Subgroups — C₂₃×#60423 #19208049

Order 5888.

2890 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3449	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3042	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3041	C₂	1	8	yes	maximal, complemented
6	C₂₃×#3036	C₂	1	8	yes	maximal, complemented
7	C₂₃×#2905	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2896	C₂	1	2	yes	maximal, complemented
9	C₂²×#41×C₂₃	C₂²	1	2	yes	—
10	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
12	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	8	yes	—
13	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
14	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
15	C₂²×D₈×C₂₃	C₂²	1	4	yes	complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#537	—	2	2	no	—
18	C₂₃×#536	C₂²	1	4	yes	complemented
19	C₂₃×#535	C₂²	1	4	yes	complemented
20	C₂₃×#534	—	2	2	no	—
21	C₂₃×#532	C₂²	1	4	yes	complemented
22	C₂₃×#525	—	2	4	no	—
23	C₂₃×#518	—	2	4	no	—
24	C₂₃×#518	C₂²	1	8	yes	complemented
25	C₂×C₄×D₄×C₂₃	—	2	2	no	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
27	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
30	C₂₃×#463	C₂²	1	32	yes	complemented
31	C₂₃×#461	C₂²	1	16	yes	complemented
32	C₂₃×#459	C₂²	1	16	yes	complemented
33	C₂₃×#424	C₂²	1	8	yes	complemented
34	C₂₃×#423	C₂²	1	4	yes	complemented
35	C₂₃×#419	C₂²	1	4	yes	complemented
36	C₂₃×#418	C₂²	1	4	yes	complemented
37	C₂₃×#417	C₂²	1	8	yes	complemented
38	C₂₃×#416	C₂²	1	8	yes	complemented
39	C₂⁵×C₂₃	—	2	1	no	—
40	C₂×#41×C₂₃	—	2	2	no	—
41	C₂×#41×C₂₃	—	2	2	no	—
42	C₂×#41×C₂₃	—	2	8	no	—
43	C₂×#41×C₂₃	—	2	8	no	—
44	C₂×#41×C₂₃	C₂³	1	4	yes	—
45	C₂×#41×C₂₃	C₂³	1	4	yes	—
46	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	2	no	—
50	C₂²×D₄×C₂₃	—	2	4	no	—
51	C₂²×D₄×C₂₃	D₄	1	8	yes	complemented
52	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
54	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
55	C₂³×C₄×C₂₃	—	2	2	no	—
56	C₂³×C₄×C₂₃	—	2	4	no	—
57	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂₃×AGL(1,ℤ/8)	—	2	32	no	—
59	C₂₃×AGL(1,ℤ/8)	—	2	32	no	—
60	C₂×SD₁₆×C₂₃	—	2	16	no	—
61	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
62	C₂×D₈×C₂₃	—	2	16	no	—
63	C₂×D₈×C₂₃	C₂³	1	16	yes	—
64	C₂×M₁₆×C₂₃	—	2	8	no	—
65	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
66	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
67	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
68	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
69	C₂₃×#127	C₂³	1	4	yes	complemented
70	C₂₃×#124	C₂³	1	8	yes	complemented
71	C₂₃×#123	—	2	8	no	—
72	C₂₃×C₂²⋊Q₈	—	2	8	no	—
73	C₂₃×#121	—	2	8	no	—
74	C₂₃×#121	—	2	8	no	—
75	C₂₃×#120	—	2	8	no	—
76	C₄×D₄×C₂₃	—	2	16	no	—
77	C₄×D₄×C₂₃	—	2	8	no	—
78	C₄×D₄×C₂₃	—	2	8	no	—
79	C₄×D₄×C₂₃	C₂³	1	16	yes	—
80	C₂₃×#117	—	2	2	no	—
81	C₂₃×#117	D₄	1	4	yes	—
82	C₂₃×#117	C₂³	1	4	yes	—
83	C₂₃×#117	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
88	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
89	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
92	C₂₃×C₄⋊C₈	C₂³	1	16	yes	complemented
93	C₂₃×#103	C₂³	1	16	yes	complemented
94	C₂₃×#102	C₂³	1	16	yes	complemented
95	C₂⁴×C₂₃	—	2	2	no	—
96	C₂⁴×C₂₃	—	2	8	no	—
97	C₂⁴×C₂₃	—	2	4	no	—
98	C₂⁴×C₂₃	—	2	2	no	—
99	C₂⁴×C₂₃	—	8	4	no	—
100	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
101	#41×C₂₃	—	2	8	no	—
