Subgroups — C₂₃×#60900 #19208526

Order 5888.

1942 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3444	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3250	C₂	1	4	yes	maximal, complemented
3	C₂₃×#3249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3248	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3247	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3246	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3245	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3076	C₂	1	4	yes	maximal, complemented
9	C₂₃×#3050	C₂	1	4	yes	maximal, complemented
10	C₂₃×#3035	C₂	1	4	yes	maximal, complemented
11	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	4	yes	—
13	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#533	C₂²	1	2	yes	—
15	C₂₃×#533	C₂²	1	2	yes	—
16	C₂₃×#533	C₂²	1	2	yes	—
17	C₂₃×#532	C₂²	1	2	yes	—
18	C₂₃×#532	C₂²	1	2	yes	—
19	C₂₃×#532	C₂²	1	2	yes	—
20	C₂₃×#529	—	2	2	no	—
21	C₂₃×#529	—	2	2	no	—
22	C₂₃×#529	—	2	2	no	—
23	C₂₃×#529	—	2	2	no	—
24	C₂₃×#529	C₂²	1	4	yes	—
25	C₂₃×#529	C₂²	1	4	yes	—
26	C₂₃×#525	—	2	2	no	—
27	C₂₃×#525	—	2	2	no	—
28	C₂₃×#525	C₂²	1	4	yes	—
29	C₄×#41×C₂₃	—	2	2	no	—
30	C₄×#41×C₂₃	—	2	2	no	—
31	C₄×#41×C₂₃	C₂²	1	4	yes	—
32	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#479	C₂²	1	8	yes	complemented
34	C₂₃×#478	C₂²	1	8	yes	complemented
35	C₂₃×#477	C₂²	1	4	yes	complemented
36	C₂₃×#476	C₂²	1	4	yes	complemented
37	C₂₃×#475	C₂²	1	4	yes	complemented
38	C₂₃×#474	C₂²	1	4	yes	complemented
39	C₂₃×#467	C₂²	1	4	yes	—
40	C₂₃×#467	C₂²	1	4	yes	—
41	C₂₃×#466	C₂²	1	4	yes	—
42	C₂₃×#466	C₂²	1	4	yes	—
43	C₂₃×#465	C₂²	1	8	yes	—
44	C₂₃×#465	C₂²	1	8	yes	—
45	C₂₃×#464	C₂²	1	8	yes	complemented
46	C₂₃×#463	C₂²	1	8	yes	complemented
47	C₂₃×#462	C₂²	1	4	yes	complemented
48	C₂₃×#461	C₂²	1	4	yes	complemented
49	C₂₃×#460	C₂²	1	4	yes	complemented
50	C₂₃×#459	C₂²	1	4	yes	complemented
51	C₂₃×#427	C₂²	1	4	yes	—
52	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
55	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
56	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
57	C₂₃×#416	C₂²	1	8	yes	—
58	C₂×#41×C₂₃	—	2	2	no	—
59	C₂×#41×C₂₃	—	2	2	no	—
60	C₂×#41×C₂₃	C₂³	1	4	yes	—
61	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×#135	—	4	8	no	—
65	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
66	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
67	C₂×D₈×C₂₃	C₂³	1	4	yes	—
68	C₂²×C₈×C₂₃	—	4	4	no	—
69	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
70	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
71	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
72	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
73	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₂₃×#127	C₂³	1	2	yes	—
77	C₂₃×#126	—	2	4	no	—
78	C₂₃×#126	—	2	4	no	—
79	C₂₃×#125	—	2	2	no	—
80	C₂₃×#125	—	2	2	no	—
81	C₂₃×#125	C₂³	1	4	yes	—
82	C₂₃×#124	—	2	2	no	—
83	C₂₃×#124	—	2	2	no	—
84	C₂₃×#124	C₂³	1	4	yes	—
85	C₂₃×#123	—	2	4	no	—
86	C₂₃×#123	—	2	4	no	—
