Subgroups — C₂₃×#60921 #19208547

Order 5888.

1758 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3455	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3261	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3260	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3258	C₂	1	8	yes	maximal, complemented
6	C₂₃×#3080	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3078	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3056	C₂	1	4	yes	maximal, complemented
9	C₂₃×#3039	C₂	1	2	yes	maximal, complemented
10	C₂₃×#3037	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2905	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#137	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#542	C₂²	1	4	yes	—
15	C₂₃×#541	—	2	4	no	—
16	C₂₃×#541	C₂²	1	4	yes	—
17	C₂₃×#541	C₂²	1	4	yes	—
18	C₂₃×#541	C₂²	1	2	yes	—
19	C₂₃×#540	C₂²	1	2	yes	—
20	C₂₃×#539	—	2	2	no	—
21	C₂₃×#538	—	2	2	no	—
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂₃×#532	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂₃×#529	—	2	4	no	—
26	C₂×C₂₃×#126	—	2	2	no	—
27	C₂×C₂₃×#125	—	2	2	no	—
28	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
29	C₂₃×#525	C₂²	1	2	yes	—
30	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
31	C₄×#41×C₂₃	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
33	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
34	C₂₃×#479	C₂²	1	8	yes	complemented
35	C₂₃×#479	C₂²	1	8	yes	complemented
36	C₂₃×#478	C₂²	1	8	yes	complemented
37	C₂₃×#477	C₂²	1	4	yes	complemented
38	C₂₃×#475	C₂²	1	4	yes	complemented
39	C₂₃×#470	C₂²	1	8	yes	—
40	C₂₃×#468	C₂²	1	8	yes	—
41	C₂₃×#467	C₂²	1	8	yes	—
42	C₂₃×#465	C₂²	1	8	yes	—
43	C₂₃×#464	C₂²	1	8	yes	complemented
44	C₂₃×#462	C₂²	1	8	yes	complemented
45	C₂₃×#462	C₂²	1	4	yes	complemented
46	C₂₃×#461	C₂²	1	8	yes	complemented
47	C₂₃×#460	C₂²	1	4	yes	complemented
48	C₂₃×#428	C₂²	1	4	yes	—
49	C₂₃×#424	C₂²	1	4	yes	—
50	C₂₃×#424	C₂²	1	4	yes	—
51	C₂₃×#423	C₂²	1	2	yes	—
52	C₂₃×#423	C₂²	1	2	yes	—
53	C₂₃×#419	C₂²	1	4	yes	—
54	C₂₃×#417	C₂²	1	4	yes	—
55	C₂×#41×C₂₃	C₂³	1	2	yes	—
56	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
57	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂₃×#137	—	4	8	no	—
59	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
60	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
61	C₂×M₁₆×C₂₃	—	4	4	no	—
62	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
63	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
64	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
65	C₂₃×C₄⋊Q₈	—	2	2	no	—
66	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
67	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₂₃×#127	C₂³	1	1	yes	characteristic
70	C₂₃×#126	—	2	4	no	—
71	C₂₃×#126	—	2	8	no	—
72	C₂₃×#126	—	2	8	no	—
73	C₂₃×#126	—	2	4	no	—
74	C₂₃×#125	—	2	8	no	—
75	C₂₃×#125	—	2	8	no	—
76	C₂₃×#125	—	2	4	no	—
77	C₂₃×#125	—	2	2	no	—
78	C₂₃×#125	C₂³	1	4	yes	—
79	C₂₃×#125	C₂³	1	2	yes	—
80	C₂₃×#124	—	2	4	no	—
81	C₂₃×#124	C₂³	1	2	yes	—
82	C₂₃×#123	—	2	2	no	—
83	C₂₃×#123	—	2	4	no	—
84	C₂₃×#123	—	2	4	no	—
85	C₂₃×C₂²⋊Q₈	—	2	4	no	—
86	C₂₃×C₂²⋊Q₈	—	2	2	no	—
87	C₂₃×C₂²⋊Q₈	—	2	4	no	—
88	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
89	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
