Subgroups — C₂₃×#60429 #19208055

Order 5888.

2730 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#577	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3046	C₂	1	6	yes	maximal, complemented
5	C₂₃×#3042	C₂	1	6	yes	maximal, complemented
6	C₂₃×#3039	C₂	1	3	yes	maximal, complemented
7	C₂₃×#3036	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3035	C₂	1	3	yes	maximal, complemented
9	C₂×C₂₃×#461	C₂	1	3	yes	maximal, complemented
10	C₂₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2898	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
14	C₂₃×#577	—	2	16	no	—
15	C₂×C₂₃×#137	C₂²	1	6	yes	—
16	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	6	yes	—
17	C₂×C₂₃×#135	C₂²	1	6	yes	—
18	C₂²×SD₁₆×C₂₃	C₂²	1	6	yes	—
19	C₂×C₂₃×#131	C₂²	1	2	yes	—
20	D₄×Q₈×C₂₃	—	2	1	no	—
21	D₄×Q₈×C₂₃	—	2	1	no	—
22	C₂₃×#547	—	2	3	no	—
23	C₂₃×#547	—	2	3	no	—
24	C₂₃×#536	C₂²	1	6	yes	complemented
25	C₂₃×#532	C₂²	1	3	yes	complemented
26	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
27	C₂₃×#520	—	2	1	no	—
28	C₂₃×#520	C₂²	1	3	yes	—
29	C₂₃×#520	C₂²	1	1	yes	characteristic
30	C₂₃×#518	—	2	3	no	—
31	C₂₃×#518	C₂²	1	3	yes	—
32	C₄×#41×C₂₃	—	2	1	no	—
33	C₄×#41×C₂₃	C₂²	1	3	yes	—
34	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₂₃	—	2	3	no	—
36	C₂×C₄×D₄×C₂₃	C₂²	1	3	yes	—
37	C₂₃×#464	C₂²	1	6	yes	complemented
38	C₂₃×#463	C₂²	1	6	yes	complemented
39	C₂₃×#463	C₂²	1	12	yes	complemented
40	C₂₃×#462	C₂²	1	3	yes	complemented
41	C₂₃×#462	C₂²	1	6	yes	complemented
42	C₂₃×#461	C₂²	1	3	yes	complemented
43	C₂₃×#461	C₂²	1	6	yes	complemented
44	C₂₃×#461	C₂²	1	12	yes	complemented
45	C₂₃×#461	C₂²	1	6	yes	complemented
46	C₂₃×#460	C₂²	1	1	yes	characteristic, complemented
47	C₂₃×#459	C₂²	1	3	yes	complemented
48	C₂₃×#424	C₂²	1	6	yes	complemented
49	C₂₃×#423	C₂²	1	3	yes	complemented
50	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	complemented
51	C₂₃×#419	C₂²	1	6	yes	complemented
52	C₂₃×#417	C₂²	1	6	yes	complemented
53	C₂₃×#416	C₂²	1	6	yes	complemented
54	C₂×C₂₃×#103	C₂²	1	6	yes	complemented
55	C₂₃×2₋¹⁺⁴	—	2	8	no	—
56	C₂₃×2₋¹⁺⁴	—	2	8	no	—
57	C₂₃×H(5,𝔽₂)	—	2	4	no	—
58	C₂₃×H(5,𝔽₂)	D₄	1	8	yes	complemented
59	C₂×#41×C₂₃	—	2	1	no	—
60	C₂×#41×C₂₃	—	2	6	no	—
61	C₂×#41×C₂₃	C₂³	1	6	yes	—
62	C₂×#41×C₂₃	C₂³	1	3	yes	—
63	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₂₃	—	2	2	no	—
65	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
66	C₂²×D₄×C₂₃	—	2	3	no	—
67	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
68	C₂³×C₄×C₂₃	—	2	3	no	—
69	C₂₃×#137	—	2	24	no	—
70	C₂₃×AGL(1,ℤ/8)	—	2	24	no	—
71	C₂₃×#135	—	2	24	no	—
72	C₂×Q₁₆×C₂₃	C₂³	1	6	yes	—
73	C₂×SD₁₆×C₂₃	—	2	24	no	—
74	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
75	C₂×SD₁₆×C₂₃	C₂³	1	12	yes	—
76	C₂×SD₁₆×C₂₃	C₂³	1	2	yes	—
77	C₂×D₈×C₂₃	C₂³	1	6	yes	—
78	C₂₃×#131	—	2	8	no	—
79	C₂×M₁₆×C₂₃	C₂³	1	6	yes	—
80	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
81	C₂₃×C₄⋊Q₈	—	2	3	no	—