102	#41×C₂₃	—	2	8	no	—
103	#41×C₂₃	—	2	8	no	—
104	#41×C₂₃	—	2	8	no	—
105	#41×C₂₃	—	4	8	no	—
106	#41×C₂₃	—	4	8	no	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	2	4	no	—
109	C₂×Q₈×C₂₃	—	2	4	no	—
110	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
111	C₂×D₄×C₂₃	—	2	4	no	—
112	C₂×D₄×C₂₃	—	2	4	no	—
113	C₂×D₄×C₂₃	—	2	16	no	—
114	C₂×D₄×C₂₃	—	2	8	no	—
115	C₂×D₄×C₂₃	—	2	4	no	—
116	C₂×D₄×C₂₃	—	2	2	no	—
117	C₂×D₄×C₂₃	—	2	4	no	—
118	C₂×D₄×C₂₃	—	2	2	no	—
119	C₂×D₄×C₂₃	—	2	8	no	—
120	C₂×D₄×C₂₃	—	2	8	no	—
121	C₂×D₄×C₂₃	—	2	4	no	—
122	C₂×D₄×C₂₃	—	2	4	no	—
123	C₂×D₄×C₂₃	—	4	8	no	—
124	C₂×D₄×C₂₃	C₂×D₄	1	16	yes	—
125	C₂×D₄×C₂₃	C₂×D₄	1	8	yes	—
126	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
127	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
128	C₂²×C₄×C₂₃	—	2	2	no	—
129	C₂²×C₄×C₂₃	—	2	2	no	—
130	C₂²×C₄×C₂₃	—	2	4	no	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	16	no	—
133	C₂²×C₄×C₂₃	—	4	8	no	—
134	C₂²×C₄×C₂₃	—	8	8	no	—
135	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
136	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₂₃	#41	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
142	SD₁₆×C₂₃	—	2	32	no	—
143	D₈×C₂₃	—	2	32	no	—
144	M₁₆×C₂₃	—	2	8	no	—
145	M₁₆×C₂₃	—	2	8	no	—
146	C₂×C₈×C₂₃	—	2	8	no	—
147	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	—	2	8	no	—
150	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
151	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
152	C₂²⋊C₄×C₂₃	—	2	4	no	—
153	C₂²⋊C₄×C₂₃	—	2	16	no	—
154	C₂²⋊C₄×C₂₃	—	2	16	no	—
155	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
156	C₄²×C₂₃	—	2	4	no	—
157	C₄²×C₂₃	C₂×D₄	1	4	yes	—
158	C₄²×C₂₃	C₂⁴	1	4	yes	—
159	#60423	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₂₃	—	2	8	no	—
161	C₂³×C₂₃	—	2	4	no	—
162	C₂³×C₂₃	—	2	16	no	—
163	C₂³×C₂₃	—	2	4	no	—
164	C₂³×C₂₃	—	4	2	no	—
165	C₂³×C₂₃	—	4	8	no	—
166	C₂³×C₂₃	—	4	4	no	—
167	C₂³×C₂₃	—	8	4	no	—
168	C₂³×C₂₃	—	8	8	no	—
169	C₂³×C₂₃	—	8	16	no	—
170	C₂³×C₂₃	#121	1	4	yes	—
171	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
172	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₂₃	—	2	4	no	—
174	Q₈×C₂₃	—	2	4	no	—
175	Q₈×C₂₃	—	4	4	no	—
176	D₄×C₂₃	—	2	8	no	—
177	D₄×C₂₃	—	2	16	no	—
178	D₄×C₂₃	—	2	4	no	—
179	D₄×C₂₃	—	2	4	no	—
180	D₄×C₂₃	—	2	4	no	—
181	D₄×C₂₃	—	4	8	no	—
182	D₄×C₂₃	—	4	8	no	—
183	D₄×C₂₃	—	4	4	no	—
184	D₄×C₂₃	—	8	8	no	—
185	D₄×C₂₃	C₂²×D₄	1	8	yes	—
186	C₂×C₄×C₂₃	—	2	4	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	4	4	no	—
189	C₂×C₄×C₂₃	—	4	4	no	—
190	C₂×C₄×C₂₃	—	4	4	no	—
191	C₂×C₄×C₂₃	—	4	8	no	—
192	C₂×C₄×C₂₃	—	4	8	no	—
193	C₂×C₄×C₂₃	—	8	16	no	—
194	C₂×C₄×C₂₃	#121	1	8	yes	—
195	C₂×C₄×C₂₃	#121	1	4	yes	—
196	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
197	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
198	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
200	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
201	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
202	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
203	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
204	C₈×C₂₃	—	2	8	no	—
205	C₂²×AGL(1,ℤ/8)	C₂×C₂₃	1	2	yes	complemented
206	#3449	C₂×C₂₃	1	1	yes	characteristic, complemented
207	C₂×#518	C₂×C₂₃	1	1	yes	characteristic, complemented
208	#3042	C₂×C₂₃	1	8	yes	complemented
