Subgroups — C₂₃×#60391 #19208017

Order 5888.

4666 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×2₋¹⁺⁶	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3586	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3022	C₂	1	6	yes	maximal, complemented
5	C₂₃×#3020	C₂	1	6	yes	maximal, complemented
6	C₂₃×#3018	C₂	1	6	yes	maximal, complemented
7	C₂×C₂₃×#456	C₂	1	6	yes	maximal, complemented
8	C₂₃×#2901	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2889	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#585	—	2	8	no	—
11	C₂₃×#585	—	2	6	no	—
12	C₂₃×#585	C₂²	1	6	yes	—
13	C₂₃×#585	C₂²	1	2	yes	—
14	C₂×C₂₃×2₋¹⁺⁴	—	2	8	no	—
15	C₂×C₂₃×2₋¹⁺⁴	—	2	2	no	—
16	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	6	yes	—
17	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#578	—	2	8	no	—
20	C₂₃×#577	—	2	8	no	—
21	C₂₃×#575	C₂²	1	12	yes	complemented
22	C₂×C₂₃×#137	C₂²	1	12	yes	complemented
23	C₂₃×#568	C₂²	1	2	yes	complemented
24	D₄×Q₈×C₂₃	—	2	2	no	—
25	C₂₃×#547	—	2	6	no	—
26	C₂₃×#536	C₂²	1	3	yes	complemented
27	C₂₃×#536	C₂²	1	3	yes	complemented
28	C₂₃×#532	C₂²	1	3	yes	complemented
29	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
30	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#456	C₂²	1	12	yes	complemented
32	C₂₃×#456	C₂²	1	24	yes	complemented
33	C₂₃×#455	C₂²	1	12	yes	complemented
34	C₂₃×#454	C₂²	1	12	yes	complemented
35	C₂₃×#453	C₂²	1	4	yes	complemented
36	C₂₃×#421	C₂²	1	12	yes	complemented
37	C₂×C₂₃×#104	C₂²	1	12	yes	complemented
38	C₂₃×#413	C₂²	1	6	yes	complemented
39	C₂×C₂₃×#101	C₂²	1	6	yes	complemented
40	C₂₃×2₋¹⁺⁴	—	2	24	no	—
41	C₂₃×2₋¹⁺⁴	—	2	16	no	—
42	C₂₃×2₋¹⁺⁴	—	2	24	no	—
43	C₂₃×2₋¹⁺⁴	—	2	8	no	—
44	C₂₃×2₋¹⁺⁴	—	2	12	no	—
45	C₂₃×2₋¹⁺⁴	—	2	2	no	—
46	C₂₃×2₋¹⁺⁴	—	2	12	no	—
47	C₂₃×2₋¹⁺⁴	—	2	8	no	—
48	C₂₃×2₋¹⁺⁴	D₄	1	8	yes	complemented
49	C₂₃×2₋¹⁺⁴	C₂³	1	12	yes	—
50	C₂₃×H(5,𝔽₂)	—	2	24	no	—
51	C₂₃×H(5,𝔽₂)	—	2	12	no	—
52	C₂₃×H(5,𝔽₂)	—	2	12	no	—
53	C₂₃×H(5,𝔽₂)	—	2	6	no	—
54	C₂₃×H(5,𝔽₂)	D₄	1	8	yes	complemented
55	C₂₃×H(5,𝔽₂)	C₂³	1	4	yes	—
56	C₂×#41×C₂₃	—	2	8	no	—
57	C₂×#41×C₂₃	—	2	48	no	—
58	C₂×#41×C₂₃	—	2	24	no	—
59	C₂×#41×C₂₃	—	2	8	no	—
60	C₂×#41×C₂₃	—	2	12	no	—
61	C₂×#41×C₂₃	—	2	3	no	—
62	C₂×#41×C₂₃	—	2	12	no	—
63	C₂×#41×C₂₃	—	2	12	no	—
64	C₂×#41×C₂₃	D₄	1	6	yes	—
65	C₂×#41×C₂₃	C₂³	1	12	yes	—
66	C₂×#41×C₂₃	C₂³	1	3	yes	—
67	C₂×#41×C₂₃	C₂³	1	3	yes	—
68	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×Q₈×C₂₃	—	2	8	no	—
