Subgroups — C₂₃×#60649 #19208275

Order 5888.

3178 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#577	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3500	C₂	1	2	yes	maximal, complemented
5	C₂₃×#3156	C₂	1	3	yes	maximal, complemented
6	C₂₃×#3155	C₂	1	3	yes	maximal, complemented
7	C₂₃×#3154	C₂	1	6	yes	maximal, complemented
8	C₂₃×#3153	C₂	1	6	yes	maximal, complemented
9	C₂₃×#3147	C₂	1	3	yes	maximal, complemented
10	C₂×C₂₃×#495	C₂	1	3	yes	maximal, complemented
11	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	2	yes	—
14	C₂₃×#578	—	2	8	no	—
15	C₂₃×#577	—	2	16	no	—
16	C₂₃×#576	—	2	8	no	—
17	C₂×C₂₃×#137	C₂²	1	6	yes	complemented
18	C₂×C₂₃×#137	C₂²	1	6	yes	complemented
19	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	6	yes	—
20	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
21	C₂×C₂₃×#135	C₂²	1	3	yes	—
22	C₂×C₂₃×#135	C₂²	1	6	yes	complemented
23	C₂×C₂₃×#135	C₂²	1	3	yes	complemented
24	C₂²×Q₁₆×C₂₃	C₂²	1	3	yes	complemented
25	C₂²×SD₁₆×C₂₃	C₂²	1	6	yes	complemented
26	C₂²×D₈×C₂₃	C₂²	1	3	yes	—
27	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
29	C₂₃×#548	—	2	4	no	—
30	C₂₃×#546	—	2	12	no	—
31	C₂₃×#536	C₂²	1	6	yes	complemented
32	C₂₃×#535	C₂²	1	6	yes	complemented
33	C₂₃×#532	C₂²	1	6	yes	complemented
34	C₂×C₂₃×#124	C₂²	1	6	yes	complemented
35	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#497	C₂²	1	12	yes	complemented
37	C₂₃×#496	C₂²	1	12	yes	complemented
38	C₂₃×#495	C₂²	1	3	yes	complemented
39	C₂₃×#495	C₂²	1	6	yes	complemented
40	C₂₃×#495	C₂²	1	6	yes	complemented
41	C₂₃×#495	C₂²	1	6	yes	complemented
42	C₂₃×#495	C₂²	1	1	yes	characteristic, complemented
43	C₂₃×#495	C₂²	1	6	yes	complemented
44	C₂₃×#494	C₂²	1	3	yes	complemented
45	C₂₃×#493	C₂²	1	3	yes	complemented
46	C₂₃×#492	C₂²	1	6	yes	complemented
47	C₂₃×#405	C₂²	1	3	yes	complemented
48	C₂₃×#405	C₂²	1	3	yes	complemented
49	C₄×M₁₆×C₂₃	C₂²	1	3	yes	complemented
50	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	complemented
51	C₂₃×2₋¹⁺⁴	—	2	8	no	—
52	C₂₃×2₋¹⁺⁴	—	2	8	no	—
53	C₂₃×H(5,𝔽₂)	—	2	8	no	—
54	C₂₃×H(5,𝔽₂)	—	2	8	no	—
55	C₂×#41×C₂₃	—	2	2	no	—
56	C₂×#41×C₂₃	—	2	6	no	—
57	C₂×#41×C₂₃	C₂³	1	6	yes	—
58	C₂×#41×C₂₃	C₂³	1	6	yes	—
59	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×Q₈×C₂₃	—	2	2	no	—
61	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
62	C₂²×D₄×C₂₃	—	2	6	no	—
63	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
64	C₂₃×#137	—	2	24	no	—
65	C₂₃×#137	—	2	24	no	—
66	C₂₃×AGL(1,ℤ/8)	—	2	24	no	—
67	C₂₃×AGL(1,ℤ/8)	—	2	24	no	—
68	C₂₃×#135	—	2	12	no	—
69	C₂₃×#135	—	2	24	no	—
70	C₂₃×#135	—	2	12	no	—
71	C₂×Q₁₆×C₂₃	—	2	12	no	—
72	C₂×Q₁₆×C₂₃	C₂³	1	3	yes	—