209	#3041	C₂×C₂₃	1	8	yes	complemented
210	#3036	C₂×C₂₃	1	8	yes	complemented
211	#2905	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#2896	C₂×C₂₃	1	2	yes	complemented
213	C₂²×C₂₃	—	2	2	no	—
214	C₂²×C₂₃	—	2	4	no	—
215	C₂²×C₂₃	—	2	8	no	—
216	C₂²×C₂₃	—	4	4	no	—
217	C₂²×C₂₃	—	4	16	no	—
218	C₂²×C₂₃	—	4	8	no	—
219	C₂²×C₂₃	—	4	8	no	—
220	C₂²×C₂₃	—	4	4	no	—
221	C₂²×C₂₃	—	8	16	no	—
222	C₂²×C₂₃	—	8	8	no	—
223	C₂²×C₂₃	C₂×#121	1	4	yes	—
224	C₂²×C₂₃	C₂×#121	1	2	yes	—
225	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
226	C₄×C₂₃	—	2	8	no	—
227	C₄×C₂₃	—	4	4	no	—
228	C₄×C₂₃	—	4	4	no	—
229	C₄×C₂₃	C₂×#121	1	2	yes	—
230	C₄×C₂₃	C₂×#121	1	4	yes	—
231	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
232	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
233	C₂²×#41	C₂²×C₂₃	1	2	yes	—
234	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
235	C₂×AGL(1,ℤ/8)	—	2	16	no	—
236	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	8	yes	—
237	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	8	yes	complemented
238	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
239	C₂²×D₈	C₂²×C₂₃	1	4	yes	complemented
240	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
241	#537	—	2	2	no	—
242	#536	C₂²×C₂₃	1	4	yes	complemented
243	#535	C₂²×C₂₃	1	4	yes	complemented
244	#534	—	2	2	no	—
245	#532	C₂²×C₂₃	1	4	yes	complemented
246	#525	—	2	4	no	—
247	#518	—	2	4	no	—
248	#518	C₂²×C₂₃	1	8	yes	complemented
249	C₂×C₄×D₄	—	2	2	no	—
250	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
251	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
253	C₂²×C₂²⋊C₄	—	2	2	no	—
254	#463	C₂²×C₂₃	1	32	yes	complemented
255	#461	C₂²×C₂₃	1	16	yes	complemented
256	#459	C₂²×C₂₃	1	16	yes	complemented
257	#424	C₂²×C₂₃	1	8	yes	complemented
258	#423	C₂²×C₂₃	1	4	yes	complemented
259	#419	C₂²×C₂₃	1	4	yes	complemented
260	#418	C₂²×C₂₃	1	4	yes	complemented
261	#417	C₂²×C₂₃	1	8	yes	complemented
262	#416	C₂²×C₂₃	1	8	yes	complemented
263	C₂×C₂₃	—	2	2	no	—
264	C₂×C₂₃	—	2	4	no	—
265	C₂×C₂₃	—	4	8	no	—
266	C₂×C₂₃	—	8	4	no	—
267	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
268	C₂×C₂₃	#3585	1	2	yes	—
269	C₂⁵	—	2	1	no	—
270	C₂×#41	—	2	2	no	—
271	C₂×#41	—	2	2	no	—
272	C₂×#41	—	2	8	no	—
273	C₂×#41	—	2	8	no	—
274	C₂×#41	C₂³×C₂₃	1	4	yes	—
275	C₂×#41	C₂³×C₂₃	1	4	yes	—
276	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
277	C₂²×D₄	—	2	2	no	—
278	C₂²×D₄	—	2	1	no	—
279	C₂²×D₄	—	2	2	no	—
280	C₂²×D₄	—	2	4	no	—
281	C₂²×D₄	D₄×C₂₃	1	8	yes	complemented
282	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
283	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
284	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
285	C₂³×C₄	—	2	2	no	—
286	C₂³×C₄	—	2	4	no	—
287	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
288	AGL(1,ℤ/8)	—	2	32	no	—
289	AGL(1,ℤ/8)	—	2	32	no	—
290	C₂×SD₁₆	—	2	16	no	—
291	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
292	C₂×D₈	—	2	16	no	—
293	C₂×D₈	C₂³×C₂₃	1	16	yes	—
294	C₂×M₁₆	—	2	8	no	—
295	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
296	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
297	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
298	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
299	#127	C₂³×C₂₃	1	4	yes	complemented
300	#124	C₂³×C₂₃	1	8	yes	complemented
301	#123	—	2	8	no	—
302	C₂²⋊Q₈	—	2	8	no	—
303	#121	—	2	8	no	—