70	C₂²×Q₈×C₂₃	—	2	12	no	—
71	C₂²×Q₈×C₂₃	—	2	1	no	—
72	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
73	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
74	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
75	C₂₃×#137	—	2	24	no	—
76	C₂₃×#137	—	2	12	no	—
77	C₂₃×#137	C₂³	1	24	yes	—
78	C₂₃×AGL(1,ℤ/8)	—	2	12	no	—
79	C₂₃×AGL(1,ℤ/8)	C₂³	1	8	yes	—
80	C₂₃×#135	—	2	24	no	—
81	C₂×Q₁₆×C₂₃	—	2	12	no	—
82	C₂×SD₁₆×C₂₃	—	2	12	no	—
83	C₂₃×#131	—	2	8	no	—
84	C₂×M₁₆×C₂₃	C₂³	1	12	yes	—
85	C₂₃×C₄⋊Q₈	—	2	6	no	—
86	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
87	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
88	C₂₃×#127	C₂³	1	1	yes	characteristic
89	C₂₃×#124	C₂³	1	3	yes	—
90	C₂₃×#124	C₂³	1	3	yes	—
91	C₂₃×#123	—	2	6	no	—
92	C₂₃×#123	—	2	6	no	—
93	C₂₃×C₂²⋊Q₈	—	2	6	no	—
94	C₂₃×C₂²⋊Q₈	—	2	6	no	—
95	C₂₃×#121	—	2	6	no	—
96	C₄×Q₈×C₂₃	—	2	1	no	—
97	C₄×D₄×C₂₃	—	2	3	no	—
98	C₄×D₄×C₂₃	—	2	6	no	—
99	C₂₃×#117	C₂³	1	3	yes	—
100	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
101	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
102	C₂₃×#104	C₂³	1	8	yes	—
103	C₂₃×#104	C₂³	1	24	yes	—
104	C₂₃×#101	C₂³	1	12	yes	—
105	C₂₃×#100	C₂³	1	4	yes	—
106	#41×C₂₃	—	2	48	no	—
107	#41×C₂₃	—	2	48	no	—
108	#41×C₂₃	—	2	48	no	—
109	#41×C₂₃	—	2	24	no	—
110	#41×C₂₃	—	2	24	no	—
111	#41×C₂₃	—	2	8	no	—
112	#41×C₂₃	—	2	8	no	—
113	#41×C₂₃	—	2	24	no	—
114	#41×C₂₃	—	2	24	no	—
115	#41×C₂₃	—	2	24	no	—
116	#41×C₂₃	—	2	24	no	—
117	#41×C₂₃	—	2	12	no	—
118	#41×C₂₃	—	2	12	no	—
119	#41×C₂₃	—	2	12	no	—
120	#41×C₂₃	—	2	12	no	—
121	#41×C₂₃	—	4	12	no	—
122	#41×C₂₃	C₂×D₄	1	24	yes	—
123	#41×C₂₃	C₂×D₄	1	8	yes	—
124	#41×C₂₃	C₂⁴	1	8	yes	—
125	C₂×Q₈×C₂₃	—	2	48	no	—
126	C₂×Q₈×C₂₃	—	2	12	no	—
127	C₂×Q₈×C₂₃	—	2	12	no	—
128	C₂×Q₈×C₂₃	—	2	8	no	—
129	C₂×Q₈×C₂₃	—	2	8	no	—
130	C₂×Q₈×C₂₃	—	2	24	no	—
131	C₂×Q₈×C₂₃	—	2	12	no	—
132	C₂×Q₈×C₂₃	—	2	3	no	—
133	C₂×Q₈×C₂₃	—	2	12	no	—
134	C₂×Q₈×C₂₃	—	4	4	no	—
135	C₂×Q₈×C₂₃	C₂×D₄	1	12	yes	—
136	C₂×Q₈×C₂₃	C₂×D₄	1	6	yes	—
137	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
139	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
140	C₂×D₄×C₂₃	—	2	48	no	—
141	C₂×D₄×C₂₃	—	2	12	no	—
142	C₂×D₄×C₂₃	—	2	12	no	—
143	C₂×D₄×C₂₃	—	2	8	no	—
144	C₂×D₄×C₂₃	—	2	12	no	—
145	C₂×D₄×C₂₃	—	2	24	no	—
146	C₂×D₄×C₂₃	—	2	8	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	2	3	no	—
149	C₂×D₄×C₂₃	C₂×D₄	1	6	yes	—
150	C₂×D₄×C₂₃	C₂×D₄	1	12	yes	—