73	C₂×Q₁₆×C₂₃	C₂³	1	6	yes	—
74	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
75	C₂×Q₁₆×C₂₃	C₂³	1	6	yes	—
76	C₂×SD₁₆×C₂₃	—	2	24	no	—
77	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
78	C₂×SD₁₆×C₂₃	C₂³	1	12	yes	—
79	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
80	C₂×SD₁₆×C₂₃	C₂³	1	2	yes	—
81	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
82	C₂×D₈×C₂₃	—	2	12	no	—
83	C₂×D₈×C₂₃	C₂³	1	3	yes	—
84	C₂×D₈×C₂₃	C₂³	1	6	yes	—
85	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×D₈×C₂₃	C₂³	1	6	yes	—
87	C₂₃×#131	D₄	1	8	yes	complemented
88	C₂₃×#131	D₄	1	8	yes	complemented
89	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
90	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
91	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
92	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
93	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
94	C₂₃×#127	C₂³	1	6	yes	—
95	C₂₃×#124	—	2	12	no	—
96	C₂₃×#124	C₂³	1	6	yes	—
97	C₂₃×#124	C₂³	1	2	yes	—
98	C₂₃×#124	C₂³	1	12	yes	—
99	C₂₃×#123	—	2	12	no	—
100	C₂₃×C₂²⋊Q₈	—	2	12	no	—
101	C₂₃×#121	—	2	12	no	—
102	C₂₃×#121	—	2	12	no	—
103	C₂₃×#120	—	2	12	no	—
104	C₄×Q₈×C₂₃	D₄	1	2	yes	—
105	C₄×D₄×C₂₃	—	2	12	no	—
106	C₄×D₄×C₂₃	D₄	1	6	yes	—
107	C₂₃×#117	C₂³	1	3	yes	—
108	C₂×C₂²⋊C₄×C₂₃	—	2	12	no	—
109	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
110	C₂₃×#97	C₂³	1	6	yes	—
111	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₂₃	C₂³	1	3	yes	—
113	C₄×C₈×C₂₃	C₂³	1	3	yes	—
114	C₄×C₈×C₂₃	C₂³	1	3	yes	—
115	C₂⁴×C₂₃	—	2	6	no	—
116	#41×C₂₃	—	2	12	no	—
117	#41×C₂₃	—	2	12	no	—
118	#41×C₂₃	—	2	12	no	—
119	#41×C₂₃	—	2	12	no	—
120	#41×C₂₃	—	4	8	no	—
121	#41×C₂₃	—	4	24	no	—
122	#41×C₂₃	C₂×D₄	1	4	yes	—
123	#41×C₂₃	C₂×D₄	1	4	yes	—
124	C₂×Q₈×C₂₃	—	2	6	no	—
125	C₂×Q₈×C₂₃	—	2	12	no	—
126	C₂×Q₈×C₂₃	—	2	12	no	—
127	C₂×Q₈×C₂₃	—	4	8	no	—
128	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Q₈×C₂₃	C₂⁴	1	6	yes	—
130	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
131	C₂×D₄×C₂₃	—	2	6	no	—
132	C₂×D₄×C₂₃	—	2	6	no	—
133	C₂×D₄×C₂₃	—	2	12	no	—
134	C₂×D₄×C₂₃	—	2	12	no	—
135	C₂×D₄×C₂₃	—	2	6	no	—
136	C₂×D₄×C₂₃	—	2	12	no	—
137	C₂×D₄×C₂₃	—	4	24	no	—
138	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
139	C₂×D₄×C₂₃	C₂⁴	1	6	yes	—
140	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₂₃	—	2	6	no	—
142	C₂²×C₄×C₂₃	—	2	6	no	—
143	C₂²×C₄×C₂₃	—	2	6	no	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
145	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
146	Q₁₆×C₂₃	—	2	6	no	—
147	Q₁₆×C₂₃	—	2	12	no	—
148	Q₁₆×C₂₃	—	2	2	no	—
149	Q₁₆×C₂₃	—	2	12	no	—
150	SD₁₆×C₂₃	—	2	12	no	—
151	SD₁₆×C₂₃	—	2	24	no	—