82	C₂₃×C₄⋊Q₈	—	2	3	no	—
83	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
84	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
85	C₂₃×#127	C₂³	1	1	yes	characteristic
86	C₂₃×#124	C₂³	1	6	yes	—
87	C₂₃×#123	—	2	12	no	—
88	C₂₃×C₂²⋊Q₈	—	2	6	no	—
89	C₂₃×C₂²⋊Q₈	—	2	6	no	—
90	C₂₃×#121	—	2	3	no	—
91	C₂₃×#121	—	2	3	no	—
92	C₄×Q₈×C₂₃	—	2	1	no	—
93	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
97	C₄×D₄×C₂₃	—	2	3	no	—
98	C₄×D₄×C₂₃	—	2	3	no	—
99	C₄×D₄×C₂₃	—	2	3	no	—
100	C₄×D₄×C₂₃	—	2	3	no	—
101	C₄×D₄×C₂₃	—	2	3	no	—
102	C₄×D₄×C₂₃	—	2	6	no	—
103	C₄×D₄×C₂₃	—	2	3	no	—
104	C₄×D₄×C₂₃	—	2	3	no	—
105	C₄×D₄×C₂₃	—	2	3	no	—
106	C₄×D₄×C₂₃	—	2	3	no	—
107	C₄×D₄×C₂₃	D₄	1	3	yes	—
108	C₄×D₄×C₂₃	D₄	1	3	yes	—
109	C₄×D₄×C₂₃	C₂³	1	3	yes	—
110	C₄×D₄×C₂₃	C₂³	1	6	yes	—
111	C₄×D₄×C₂₃	C₂³	1	3	yes	—
112	C₂₃×#117	—	2	3	no	—
113	C₂₃×#117	—	2	3	no	—
114	C₂₃×#117	C₂³	1	3	yes	—
115	C₂₃×#117	C₂³	1	3	yes	—
116	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
117	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
118	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
119	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
120	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	6	no	—
122	C₂×C₄²×C₂₃	—	2	3	no	—
123	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
124	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
125	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
126	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
127	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
128	C₂₃×#103	C₂³	1	6	yes	—
129	C₂₃×#103	C₂³	1	12	yes	—
130	C₂₃×#103	C₂³	1	6	yes	—
131	C₂₃×#102	C₂³	1	2	yes	—
132	C₂₃×#102	C₂³	1	6	yes	—
133	C₂⁴×C₂₃	—	2	3	no	—
134	#41×C₂₃	—	2	8	no	—
135	#41×C₂₃	—	2	6	no	—
136	#41×C₂₃	—	2	12	no	—
137	#41×C₂₃	—	2	12	no	—
138	#41×C₂₃	—	4	24	no	—
139	#41×C₂₃	C₂×D₄	1	12	yes	—
140	#41×C₂₃	C₂×D₄	1	4	yes	—
141	#41×C₂₃	#41	1	4	yes	—
142	C₂×Q₈×C₂₃	—	2	6	no	—
143	C₂×Q₈×C₂₃	—	2	12	no	—
144	C₂×Q₈×C₂₃	—	2	12	no	—
145	C₂×Q₈×C₂₃	—	4	8	no	—
146	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂×Q₈×C₂₃	C₂⁴	1	6	yes	—
148	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
149	C₂×D₄×C₂₃	—	2	3	no	—
150	C₂×D₄×C₂₃	—	2	24	no	—
151	C₂×D₄×C₂₃	—	2	6	no	—
152	C₂×D₄×C₂₃	—	2	3	no	—
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	—	2	6	no	—
156	C₂×D₄×C₂₃	C₂×D₄	1	12	yes	—
157	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
158	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
159	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
160	C₂²×C₄×C₂₃	—	2	3	no	—
161	C₂²×C₄×C₂₃	—	2	6	no	—
162	C₂²×C₄×C₂₃	—	2	3	no	—
163	C₂²×C₄×C₂₃	—	2	3	no	—
164	C₂²×C₄×C₂₃	—	2	6	no	—
165	C₂²×C₄×C₂₃	—	2	3	no	—
166	C₂²×C₄×C₂₃	—	2	6	no	—
167	C₂²×C₄×C₂₃	—	8	6	no	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
170	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
171	Q₁₆×C₂₃	—	2	12	no	—
172	SD₁₆×C₂₃	—	2	4	no	—
173	SD₁₆×C₂₃	—	2	12	no	—