151	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
152	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
153	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	—	2	24	no	—
155	C₂²×C₄×C₂₃	—	2	24	no	—
156	C₂²×C₄×C₂₃	—	2	12	no	—
157	C₂²×C₄×C₂₃	—	2	3	no	—
158	C₂²×C₄×C₂₃	—	2	3	no	—
159	C₂²×C₄×C₂₃	—	2	3	no	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
162	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
163	Q₁₆×C₂₃	—	2	24	no	—
164	SD₁₆×C₂₃	—	2	8	no	—
165	SD₁₆×C₂₃	—	2	24	no	—
166	D₈×C₂₃	—	2	8	no	—
167	M₁₆×C₂₃	—	2	12	no	—
168	M₁₆×C₂₃	C₂⁴	1	8	yes	—
169	C₂×C₈×C₂₃	—	2	12	no	—
170	C₄⋊C₄×C₂₃	—	2	6	no	—
171	C₄⋊C₄×C₂₃	—	2	12	no	—
172	C₄⋊C₄×C₂₃	—	2	3	no	—
173	C₂²⋊C₄×C₂₃	—	2	3	no	—
174	C₂²⋊C₄×C₂₃	—	2	6	no	—
175	C₂²⋊C₄×C₂₃	—	2	6	no	—
176	C₄²×C₂₃	—	2	3	no	—
177	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₄²×C₂₃	C₂⁴	1	3	yes	—
179	#60391	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₂₃	—	2	8	no	—
181	C₂³×C₂₃	—	2	12	no	—
182	C₂³×C₂₃	—	2	1	no	—
183	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
184	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
185	Q₈×C₂₃	—	2	24	no	—
186	Q₈×C₂₃	—	2	12	no	—
187	Q₈×C₂₃	—	2	12	no	—
188	Q₈×C₂₃	—	2	6	no	—
189	Q₈×C₂₃	—	2	6	no	—
190	Q₈×C₂₃	—	4	6	no	—
191	Q₈×C₂₃	#120	1	4	yes	—
192	Q₈×C₂₃	C₂²×D₄	1	8	yes	—
193	D₄×C₂₃	—	2	16	no	—
194	D₄×C₂₃	—	2	24	no	—
195	D₄×C₂₃	—	2	24	no	—
196	D₄×C₂₃	—	2	12	no	—
197	D₄×C₂₃	—	2	8	no	—
198	D₄×C₂₃	—	2	12	no	—
199	D₄×C₂₃	—	2	2	no	—
200	D₄×C₂₃	—	2	2	no	—
201	D₄×C₂₃	—	4	4	no	—
202	D₄×C₂₃	—	4	6	no	—
203	D₄×C₂₃	#120	1	12	yes	—
204	D₄×C₂₃	C₂²×D₄	1	8	yes	—
205	C₂×C₄×C₂₃	—	2	8	no	—
206	C₂×C₄×C₂₃	—	2	48	no	—
207	C₂×C₄×C₂₃	—	2	12	no	—
208	C₂×C₄×C₂₃	—	2	12	no	—
209	C₂×C₄×C₂₃	—	2	1	no	—
210	C₂×C₄×C₂₃	—	2	24	no	—
211	C₂×C₄×C₂₃	—	2	12	no	—
212	C₂×C₄×C₂₃	—	2	8	no	—
213	C₂×C₄×C₂₃	—	2	3	no	—
214	C₂×C₄×C₂₃	—	2	3	no	—
215	C₂×C₄×C₂₃	—	4	6	no	—
216	C₂×C₄×C₂₃	—	4	6	no	—
217	C₂×C₄×C₂₃	—	4	6	no	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
220	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
221	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
222	C₈×C₂₃	—	2	8	no	—
223	2₋¹⁺⁶	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#3586	C₂×C₂₃	1	2	yes	complemented
225	#3499	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#3022	C₂×C₂₃	1	6	yes	complemented
227	#3020	C₂×C₂₃	1	6	yes	complemented
228	#3018	C₂×C₂₃	1	6	yes	complemented
229	C₂×#456	C₂×C₂₃	1	6	yes	complemented
