Subgroups — C₂×C₂₃×#3210 #19208406

Order 5888.

2186 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3210	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#464	C₂	1	4	yes	maximal, complemented
9	C₂×C₂₃×#452	C₂	1	4	yes	maximal, complemented
10	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₂₃	C₂²	1	2	yes	—
12	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	—
13	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
14	C₂₃×#533	—	2	4	no	—
15	C₂₃×#533	C₂²	1	8	yes	complemented
16	C₂₃×#532	—	2	4	no	—
17	C₂₃×#532	C₂²	1	8	yes	—
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#125	—	2	1	no	—
23	C₂×C₂₃×#124	C₂²	1	2	yes	—
24	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₂₃×#121	—	2	1	no	—
27	C₂×C₂₃×#121	—	2	1	no	—
28	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
29	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₂₃	—	2	1	no	—
31	C₂×C₄×D₄×C₂₃	—	2	1	no	—
32	C₂×C₂₃×#117	—	2	1	no	—
33	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#494	C₂²	1	8	yes	complemented
35	C₂₃×#492	C₂²	1	8	yes	complemented
36	C₂₃×#487	C₂²	1	8	yes	complemented
37	C₂₃×#486	C₂²	1	8	yes	complemented
38	C₂₃×#464	C₂²	1	32	yes	complemented
39	C₂₃×#452	C₂²	1	32	yes	complemented
40	C₂₃×#431	C₂²	1	8	yes	complemented
41	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
42	C₂×C₂₃×#103	C₂²	1	4	yes	—
43	C₂×C₂₃×#102	C₂²	1	4	yes	—
44	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
47	C₂×#41×C₂₃	—	2	4	no	—
48	C₂×#41×C₂₃	—	4	8	no	—
49	C₂×#41×C₂₃	C₂³	1	8	yes	—
50	C₂²×Q₈×C₂₃	—	2	1	no	—
51	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
53	C₂²×D₄×C₂₃	—	2	2	no	—
54	C₂²×D₄×C₂₃	—	2	1	no	—
55	C₂²×D₄×C₂₃	—	2	1	no	—
56	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
57	C₂³×C₄×C₂₃	—	2	2	no	—
58	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×Q₁₆×C₂₃	—	4	8	no	—
62	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
63	C₂×SD₁₆×C₂₃	—	4	8	no	—
64	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
65	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
66	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
67	C₂₃×C₄⋊Q₈	—	2	2	no	—
68	C₂₃×C₄⋊Q₈	—	2	2	no	—
69	C₂₃×C₄⋊Q₈	—	2	2	no	—
70	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
71	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
72	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
73	C₂₃×#127	—	2	2	no	—
74	C₂₃×#127	C₂³	1	4	yes	—
75	C₂₃×#125	—	2	8	no	—
76	C₂₃×#124	—	2	4	no	—
77	C₂₃×#124	C₂³	1	8	yes	—
78	C₂₃×C₂²⋊Q₈	—	2	8	no	—
79	C₂₃×C₂²⋊Q₈	—	2	4	no	—
80	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
81	C₂₃×#121	—	2	8	no	—
82	C₂₃×#121	—	2	8	no	—
83	C₄×Q₈×C₂₃	—	2	8	no	—
84	C₄×Q₈×C₂₃	—	2	4	no	—
85	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
86	C₄×D₄×C₂₃	—	2	8	no	—
87	C₄×D₄×C₂₃	—	2	8	no	—
88	C₂₃×#117	—	2	8	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
100	C₂×C₄²×C₂₃	—	2	2	no	—
101	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	D₄	1	4	yes	—
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
107	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
108	C₂₃×#103	C₂³	1	16	yes	—
109	C₂₃×#102	C₂³	1	16	yes	—
110	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
111	C₄×C₈×C₂₃	C₂³	1	4	yes	—
112	C₄×C₈×C₂₃	C₂³	1	4	yes	—
113	C₂⁴×C₂₃	—	2	2	no	—
114	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
115	#41×C₂₃	—	4	16	no	—
116	#41×C₂₃	—	4	16	no	—
117	C₂×Q₈×C₂₃	—	2	2	no	—
118	C₂×Q₈×C₂₃	—	2	4	no	—
119	C₂×Q₈×C₂₃	—	2	2	no	—
120	C₂×Q₈×C₂₃	—	2	8	no	—
121	C₂×Q₈×C₂₃	—	2	8	no	—
122	C₂×Q₈×C₂₃	—	4	4	no	—
123	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
124	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
125	C₂×D₄×C₂₃	—	2	8	no	—
