Subgroups — C₂₃×#29208 #19176834

Order 5888.

810 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2057	C₂	1	4	yes	maximal
4	C₂₃×#1990	C₂	1	2	yes	maximal
5	C₂₃×#1947	C₂	1	4	yes	maximal
6	C₂₃×#1845	C₂	1	2	yes	maximal
7	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
13	C₂×C₂₃×#107	C₂²	1	8	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#391	—	2	1	no	—
17	C₂₃×#389	—	2	1	no	—
18	C₂₃×#389	C₂²	1	2	yes	—
19	C₂₃×#384	—	2	1	no	—
20	C₂₃×#384	—	2	1	no	—
21	C₂₃×#384	C₂²	1	2	yes	—
22	C₂₃×#384	C₂²	1	2	yes	—
23	C₂₃×#384	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₄³×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#340	C₂²	1	8	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₂₃	—	2	1	no	—
30	C₂²×Q₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
32	C₂₃×C₄⋊Q₈	—	2	2	no	—
33	C₂₃×C₄⋊Q₈	—	4	4	no	—
34	C₂₃×C₄⋊Q₈	—	4	4	no	—
35	C₂₃×C₄⋊Q₈	—	4	4	no	—
36	C₂₃×C₄⋊Q₈	Q₈	1	4	yes	complemented
37	C₄×Q₈×C₂₃	—	4	4	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₂₃	—	2	2	no	—
51	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
58	C₂₃×#107	—	4	8	no	—
59	C₂₃×C₄⋊C₈	Q₈	1	8	yes	complemented
60	C₄×C₈×C₂₃	—	2	4	no	—
61	C₂₃×#95	—	2	2	no	—
62	C₂₃×#95	—	2	2	no	—
63	C₂×Q₈×C₂₃	—	2	4	no	—
64	C₂×Q₈×C₂₃	—	4	4	no	—
65	C₂×Q₈×C₂₃	—	4	4	no	—
66	C₂×Q₈×C₂₃	—	4	2	no	—
67	C₂×Q₈×C₂₃	—	4	4	no	—
68	C₂×Q₈×C₂₃	—	4	2	no	—
69	C₂²×C₄×C₂₃	—	2	2	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	4	no	—
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	C₂×Q₈	1	8	yes	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₄⋊C₄×C₂₃	—	2	4	no	—
85	C₄⋊C₄×C₂₃	—	2	8	no	—
86	C₄⋊C₄×C₂₃	—	4	4	no	—
87	C₄⋊C₄×C₂₃	—	4	4	no	—
88	C₄⋊C₄×C₂₃	—	4	4	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄⋊C₄×C₂₃	—	4	4	no	—
91	C₄⋊C₄×C₂₃	—	4	4	no	—
92	C₄⋊C₄×C₂₃	—	4	2	no	—
93	C₄⋊C₄×C₂₃	—	4	4	no	—
94	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	4	4	no	—
99	C₄²×C₂₃	D₈	1	4	yes	—
100	C₄²×C₂₃	Q₁₆	1	4	yes	—
101	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
102	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
103	C₄²×C₂₃	#41	1	2	yes	—
104	#29208	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₂₃	—	4	8	no	—
107	Q₈×C₂₃	—	8	8	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
131	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
132	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
133	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₂₃	#137	1	2	yes	—
135	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
136	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
139	C₈×C₂₃	—	2	8	no	—
140	#2868	C₂×C₂₃	1	1	yes	characteristic
141	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
142	#2057	C₂×C₂₃	1	4	yes	—
143	#1990	C₂×C₂₃	1	2	yes	—
144	#1947	C₂×C₂₃	1	4	yes	—
145	#1845	C₂×C₂₃	1	2	yes	—
146	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
147	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
148	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₂₃	#535	1	1	yes	characteristic
150	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
151	C₂²×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
152	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
154	C₄×C₂₃	—	2	2	no	—
155	C₄×C₂₃	—	2	2	no	—
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	4	4	no	—
158	C₄×C₂₃	—	8	2	no	—
159	C₄×C₂₃	—	8	4	no	—
160	C₄×C₂₃	#474	1	4	yes	—
161	C₄×C₂₃	#477	1	4	yes	—
162	C₄×C₂₃	#500	1	4	yes	—
163	C₄×C₂₃	#554	1	2	yes	—
164	C₄×C₂₃	Q₈²	1	2	yes	—
165	C₂×C₄⋊Q₈	—	2	1	no	—
166	C₂×C₄⋊Q₈	—	2	1	no	—
167	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
169	C₂×C₄×Q₈	—	2	1	no	—
170	C₂×#107	C₂²×C₂₃	1	8	yes	—
171	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