230	#2901	C₂×C₂₃	1	2	yes	complemented
231	#2889	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂²×C₂₃	—	2	2	no	—
233	C₂²×C₂₃	—	2	8	no	—
234	C₂²×C₂₃	—	4	3	no	—
235	C₂²×C₂₃	—	4	2	no	—
236	C₂²×C₂₃	—	4	2	no	—
237	C₂²×C₂₃	—	4	1	no	—
238	C₂²×C₂₃	—	8	12	no	—
239	C₂²×C₂₃	—	8	8	no	—
240	C₂²×C₂₃	C₂×#120	1	6	yes	—
241	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₂₃	—	2	2	no	—
243	C₄×C₂₃	—	2	8	no	—
244	C₄×C₂₃	—	2	6	no	—
245	C₄×C₂₃	—	4	3	no	—
246	C₄×C₂₃	—	4	3	no	—
247	C₄×C₂₃	C₂×#120	1	6	yes	—
248	C₄×C₂₃	C₂³×D₄	1	2	yes	—
249	#585	—	2	8	no	—
250	#585	—	2	6	no	—
251	#585	C₂²×C₂₃	1	6	yes	—
252	#585	C₂²×C₂₃	1	2	yes	—
253	C₂×2₋¹⁺⁴	—	2	8	no	—
254	C₂×2₋¹⁺⁴	—	2	2	no	—
255	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	6	yes	—
256	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic, complemented
258	#578	—	2	8	no	—
259	#577	—	2	8	no	—
260	#575	C₂²×C₂₃	1	12	yes	complemented
261	C₂×#137	C₂²×C₂₃	1	12	yes	complemented
262	#568	C₂²×C₂₃	1	2	yes	complemented
263	D₄×Q₈	—	2	2	no	—
264	#547	—	2	6	no	—
265	#536	C₂²×C₂₃	1	3	yes	complemented
266	#536	C₂²×C₂₃	1	3	yes	complemented
267	#532	C₂²×C₂₃	1	3	yes	complemented
268	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	complemented
269	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic, complemented
270	#456	C₂²×C₂₃	1	12	yes	complemented
271	#456	C₂²×C₂₃	1	24	yes	complemented
272	#455	C₂²×C₂₃	1	12	yes	complemented
273	#454	C₂²×C₂₃	1	12	yes	complemented
274	#453	C₂²×C₂₃	1	4	yes	complemented
275	#421	C₂²×C₂₃	1	12	yes	complemented
276	C₂×#104	C₂²×C₂₃	1	12	yes	complemented
277	#413	C₂²×C₂₃	1	6	yes	complemented
278	C₂×#101	C₂²×C₂₃	1	6	yes	complemented
279	C₂×C₂₃	—	2	1	no	—
280	C₂×C₂₃	—	2	6	no	—
281	C₂×C₂₃	—	4	2	no	—
282	C₂×C₂₃	—	4	8	no	—
283	C₂×C₂₃	—	8	1	no	—
284	C₂×C₂₃	C₂²×#120	1	1	yes	characteristic
285	2₋¹⁺⁴	—	2	24	no	—
286	2₋¹⁺⁴	—	2	16	no	—
287	2₋¹⁺⁴	—	2	24	no	—
288	2₋¹⁺⁴	—	2	8	no	—
289	2₋¹⁺⁴	—	2	12	no	—
290	2₋¹⁺⁴	—	2	2	no	—
291	2₋¹⁺⁴	—	2	12	no	—
292	2₋¹⁺⁴	—	2	8	no	—
293	2₋¹⁺⁴	D₄×C₂₃	1	8	yes	complemented
294	2₋¹⁺⁴	C₂³×C₂₃	1	12	yes	—
295	H(5,𝔽₂)	—	2	24	no	—
296	H(5,𝔽₂)	—	2	12	no	—
297	H(5,𝔽₂)	—	2	12	no	—
298	H(5,𝔽₂)	—	2	6	no	—
299	H(5,𝔽₂)	D₄×C₂₃	1	8	yes	complemented
300	H(5,𝔽₂)	C₂³×C₂₃	1	4	yes	—
301	C₂×#41	—	2	8	no	—
302	C₂×#41	—	2	48	no	—
303	C₂×#41	—	2	24	no	—
304	C₂×#41	—	2	8	no	—
305	C₂×#41	—	2	12	no	—
306	C₂×#41	—	2	3	no	—
307	C₂×#41	—	2	12	no	—
308	C₂×#41	—	2	12	no	—