87	C₂₃×#123	—	2	4	no	—
88	C₂₃×C₂²⋊Q₈	—	2	2	no	—
89	C₂₃×C₂²⋊Q₈	—	2	2	no	—
90	C₂₃×C₂²⋊Q₈	—	2	2	no	—
91	C₂₃×C₂²⋊Q₈	—	2	2	no	—
92	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
93	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
94	C₂₃×#121	—	2	2	no	—
95	C₂₃×#121	—	2	2	no	—
96	C₂₃×#121	—	2	2	no	—
97	C₂₃×#121	—	2	2	no	—
98	C₂₃×#121	C₂³	1	4	yes	—
99	C₂₃×#121	C₂³	1	4	yes	—
100	C₂₃×#120	—	2	4	no	—
101	C₄×Q₈×C₂₃	—	2	2	no	—
102	C₄×Q₈×C₂₃	—	2	2	no	—
103	C₄×Q₈×C₂₃	—	2	2	no	—
104	C₄×Q₈×C₂₃	—	2	2	no	—
105	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
106	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
107	C₄×D₄×C₂₃	—	2	4	no	—
108	C₄×D₄×C₂₃	—	2	4	no	—
109	C₄×D₄×C₂₃	—	2	4	no	—
110	C₄×D₄×C₂₃	—	2	4	no	—
111	C₄×D₄×C₂₃	—	2	4	no	—
112	C₄×D₄×C₂₃	—	2	4	no	—
113	C₄×D₄×C₂₃	—	2	2	no	—
114	C₄×D₄×C₂₃	—	2	2	no	—
115	C₄×D₄×C₂₃	—	2	2	no	—
116	C₄×D₄×C₂₃	—	2	2	no	—
117	C₄×D₄×C₂₃	C₂³	1	4	yes	—
118	C₄×D₄×C₂₃	C₂³	1	4	yes	—
119	C₂₃×#117	—	2	2	no	—
120	C₂₃×#117	—	2	2	no	—
121	C₂₃×#117	—	2	4	no	—
122	C₂₃×#117	—	2	4	no	—
123	C₂₃×#117	—	2	4	no	—
124	C₂₃×#117	C₂³	1	4	yes	—
125	C₂×C₄²×C₂₃	—	2	4	no	—
126	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₂₃	D₄	1	2	yes	—
128	C₂×C₄²×C₂₃	D₄	1	2	yes	—
129	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
130	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
131	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
133	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
134	C₂₃×#107	C₂³	1	4	yes	—
135	C₂₃×#107	C₂³	1	4	yes	—
136	C₂₃×#106	C₂³	1	8	yes	—
137	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
138	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
139	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
140	C₂₃×#103	C₂³	1	8	yes	—
141	C₂₃×#103	C₂³	1	8	yes	—
142	C₂₃×#102	C₂³	1	8	yes	—
143	C₂₃×#102	C₂³	1	8	yes	—
144	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
145	#41×C₂₃	—	2	8	no	—
146	#41×C₂₃	—	2	8	no	—
147	#41×C₂₃	—	4	8	no	—
148	#41×C₂₃	—	4	8	no	—
149	C₂×Q₈×C₂₃	—	2	2	no	—
150	C₂×Q₈×C₂₃	—	2	2	no	—
151	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
152	C₂×D₄×C₂₃	—	2	4	no	—
153	C₂×D₄×C₂₃	—	2	4	no	—
154	C₂×D₄×C₂₃	—	2	4	no	—
155	C₂×D₄×C₂₃	—	2	2	no	—
156	C₂×D₄×C₂₃	—	2	2	no	—
157	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	2	2	no	—
160	C₂²×C₄×C₂₃	—	2	2	no	—
161	C₂²×C₄×C₂₃	—	2	4	no	—
162	C₂²×C₄×C₂₃	—	2	4	no	—
163	C₂²×C₄×C₂₃	—	2	4	no	—
164	C₂²×C₄×C₂₃	—	4	2	no	—
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
179	Q₁₆×C₂₃	—	4	4	no	—
180	SD₁₆×C₂₃	—	4	8	no	—
181	D₈×C₂₃	—	4	4	no	—
182	C₂×C₈×C₂₃	—	2	8	no	—
183	C₂×C₈×C₂₃	—	4	2	no	—
184	C₂×C₈×C₂₃	—	4	4	no	—
185	C₂×C₈×C₂₃	—	4	2	no	—
186	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
187	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	4	no	—
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	—	2	4	no	—
195	C₄⋊C₄×C₂₃	—	2	4	no	—
196	C₄⋊C₄×C₂₃	—	2	4	no	—
197	C₄⋊C₄×C₂₃	—	2	4	no	—
198	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