174	SD₁₆×C₂₃	—	2	24	no	—
175	D₈×C₂₃	—	2	12	no	—
176	M₁₆×C₂₃	—	2	12	no	—
177	C₂×C₈×C₂₃	—	2	12	no	—
178	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
179	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
180	C₄⋊C₄×C₂₃	—	2	3	no	—
181	C₄⋊C₄×C₂₃	—	2	3	no	—
182	C₄⋊C₄×C₂₃	—	2	3	no	—
183	C₄⋊C₄×C₂₃	—	2	3	no	—
184	C₄⋊C₄×C₂₃	—	2	6	no	—
185	C₄⋊C₄×C₂₃	—	2	6	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
188	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
189	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
190	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
191	C₂²⋊C₄×C₂₃	—	2	3	no	—
192	C₂²⋊C₄×C₂₃	—	2	3	no	—
193	C₂²⋊C₄×C₂₃	—	2	6	no	—
194	C₂²⋊C₄×C₂₃	—	2	3	no	—
195	C₂²⋊C₄×C₂₃	—	2	6	no	—
196	C₂²⋊C₄×C₂₃	—	2	6	no	—
197	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
198	C₄²×C₂₃	—	2	1	no	—
199	C₄²×C₂₃	—	2	3	no	—
200	C₄²×C₂₃	—	2	3	no	—
201	C₄²×C₂₃	C₂×D₄	1	3	yes	—
202	C₄²×C₂₃	C₂×D₄	1	3	yes	—
203	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
204	C₄²×C₂₃	C₂⁴	1	3	yes	—
205	#60429	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₂₃	—	2	3	no	—
207	C₂³×C₂₃	—	2	12	no	—
208	C₂³×C₂₃	—	2	2	no	—
209	C₂³×C₂₃	—	8	3	no	—
210	C₂³×C₂₃	—	8	3	no	—
211	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
212	Q₈×C₂₃	—	2	1	no	—
213	Q₈×C₂₃	—	2	6	no	—
214	Q₈×C₂₃	—	2	6	no	—
215	Q₈×C₂₃	—	4	12	no	—
216	Q₈×C₂₃	#121	1	4	yes	—
217	Q₈×C₂₃	C₂²×D₄	1	2	yes	—
218	D₄×C₂₃	—	2	12	no	—
219	D₄×C₂₃	—	2	12	no	—
220	D₄×C₂₃	—	2	2	no	—
221	D₄×C₂₃	—	2	3	no	—
222	D₄×C₂₃	—	2	2	no	—
223	D₄×C₂₃	—	4	2	no	—
224	D₄×C₂₃	—	4	12	no	—
225	D₄×C₂₃	—	4	2	no	—
226	D₄×C₂₃	#121	1	12	yes	—
227	D₄×C₂₃	C₂²×D₄	1	6	yes	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	12	no	—
230	C₂×C₄×C₂₃	—	2	6	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	3	no	—
233	C₂×C₄×C₂₃	—	2	6	no	—
234	C₂×C₄×C₂₃	—	4	3	no	—
235	C₂×C₄×C₂₃	—	4	12	no	—
236	C₂×C₄×C₂₃	—	4	12	no	—
237	C₂×C₄×C₂₃	—	4	6	no	—
238	C₂×C₄×C₂₃	—	8	3	no	—
239	C₂×C₄×C₂₃	—	8	6	no	—
240	C₂×C₄×C₂₃	—	8	3	no	—
241	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
242	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
244	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
245	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
246	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
248	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
249	C₈×C₂₃	—	2	2	no	—
250	C₈×C₂₃	—	2	6	no	—
251	C₂×#577	C₂×C₂₃	1	2	yes	complemented
252	#3499	C₂×C₂₃	1	1	yes	characteristic, complemented
253	C₄×H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic, complemented
254	#3046	C₂×C₂₃	1	6	yes	complemented
255	#3042	C₂×C₂₃	1	6	yes	complemented
256	#3039	C₂×C₂₃	1	3	yes	complemented
257	#3036	C₂×C₂₃	1	3	yes	complemented
258	#3035	C₂×C₂₃	1	3	yes	complemented
259	C₂×#461	C₂×C₂₃	1	3	yes	complemented
260	#2906	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#2898	C₂×C₂₃	1	2	yes	complemented
262	C₂²×C₂₃	—	2	3	no	—