126	C₂×D₄×C₂₃	—	2	4	no	—
127	C₂×D₄×C₂₃	—	2	4	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	2	8	no	—
130	C₂×D₄×C₂₃	—	4	8	no	—
131	C₂×D₄×C₂₃	—	4	4	no	—
132	C₂×D₄×C₂₃	—	4	4	no	—
133	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	4	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	2	2	no	—
140	C₂²×C₄×C₂₃	—	2	8	no	—
141	C₂²×C₄×C₂₃	—	4	8	no	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
152	Q₁₆×C₂₃	—	4	16	no	—
153	SD₁₆×C₂₃	—	4	16	no	—
154	C₂×C₈×C₂₃	—	2	4	no	—
155	C₂×C₈×C₂₃	—	2	4	no	—
156	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
157	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
158	C₄⋊C₄×C₂₃	—	2	4	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	4	no	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	—	2	4	no	—
163	C₄⋊C₄×C₂₃	—	2	8	no	—
164	C₄⋊C₄×C₂₃	—	2	8	no	—
165	C₄⋊C₄×C₂₃	—	2	2	no	—
166	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
167	C₂²⋊C₄×C₂₃	—	2	8	no	—
168	C₂²⋊C₄×C₂₃	—	2	8	no	—
169	C₂²⋊C₄×C₂₃	—	2	8	no	—
170	C₄²×C₂₃	—	2	8	no	—
171	C₄²×C₂₃	C₂×D₄	1	4	yes	—
172	C₄²×C₂₃	C₂×D₄	1	4	yes	—
173	C₄²×C₂₃	C₂×D₄	1	2	yes	—
174	C₄²×C₂₃	C₂×D₄	1	2	yes	—
175	C₄²×C₂₃	C₂⁴	1	2	yes	—
176	C₄²×C₂₃	C₂⁴	1	2	yes	—
177	C₂×#3210	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂³×C₂₃	—	2	2	no	—
179	C₂³×C₂₃	—	2	2	no	—
180	C₂³×C₂₃	—	2	2	no	—
181	C₂³×C₂₃	—	2	4	no	—
182	C₂³×C₂₃	—	4	4	no	—
183	C₂³×C₂₃	—	4	4	no	—
184	C₂³×C₂₃	—	4	4	no	—
185	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
186	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₂₃	—	2	8	no	—
188	Q₈×C₂₃	—	2	8	no	—
189	Q₈×C₂₃	—	4	4	no	—
190	Q₈×C₂₃	—	4	4	no	—
191	Q₈×C₂₃	—	4	8	no	—
192	D₄×C₂₃	—	2	8	no	—
193	D₄×C₂₃	—	4	16	no	—
194	D₄×C₂₃	—	4	4	no	—
195	D₄×C₂₃	—	4	8	no	—
196	D₄×C₂₃	—	4	8	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	8	no	—
202	C₂×C₄×C₂₃	—	2	8	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	4	16	no	—
208	C₂×C₄×C₂₃	—	4	4	no	—
209	C₂×C₄×C₂₃	—	4	4	no	—
210	C₂×C₄×C₂₃	—	4	4	no	—
211	C₂×C₄×C₂₃	#135	1	4	yes	—
212	C₂×C₄×C₂₃	#137	1	4	yes	—
213	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
220	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
221	C₈×C₂₃	—	2	4	no	—
222	C₈×C₂₃	—	2	4	no	—
223	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
224	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
225	#3210	C₂×C₂₃	1	16	yes	direct product factor
226	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
227	C₂×#492	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
230	C₂×#464	C₂×C₂₃	1	4	yes	complemented
231	C₂×#452	C₂×C₂₃	1	4	yes	complemented
232	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
233	C₂²×C₂₃	—	2	2	no	—
234	C₂²×C₂₃	—	2	2	no	—
235	C₂²×C₂₃	—	2	2	no	—
236	C₂²×C₂₃	—	4	4	no	—
237	C₂²×C₂₃	—	4	4	no	—
238	C₂²×C₂₃	—	4	4	no	—
239	C₂²×C₂₃	—	4	4	no	—
240	C₂²×C₂₃	—	8	8	no	—
241	C₂²×C₂₃	#535	1	2	yes	—
242	C₂²×C₂₃	C₂×#135	1	2	yes	—
243	C₂²×C₂₃	C₂×#137	1	2	yes	—
244	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	2	2	no	—
248	C₄×C₂₃	—	4	4	no	—
249	C₄×C₂₃	—	4	4	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	#535	1	2	yes	—
252	C₄×C₂₃	C₂×#135	1	4	yes	—
253	C₄×C₂₃	C₂×#137	1	4	yes	—
254	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
255	C₂²×#41	C₂²×C₂₃	1	2	yes	—
256	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	—
257	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
258	#533	—	2	4	no	—
259	#533	C₂²×C₂₃	1	8	yes	complemented
260	#532	—	2	4	no	—
261	#532	C₂²×C₂₃	1	8	yes	—
262	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
263	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×#125	—	2	1	no	—
267	C₂×#124	C₂²×C₂₃	1	2	yes	—
268	C₂×C₂²⋊Q₈	—	2	1	no	—