199	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
200	C₂²⋊C₄×C₂₃	—	2	4	no	—
201	C₂²⋊C₄×C₂₃	—	2	4	no	—
202	C₂²⋊C₄×C₂₃	—	2	4	no	—
203	C₂²⋊C₄×C₂₃	—	2	4	no	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	4	no	—
207	C₂²⋊C₄×C₂₃	—	2	4	no	—
208	C₂²⋊C₄×C₂₃	—	2	4	no	—
209	C₄²×C₂₃	—	2	4	no	—
210	C₄²×C₂₃	—	2	4	no	—
211	C₄²×C₂₃	—	2	4	no	—
212	C₄²×C₂₃	C₂×D₄	1	2	yes	—
213	C₄²×C₂₃	C₂×D₄	1	2	yes	—
214	C₄²×C₂₃	C₂×D₄	1	2	yes	—
215	C₄²×C₂₃	C₂×D₄	1	2	yes	—
216	C₄²×C₂₃	C₂×D₄	1	2	yes	—
217	C₄²×C₂₃	C₂×D₄	1	2	yes	—
218	C₄²×C₂₃	C₂⁴	1	2	yes	—
219	C₄²×C₂₃	C₂⁴	1	2	yes	—
220	#60900	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	2	2	no	—
224	C₂³×C₂₃	—	2	4	no	—
225	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
227	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₂₃	—	2	4	no	—
229	Q₈×C₂₃	—	2	4	no	—
230	Q₈×C₂₃	—	4	4	no	—
231	Q₈×C₂₃	—	4	4	no	—
232	D₄×C₂₃	—	2	4	no	—
233	D₄×C₂₃	—	2	4	no	—
234	D₄×C₂₃	—	4	4	no	—
235	D₄×C₂₃	—	4	8	no	—
236	D₄×C₂₃	—	4	8	no	—
237	D₄×C₂₃	—	4	8	no	—
238	D₄×C₂₃	—	4	4	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	4	no	—
242	C₂×C₄×C₂₃	—	2	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	4	no	—
245	C₂×C₄×C₂₃	—	2	4	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	4	no	—
251	C₂×C₄×C₂₃	—	2	4	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	2	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	8	no	—
257	C₂×C₄×C₂₃	—	4	8	no	—
258	C₂×C₄×C₂₃	—	4	8	no	—
259	C₂×C₄×C₂₃	#135	1	4	yes	complemented
260	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
268	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
269	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
270	C₈×C₂₃	—	2	4	no	—
271	C₈×C₂₃	—	2	4	no	—
272	#3444	C₂×C₂₃	1	2	yes	complemented
273	#3250	C₂×C₂₃	1	4	yes	complemented
274	#3249	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#3248	C₂×C₂₃	1	4	yes	complemented
276	#3247	C₂×C₂₃	1	4	yes	complemented
277	#3246	C₂×C₂₃	1	2	yes	complemented
278	#3245	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#3076	C₂×C₂₃	1	4	yes	complemented
280	#3050	C₂×C₂₃	1	4	yes	complemented
281	#3035	C₂×C₂₃	1	4	yes	complemented
282	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
283	C₂²×C₂₃	—	2	8	no	—
284	C₂²×C₂₃	—	2	1	no	—
285	C₂²×C₂₃	—	2	1	no	—
286	C₂²×C₂₃	—	2	1	no	—
287	C₂²×C₂₃	—	2	1	no	—
288	C₂²×C₂₃	—	2	1	no	—
289	C₂²×C₂₃	—	2	1	no	—
290	C₂²×C₂₃	—	4	4	no	—
291	C₂²×C₂₃	—	4	4	no	—
292	C₂²×C₂₃	—	4	4	no	—
293	C₂²×C₂₃	#537	1	2	yes	—
294	C₂²×C₂₃	C₂×#135	1	2	yes	—
295	C₂²×C₂₃	#575	1	2	yes	—
296	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
297	C₄×C₂₃	—	2	1	no	—
298	C₄×C₂₃	—	2	1	no	—
299	C₄×C₂₃	—	2	2	no	—
300	C₄×C₂₃	—	2	2	no	—
301	C₄×C₂₃	—	2	2	no	—
302	C₄×C₂₃	—	4	4	no	—
303	C₄×C₂₃	—	4	4	no	—
304	C₄×C₂₃	—	4	4	no	—
305	C₄×C₂₃	#537	1	2	yes	—
306	C₄×C₂₃	C₂×#135	1	4	yes	—
307	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
308	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
309	C₂×#135	C₂²×C₂₃	1	4	yes	—