152	SD₁₆×C₂₃	—	2	4	no	—
153	SD₁₆×C₂₃	—	2	12	no	—
154	SD₁₆×C₂₃	—	2	12	no	—
155	D₈×C₂₃	—	2	6	no	—
156	D₈×C₂₃	—	2	12	no	—
157	D₈×C₂₃	—	2	2	no	—
158	D₈×C₂₃	—	2	12	no	—
159	M₁₆×C₂₃	C₂×D₄	1	12	yes	—
160	M₁₆×C₂₃	C₂×D₄	1	12	yes	—
161	C₂×C₈×C₂₃	C₂×D₄	1	12	yes	—
162	C₂×C₈×C₂₃	C₂×D₄	1	12	yes	—
163	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
164	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
165	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
166	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
167	C₄⋊C₄×C₂₃	—	2	12	no	—
168	C₄⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
169	C₂²⋊C₄×C₂₃	—	2	12	no	—
170	C₂²⋊C₄×C₂₃	—	2	24	no	—
171	C₂²⋊C₄×C₂₃	—	2	4	no	—
172	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
173	C₄²×C₂₃	C₂×D₄	1	6	yes	—
174	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
175	C₄²×C₂₃	C₂⁴	1	3	yes	—
176	#60649	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	6	no	—
178	C₂³×C₂₃	—	2	2	no	—
179	C₂³×C₂₃	—	4	12	no	—
180	C₂³×C₂₃	—	8	12	no	—
181	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
182	Q₈×C₂₃	—	2	6	no	—
183	Q₈×C₂₃	—	2	2	no	—
184	Q₈×C₂₃	—	2	6	no	—
185	Q₈×C₂₃	—	2	2	no	—
186	Q₈×C₂₃	—	4	12	no	—
187	Q₈×C₂₃	#127	1	4	yes	—
188	D₄×C₂₃	—	2	6	no	—
189	D₄×C₂₃	—	2	2	no	—
190	D₄×C₂₃	—	2	6	no	—
191	D₄×C₂₃	—	2	2	no	—
192	D₄×C₂₃	—	4	12	no	—
193	D₄×C₂₃	—	4	12	no	—
194	D₄×C₂₃	—	4	12	no	—
195	D₄×C₂₃	—	8	12	no	—
196	D₄×C₂₃	#127	1	12	yes	—
197	C₂×C₄×C₂₃	—	2	6	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	4	6	no	—
200	C₂×C₄×C₂₃	—	4	12	no	—
201	C₂×C₄×C₂₃	—	4	12	no	—
202	C₂×C₄×C₂₃	—	4	12	no	—
203	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
204	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
205	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
206	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
208	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
209	C₈×C₂₃	C₂²×D₄	1	2	yes	—
210	C₈×C₂₃	C₂²×D₄	1	2	yes	—
211	C₈×C₂₃	C₂²×D₄	1	6	yes	—
212	C₈×C₂₃	C₂²×D₄	1	6	yes	—
213	C₂×#578	C₂×C₂₃	1	1	yes	characteristic, complemented
214	C₂×#577	C₂×C₂₃	1	2	yes	complemented
215	C₂×#576	C₂×C₂₃	1	1	yes	characteristic, complemented
216	#3500	C₂×C₂₃	1	2	yes	complemented
217	#3156	C₂×C₂₃	1	3	yes	complemented
218	#3155	C₂×C₂₃	1	3	yes	complemented
219	#3154	C₂×C₂₃	1	6	yes	complemented
220	#3153	C₂×C₂₃	1	6	yes	complemented
221	#3147	C₂×C₂₃	1	3	yes	complemented
222	C₂×#495	C₂×C₂₃	1	3	yes	complemented
223	C₄×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
224	C₂²×C₂₃	—	2	6	no	—
225	C₂²×C₂₃	—	4	4	no	—
226	C₂²×C₂₃	—	4	12	no	—
227	C₂²×C₂₃	—	4	2	no	—
228	C₂²×C₂₃	—	4	6	no	—
229	C₂²×C₂₃	—	16	12	no	—