263	C₂²×C₂₃	—	2	3	no	—
264	C₂²×C₂₃	—	2	6	no	—
265	C₂²×C₂₃	—	4	3	no	—
266	C₂²×C₂₃	—	4	3	no	—
267	C₂²×C₂₃	—	4	2	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	—	4	2	no	—
270	C₂²×C₂₃	—	8	12	no	—
271	C₂²×C₂₃	C₂×#121	1	6	yes	—
272	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
273	C₄×C₂₃	—	2	1	no	—
274	C₄×C₂₃	—	2	3	no	—
275	C₄×C₂₃	—	2	2	no	—
276	C₄×C₂₃	—	2	3	no	—
277	C₄×C₂₃	—	4	1	no	—
278	C₄×C₂₃	—	4	6	no	—
279	C₄×C₂₃	—	4	3	no	—
280	C₄×C₂₃	C₂×#121	1	6	yes	—
281	C₄×C₂₃	#577	1	2	yes	complemented
282	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
283	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
284	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	—
285	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	1	yes	characteristic
286	#577	—	2	16	no	—
287	C₂×#137	C₂²×C₂₃	1	6	yes	—
288	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	6	yes	—
289	C₂×#135	C₂²×C₂₃	1	6	yes	—
290	C₂²×SD₁₆	C₂²×C₂₃	1	6	yes	—
291	C₂×#131	C₂²×C₂₃	1	2	yes	—
292	D₄×Q₈	—	2	1	no	—
293	D₄×Q₈	—	2	1	no	—
294	#547	—	2	3	no	—
295	#547	—	2	3	no	—
296	#536	C₂²×C₂₃	1	6	yes	complemented
297	#532	C₂²×C₂₃	1	3	yes	complemented
298	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	complemented
299	#520	—	2	1	no	—
300	#520	C₂²×C₂₃	1	3	yes	—
301	#520	C₂²×C₂₃	1	1	yes	characteristic
302	#518	—	2	3	no	—
303	#518	C₂²×C₂₃	1	3	yes	—
304	C₄×#41	—	2	1	no	—
305	C₄×#41	C₂²×C₂₃	1	3	yes	—
306	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₄×D₄	—	2	3	no	—
308	C₂×C₄×D₄	C₂²×C₂₃	1	3	yes	—
309	#464	C₂²×C₂₃	1	6	yes	complemented
310	#463	C₂²×C₂₃	1	6	yes	complemented
311	#463	C₂²×C₂₃	1	12	yes	complemented
312	#462	C₂²×C₂₃	1	3	yes	complemented
313	#462	C₂²×C₂₃	1	6	yes	complemented
314	#461	C₂²×C₂₃	1	3	yes	complemented
315	#461	C₂²×C₂₃	1	6	yes	complemented
316	#461	C₂²×C₂₃	1	12	yes	complemented
317	#461	C₂²×C₂₃	1	6	yes	complemented
318	#460	C₂²×C₂₃	1	1	yes	characteristic, complemented
319	#459	C₂²×C₂₃	1	3	yes	complemented
320	#424	C₂²×C₂₃	1	6	yes	complemented
321	#423	C₂²×C₂₃	1	3	yes	complemented
322	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	complemented
323	#419	C₂²×C₂₃	1	6	yes	complemented
324	#417	C₂²×C₂₃	1	6	yes	complemented
325	#416	C₂²×C₂₃	1	6	yes	complemented
326	C₂×#103	C₂²×C₂₃	1	6	yes	complemented
327	C₂×C₂₃	—	2	6	no	—
328	C₂×C₂₃	—	4	6	no	—
329	C₂×C₂₃	—	8	2	no	—
330	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
331	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
332	C₂×C₂₃	#3586	1	1	yes	characteristic
333	2₋¹⁺⁴	—	2	8	no	—
334	2₋¹⁺⁴	—	2	8	no	—
335	H(5,𝔽₂)	—	2	4	no	—
336	H(5,𝔽₂)	D₄×C₂₃	1	8	yes	complemented
337	C₂×#41	—	2	1	no	—
338	C₂×#41	—	2	6	no	—
339	C₂×#41	C₂³×C₂₃	1	6	yes	—
340	C₂×#41	C₂³×C₂₃	1	3	yes	—
341	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
342	C₂²×Q₈	—	2	2	no	—
343	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
344	C₂²×D₄	—	2	3	no	—
345	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