173	#391	—	2	1	no	—
174	#389	—	2	1	no	—
175	#389	C₂²×C₂₃	1	2	yes	—
176	#384	—	2	1	no	—
177	#384	—	2	1	no	—
178	#384	C₂²×C₂₃	1	1	yes	characteristic
179	#384	C₂²×C₂₃	1	2	yes	—
180	#384	C₂²×C₂₃	1	2	yes	—
181	C₄×C₄⋊C₄	—	2	1	no	—
182	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
183	C₄³	C₂²×C₂₃	1	1	yes	characteristic
184	#340	C₂²×C₂₃	1	8	yes	—
185	#336	C₂²×C₂₃	1	2	yes	—
186	C₂×C₂₃	#2567	1	1	yes	characteristic
187	C₂×C₂₃	C₂×#474	1	1	yes	characteristic
188	C₂×C₂₃	C₂×#477	1	1	yes	characteristic
189	C₂×C₂₃	C₂×#500	1	1	yes	characteristic
190	C₂×C₂₃	#3165	1	1	yes	characteristic
191	C₂×C₂₃	#3205	1	1	yes	characteristic
192	C₂×C₂₃	#3209	1	1	yes	characteristic
193	C₂²×Q₈	—	2	1	no	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	4	4	no	—
198	C₄⋊Q₈	—	4	4	no	—
199	C₄⋊Q₈	—	4	4	no	—
200	C₄⋊Q₈	Q₈×C₂₃	1	4	yes	complemented
201	C₄×Q₈	—	4	4	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
212	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
222	#107	—	4	8	no	—
223	C₄⋊C₈	Q₈×C₂₃	1	8	yes	complemented
224	C₄×C₈	—	2	4	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	C₂₃	#29208	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	C₂×Q₈×C₂₃	1	8	yes	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	4	4	no	—
264	C₄²	D₈×C₂₃	1	4	yes	—
265	C₄²	Q₁₆×C₂₃	1	4	yes	—
266	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
267	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
268	C₄²	#41×C₂₃	1	2	yes	—
269	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
270	Q₈	—	4	8	no	—
271	Q₈	—	8	8	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
288	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
289	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
290	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
291	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
292	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
293	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
294	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
295	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
296	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
297	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
298	C₂×C₄	C₂₃×#137	1	2	yes	—
299	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
300	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
302	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
303	C₈	—	2	8	no	—
304	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
305	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
306	C₂²	C₂₃×#535	1	1	yes	characteristic
307	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
308	C₂²	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
309	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	8	4	no	—
316	C₄	—	8	2	no	—
317	C₄	C₂₃×#474	1	4	yes	—
318	C₄	C₂₃×#477	1	4	yes	—
319	C₄	C₂₃×#500	1	4	yes	—
320	C₄	C₂₃×#554	1	2	yes	—
321	C₄	Q₈²×C₂₃	1	2	yes	—
322	C₂	C₂₃×#2567	1	1	yes	characteristic
323	C₂	C₂×C₂₃×#474	1	1	yes	characteristic
324	C₂	C₂×C₂₃×#477	1	1	yes	characteristic
325	C₂	C₂×C₂₃×#500	1	1	yes	characteristic
326	C₂	C₂₃×#3165	1	1	yes	characteristic
327	C₂	C₂₃×#3205	1	1	yes	characteristic
328	C₂	C₂₃×#3209	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
36. C₂₃×C₄⋊Q₈	Q₈	semidirect
59. C₂₃×C₄⋊C₈	Q₈	semidirect
104. #29208	C₂₃	direct (N × H)
200. C₄⋊Q₈	Q₈×C₂₃	semidirect
223. C₄⋊C₈	Q₈×C₂₃	semidirect
227. C₂₃	#29208	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×#2868	2
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#2057	2
C₂₃×#1990	2
C₂₃×#1947	2
C₂₃×#1845	2
C₄×C₂₃×C₄⋊C₈	2
#29208	23

Subgroups — C23×#29208 #19176834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29208 #19176834