230	C₂²×C₂₃	C₂×#127	1	6	yes	—
231	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
232	C₄×C₂₃	—	2	2	no	—
233	C₄×C₂₃	—	2	6	no	—
234	C₄×C₂₃	—	4	2	no	—
235	C₄×C₂₃	—	4	6	no	—
236	C₄×C₂₃	C₂×#127	1	6	yes	—
237	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
238	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
239	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	—
240	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	2	yes	—
241	#578	—	2	8	no	—
242	#577	—	2	16	no	—
243	#576	—	2	8	no	—
244	C₂×#137	C₂²×C₂₃	1	6	yes	complemented
245	C₂×#137	C₂²×C₂₃	1	6	yes	complemented
246	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	6	yes	—
247	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	6	yes	complemented
248	C₂×#135	C₂²×C₂₃	1	3	yes	—
249	C₂×#135	C₂²×C₂₃	1	6	yes	complemented
250	C₂×#135	C₂²×C₂₃	1	3	yes	complemented
251	C₂²×Q₁₆	C₂²×C₂₃	1	3	yes	complemented
252	C₂²×SD₁₆	C₂²×C₂₃	1	6	yes	complemented
253	C₂²×D₈	C₂²×C₂₃	1	3	yes	—
254	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
256	#548	—	2	4	no	—
257	#546	—	2	12	no	—
258	#536	C₂²×C₂₃	1	6	yes	complemented
259	#535	C₂²×C₂₃	1	6	yes	complemented
260	#532	C₂²×C₂₃	1	6	yes	complemented
261	C₂×#124	C₂²×C₂₃	1	6	yes	complemented
262	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
263	#497	C₂²×C₂₃	1	12	yes	complemented
264	#496	C₂²×C₂₃	1	12	yes	complemented
265	#495	C₂²×C₂₃	1	3	yes	complemented
266	#495	C₂²×C₂₃	1	6	yes	complemented
267	#495	C₂²×C₂₃	1	6	yes	complemented
268	#495	C₂²×C₂₃	1	6	yes	complemented
269	#495	C₂²×C₂₃	1	1	yes	characteristic, complemented
270	#495	C₂²×C₂₃	1	6	yes	complemented
271	#494	C₂²×C₂₃	1	3	yes	complemented
272	#493	C₂²×C₂₃	1	3	yes	complemented
273	#492	C₂²×C₂₃	1	6	yes	complemented
274	#405	C₂²×C₂₃	1	3	yes	complemented
275	#405	C₂²×C₂₃	1	3	yes	complemented
276	C₄×M₁₆	C₂²×C₂₃	1	3	yes	complemented
277	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	complemented
278	C₂×C₂₃	—	2	6	no	—
279	C₂×C₂₃	—	8	2	no	—
280	C₂×C₂₃	—	8	6	no	—
281	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
282	C₂×C₂₃	#3584	1	2	yes	—
283	2₋¹⁺⁴	—	2	8	no	—
284	2₋¹⁺⁴	—	2	8	no	—
285	H(5,𝔽₂)	—	2	8	no	—
286	H(5,𝔽₂)	—	2	8	no	—
287	C₂×#41	—	2	2	no	—
288	C₂×#41	—	2	6	no	—
289	C₂×#41	C₂³×C₂₃	1	6	yes	—
290	C₂×#41	C₂³×C₂₃	1	6	yes	—
291	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
292	C₂²×Q₈	—	2	2	no	—
293	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
294	C₂²×D₄	—	2	6	no	—
295	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
296	#137	—	2	24	no	—
297	#137	—	2	24	no	—
298	AGL(1,ℤ/8)	—	2	24	no	—
299	AGL(1,ℤ/8)	—	2	24	no	—
300	#135	—	2	12	no	—
301	#135	—	2	24	no	—