346	C₂³×C₄	—	2	3	no	—
347	#137	—	2	24	no	—
348	AGL(1,ℤ/8)	—	2	24	no	—
349	#135	—	2	24	no	—
350	C₂×Q₁₆	C₂³×C₂₃	1	6	yes	—
351	C₂×SD₁₆	—	2	24	no	—
352	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
353	C₂×SD₁₆	C₂³×C₂₃	1	12	yes	—
354	C₂×SD₁₆	C₂³×C₂₃	1	2	yes	—
355	C₂×D₈	C₂³×C₂₃	1	6	yes	—
356	#131	—	2	8	no	—
357	C₂×M₁₆	C₂³×C₂₃	1	6	yes	—
358	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
359	C₄⋊Q₈	—	2	3	no	—
360	C₄⋊Q₈	—	2	3	no	—
361	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
362	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
363	#127	C₂³×C₂₃	1	1	yes	characteristic
364	#124	C₂³×C₂₃	1	6	yes	—
365	#123	—	2	12	no	—
366	C₂²⋊Q₈	—	2	6	no	—
367	C₂²⋊Q₈	—	2	6	no	—
368	#121	—	2	3	no	—
369	#121	—	2	3	no	—
370	C₄×Q₈	—	2	1	no	—
371	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
372	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
373	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
374	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
375	C₄×D₄	—	2	3	no	—
376	C₄×D₄	—	2	3	no	—
377	C₄×D₄	—	2	3	no	—
378	C₄×D₄	—	2	3	no	—
379	C₄×D₄	—	2	3	no	—
380	C₄×D₄	—	2	6	no	—
381	C₄×D₄	—	2	3	no	—
382	C₄×D₄	—	2	3	no	—
383	C₄×D₄	—	2	3	no	—
384	C₄×D₄	—	2	3	no	—
385	C₄×D₄	D₄×C₂₃	1	3	yes	—
386	C₄×D₄	D₄×C₂₃	1	3	yes	—
387	C₄×D₄	C₂³×C₂₃	1	3	yes	—
388	C₄×D₄	C₂³×C₂₃	1	6	yes	—
389	C₄×D₄	C₂³×C₂₃	1	3	yes	—
390	#117	—	2	3	no	—
391	#117	—	2	3	no	—
392	#117	C₂³×C₂₃	1	3	yes	—
393	#117	C₂³×C₂₃	1	3	yes	—
394	C₂×C₄⋊C₄	—	2	3	no	—
395	C₂×C₄⋊C₄	—	2	3	no	—
396	C₂×C₄⋊C₄	—	2	3	no	—
397	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
398	C₂×C₂²⋊C₄	—	2	3	no	—
399	C₂×C₂²⋊C₄	—	2	6	no	—
400	C₂×C₄²	—	2	3	no	—
401	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
402	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
403	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
404	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
405	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
406	#103	C₂³×C₂₃	1	6	yes	—
407	#103	C₂³×C₂₃	1	12	yes	—
408	#103	C₂³×C₂₃	1	6	yes	—
409	#102	C₂³×C₂₃	1	2	yes	—
410	#102	C₂³×C₂₃	1	6	yes	—
411	C₂₃	#60429	1	1	yes	characteristic, direct product factor, Sylow
412	C₂⁴	—	2	3	no	—
413	#41	—	2	8	no	—
414	#41	—	2	6	no	—
415	#41	—	2	12	no	—
416	#41	—	2	12	no	—
417	#41	—	4	24	no	—
418	#41	C₂×D₄×C₂₃	1	12	yes	—
419	#41	C₂×D₄×C₂₃	1	4	yes	—
420	#41	#41×C₂₃	1	4	yes	—
421	C₂×Q₈	—	2	6	no	—
422	C₂×Q₈	—	2	12	no	—
423	C₂×Q₈	—	2	12	no	—
424	C₂×Q₈	—	4	8	no	—
425	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
426	C₂×Q₈	C₂⁴×C₂₃	1	6	yes	—
427	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
428	C₂×D₄	—	2	3	no	—
429	C₂×D₄	—	2	24	no	—
430	C₂×D₄	—	2	6	no	—
431	C₂×D₄	—	2	3	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	6	no	—
435	C₂×D₄	C₂×D₄×C₂₃	1	12	yes	—
436	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
437	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
438	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