310	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
311	#533	C₂²×C₂₃	1	2	yes	—
312	#533	C₂²×C₂₃	1	2	yes	—
313	#533	C₂²×C₂₃	1	2	yes	—
314	#532	C₂²×C₂₃	1	2	yes	—
315	#532	C₂²×C₂₃	1	2	yes	—
316	#532	C₂²×C₂₃	1	2	yes	—
317	#529	—	2	2	no	—
318	#529	—	2	2	no	—
319	#529	—	2	2	no	—
320	#529	—	2	2	no	—
321	#529	C₂²×C₂₃	1	4	yes	—
322	#529	C₂²×C₂₃	1	4	yes	—
323	#525	—	2	2	no	—
324	#525	—	2	2	no	—
325	#525	C₂²×C₂₃	1	4	yes	—
326	C₄×#41	—	2	2	no	—
327	C₄×#41	—	2	2	no	—
328	C₄×#41	C₂²×C₂₃	1	4	yes	—
329	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
330	#479	C₂²×C₂₃	1	8	yes	complemented
331	#478	C₂²×C₂₃	1	8	yes	complemented
332	#477	C₂²×C₂₃	1	4	yes	complemented
333	#476	C₂²×C₂₃	1	4	yes	complemented
334	#475	C₂²×C₂₃	1	4	yes	complemented
335	#474	C₂²×C₂₃	1	4	yes	complemented
336	#467	C₂²×C₂₃	1	4	yes	—
337	#467	C₂²×C₂₃	1	4	yes	—
338	#466	C₂²×C₂₃	1	4	yes	—
339	#466	C₂²×C₂₃	1	4	yes	—
340	#465	C₂²×C₂₃	1	8	yes	—
341	#465	C₂²×C₂₃	1	8	yes	—
342	#464	C₂²×C₂₃	1	8	yes	complemented
343	#463	C₂²×C₂₃	1	8	yes	complemented
344	#462	C₂²×C₂₃	1	4	yes	complemented
345	#461	C₂²×C₂₃	1	4	yes	complemented
346	#460	C₂²×C₂₃	1	4	yes	complemented
347	#459	C₂²×C₂₃	1	4	yes	complemented
348	#427	C₂²×C₂₃	1	4	yes	—
349	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
350	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
353	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
354	#416	C₂²×C₂₃	1	8	yes	—
355	C₂×C₂₃	—	2	2	no	—
356	C₂×C₂₃	—	2	2	no	—
357	C₂×C₂₃	—	2	2	no	—
358	C₂×C₂₃	—	8	4	no	—
359	C₂×C₂₃	C₂×#537	1	1	yes	characteristic
360	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
361	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
362	C₂×#41	—	2	2	no	—
363	C₂×#41	—	2	2	no	—
364	C₂×#41	C₂³×C₂₃	1	4	yes	—
365	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
366	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
368	#135	—	4	8	no	—
369	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
370	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
371	C₂×D₈	C₂³×C₂₃	1	4	yes	—
372	C₂²×C₈	—	4	4	no	—
373	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
374	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
375	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
376	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
377	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
378	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
379	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
380	#127	C₂³×C₂₃	1	2	yes	—
381	#126	—	2	4	no	—
382	#126	—	2	4	no	—
383	#125	—	2	2	no	—
384	#125	—	2	2	no	—
385	#125	C₂³×C₂₃	1	4	yes	—
386	#124	—	2	2	no	—
387	#124	—	2	2	no	—
388	#124	C₂³×C₂₃	1	4	yes	—
389	#123	—	2	4	no	—
390	#123	—	2	4	no	—
391	#123	—	2	4	no	—
392	C₂²⋊Q₈	—	2	2	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	—	2	2	no	—
396	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
397	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
398	#121	—	2	2	no	—
399	#121	—	2	2	no	—