302	#135	—	2	12	no	—
303	C₂×Q₁₆	—	2	12	no	—
304	C₂×Q₁₆	C₂³×C₂₃	1	3	yes	—
305	C₂×Q₁₆	C₂³×C₂₃	1	6	yes	—
306	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
307	C₂×Q₁₆	C₂³×C₂₃	1	6	yes	—
308	C₂×SD₁₆	—	2	24	no	—
309	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
310	C₂×SD₁₆	C₂³×C₂₃	1	12	yes	—
311	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
312	C₂×SD₁₆	C₂³×C₂₃	1	2	yes	—
313	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
314	C₂×D₈	—	2	12	no	—
315	C₂×D₈	C₂³×C₂₃	1	3	yes	—
316	C₂×D₈	C₂³×C₂₃	1	6	yes	—
317	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
318	C₂×D₈	C₂³×C₂₃	1	6	yes	—
319	#131	D₄×C₂₃	1	8	yes	complemented
320	#131	D₄×C₂₃	1	8	yes	complemented
321	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
322	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
323	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
324	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
325	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
326	#127	C₂³×C₂₃	1	6	yes	—
327	#124	—	2	12	no	—
328	#124	C₂³×C₂₃	1	6	yes	—
329	#124	C₂³×C₂₃	1	12	yes	—
330	#124	C₂³×C₂₃	1	2	yes	—
331	#123	—	2	12	no	—
332	C₂²⋊Q₈	—	2	12	no	—
333	#121	—	2	12	no	—
334	#121	—	2	12	no	—
335	#120	—	2	12	no	—
336	C₄×Q₈	D₄×C₂₃	1	2	yes	—
337	C₄×D₄	—	2	12	no	—
338	C₄×D₄	D₄×C₂₃	1	6	yes	—
339	#117	C₂³×C₂₃	1	3	yes	—
340	C₂×C₂²⋊C₄	—	2	12	no	—
341	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
342	#97	C₂³×C₂₃	1	6	yes	—
343	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
344	C₄×C₈	C₂³×C₂₃	1	3	yes	—
345	C₄×C₈	C₂³×C₂₃	1	3	yes	—
346	C₄×C₈	C₂³×C₂₃	1	3	yes	—
347	C₂₃	#60649	1	1	yes	characteristic, direct product factor, Sylow
348	C₂⁴	—	2	6	no	—
349	#41	—	2	12	no	—
350	#41	—	2	12	no	—
351	#41	—	2	12	no	—
352	#41	—	2	12	no	—
353	#41	—	4	8	no	—
354	#41	—	4	24	no	—
355	#41	C₂×D₄×C₂₃	1	4	yes	—
356	#41	C₂×D₄×C₂₃	1	4	yes	—
357	C₂×Q₈	—	2	6	no	—
358	C₂×Q₈	—	2	12	no	—
359	C₂×Q₈	—	2	12	no	—
360	C₂×Q₈	—	4	8	no	—
361	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
362	C₂×Q₈	C₂⁴×C₂₃	1	6	yes	—
363	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
364	C₂×D₄	—	2	6	no	—
365	C₂×D₄	—	2	6	no	—
366	C₂×D₄	—	2	12	no	—
367	C₂×D₄	—	2	12	no	—
368	C₂×D₄	—	2	6	no	—
369	C₂×D₄	—	2	12	no	—
370	C₂×D₄	—	4	24	no	—
371	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
372	C₂×D₄	C₂⁴×C₂₃	1	6	yes	—
373	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
374	C₂²×C₄	—	2	6	no	—
375	C₂²×C₄	—	2	6	no	—
376	C₂²×C₄	—	2	6	no	—
377	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
378	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
379	Q₁₆	—	2	6	no	—
380	Q₁₆	—	2	12	no	—