439	C₂²×C₄	—	2	3	no	—
440	C₂²×C₄	—	2	6	no	—
441	C₂²×C₄	—	2	3	no	—
442	C₂²×C₄	—	2	3	no	—
443	C₂²×C₄	—	2	6	no	—
444	C₂²×C₄	—	2	3	no	—
445	C₂²×C₄	—	2	6	no	—
446	C₂²×C₄	—	8	6	no	—
447	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
448	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
449	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
450	Q₁₆	—	2	12	no	—
451	SD₁₆	—	2	4	no	—
452	SD₁₆	—	2	12	no	—
453	SD₁₆	—	2	24	no	—
454	D₈	—	2	12	no	—
455	M₁₆	—	2	12	no	—
456	C₂×C₈	—	2	12	no	—
457	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
458	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
459	C₄⋊C₄	—	2	3	no	—
460	C₄⋊C₄	—	2	3	no	—
461	C₄⋊C₄	—	2	3	no	—
462	C₄⋊C₄	—	2	3	no	—
463	C₄⋊C₄	—	2	6	no	—
464	C₄⋊C₄	—	2	6	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
467	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
468	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
469	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
470	C₂²⋊C₄	—	2	3	no	—
471	C₂²⋊C₄	—	2	3	no	—
472	C₂²⋊C₄	—	2	6	no	—
473	C₂²⋊C₄	—	2	3	no	—
474	C₂²⋊C₄	—	2	6	no	—
475	C₂²⋊C₄	—	2	6	no	—
476	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	3	no	—
479	C₄²	—	2	3	no	—
480	C₄²	C₂×D₄×C₂₃	1	3	yes	—
481	C₄²	C₂×D₄×C₂₃	1	3	yes	—
482	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
483	C₄²	C₂⁴×C₂₃	1	3	yes	—
484	C₂³	—	2	3	no	—
485	C₂³	—	2	12	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	8	3	no	—
488	C₂³	—	8	3	no	—
489	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
490	Q₈	—	2	1	no	—
491	Q₈	—	2	6	no	—
492	Q₈	—	2	6	no	—
493	Q₈	—	4	12	no	—
494	Q₈	C₂₃×#121	1	4	yes	—
495	Q₈	C₂²×D₄×C₂₃	1	2	yes	—
496	D₄	—	2	12	no	—
497	D₄	—	2	2	no	—
498	D₄	—	2	3	no	—
499	D₄	—	2	2	no	—
500	D₄	—	2	12	no	—
501	D₄	—	4	2	no	—
502	D₄	—	4	12	no	—
503	D₄	—	4	2	no	—
504	D₄	C₂₃×#121	1	12	yes	—
505	D₄	C₂²×D₄×C₂₃	1	6	yes	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	6	no	—
508	C₂×C₄	—	2	12	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	3	no	—
511	C₂×C₄	—	2	6	no	—
512	C₂×C₄	—	4	3	no	—
513	C₂×C₄	—	4	12	no	—
514	C₂×C₄	—	4	6	no	—
515	C₂×C₄	—	4	12	no	—
516	C₂×C₄	—	8	6	no	—
517	C₂×C₄	—	8	3	no	—
518	C₂×C₄	—	8	3	no	—
519	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
520	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
522	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
524	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
525	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
526	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
527	C₈	—	2	2	no	—
528	C₈	—	2	6	no	—
529	C₂²	—	2	3	no	—
530	C₂²	—	2	3	no	—
531	C₂²	—	2	6	no	—
532	C₂²	—	4	3	no	—
533	C₂²	—	4	3	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	8	12	no	—
538	C₂²	C₂×C₂₃×#121	1	6	yes	—
539	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
540	C₄	—	2	2	no	—
541	C₄	—	2	3	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	3	no	—