304	#121	—	2	8	no	—
305	#120	—	2	8	no	—
306	C₄×D₄	—	2	16	no	—
307	C₄×D₄	—	2	8	no	—
308	C₄×D₄	—	2	8	no	—
309	C₄×D₄	C₂³×C₂₃	1	16	yes	—
310	#117	—	2	2	no	—
311	#117	D₄×C₂₃	1	4	yes	—
312	#117	C₂³×C₂₃	1	4	yes	—
313	#117	C₂³×C₂₃	1	4	yes	—
314	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
316	C₂×C₂²⋊C₄	—	2	1	no	—
317	C₂×C₂²⋊C₄	—	2	4	no	—
318	C₂×C₂²⋊C₄	—	2	16	no	—
319	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
322	C₄⋊C₈	C₂³×C₂₃	1	16	yes	complemented
323	#103	C₂³×C₂₃	1	16	yes	complemented
324	#102	C₂³×C₂₃	1	16	yes	complemented
325	C₂₃	#60423	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	—	2	2	no	—
327	C₂⁴	—	2	8	no	—
328	C₂⁴	—	2	4	no	—
329	C₂⁴	—	2	2	no	—
330	C₂⁴	—	8	4	no	—
331	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
332	#41	—	2	8	no	—
333	#41	—	2	8	no	—
334	#41	—	2	8	no	—
335	#41	—	2	8	no	—
336	#41	—	4	8	no	—
337	#41	—	4	8	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	4	no	—
340	C₂×Q₈	—	2	4	no	—
341	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
342	C₂×D₄	—	2	4	no	—
343	C₂×D₄	—	2	4	no	—
344	C₂×D₄	—	2	16	no	—
345	C₂×D₄	—	2	8	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	8	no	—
351	C₂×D₄	—	2	8	no	—
352	C₂×D₄	—	2	4	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	4	8	no	—
355	C₂×D₄	C₂×D₄×C₂₃	1	16	yes	—
356	C₂×D₄	C₂×D₄×C₂₃	1	8	yes	—
357	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
358	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	16	no	—
364	C₂²×C₄	—	4	8	no	—
365	C₂²×C₄	—	8	8	no	—
366	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
367	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
370	C₂²×C₄	#41×C₂₃	1	4	yes	—
371	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
372	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
373	SD₁₆	—	2	32	no	—
374	D₈	—	2	32	no	—
375	M₁₆	—	2	8	no	—
376	M₁₆	—	2	8	no	—
377	C₂×C₈	—	2	8	no	—
378	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	8	no	—
381	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
382	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
383	C₂²⋊C₄	—	2	4	no	—
384	C₂²⋊C₄	—	2	16	no	—
385	C₂²⋊C₄	—	2	16	no	—
386	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
387	C₄²	—	2	4	no	—
388	C₄²	C₂×D₄×C₂₃	1	4	yes	—
389	C₄²	C₂⁴×C₂₃	1	4	yes	—
390	C₂³	—	2	4	no	—
391	C₂³	—	2	8	no	—
392	C₂³	—	2	16	no	—
393	C₂³	—	2	4	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	8	no	—
396	C₂³	—	4	4	no	—
397	C₂³	—	8	8	no	—
398	C₂³	—	8	4	no	—
399	C₂³	—	8	16	no	—
400	C₂³	C₂₃×#121	1	4	yes	—
401	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
402	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
403	Q₈	—	2	4	no	—
404	Q₈	—	2	4	no	—
405	Q₈	—	4	4	no	—
406	D₄	—	2	4	no	—
407	D₄	—	2	4	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	8	no	—
410	D₄	—	2	16	no	—
411	D₄	—	4	8	no	—
412	D₄	—	4	8	no	—
413	D₄	—	4	4	no	—
414	D₄	—	8	8	no	—
415	D₄	C₂²×D₄×C₂₃	1	8	yes	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	4	4	no	—
419	C₂×C₄	—	4	4	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	—	4	8	no	—
423	C₂×C₄	—	8	16	no	—
424	C₂×C₄	C₂₃×#121	1	8	yes	—
425	C₂×C₄	C₂₃×#121	1	4	yes	—
426	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
427	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
428	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
429	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