90	C₂₃×#121	—	2	2	no	—
91	C₂₃×#121	—	2	4	no	—
92	C₂₃×#121	C₂³	1	4	yes	—
93	C₂₃×#120	—	2	2	no	—
94	C₄×Q₈×C₂₃	—	2	2	no	—
95	C₄×Q₈×C₂₃	—	2	2	no	—
96	C₄×Q₈×C₂₃	—	2	2	no	—
97	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
98	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
99	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
100	C₄×D₄×C₂₃	—	2	4	no	—
101	C₄×D₄×C₂₃	—	2	4	no	—
102	C₄×D₄×C₂₃	—	2	2	no	—
103	C₄×D₄×C₂₃	C₂³	1	4	yes	—
104	C₂₃×#117	—	2	2	no	—
105	C₂₃×#117	D₄	1	4	yes	—
106	C₂₃×#117	C₂³	1	2	yes	—
107	C₂₃×#117	C₂³	1	2	yes	—
108	C₂₃×#117	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
110	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
111	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
113	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
117	C₂×C₄²×C₂₃	—	2	2	no	—
118	C₂×C₄²×C₂₃	—	2	2	no	—
119	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂₃×#107	C₂³	1	8	yes	—
122	C₂₃×#106	C₂³	1	8	yes	—
123	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
124	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
125	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
126	C₂₃×#103	C₂³	1	8	yes	—
127	C₂₃×#103	C₂³	1	8	yes	—
128	C₂₃×#103	C₂³	1	8	yes	—
129	C₂₃×#102	C₂³	1	8	yes	—
130	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
131	#41×C₂₃	—	2	4	no	—
132	#41×C₂₃	—	2	4	no	—
133	C₂×Q₈×C₂₃	—	2	2	no	—
134	C₂×Q₈×C₂₃	—	2	2	no	—
135	C₂×Q₈×C₂₃	—	2	4	no	—
136	C₂×Q₈×C₂₃	—	2	4	no	—
137	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
138	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
139	C₂×D₄×C₂₃	—	2	2	no	—
140	C₂×D₄×C₂₃	—	2	4	no	—
141	C₂×D₄×C₂₃	—	2	2	no	—
142	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₂₃	—	2	2	no	—
144	C₂²×C₄×C₂₃	—	2	2	no	—
145	C₂²×C₄×C₂₃	—	2	2	no	—
146	C₂²×C₄×C₂₃	—	2	2	no	—
147	C₂²×C₄×C₂₃	—	2	2	no	—
148	C₂²×C₄×C₂₃	—	2	4	no	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
155	Q₁₆×C₂₃	—	4	8	no	—
156	SD₁₆×C₂₃	—	4	8	no	—
157	M₁₆×C₂₃	—	2	8	no	—
158	M₁₆×C₂₃	—	4	4	no	—
159	C₂×C₈×C₂₃	—	4	4	no	—
160	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
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	8	no	—
166	C₄⋊C₄×C₂₃	—	2	2	no	—
167	C₄⋊C₄×C₂₃	—	2	4	no	—
168	C₄⋊C₄×C₂₃	—	2	2	no	—
169	C₄⋊C₄×C₂₃	—	2	4	no	—
170	C₄⋊C₄×C₂₃	—	2	4	no	—
171	C₄⋊C₄×C₂₃	—	2	2	no	—
172	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
173	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
174	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
175	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
176	C₂²⋊C₄×C₂₃	—	2	8	no	—
177	C₂²⋊C₄×C₂₃	—	2	2	no	—
178	C₂²⋊C₄×C₂₃	—	2	4	no	—
179	C₂²⋊C₄×C₂₃	—	2	4	no	—
180	C₂²⋊C₄×C₂₃	—	2	4	no	—
181	C₂²⋊C₄×C₂₃	—	2	4	no	—
182	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
183	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
184	C₄²×C₂₃	—	2	2	no	—
185	C₄²×C₂₃	—	2	4	no	—
186	C₄²×C₂₃	—	2	4	no	—
187	C₄²×C₂₃	—	2	4	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	1	yes	characteristic
192	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
193	#60921	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₂₃	—	2	2	no	—
195	C₂³×C₂₃	—	2	2	no	—
196	C₂³×C₂₃	—	4	2	no	—
197	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