544	C₄	—	4	6	no	—
545	C₄	—	4	3	no	—
546	C₄	—	4	1	no	—
547	C₄	C₂×C₂₃×#121	1	6	yes	—
548	C₄	C₂₃×#577	1	2	yes	complemented
549	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
550	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
551	C₂	—	2	6	no	—
552	C₂	—	4	6	no	—
553	C₂	—	8	2	no	—
554	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
555	C₂	C₂×C₂₃×#577	1	1	yes	characteristic
556	C₂	C₂₃×#3586	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#577	C₂	semidirect
2. C₂₃×#3499	C₂	semidirect
3. C₄×C₂₃×H(5,𝔽₂)	C₂	semidirect
4. C₂₃×#3046	C₂	semidirect
5. C₂₃×#3042	C₂	semidirect
6. C₂₃×#3039	C₂	semidirect
7. C₂₃×#3036	C₂	semidirect
8. C₂₃×#3035	C₂	semidirect
9. C₂×C₂₃×#461	C₂	semidirect
10. C₂₃×#2906	C₂	semidirect
11. C₂₃×#2898	C₂	semidirect
24. C₂₃×#536	C₂²	semidirect
25. C₂₃×#532	C₂²	semidirect
26. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
37. C₂₃×#464	C₂²	semidirect
38. C₂₃×#463	C₂²	semidirect
39. C₂₃×#463	C₂²	semidirect
40. C₂₃×#462	C₂²	semidirect
41. C₂₃×#462	C₂²	semidirect
42. C₂₃×#461	C₂²	semidirect
43. C₂₃×#461	C₂²	semidirect
44. C₂₃×#461	C₂²	semidirect
45. C₂₃×#461	C₂²	semidirect
46. C₂₃×#460	C₂²	semidirect
47. C₂₃×#459	C₂²	semidirect
48. C₂₃×#424	C₂²	semidirect
49. C₂₃×#423	C₂²	semidirect
50. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
51. C₂₃×#419	C₂²	semidirect
52. C₂₃×#417	C₂²	semidirect
53. C₂₃×#416	C₂²	semidirect
54. C₂×C₂₃×#103	C₂²	semidirect
58. C₂₃×H(5,𝔽₂)	D₄	semidirect
205. #60429	C₂₃	direct (N × H)
251. C₂×#577	C₂×C₂₃	semidirect
252. #3499	C₂×C₂₃	semidirect
253. C₄×H(5,𝔽₂)	C₂×C₂₃	semidirect
254. #3046	C₂×C₂₃	semidirect
255. #3042	C₂×C₂₃	semidirect
256. #3039	C₂×C₂₃	semidirect
257. #3036	C₂×C₂₃	semidirect
258. #3035	C₂×C₂₃	semidirect
259. C₂×#461	C₂×C₂₃	semidirect
260. #2906	C₂×C₂₃	semidirect
261. #2898	C₂×C₂₃	semidirect
281. C₄×C₂₃	#577	semidirect
296. #536	C₂²×C₂₃	semidirect
297. #532	C₂²×C₂₃	semidirect
298. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
309. #464	C₂²×C₂₃	semidirect
310. #463	C₂²×C₂₃	semidirect
311. #463	C₂²×C₂₃	semidirect
312. #462	C₂²×C₂₃	semidirect
313. #462	C₂²×C₂₃	semidirect
314. #461	C₂²×C₂₃	semidirect
315. #461	C₂²×C₂₃	semidirect
316. #461	C₂²×C₂₃	semidirect
317. #461	C₂²×C₂₃	semidirect
318. #460	C₂²×C₂₃	semidirect
319. #459	C₂²×C₂₃	semidirect
320. #424	C₂²×C₂₃	semidirect
321. #423	C₂²×C₂₃	semidirect
322. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
323. #419	C₂²×C₂₃	semidirect
324. #417	C₂²×C₂₃	semidirect
325. #416	C₂²×C₂₃	semidirect
326. C₂×#103	C₂²×C₂₃	semidirect
336. H(5,𝔽₂)	D₄×C₂₃	semidirect
411. C₂₃	#60429	direct (N × H)
548. C₄	C₂₃×#577	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#577	2
C₂₃×#3499	2
C₄×C₂₃×H(5,𝔽₂)	2
C₂₃×#3046	2
C₂₃×#3042	2
C₂₃×#3039	2
C₂₃×#3036	2
C₂₃×#3035	2
C₂×C₂₃×#461	2
C₂₃×#2906	2
C₂₃×#2898	2
#60429	23

Subgroups — C23×#60429 #19208055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60429 #19208055