400	#121	—	2	2	no	—
401	#121	—	2	2	no	—
402	#121	C₂³×C₂₃	1	4	yes	—
403	#121	C₂³×C₂₃	1	4	yes	—
404	#120	—	2	4	no	—
405	C₄×Q₈	—	2	2	no	—
406	C₄×Q₈	—	2	2	no	—
407	C₄×Q₈	—	2	2	no	—
408	C₄×Q₈	—	2	2	no	—
409	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
410	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
411	C₄×D₄	—	2	4	no	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	4	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	4	no	—
416	C₄×D₄	—	2	4	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	C₂³×C₂₃	1	4	yes	—
422	C₄×D₄	C₂³×C₂₃	1	4	yes	—
423	#117	—	2	2	no	—
424	#117	—	2	2	no	—
425	#117	—	2	4	no	—
426	#117	—	2	4	no	—
427	#117	—	2	4	no	—
428	#117	C₂³×C₂₃	1	4	yes	—
429	C₂×C₄²	—	2	4	no	—
430	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
431	C₂×C₄²	D₄×C₂₃	1	2	yes	—
432	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
433	C₂×C₄²	D₄×C₂₃	1	2	yes	—
434	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
435	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
437	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
438	#107	C₂³×C₂₃	1	4	yes	—
439	#107	C₂³×C₂₃	1	4	yes	—
440	#106	C₂³×C₂₃	1	8	yes	—
441	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
442	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
443	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
444	#103	C₂³×C₂₃	1	8	yes	—
445	#103	C₂³×C₂₃	1	8	yes	—
446	#102	C₂³×C₂₃	1	8	yes	—
447	#102	C₂³×C₂₃	1	8	yes	—
448	C₂₃	#60900	1	1	yes	characteristic, direct product factor, Sylow
449	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
450	#41	—	2	8	no	—
451	#41	—	2	8	no	—
452	#41	—	4	8	no	—
453	#41	—	4	8	no	—
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
474	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
484	Q₁₆	—	4	4	no	—
485	SD₁₆	—	4	8	no	—
486	D₈	—	4	4	no	—
487	C₂×C₈	—	2	8	no	—
488	C₂×C₈	—	4	2	no	—
489	C₂×C₈	—	4	4	no	—
490	C₂×C₈	—	4	2	no	—
491	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
492	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
504	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₄²	—	2	4	no	—
515	C₄²	—	2	4	no	—
516	C₄²	—	2	4	no	—
517	C₄²	C₂×D₄×C₂₃	1	2	yes	—
518	C₄²	C₂×D₄×C₂₃	1	2	yes	—
519	C₄²	C₂×D₄×C₂₃	1	2	yes	—
520	C₄²	C₂×D₄×C₂₃	1	2	yes	—
521	C₄²	C₂×D₄×C₂₃	1	2	yes	—
522	C₄²	C₂×D₄×C₂₃	1	2	yes	—
523	C₄²	C₂⁴×C₂₃	1	2	yes	—
524	C₄²	C₂⁴×C₂₃	1	2	yes	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	4	no	—
529	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
530	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
531	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
532	Q₈	—	2	4	no	—
533	Q₈	—	2	4	no	—
534	Q₈	—	4	4	no	—
535	Q₈	—	4	4	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	4	no	—
538	D₄	—	4	4	no	—
539	D₄	—	4	8	no	—
540	D₄	—	4	8	no	—
541	D₄	—	4	8	no	—
542	D₄	—	4	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	8	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	—	4	8	no	—
563	C₂×C₄	C₂₃×#135	1	4	yes	complemented