198	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₂₃	—	2	4	no	—
200	Q₈×C₂₃	—	2	2	no	—
201	Q₈×C₂₃	—	2	4	no	—
202	Q₈×C₂₃	—	2	2	no	—
203	Q₈×C₂₃	—	4	4	no	—
204	Q₈×C₂₃	—	8	2	no	—
205	D₄×C₂₃	—	2	2	no	—
206	D₄×C₂₃	—	2	2	no	—
207	D₄×C₂₃	—	4	4	no	—
208	D₄×C₂₃	—	8	2	no	—
209	D₄×C₂₃	—	8	4	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	4	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	2	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	—	4	2	no	—
219	C₂×C₄×C₂₃	—	4	2	no	—
220	C₂×C₄×C₂₃	—	4	4	no	—
221	C₂×C₄×C₂₃	—	4	4	no	—
222	C₂×C₄×C₂₃	—	4	4	no	—
223	C₂×C₄×C₂₃	—	8	4	no	—
224	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
230	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
231	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
232	C₈×C₂₃	—	2	8	no	—
233	#3455	C₂×C₂₃	1	1	yes	characteristic
234	C₂×#541	C₂×C₂₃	1	1	yes	characteristic, complemented
235	#3261	C₂×C₂₃	1	4	yes	complemented
236	#3260	C₂×C₂₃	1	4	yes	complemented
237	#3258	C₂×C₂₃	1	8	yes	complemented
238	#3080	C₂×C₂₃	1	2	yes	complemented
239	#3078	C₂×C₂₃	1	2	yes	complemented
240	#3056	C₂×C₂₃	1	4	yes	complemented
241	#3039	C₂×C₂₃	1	2	yes	complemented
242	#3037	C₂×C₂₃	1	2	yes	complemented
243	#2905	C₂×C₂₃	1	1	yes	characteristic, complemented
244	C₂²×C₂₃	—	2	2	no	—
245	C₂²×C₂₃	—	2	4	no	—
246	C₂²×C₂₃	—	2	2	no	—
247	C₂²×C₂₃	—	4	4	no	—
248	C₂²×C₂₃	—	4	2	no	—
249	C₂²×C₂₃	—	8	2	no	—
250	C₂²×C₂₃	#537	1	2	yes	—
251	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₂₃	—	2	4	no	—
253	C₄×C₂₃	—	2	2	no	—
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	2	2	no	—
256	C₄×C₂₃	—	4	4	no	—
257	C₄×C₂₃	—	4	4	no	—
258	C₄×C₂₃	—	4	2	no	—
259	C₄×C₂₃	—	4	4	no	—
260	C₄×C₂₃	#537	1	2	yes	—
261	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
262	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
263	C₂×#137	C₂²×C₂₃	1	4	yes	—
264	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
265	#542	C₂²×C₂₃	1	4	yes	—
266	#541	—	2	4	no	—
267	#541	C₂²×C₂₃	1	4	yes	—
268	#541	C₂²×C₂₃	1	4	yes	—
269	#541	C₂²×C₂₃	1	2	yes	—
270	#540	C₂²×C₂₃	1	2	yes	—
271	#539	—	2	2	no	—
272	#538	—	2	2	no	—
273	#533	C₂²×C₂₃	1	1	yes	characteristic
274	#532	C₂²×C₂₃	1	1	yes	characteristic
275	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
276	#529	—	2	4	no	—
277	C₂×#126	—	2	2	no	—
278	C₂×#125	—	2	2	no	—
279	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
280	#525	C₂²×C₂₃	1	2	yes	—
281	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
282	C₄×#41	C₂²×C₂₃	1	2	yes	—
283	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
284	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
285	#479	C₂²×C₂₃	1	8	yes	complemented
286	#479	C₂²×C₂₃	1	8	yes	complemented
287	#478	C₂²×C₂₃	1	8	yes	complemented
288	#477	C₂²×C₂₃	1	4	yes	complemented
289	#475	C₂²×C₂₃	1	4	yes	complemented
290	#470	C₂²×C₂₃	1	8	yes	—
291	#468	C₂²×C₂₃	1	8	yes	—
292	#467	C₂²×C₂₃	1	8	yes	—
293	#465	C₂²×C₂₃	1	8	yes	—
294	#464	C₂²×C₂₃	1	8	yes	complemented
295	#462	C₂²×C₂₃	1	8	yes	complemented
296	#462	C₂²×C₂₃	1	4	yes	complemented
297	#461	C₂²×C₂₃	1	8	yes	complemented
298	#460	C₂²×C₂₃	1	4	yes	complemented
299	#428	C₂²×C₂₃	1	4	yes	—
300	#424	C₂²×C₂₃	1	4	yes	—
301	#424	C₂²×C₂₃	1	4	yes	—
302	#423	C₂²×C₂₃	1	2	yes	—
303	#423	C₂²×C₂₃	1	2	yes	—
304	#419	C₂²×C₂₃	1	4	yes	—
