₂₃	1	2	yes	—
452	C₄²	C₂⁴×C₂₃	1	2	yes	—
453	C₄²	C₂⁴×C₂₃	1	2	yes	—
454	C₂³	—	2	4	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	4	4	no	—
458	C₂³	—	4	2	no	—
459	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
460	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
461	Q₈	—	2	4	no	—
462	Q₈	—	2	4	no	—
463	Q₈	—	2	4	no	—
464	Q₈	—	2	4	no	—
465	Q₈	—	2	4	no	—
466	Q₈	—	4	8	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	4	2	no	—
469	Q₈	—	4	2	no	—
470	Q₈	—	4	4	no	—
471	D₄	—	2	4	no	—
472	D₄	—	4	8	no	—
473	D₄	—	4	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	8	no	—
485	C₂×C₄	—	4	8	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	4	no	—
491	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
493	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
495	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
496	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
497	C₈	—	2	4	no	—
498	C₈	—	2	4	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	4	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	4	4	no	—
504	C₂²	—	8	4	no	—
505	C₂²	C₂₃×#535	1	2	yes	—
506	C₂²	C₂₃×#578	1	4	yes	—
507	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
508	C₄	—	2	2	no	—
509	C₄	—	2	4	no	—
510	C₄	—	2	4	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	2	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	4	no	—
516	C₄	C₂₃×#535	1	2	yes	—
517	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
518	C₂	—	4	2	no	—
519	C₂	—	8	2	no	—
520	C₂	C₂₃×#3223	1	4	yes	direct product factor
521	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
522	C₂	C₂×C₂₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#541	C₂	semidirect
3. C₂₃×#3223	C₂	direct (N × H)
4. C₂×C₂₃×#497	C₂	semidirect
5. C₂×C₂₃×#494	C₂	semidirect
6. C₂×C₂₃×#491	C₂	semidirect
7. C₂×C₂₃×#488	C₂	semidirect
8. C₂×C₂₃×#464	C₂	semidirect
9. C₂×C₂₃×#462	C₂	semidirect
11. C₂×C₂₃×#451	C₂	semidirect
12. C₂×C₂₃×#433	C₂	semidirect
20. C₂₃×#541	C₂²	semidirect
36. C₂₃×#497	C₂²	semidirect
37. C₂₃×#494	C₂²	semidirect
38. C₂₃×#491	C₂²	semidirect
39. C₂₃×#488	C₂²	semidirect
40. C₂₃×#464	C₂²	semidirect
41. C₂₃×#462	C₂²	semidirect
43. C₂₃×#451	C₂²	semidirect
44. C₂₃×#433	C₂²	semidirect
192. C₂×#3223	C₂₃	direct (N × H)
238. C₂×#541	C₂×C₂₃	semidirect
240. #3223	C₂×C₂₃	direct (N × H)
241. C₂×#497	C₂×C₂₃	semidirect
242. C₂×#494	C₂×C₂₃	semidirect
243. C₂×#491	C₂×C₂₃	semidirect
244. C₂×#488	C₂×C₂₃	semidirect
245. C₂×#464	C₂×C₂₃	semidirect
246. C₂×#462	C₂×C₂₃	semidirect
248. C₂×#451	C₂×C₂₃	semidirect
249. C₂×#433	C₂×C₂₃	semidirect
276. #541	C₂²×C₂₃	semidirect
292. #497	C₂²×C₂₃	semidirect
293. #494	C₂²×C₂₃	semidirect
294. #491	C₂²×C₂₃	semidirect
295. #488	C₂²×C₂₃	semidirect
296. #464	C₂²×C₂₃	semidirect
297. #462	C₂²×C₂₃	semidirect
299. #451	C₂²×C₂₃	semidirect
300. #433	C₂²×C₂₃	semidirect
311. C₂×C₂₃	#3223	direct (N × H)
385. C₂₃	C₂×#3223	direct (N × H)
520. C₂	C₂₃×#3223	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×#541	2
C₂×C₂₃×#536	2
C₂₃×#3223	2
C₂×C₂₃×#497	2
C₂×C₂₃×#494	2
C₂×C₂₃×#491	2
C₂×C₂₃×#488	2
C₂×C₂₃×#464	2
C₂×C₂₃×#462	2
C₂×C₂₃×#452	2
C₂×C₂₃×#451	2
C₂×C₂₃×#433	2
C₂×#3223	23

Subgroups — C2×C23×#3223 #19208421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3223 #19208421