381	Q₁₆	—	2	2	no	—
382	Q₁₆	—	2	12	no	—
383	SD₁₆	—	2	12	no	—
384	SD₁₆	—	2	24	no	—
385	SD₁₆	—	2	4	no	—
386	SD₁₆	—	2	12	no	—
387	SD₁₆	—	2	12	no	—
388	D₈	—	2	6	no	—
389	D₈	—	2	12	no	—
390	D₈	—	2	2	no	—
391	D₈	—	2	12	no	—
392	M₁₆	C₂×D₄×C₂₃	1	12	yes	—
393	M₁₆	C₂×D₄×C₂₃	1	12	yes	—
394	C₂×C₈	C₂×D₄×C₂₃	1	12	yes	—
395	C₂×C₈	C₂×D₄×C₂₃	1	12	yes	—
396	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
397	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
398	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
399	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
400	C₄⋊C₄	—	2	12	no	—
401	C₄⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
402	C₂²⋊C₄	—	2	12	no	—
403	C₂²⋊C₄	—	2	24	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
406	C₄²	C₂×D₄×C₂₃	1	6	yes	—
407	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
408	C₄²	C₂⁴×C₂₃	1	3	yes	—
409	C₂³	—	2	6	no	—
410	C₂³	—	2	2	no	—
411	C₂³	—	4	12	no	—
412	C₂³	—	8	12	no	—
413	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
414	Q₈	—	2	6	no	—
415	Q₈	—	2	2	no	—
416	Q₈	—	2	6	no	—
417	Q₈	—	2	2	no	—
418	Q₈	—	4	12	no	—
419	Q₈	C₂₃×#127	1	4	yes	—
420	D₄	—	2	6	no	—
421	D₄	—	2	2	no	—
422	D₄	—	2	6	no	—
423	D₄	—	2	2	no	—
424	D₄	—	4	12	no	—
425	D₄	—	4	12	no	—
426	D₄	—	4	12	no	—
427	D₄	—	8	12	no	—
428	D₄	C₂₃×#127	1	12	yes	—
429	C₂×C₄	—	2	6	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	6	no	—
432	C₂×C₄	—	4	12	no	—
433	C₂×C₄	—	4	12	no	—
434	C₂×C₄	—	4	12	no	—
435	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
437	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
438	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
439	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
440	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
441	C₈	C₂²×D₄×C₂₃	1	2	yes	—
442	C₈	C₂²×D₄×C₂₃	1	2	yes	—
443	C₈	C₂²×D₄×C₂₃	1	6	yes	—
444	C₈	C₂²×D₄×C₂₃	1	6	yes	—
445	C₂²	—	2	6	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	6	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	12	no	—
450	C₂²	—	16	12	no	—
451	C₂²	C₂×C₂₃×#127	1	6	yes	—
452	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
453	C₄	—	2	2	no	—
454	C₄	—	2	6	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	6	no	—
457	C₄	C₂×C₂₃×#127	1	6	yes	—
458	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
459	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
460	C₂	—	2	6	no	—
461	C₂	—	8	6	no	—
462	C₂	—	8	2	no	—
463	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
464	C₂	C₂₃×#3584	1	2	yes	—

	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×#578	C₂	semidirect
2. C₂×C₂₃×#577	C₂	semidirect
3. C₂×C₂₃×#576	C₂	semidirect
4. C₂₃×#3500	C₂	semidirect