305	#417	C₂²×C₂₃	1	4	yes	—
306	C₂×C₂₃	—	2	2	no	—
307	C₂×C₂₃	—	4	2	no	—
308	C₂×C₂₃	—	8	2	no	—
309	C₂×C₂₃	C₂×#537	1	1	yes	characteristic
310	C₂×C₂₃	#3586	1	2	yes	—
311	C₂×#41	C₂³×C₂₃	1	2	yes	—
312	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
313	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
314	#137	—	4	8	no	—
315	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
316	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
317	C₂×M₁₆	—	4	4	no	—
318	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
319	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
320	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
321	C₄⋊Q₈	—	2	2	no	—
322	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
323	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
324	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
325	#127	C₂³×C₂₃	1	1	yes	characteristic
326	#126	—	2	4	no	—
327	#126	—	2	8	no	—
328	#126	—	2	8	no	—
329	#126	—	2	4	no	—
330	#125	—	2	8	no	—
331	#125	—	2	8	no	—
332	#125	—	2	4	no	—
333	#125	—	2	2	no	—
334	#125	C₂³×C₂₃	1	4	yes	—
335	#125	C₂³×C₂₃	1	2	yes	—
336	#124	—	2	4	no	—
337	#124	C₂³×C₂₃	1	2	yes	—
338	#123	—	2	2	no	—
339	#123	—	2	4	no	—
340	#123	—	2	4	no	—
341	C₂²⋊Q₈	—	2	4	no	—
342	C₂²⋊Q₈	—	2	2	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
345	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
346	#121	—	2	2	no	—
347	#121	—	2	4	no	—
348	#121	C₂³×C₂₃	1	4	yes	—
349	#120	—	2	2	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	2	no	—
352	C₄×Q₈	—	2	2	no	—
353	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
354	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
355	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	4	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	C₂³×C₂₃	1	4	yes	—
360	#117	—	2	2	no	—
361	#117	D₄×C₂₃	1	4	yes	—
362	#117	C₂³×C₂₃	1	2	yes	—
363	#117	C₂³×C₂₃	1	2	yes	—
364	#117	C₂³×C₂₃	1	2	yes	—
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	—	2	2	no	—
369	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
373	C₂×C₄²	—	2	2	no	—
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	#107	C₂³×C₂₃	1	8	yes	—
378	#106	C₂³×C₂₃	1	8	yes	—
379	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
380	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
381	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
382	#103	C₂³×C₂₃	1	8	yes	—
383	#103	C₂³×C₂₃	1	8	yes	—
384	#103	C₂³×C₂₃	1	8	yes	—
385	#102	C₂³×C₂₃	1	8	yes	—
386	C₂₃	#60921	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
388	#41	—	2	4	no	—
389	#41	—	2	4	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	2	no	—
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	—	2	4	no	—
394	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
395	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
409	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
411	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
412	Q₁₆	—	4	8	no	—
413	SD₁₆	—	4	8	no	—
414	M₁₆	—	2	8	no	—
415	M₁₆	—	4	4	no	—
416	C₂×C₈	—	4	4	no	—
417	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	4	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
430	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
431	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