269	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
270	C₂×#121	—	2	1	no	—
271	C₂×#121	—	2	1	no	—
272	C₂×C₄×Q₈	—	2	1	no	—
273	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
274	C₂×C₄×D₄	—	2	1	no	—
275	C₂×C₄×D₄	—	2	1	no	—
276	C₂×#117	—	2	1	no	—
277	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
278	#494	C₂²×C₂₃	1	8	yes	complemented
279	#492	C₂²×C₂₃	1	8	yes	complemented
280	#487	C₂²×C₂₃	1	8	yes	complemented
281	#486	C₂²×C₂₃	1	8	yes	complemented
282	#464	C₂²×C₂₃	1	32	yes	complemented
283	#452	C₂²×C₂₃	1	32	yes	complemented
284	#431	C₂²×C₂₃	1	8	yes	complemented
285	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
286	C₂×#103	C₂²×C₂₃	1	4	yes	—
287	C₂×#102	C₂²×C₂₃	1	4	yes	—
288	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
289	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
290	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
291	C₂×C₂₃	—	4	2	no	—
292	C₂×C₂₃	—	8	4	no	—
293	C₂×C₂₃	#3210	1	4	yes	direct product factor
294	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
295	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
296	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
297	C₂×#41	—	2	4	no	—
298	C₂×#41	—	4	8	no	—
299	C₂×#41	C₂³×C₂₃	1	8	yes	—
300	C₂²×Q₈	—	2	1	no	—
301	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
302	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
303	C₂²×D₄	—	2	2	no	—
304	C₂²×D₄	—	2	1	no	—
305	C₂²×D₄	—	2	1	no	—
306	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
307	C₂³×C₄	—	2	2	no	—
308	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
309	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
311	C₂×Q₁₆	—	4	8	no	—
312	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
313	C₂×SD₁₆	—	4	8	no	—
314	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
315	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
316	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
317	C₄⋊Q₈	—	2	2	no	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
321	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
322	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
323	#127	—	2	2	no	—
324	#127	C₂³×C₂₃	1	4	yes	—
325	#125	—	2	8	no	—
326	#124	—	2	4	no	—
327	#124	C₂³×C₂₃	1	8	yes	—
328	C₂²⋊Q₈	—	2	8	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
331	#121	—	2	8	no	—
332	#121	—	2	8	no	—
333	C₄×Q₈	—	2	8	no	—
334	C₄×Q₈	—	2	4	no	—
335	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
336	C₄×D₄	—	2	8	no	—
337	C₄×D₄	—	2	8	no	—
338	#117	—	2	8	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	2	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
352	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
353	C₂×C₄²	D₄×C₂₃	1	4	yes	—
354	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
357	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
358	#103	C₂³×C₂₃	1	16	yes	—
359	#102	C₂³×C₂₃	1	16	yes	—
360	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
361	C₄×C₈	C₂³×C₂₃	1	4	yes	—
362	C₄×C₈	C₂³×C₂₃	1	4	yes	—
363	C₂₃	C₂×#3210	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	2	no	—
365	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
366	#41	—	4	16	no	—
367	#41	—	4	16	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	8	no	—
372	C₂×Q₈	—	2	8	no	—
373	C₂×Q₈	—	4	4	no	—
374	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
375	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	2	4	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	8	no	—
381	C₂×D₄	—	4	8	no	—
382	C₂×D₄	—	4	4	no	—
383	C₂×D₄	—	4	4	no	—
384	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	8	no	—
392	C₂²×C₄	—	4	8	no	—
393	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