309	C₂×#41	D₄×C₂₃	1	6	yes	—
310	C₂×#41	C₂³×C₂₃	1	12	yes	—
311	C₂×#41	C₂³×C₂₃	1	3	yes	—
312	C₂×#41	C₂³×C₂₃	1	3	yes	—
313	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
314	C₂²×Q₈	—	2	8	no	—
315	C₂²×Q₈	—	2	12	no	—
316	C₂²×Q₈	—	2	1	no	—
317	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
318	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
319	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
320	#137	—	2	24	no	—
321	#137	—	2	12	no	—
322	#137	C₂³×C₂₃	1	24	yes	—
323	AGL(1,ℤ/8)	—	2	12	no	—
324	AGL(1,ℤ/8)	C₂³×C₂₃	1	8	yes	—
325	#135	—	2	24	no	—
326	C₂×Q₁₆	—	2	12	no	—
327	C₂×SD₁₆	—	2	12	no	—
328	#131	—	2	8	no	—
329	C₂×M₁₆	C₂³×C₂₃	1	12	yes	—
330	C₄⋊Q₈	—	2	6	no	—
331	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
332	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
333	#127	C₂³×C₂₃	1	1	yes	characteristic
334	#124	C₂³×C₂₃	1	3	yes	—
335	#124	C₂³×C₂₃	1	3	yes	—
336	#123	—	2	6	no	—
337	#123	—	2	6	no	—
338	C₂²⋊Q₈	—	2	6	no	—
339	C₂²⋊Q₈	—	2	6	no	—
340	#121	—	2	6	no	—
341	C₄×Q₈	—	2	1	no	—
342	C₄×D₄	—	2	3	no	—
343	C₄×D₄	—	2	6	no	—
344	#117	C₂³×C₂₃	1	3	yes	—
345	C₂×C₄⋊C₄	—	2	6	no	—
346	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
347	#104	C₂³×C₂₃	1	8	yes	—
348	#104	C₂³×C₂₃	1	24	yes	—
349	#101	C₂³×C₂₃	1	12	yes	—
350	#100	C₂³×C₂₃	1	4	yes	—
351	C₂₃	#60391	1	1	yes	characteristic, direct product factor, Sylow
352	#41	—	2	48	no	—
353	#41	—	2	48	no	—
354	#41	—	2	48	no	—
355	#41	—	2	24	no	—
356	#41	—	2	24	no	—
357	#41	—	2	8	no	—
358	#41	—	2	8	no	—
359	#41	—	2	24	no	—
360	#41	—	2	24	no	—
361	#41	—	2	24	no	—
362	#41	—	2	24	no	—
363	#41	—	2	12	no	—
364	#41	—	2	12	no	—
365	#41	—	2	12	no	—
366	#41	—	2	12	no	—
367	#41	—	4	12	no	—
368	#41	C₂×D₄×C₂₃	1	24	yes	—
369	#41	C₂×D₄×C₂₃	1	8	yes	—
370	#41	C₂⁴×C₂₃	1	8	yes	—
371	C₂×Q₈	—	2	48	no	—
372	C₂×Q₈	—	2	12	no	—
373	C₂×Q₈	—	2	12	no	—
374	C₂×Q₈	—	2	8	no	—
375	C₂×Q₈	—	2	8	no	—
376	C₂×Q₈	—	2	24	no	—
377	C₂×Q₈	—	2	12	no	—
378	C₂×Q₈	—	2	3	no	—
379	C₂×Q₈	—	2	12	no	—
380	C₂×Q₈	—	4	4	no	—
381	C₂×Q₈	C₂×D₄×C₂₃	1	12	yes	—
382	C₂×Q₈	C₂×D₄×C₂₃	1	6	yes	—
383	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
384	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
385	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
386	C₂×D₄	—	2	48	no	—
387	C₂×D₄	—	2	12	no	—
388	C₂×D₄	—	2	12	no	—
389	C₂×D₄	—	2	8	no	—
390	C₂×D₄	—	2	12	no	—
391	C₂×D₄	—	2	24	no	—