564	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
569	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
570	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
571	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
572	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
573	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
574	C₈	—	2	4	no	—
575	C₈	—	2	4	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	2	1	no	—
582	C₂²	—	2	1	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	4	no	—
586	C₂²	C₂₃×#537	1	2	yes	—
587	C₂²	C₂×C₂₃×#135	1	2	yes	—
588	C₂²	C₂₃×#575	1	2	yes	—
589	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
590	C₄	—	2	1	no	—
591	C₄	—	2	1	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	4	4	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	4	no	—
598	C₄	C₂₃×#537	1	2	yes	—
599	C₄	C₂×C₂₃×#135	1	4	yes	—
600	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
601	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
602	C₂	—	2	2	no	—
603	C₂	—	2	2	no	—
604	C₂	—	2	2	no	—
605	C₂	—	8	4	no	—
606	C₂	C₂×C₂₃×#537	1	1	yes	characteristic
607	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
608	C₂	C₂×C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3444	C₂	semidirect
2. C₂₃×#3250	C₂	semidirect
3. C₂₃×#3249	C₂	semidirect
4. C₂₃×#3248	C₂	semidirect
5. C₂₃×#3247	C₂	semidirect
6. C₂₃×#3246	C₂	semidirect
7. C₂₃×#3245	C₂	semidirect
8. C₂₃×#3076	C₂	semidirect
9. C₂₃×#3050	C₂	semidirect
10. C₂₃×#3035	C₂	semidirect
11. C₂²×C₂₃×C₄⋊C₈	C₂	semidirect
33. C₂₃×#479	C₂²	semidirect
34. C₂₃×#478	C₂²	semidirect
35. C₂₃×#477	C₂²	semidirect
36. C₂₃×#476	C₂²	semidirect
37. C₂₃×#475	C₂²	semidirect
38. C₂₃×#474	C₂²	semidirect
45. C₂₃×#464	C₂²	semidirect
46. C₂₃×#463	C₂²	semidirect
47. C₂₃×#462	C₂²	semidirect
48. C₂₃×#461	C₂²	semidirect
49. C₂₃×#460	C₂²	semidirect
50. C₂₃×#459	C₂²	semidirect
220. #60900	C₂₃	direct (N × H)
259. C₂×C₄×C₂₃	#135	semidirect
272. #3444	C₂×C₂₃	semidirect
273. #3250	C₂×C₂₃	semidirect
274. #3249	C₂×C₂₃	semidirect
275. #3248	C₂×C₂₃	semidirect
276. #3247	C₂×C₂₃	semidirect
277. #3246	C₂×C₂₃	semidirect
278. #3245	C₂×C₂₃	semidirect
279. #3076	C₂×C₂₃	semidirect
280. #3050	C₂×C₂₃	semidirect
281. #3035	C₂×C₂₃	semidirect
282. C₂²×C₄⋊C₈	C₂×C₂₃	semidirect
330. #479	C₂²×C₂₃	semidirect
331. #478	C₂²×C₂₃	semidirect
332. #477	C₂²×C₂₃	semidirect
333. #476	C₂²×C₂₃	semidirect
334. #475	C₂²×C₂₃	semidirect
335. #474	C₂²×C₂₃	semidirect
342. #464	C₂²×C₂₃	semidirect
343. #463	C₂²×C₂₃	semidirect
344. #462	C₂²×C₂₃	semidirect
345. #461	C₂²×C₂₃	semidirect
346. #460	C₂²×C₂₃	semidirect
347. #459	C₂²×C₂₃	semidirect
448. C₂₃	#60900	direct (N × H)
563. C₂×C₄	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3444	2
C₂₃×#3250	2
C₂₃×#3249	2
C₂₃×#3248	2
C₂₃×#3247	2
C₂₃×#3246	2
C₂₃×#3245	2
C₂₃×#3076	2
C₂₃×#3050	2
C₂₃×#3035	2
C₂²×C₂₃×C₄⋊C₈	2
#60900	23

Subgroups — C23×#60900 #19208526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60900 #19208526