398	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
399	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
402	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
403	Q₁₆	—	4	16	no	—
404	SD₁₆	—	4	16	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	—	2	4	no	—
407	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
408	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₄²	—	2	8	no	—
422	C₄²	C₂×D₄×C₂₃	1	4	yes	—
423	C₄²	C₂×D₄×C₂₃	1	4	yes	—
424	C₄²	C₂×D₄×C₂₃	1	2	yes	—
425	C₄²	C₂×D₄×C₂₃	1	2	yes	—
426	C₄²	C₂⁴×C₂₃	1	2	yes	—
427	C₄²	C₂⁴×C₂₃	1	2	yes	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	4	4	no	—
433	C₂³	—	4	4	no	—
434	C₂³	—	4	4	no	—
435	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
436	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
437	Q₈	—	2	8	no	—
438	Q₈	—	2	8	no	—
439	Q₈	—	4	8	no	—
440	Q₈	—	4	4	no	—
441	Q₈	—	4	4	no	—
442	D₄	—	2	8	no	—
443	D₄	—	4	16	no	—
444	D₄	—	4	8	no	—
445	D₄	—	4	8	no	—
446	D₄	—	4	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	8	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	4	16	no	—
458	C₂×C₄	—	4	4	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	C₂₃×#135	1	4	yes	—
462	C₂×C₄	C₂₃×#137	1	4	yes	—
463	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
464	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
469	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
470	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
471	C₈	—	2	4	no	—
472	C₈	—	2	4	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	8	8	no	—
481	C₂²	C₂₃×#535	1	2	yes	—
482	C₂²	C₂×C₂₃×#135	1	2	yes	—
483	C₂²	C₂×C₂₃×#137	1	2	yes	—
484	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	4	4	no	—
489	C₄	—	4	4	no	—
490	C₄	—	4	4	no	—
491	C₄	C₂₃×#535	1	2	yes	—
492	C₄	C₂×C₂₃×#135	1	4	yes	—
493	C₄	C₂×C₂₃×#137	1	4	yes	—
494	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
495	C₂	—	4	2	no	—
496	C₂	—	8	4	no	—
497	C₂	C₂₃×#3210	1	4	yes	direct product factor
498	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
499	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
500	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#532	C₂
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. C₂×C₂₃×#533	C₂	semidirect
3. C₂₃×#3210	C₂	direct (N × H)
4. C₂×C₂₃×#494	C₂	semidirect
5. C₂×C₂₃×#492	C₂	semidirect
6. C₂×C₂₃×#487	C₂	semidirect
7. C₂×C₂₃×#486	C₂	semidirect
8. C₂×C₂₃×#464	C₂	semidirect
9. C₂×C₂₃×#452	C₂	semidirect
10. C₂×C₂₃×#431	C₂	semidirect
15. C₂₃×#533	C₂²	semidirect
34. C₂₃×#494	C₂²	semidirect
35. C₂₃×#492	C₂²	semidirect
36. C₂₃×#487	C₂²	semidirect
37. C₂₃×#486	C₂²	semidirect
38. C₂₃×#464	C₂²	semidirect
39. C₂₃×#452	C₂²	semidirect
40. C₂₃×#431	C₂²	semidirect
177. C₂×#3210	C₂₃	direct (N × H)
223. C₂×#533	C₂×C₂₃	semidirect
225. #3210	C₂×C₂₃	direct (N × H)
226. C₂×#494	C₂×C₂₃	semidirect
227. C₂×#492	C₂×C₂₃	semidirect
228. C₂×#487	C₂×C₂₃	semidirect
229. C₂×#486	C₂×C₂₃	semidirect
230. C₂×#464	C₂×C₂₃	semidirect
231. C₂×#452	C₂×C₂₃	semidirect
232. C₂×#431	C₂×C₂₃	semidirect
259. #533	C₂²×C₂₃	semidirect
278. #494	C₂²×C₂₃	semidirect
279. #492	C₂²×C₂₃	semidirect
280. #487	C₂²×C₂₃	semidirect
281. #486	C₂²×C₂₃	semidirect
282. #464	C₂²×C₂₃	semidirect
283. #452	C₂²×C₂₃	semidirect
284. #431	C₂²×C₂₃	semidirect
293. C₂×C₂₃	#3210	direct (N × H)
363. C₂₃	C₂×#3210	direct (N × H)
497. C₂	C₂₃×#3210	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×#533	2
C₂×C₂₃×#532	2
C₂₃×#3210	2
C₂×C₂₃×#494	2
C₂×C₂₃×#492	2
C₂×C₂₃×#487	2
C₂×C₂₃×#486	2
C₂×C₂₃×#464	2
C₂×C₂₃×#452	2
C₂×C₂₃×#431	2
C₂×#3210	23

Subgroups — C2×C23×#3210 #19208406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3210 #19208406