392	C₂×D₄	—	2	8	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	2	3	no	—
395	C₂×D₄	C₂×D₄×C₂₃	1	6	yes	—
396	C₂×D₄	C₂×D₄×C₂₃	1	12	yes	—
397	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
398	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
399	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
400	C₂²×C₄	—	2	24	no	—
401	C₂²×C₄	—	2	24	no	—
402	C₂²×C₄	—	2	12	no	—
403	C₂²×C₄	—	2	3	no	—
404	C₂²×C₄	—	2	3	no	—
405	C₂²×C₄	—	2	3	no	—
406	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
407	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
408	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
409	Q₁₆	—	2	24	no	—
410	SD₁₆	—	2	8	no	—
411	SD₁₆	—	2	24	no	—
412	D₈	—	2	8	no	—
413	M₁₆	—	2	12	no	—
414	M₁₆	C₂⁴×C₂₃	1	8	yes	—
415	C₂×C₈	—	2	12	no	—
416	C₄⋊C₄	—	2	6	no	—
417	C₄⋊C₄	—	2	12	no	—
418	C₄⋊C₄	—	2	3	no	—
419	C₂²⋊C₄	—	2	3	no	—
420	C₂²⋊C₄	—	2	6	no	—
421	C₂²⋊C₄	—	2	6	no	—
422	C₄²	—	2	3	no	—
423	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
424	C₄²	C₂⁴×C₂₃	1	3	yes	—
425	C₂³	—	2	8	no	—
426	C₂³	—	2	12	no	—
427	C₂³	—	2	1	no	—
428	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
429	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
430	Q₈	—	2	24	no	—
431	Q₈	—	2	6	no	—
432	Q₈	—	2	12	no	—
433	Q₈	—	2	12	no	—
434	Q₈	—	2	6	no	—
435	Q₈	—	4	6	no	—
436	Q₈	C₂₃×#120	1	4	yes	—
437	Q₈	C₂²×D₄×C₂₃	1	8	yes	—
438	D₄	—	2	16	no	—
439	D₄	—	2	24	no	—
440	D₄	—	2	8	no	—
441	D₄	—	2	24	no	—
442	D₄	—	2	12	no	—
443	D₄	—	2	2	no	—
444	D₄	—	2	2	no	—
445	D₄	—	2	12	no	—
446	D₄	—	4	4	no	—
447	D₄	—	4	6	no	—
448	D₄	C₂₃×#120	1	12	yes	—
449	D₄	C₂²×D₄×C₂₃	1	8	yes	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	48	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	12	no	—
454	C₂×C₄	—	2	12	no	—
455	C₂×C₄	—	2	8	no	—
456	C₂×C₄	—	2	24	no	—
457	C₂×C₄	—	2	12	no	—
458	C₂×C₄	—	2	3	no	—
459	C₂×C₄	—	2	3	no	—
460	C₂×C₄	—	4	6	no	—
461	C₂×C₄	—	4	6	no	—
462	C₂×C₄	—	4	6	no	—
463	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
464	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
465	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
466	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
467	C₈	—	2	8	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	8	no	—
470	C₂²	—	4	3	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	8	12	no	—
475	C₂²	—	8	8	no	—
476	C₂²	C₂×C₂₃×#120	1	6	yes	—
477	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	2	8	no	—
