Subgroups — C₂₃×#8593 #19156219

Order 5888.

476 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#332	C₂	1	2	yes	maximal
5	C₂₃×#1296	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂₃×#391	—	2	1	no	—
11	C₂₃×#389	—	2	1	no	—
12	C₂₃×#389	C₄	1	2	yes	—
13	C₂₃×#384	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
15	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#336	—	2	1	no	—
19	C₂₃×#336	—	2	1	no	—
20	C₂₃×#336	—	2	1	no	—
21	C₂₃×#332	—	2	4	no	—
22	C₂²×Q₈×C₂₃	—	2	1	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	—	2	1	no	—
26	C₂₃×C₄⋊Q₈	—	2	1	no	—
27	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
28	C₂₃×C₄⋊Q₈	Q₈	1	2	yes	—
29	C₄×Q₈×C₂₃	—	4	4	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	—	2	2	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	—	2	2	no	—
47	C₂₃×C₄⋊C₈	—	2	2	no	—
48	C₂₃×#95	—	2	2	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂×Q₈×C₂₃	—	2	2	no	—
51	C₂×Q₈×C₂₃	—	2	2	no	—
52	C₂×Q₈×C₂₃	—	4	2	no	—
53	C₂×Q₈×C₂₃	—	4	2	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	—	4	2	no	—
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₄⋊C₄×C₂₃	—	2	2	no	—
71	C₄⋊C₄×C₂₃	—	2	2	no	—
72	C₄⋊C₄×C₂₃	—	4	2	no	—
73	C₄⋊C₄×C₂₃	—	4	4	no	—
74	C₄⋊C₄×C₂₃	—	4	2	no	—
75	C₄⋊C₄×C₂₃	—	4	2	no	—
76	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	—	2	2	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	2	2	no	—
81	C₄²×C₂₃	—	2	2	no	—
82	C₄²×C₂₃	—	4	1	no	—
83	C₄²×C₂₃	—	4	4	no	—
84	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₂₃	#41	1	1	yes	characteristic
87	C₄²×C₂₃	#41	1	1	yes	characteristic
88	#8593	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₂₃	—	4	4	no	—
91	Q₈×C₂₃	—	4	4	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	#101	1	2	yes	—
112	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#135	1	2	yes	—
119	C₂×C₄×C₂₃	#135	1	2	yes	—
120	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₂₃	#137	1	2	yes	—
122	C₈×C₂₃	—	8	2	no	—
123	C₈×C₂₃	—	8	2	no	—
124	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
125	#1831	C₂×C₂₃	1	1	yes	characteristic
126	#1831	C₂×C₂₃	1	1	yes	characteristic
127	C₂×#332	C₂×C₂₃	1	2	yes	—
128	#1296	C₂×C₂₃	1	2	yes	—
129	C₂²×C₂₃	#380	1	1	yes	characteristic
130	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
131	C₂²×C₂₃	#413	1	1	yes	characteristic
132	C₂²×C₂₃	#416	1	1	yes	characteristic
133	C₂²×C₂₃	#416	1	1	yes	characteristic
134	C₂²×C₂₃	#418	1	1	yes	characteristic
135	C₂²×C₂₃	#419	1	1	yes	characteristic
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	—	4	4	no	—
141	C₄×C₂₃	—	4	1	no	—
142	C₄×C₂₃	—	4	1	no	—
143	C₄×C₂₃	—	8	2	no	—
144	C₄×C₂₃	#463	1	2	yes	—
145	C₄×C₂₃	#464	1	2	yes	—
146	C₄×C₂₃	#478	1	2	yes	—
147	C₄×C₂₃	#479	1	2	yes	—
148	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄×Q₈	—	2	1	no	—
150	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	#391	—	2	1	no	—
153	#389	—	2	1	no	—
154	#389	C₄×C₂₃	1	2	yes	—
155	#384	C₂²×C₂₃	1	1	yes	characteristic
156	C₄×C₄⋊C₄	—	2	1	no	—
157	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
158	C₄³	C₂²×C₂₃	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	#336	—	2	1	no	—
161	#336	—	2	1	no	—
162	#336	—	2	1	no	—
163	#332	—	2	4	no	—
164	C₂×C₂₃	#1544	1	1	yes	characteristic
165	C₂×C₂₃	#1553	1	1	yes	characteristic
166	C₂×C₂₃	#1554	1	1	yes	characteristic
167	C₂×C₂₃	#1556	1	1	yes	characteristic
168	C₂×C₂₃	#1801	1	1	yes	characteristic
169	C₂×C₂₃	#1804	1	1	yes	characteristic
170	C₂×C₂₃	#1805	1	1	yes	characteristic
171	C₂²×Q₈	—	2	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	1	no	—
175	C₄⋊Q₈	—	2	1	no	—
176	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
177	C₄⋊Q₈	Q₈×C₂₃	1	2	yes	—
178	C₄×Q₈	—	4	4	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
191	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	C₂₃	#8593	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
236	C₄²	#41×C₂₃	1	1	yes	characteristic
237	C₄²	#41×C₂₃	1	1	yes	characteristic
238	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	Q₈	—	4	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₂₃×#101	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
264	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
265	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
266	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂×C₄	C₂₃×#135	1	2	yes	—
268	C₂×C₄	C₂₃×#135	1	2	yes	—
269	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
270	C₂×C₄	C₂₃×#137	1	2	yes	—
271	C₈	—	8	2	no	—
272	C₈	—	8	2	no	—
273	C₂²	C₂₃×#380	1	1	yes	characteristic
274	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
275	C₂²	C₂₃×#413	1	1	yes	characteristic
276	C₂²	C₂₃×#416	1	1	yes	characteristic
277	C₂²	C₂₃×#416	1	1	yes	characteristic
278	C₂²	C₂₃×#418	1	1	yes	characteristic
279	C₂²	C₂₃×#419	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	4	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	8	2	no	—
288	C₄	C₂₃×#463	1	2	yes	—
289	C₄	C₂₃×#464	1	2	yes	—
290	C₄	C₂₃×#478	1	2	yes	—
291	C₄	C₂₃×#479	1	2	yes	—
292	C₂	C₂₃×#1544	1	1	yes	characteristic
293	C₂	C₂₃×#1553	1	1	yes	characteristic
294	C₂	C₂₃×#1554	1	1	yes	characteristic
295	C₂	C₂₃×#1556	1	1	yes	characteristic
296	C₂	C₂₃×#1801	1	1	yes	characteristic
297	C₂	C₂₃×#1804	1	1	yes	characteristic
298	C₂	C₂₃×#1805	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
88. #8593	C₂₃	direct (N × H)
199. C₂₃	#8593	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#1831	2
C₂×C₂₃×#332	2
C₂₃×#1296	2
#8593	23

Subgroups — C23×#8593 #19156219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8593 #19156219