432	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
433	C₂²⋊C₄	—	2	8	no	—
434	C₂²⋊C₄	—	2	2	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	4	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	4	no	—
439	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
440	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	4	no	—
443	C₄²	—	2	4	no	—
444	C₄²	—	2	4	no	—
445	C₄²	C₂×D₄×C₂₃	1	2	yes	—
446	C₄²	C₂×D₄×C₂₃	1	2	yes	—
447	C₄²	C₂⁴×C₂₃	1	2	yes	—
448	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
449	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
454	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
455	Q₈	—	2	4	no	—
456	Q₈	—	2	2	no	—
457	Q₈	—	2	4	no	—
458	Q₈	—	2	2	no	—
459	Q₈	—	4	4	no	—
460	Q₈	—	8	2	no	—
461	D₄	—	2	2	no	—
462	D₄	—	2	2	no	—
463	D₄	—	4	4	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
483	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
485	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
486	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
487	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
488	C₈	—	2	8	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	8	2	no	—
495	C₂²	C₂₃×#537	1	2	yes	—
496	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
497	C₄	—	2	2	no	—
498	C₄	—	2	4	no	—
499	C₄	—	2	2	no	—
500	C₄	—	2	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	4	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	4	no	—
505	C₄	C₂₃×#537	1	2	yes	—
506	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
507	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
508	C₂	—	2	2	no	—
509	C₂	—	4	2	no	—
510	C₂	—	8	2	no	—
511	C₂	C₂×C₂₃×#537	1	1	yes	characteristic
512	C₂	C₂₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#541	C₂	semidirect
3. C₂₃×#3261	C₂	semidirect
4. C₂₃×#3260	C₂	semidirect
5. C₂₃×#3258	C₂	semidirect
6. C₂₃×#3080	C₂	semidirect
7. C₂₃×#3078	C₂	semidirect
8. C₂₃×#3056	C₂	semidirect
9. C₂₃×#3039	C₂	semidirect
10. C₂₃×#3037	C₂	semidirect
11. C₂₃×#2905	C₂	semidirect
34. C₂₃×#479	C₂²	semidirect
35. C₂₃×#479	C₂²	semidirect
36. C₂₃×#478	C₂²	semidirect
37. C₂₃×#477	C₂²	semidirect
38. C₂₃×#475	C₂²	semidirect
43. C₂₃×#464	C₂²	semidirect
44. C₂₃×#462	C₂²	semidirect
45. C₂₃×#462	C₂²	semidirect
46. C₂₃×#461	C₂²	semidirect
47. C₂₃×#460	C₂²	semidirect
193. #60921	C₂₃	direct (N × H)
234. C₂×#541	C₂×C₂₃	semidirect
235. #3261	C₂×C₂₃	semidirect
236. #3260	C₂×C₂₃	semidirect
237. #3258	C₂×C₂₃	semidirect
238. #3080	C₂×C₂₃	semidirect
239. #3078	C₂×C₂₃	semidirect
240. #3056	C₂×C₂₃	semidirect
241. #3039	C₂×C₂₃	semidirect
242. #3037	C₂×C₂₃	semidirect
243. #2905	C₂×C₂₃	semidirect
285. #479	C₂²×C₂₃	semidirect
286. #479	C₂²×C₂₃	semidirect
287. #478	C₂²×C₂₃	semidirect
288. #477	C₂²×C₂₃	semidirect
289. #475	C₂²×C₂₃	semidirect
294. #464	C₂²×C₂₃	semidirect
295. #462	C₂²×C₂₃	semidirect
296. #462	C₂²×C₂₃	semidirect
297. #461	C₂²×C₂₃	semidirect
298. #460	C₂²×C₂₃	semidirect
386. C₂₃	#60921	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3455	2
C₂×C₂₃×#541	2
C₂₃×#3261	2
C₂₃×#3260	2
C₂₃×#3258	2
C₂₃×#3080	2
C₂₃×#3078	2
C₂₃×#3056	2
C₂₃×#3039	2
C₂₃×#3037	2
C₂₃×#2905	2
#60921	23

Subgroups — C23×#60921 #19208547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60921 #19208547