480	C₄	—	2	6	no	—
481	C₄	—	4	3	no	—
482	C₄	—	4	3	no	—
483	C₄	C₂×C₂₃×#120	1	6	yes	—
484	C₄	C₂³×D₄×C₂₃	1	2	yes	—
485	C₂	—	2	1	no	—
486	C₂	—	2	6	no	—
487	C₂	—	4	2	no	—
488	C₂	—	4	8	no	—
489	C₂	—	8	1	no	—
490	C₂	C₂²×C₂₃×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	C₂²×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×2₋¹⁺⁶	C₂	semidirect
2. C₂₃×#3586	C₂	semidirect
3. C₂₃×#3499	C₂	semidirect
4. C₂₃×#3022	C₂	semidirect
5. C₂₃×#3020	C₂	semidirect
6. C₂₃×#3018	C₂	semidirect
7. C₂×C₂₃×#456	C₂	semidirect
8. C₂₃×#2901	C₂	semidirect
9. C₂₃×#2889	C₂	semidirect
18. C₂×C₂₃×2₋¹⁺⁴	C₂²	semidirect
21. C₂₃×#575	C₂²	semidirect
22. C₂×C₂₃×#137	C₂²	semidirect
23. C₂₃×#568	C₂²	semidirect
26. C₂₃×#536	C₂²	semidirect
27. C₂₃×#536	C₂²	semidirect
28. C₂₃×#532	C₂²	semidirect
29. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
30. C₄×#41×C₂₃	C₂²	semidirect
31. C₂₃×#456	C₂²	semidirect
32. C₂₃×#456	C₂²	semidirect
33. C₂₃×#455	C₂²	semidirect
34. C₂₃×#454	C₂²	semidirect
35. C₂₃×#453	C₂²	semidirect
36. C₂₃×#421	C₂²	semidirect
37. C₂×C₂₃×#104	C₂²	semidirect
38. C₂₃×#413	C₂²	semidirect
39. C₂×C₂₃×#101	C₂²	semidirect
48. C₂₃×2₋¹⁺⁴	D₄	semidirect
54. C₂₃×H(5,𝔽₂)	D₄	semidirect
179. #60391	C₂₃	direct (N × H)
223. 2₋¹⁺⁶	C₂×C₂₃	semidirect
224. #3586	C₂×C₂₃	semidirect
225. #3499	C₂×C₂₃	semidirect
226. #3022	C₂×C₂₃	semidirect
227. #3020	C₂×C₂₃	semidirect
228. #3018	C₂×C₂₃	semidirect
229. C₂×#456	C₂×C₂₃	semidirect
230. #2901	C₂×C₂₃	semidirect
231. #2889	C₂×C₂₃	semidirect
257. C₂×2₋¹⁺⁴	C₂²×C₂₃	semidirect
260. #575	C₂²×C₂₃	semidirect
261. C₂×#137	C₂²×C₂₃	semidirect
262. #568	C₂²×C₂₃	semidirect
265. #536	C₂²×C₂₃	semidirect
266. #536	C₂²×C₂₃	semidirect
267. #532	C₂²×C₂₃	semidirect
268. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
269. C₄×#41	C₂²×C₂₃	semidirect
270. #456	C₂²×C₂₃	semidirect
271. #456	C₂²×C₂₃	semidirect
272. #455	C₂²×C₂₃	semidirect
273. #454	C₂²×C₂₃	semidirect
274. #453	C₂²×C₂₃	semidirect
275. #421	C₂²×C₂₃	semidirect
276. C₂×#104	C₂²×C₂₃	semidirect
277. #413	C₂²×C₂₃	semidirect
278. C₂×#101	C₂²×C₂₃	semidirect
293. 2₋¹⁺⁴	D₄×C₂₃	semidirect
299. H(5,𝔽₂)	D₄×C₂₃	semidirect
351. C₂₃	#60391	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×2₋¹⁺⁶	2
C₂₃×#3586	2
C₂₃×#3499	2
C₂₃×#3022	2
C₂₃×#3020	2
C₂₃×#3018	2
C₂×C₂₃×#456	2
C₂₃×#2901	2
C₂₃×#2889	2
#60391	23

Subgroups — C23×#60391 #19208017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60391 #19208017