5. C₂₃×#3156	C₂	semidirect
6. C₂₃×#3155	C₂	semidirect
7. C₂₃×#3154	C₂	semidirect
8. C₂₃×#3153	C₂	semidirect
9. C₂₃×#3147	C₂	semidirect
10. C₂×C₂₃×#495	C₂	semidirect
11. C₄×C₂₃×#131	C₂	semidirect
17. C₂×C₂₃×#137	C₂²	semidirect
18. C₂×C₂₃×#137	C₂²	semidirect
20. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
22. C₂×C₂₃×#135	C₂²	semidirect
23. C₂×C₂₃×#135	C₂²	semidirect
24. C₂²×Q₁₆×C₂₃	C₂²	semidirect
25. C₂²×SD₁₆×C₂₃	C₂²	semidirect
31. C₂₃×#536	C₂²	semidirect
32. C₂₃×#535	C₂²	semidirect
33. C₂₃×#532	C₂²	semidirect
34. C₂×C₂₃×#124	C₂²	semidirect
36. C₂₃×#497	C₂²	semidirect
37. C₂₃×#496	C₂²	semidirect
38. C₂₃×#495	C₂²	semidirect
39. C₂₃×#495	C₂²	semidirect
40. C₂₃×#495	C₂²	semidirect
41. C₂₃×#495	C₂²	semidirect
42. C₂₃×#495	C₂²	semidirect
43. C₂₃×#495	C₂²	semidirect
44. C₂₃×#494	C₂²	semidirect
45. C₂₃×#493	C₂²	semidirect
46. C₂₃×#492	C₂²	semidirect
47. C₂₃×#405	C₂²	semidirect
48. C₂₃×#405	C₂²	semidirect
49. C₄×M₁₆×C₂₃	C₂²	semidirect
50. C₂×C₄×C₈×C₂₃	C₂²	semidirect
87. C₂₃×#131	D₄	semidirect
88. C₂₃×#131	D₄	semidirect
176. #60649	C₂₃	direct (N × H)
213. C₂×#578	C₂×C₂₃	semidirect
214. C₂×#577	C₂×C₂₃	semidirect
215. C₂×#576	C₂×C₂₃	semidirect
216. #3500	C₂×C₂₃	semidirect
217. #3156	C₂×C₂₃	semidirect
218. #3155	C₂×C₂₃	semidirect
219. #3154	C₂×C₂₃	semidirect
220. #3153	C₂×C₂₃	semidirect
221. #3147	C₂×C₂₃	semidirect
222. C₂×#495	C₂×C₂₃	semidirect
223. C₄×#131	C₂×C₂₃	semidirect
244. C₂×#137	C₂²×C₂₃	semidirect
245. C₂×#137	C₂²×C₂₃	semidirect
247. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
249. C₂×#135	C₂²×C₂₃	semidirect
250. C₂×#135	C₂²×C₂₃	semidirect
251. C₂²×Q₁₆	C₂²×C₂₃	semidirect
252. C₂²×SD₁₆	C₂²×C₂₃	semidirect
258. #536	C₂²×C₂₃	semidirect
259. #535	C₂²×C₂₃	semidirect
260. #532	C₂²×C₂₃	semidirect
261. C₂×#124	C₂²×C₂₃	semidirect
263. #497	C₂²×C₂₃	semidirect
264. #496	C₂²×C₂₃	semidirect
265. #495	C₂²×C₂₃	semidirect
266. #495	C₂²×C₂₃	semidirect
267. #495	C₂²×C₂₃	semidirect
268. #495	C₂²×C₂₃	semidirect
269. #495	C₂²×C₂₃	semidirect
270. #495	C₂²×C₂₃	semidirect
271. #494	C₂²×C₂₃	semidirect
272. #493	C₂²×C₂₃	semidirect
273. #492	C₂²×C₂₃	semidirect
274. #405	C₂²×C₂₃	semidirect
275. #405	C₂²×C₂₃	semidirect
276. C₄×M₁₆	C₂²×C₂₃	semidirect
277. C₂×C₄×C₈	C₂²×C₂₃	semidirect
319. #131	D₄×C₂₃	semidirect
320. #131	D₄×C₂₃	semidirect
347. C₂₃	#60649	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×#578	2
C₂×C₂₃×#577	2
C₂×C₂₃×#576	2
C₂₃×#3500	2
C₂₃×#3156	2
C₂₃×#3155	2
C₂₃×#3154	2
C₂₃×#3153	2
C₂₃×#3147	2
C₂×C₂₃×#495	2
C₄×C₂₃×#131	2
#60649	23

Subgroups — C23×#60649 #19208275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60649 #19208275