Subgroups — C₂₃×#8660 #19156286

Order 5888.

466 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1486	C₂	1	2	yes	maximal
5	C₂₃×#1301	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₂₃	—	2	1	no	—
7	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₂₃	—	2	1	no	—
9	C₈×D₄×C₂₃	—	2	1	no	—
10	C₈×D₄×C₂₃	C₄	1	2	yes	—
11	C₈×D₄×C₂₃	C₄	1	2	yes	—
12	C₂₃×#431	C₂²	1	2	yes	—
13	C₂₃×#423	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#341	—	2	1	no	—
18	C₂₃×#341	—	2	1	no	—
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#334	—	2	1	no	—
21	C₂₃×D₄⋊C₈	—	2	4	no	—
22	C₂²×D₄×C₂₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	—	2	1	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂²×C₈×C₂₃	—	4	4	no	—
29	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₄×D₄×C₂₃	—	2	1	no	—
32	C₄×D₄×C₂₃	—	2	1	no	—
33	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₄×D₄×C₂₃	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₂₃	Q₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
37	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×C₄⋊C₈	—	2	1	no	—
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₂₃×C₂²⋊C₈	—	2	2	no	—
44	C₂₃×C₂²⋊C₈	—	2	2	no	—
45	C₂₃×C₂²⋊C₈	—	2	2	no	—
46	C₄×C₈×C₂₃	—	2	2	no	—
47	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₂⁴×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	—	2	4	no	—
53	C₂×D₄×C₂₃	—	2	1	no	—
54	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	—	4	2	no	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
63	M₁₆×C₂₃	—	4	2	no	—
64	M₁₆×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	4	4	no	—
69	C₂×C₈×C₂₃	—	4	4	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
79	C₂²⋊C₄×C₂₃	—	2	2	no	—
80	C₂²⋊C₄×C₂₃	—	2	2	no	—
81	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₂₃	#41	1	1	yes	characteristic
85	#8660	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	2	4	no	—
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	—	4	2	no	—
90	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
91	D₄×C₂₃	—	2	4	no	—
92	D₄×C₂₃	—	2	2	no	—
93	D₄×C₂₃	#108	1	4	yes	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	4	1	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#135	1	2	yes	—
108	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
109	C₈×C₂₃	—	2	1	no	—
110	C₈×C₂₃	—	2	1	no	—
111	C₈×C₂₃	—	2	1	no	—
112	C₈×C₂₃	—	2	1	no	—
113	C₈×C₂₃	—	4	2	no	—
114	C₈×C₂₃	—	8	2	no	—
115	C₈×C₂₃	—	8	2	no	—
116	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
117	#1837	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#1486	C₂×C₂₃	1	2	yes	—
120	#1301	C₂×C₂₃	1	2	yes	complemented
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	2	4	no	—
124	C₂²×C₂₃	—	4	8	no	—
125	C₂²×C₂₃	—	4	2	no	—
126	C₂²×C₂₃	—	4	2	no	—
127	C₂²×C₂₃	#380	1	1	yes	characteristic
128	C₂²×C₂₃	#478	1	2	yes	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	4	2	no	—
133	C₄×C₂₃	#416	1	1	yes	characteristic
134	C₄×C₂₃	#418	1	1	yes	characteristic
135	C₄×C₂₃	C₂×#108	1	2	yes	—
136	C₄×C₂₃	#463	1	2	yes	—
137	C₂³×C₈	—	2	1	no	—
138	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
139	C₈×D₄	—	2	1	no	—
140	C₈×D₄	—	2	1	no	—
141	C₈×D₄	C₄×C₂₃	1	2	yes	—
142	C₈×D₄	C₄×C₂₃	1	2	yes	—
143	#431	C₂²×C₂₃	1	2	yes	—
144	#423	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	#341	—	2	1	no	—
149	#341	—	2	1	no	—
150	#335	—	2	1	no	—
151	#334	—	2	1	no	—
152	D₄⋊C₈	—	2	4	no	—
153	C₂×C₂₃	—	2	2	no	—
154	C₂×C₂₃	—	4	4	no	—
155	C₂×C₂₃	#1804	1	1	yes	characteristic
156	C₂×C₂₃	#1810	1	1	yes	characteristic
157	C₂×C₂₃	#1814	1	1	yes	characteristic
158	C₂²×D₄	Q₈×C₂₃	1	1	yes	characteristic
159	C₂³×C₄	—	2	1	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	4	4	no	—
165	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
170	C₄×D₄	D₄×C₂₃	1	2	yes	—
171	C₂×C₄⋊C₄	Q₈×C₂₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
185	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
186	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
187	C₂₃	#8660	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	—	2	1	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
200	M₁₆	—	4	2	no	—
201	M₁₆	—	4	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₄²	#41×C₂₃	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	C₂₃×#108	1	4	yes	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
240	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
241	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
242	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₄	C₂₃×#135	1	2	yes	—
244	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	8	2	no	—
251	C₈	—	8	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	C₂₃×#380	1	1	yes	characteristic
259	C₂²	C₂₃×#478	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	C₂₃×#416	1	1	yes	characteristic
265	C₄	C₂₃×#418	1	1	yes	characteristic
266	C₄	C₂×C₂₃×#108	1	2	yes	—
267	C₄	C₂₃×#463	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	C₂₃×#1804	1	1	yes	characteristic
271	C₂	C₂₃×#1810	1	1	yes	characteristic
272	C₂	C₂₃×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#380
prime radical	C₂²×D₄×C₂₃	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1837	C₂	semidirect
3. C₂₃×#1564	C₂	semidirect
5. C₂₃×#1301	C₂	semidirect
85. #8660	C₂₃	direct (N × H)
117. #1837	C₂×C₂₃	semidirect
118. #1564	C₂×C₂₃	semidirect
120. #1301	C₂×C₂₃	semidirect
187. C₂₃	#8660	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×C₈×D₄×C₂₃	2
C₂₃×#1837	2
C₂₃×#1564	2
C₂₃×#1486	2
C₂₃×#1301	2
#8660	23

Subgroups — C23×#8660 #19156286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8660 #19156286