430	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
432	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
433	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
434	C₈	—	2	8	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	4	no	—
437	C₂²	—	2	8	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	16	no	—
440	C₂²	—	4	8	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	8	no	—
443	C₂²	—	8	16	no	—
444	C₂²	—	8	8	no	—
445	C₂²	C₂×C₂₃×#121	1	4	yes	—
446	C₂²	C₂×C₂₃×#121	1	2	yes	—
447	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
448	C₄	—	2	8	no	—
449	C₄	—	4	4	no	—
450	C₄	—	4	4	no	—
451	C₄	C₂×C₂₃×#121	1	2	yes	—
452	C₄	C₂×C₂₃×#121	1	4	yes	—
453	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
454	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
455	C₂	—	2	2	no	—
456	C₂	—	2	4	no	—
457	C₂	—	4	8	no	—
458	C₂	—	8	4	no	—
459	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
460	C₂	C₂₃×#3585	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 ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₂₃×#3449	C₂	semidirect
3. C₂×C₂₃×#518	C₂	semidirect
4. C₂₃×#3042	C₂	semidirect
5. C₂₃×#3041	C₂	semidirect
6. C₂₃×#3036	C₂	semidirect
7. C₂₃×#2905	C₂	semidirect
8. C₂₃×#2896	C₂	semidirect
13. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
14. C₂²×SD₁₆×C₂₃	C₂²	semidirect
15. C₂²×D₈×C₂₃	C₂²	semidirect
18. C₂₃×#536	C₂²	semidirect
19. C₂₃×#535	C₂²	semidirect
21. C₂₃×#532	C₂²	semidirect
24. C₂₃×#518	C₂²	semidirect
26. C₂×C₄×D₄×C₂₃	C₂²	semidirect
30. C₂₃×#463	C₂²	semidirect
31. C₂₃×#461	C₂²	semidirect
32. C₂₃×#459	C₂²	semidirect
33. C₂₃×#424	C₂²	semidirect
34. C₂₃×#423	C₂²	semidirect
35. C₂₃×#419	C₂²	semidirect
36. C₂₃×#418	C₂²	semidirect
37. C₂₃×#417	C₂²	semidirect
38. C₂₃×#416	C₂²	semidirect
51. C₂²×D₄×C₂₃	D₄	semidirect
68. C₂₃×C₄⋊Q₈	C₂³	semidirect
69. C₂₃×#127	C₂³	semidirect
70. C₂₃×#124	C₂³	semidirect
92. C₂₃×C₄⋊C₈	C₂³	semidirect
93. C₂₃×#103	C₂³	semidirect
94. C₂₃×#102	C₂³	semidirect
159. #60423	C₂₃	direct (N × H)
205. C₂²×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
206. #3449	C₂×C₂₃	semidirect
207. C₂×#518	C₂×C₂₃	semidirect
208. #3042	C₂×C₂₃	semidirect
209. #3041	C₂×C₂₃	semidirect
210. #3036	C₂×C₂₃	semidirect
211. #2905	C₂×C₂₃	semidirect
212. #2896	C₂×C₂₃	semidirect
237. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
238. C₂²×SD₁₆	C₂²×C₂₃	semidirect
239. C₂²×D₈	C₂²×C₂₃	semidirect
242. #536	C₂²×C₂₃	semidirect
243. #535	C₂²×C₂₃	semidirect
245. #532	C₂²×C₂₃	semidirect
248. #518	C₂²×C₂₃	semidirect
250. C₂×C₄×D₄	C₂²×C₂₃	semidirect
254. #463	C₂²×C₂₃	semidirect
255. #461	C₂²×C₂₃	semidirect
256. #459	C₂²×C₂₃	semidirect
257. #424	C₂²×C₂₃	semidirect
258. #423	C₂²×C₂₃	semidirect
259. #419	C₂²×C₂₃	semidirect
260. #418	C₂²×C₂₃	semidirect
261. #417	C₂²×C₂₃	semidirect
262. #416	C₂²×C₂₃	semidirect
281. C₂²×D₄	D₄×C₂₃	semidirect
298. C₄⋊Q₈	C₂³×C₂₃	semidirect
299. #127	C₂³×C₂₃	semidirect
300. #124	C₂³×C₂₃	semidirect
322. C₄⋊C₈	C₂³×C₂₃	semidirect
323. #103	C₂³×C₂₃	semidirect
324. #102	C₂³×C₂₃	semidirect
325. C₂₃	#60423	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×AGL(1,ℤ/8)	2
C₂₃×#3449	2
C₂×C₂₃×#518	2
C₂₃×#3042	2
C₂₃×#3041	2
C₂₃×#3036	2
C₂₃×#2905	2
C₂₃×#2896	2
#60423	23

Subgroups — C23×#60423 #19208049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60423 #19208049