Subgroups — C₂×C₂₃×#1715 #19167232

Order 5888.

1030 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#492	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#486	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1715	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×SD₁₆×C₂₃	—	2	4	no	—
8	C₂²×D₈×C₂₃	—	2	2	no	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
12	C₂₃×#498	—	2	2	no	—
13	C₂₃×#498	C₂²	1	4	yes	complemented
14	C₂₃×#493	—	2	2	no	—
15	C₂₃×#493	C₂²	1	4	yes	—
16	C₂₃×#492	—	2	4	no	—
17	C₂₃×#492	C₂²	1	8	yes	complemented
18	C₂₃×#486	—	2	4	no	—
19	C₂₃×#486	C₂²	1	8	yes	complemented
20	C₂×C₂₃×#106	—	2	2	no	—
21	C₂×C₂₃×#102	—	2	4	no	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#322	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₂₃	—	2	2	no	—
26	C₂²×D₄×C₂₃	—	2	1	no	—
27	C₂²×D₄×C₂₃	—	2	1	no	—
28	C₂²×D₄×C₂₃	—	2	2	no	—
29	C₂²×D₄×C₂₃	—	2	2	no	—
30	C₂×SD₁₆×C₂₃	—	2	16	no	—
31	C₂×SD₁₆×C₂₃	—	4	16	no	—
32	C₂×D₈×C₂₃	—	2	8	no	—
33	C₂×D₈×C₂₃	—	4	8	no	—
34	C₂²×C₈×C₂₃	D₄	1	4	yes	—
35	C₂²×C₈×C₂₃	D₄	1	2	yes	—
36	C₂₃×C₄⋊Q₈	—	2	2	no	—
37	C₂₃×C₄⋊Q₈	—	2	2	no	—
38	C₂₃×C₄⋊Q₈	—	2	2	no	—
39	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₂₃×#127	—	2	2	no	—
41	C₂₃×#127	—	2	2	no	—
42	C₂₃×#127	—	2	1	no	—
43	C₂₃×#127	—	2	1	no	—
44	C₂₃×#127	C₂³	1	2	yes	—
45	C₂₃×#127	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×#106	—	2	8	no	—
50	C₂₃×#102	—	2	16	no	—
51	C₄×C₈×C₂₃	D₄	1	4	yes	—
52	C₄×C₈×C₂₃	D₄	1	2	yes	—
53	C₄×C₈×C₂₃	C₂³	1	4	yes	—
54	C₄×C₈×C₂₃	C₂³	1	2	yes	—
55	C₂⁴×C₂₃	—	4	1	no	—
56	C₂⁴×C₂₃	—	4	1	no	—
57	C₂×Q₈×C₂₃	—	2	4	no	—
58	C₂×Q₈×C₂₃	—	4	2	no	—
59	C₂×Q₈×C₂₃	—	4	8	no	—
60	C₂×D₄×C₂₃	—	2	4	no	—
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	2	2	no	—
63	C₂×D₄×C₂₃	—	2	4	no	—
64	C₂×D₄×C₂₃	—	4	1	no	—
65	C₂×D₄×C₂₃	—	4	1	no	—
66	C₂×D₄×C₂₃	—	4	2	no	—
67	C₂×D₄×C₂₃	—	4	2	no	—
68	C₂×D₄×C₂₃	—	4	4	no	—
69	C₂×D₄×C₂₃	—	4	4	no	—
70	C₂×D₄×C₂₃	—	4	8	no	—
71	C₂×D₄×C₂₃	—	4	8	no	—
72	C₂²×C₄×C₂₃	—	4	2	no	—
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
75	SD₁₆×C₂₃	—	4	32	no	—
76	D₈×C₂₃	—	4	16	no	—
77	C₂×C₈×C₂₃	—	2	8	no	—
78	C₂×C₈×C₂₃	SD₁₆	1	8	yes	complemented
79	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
80	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	—	4	2	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄²×C₂₃	C₂×D₄	1	2	yes	—
86	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂×#1715	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	—	4	2	no	—
90	C₂³×C₂₃	—	4	2	no	—
91	C₂³×C₂₃	—	8	2	no	—
92	C₂³×C₂₃	—	8	2	no	—
93	C₂³×C₂₃	—	8	4	no	—
94	C₂³×C₂₃	—	8	4	no	—
95	C₂³×C₂₃	#127	1	1	yes	characteristic
96	Q₈×C₂₃	—	4	8	no	—
97	Q₈×C₂₃	—	8	4	no	—
98	D₄×C₂₃	—	4	4	no	—
99	D₄×C₂₃	—	4	4	no	—
100	D₄×C₂₃	—	4	8	no	—
101	D₄×C₂₃	—	4	8	no	—
102	D₄×C₂₃	—	8	2	no	—
103	D₄×C₂₃	—	8	2	no	—
104	D₄×C₂₃	—	8	4	no	—
105	D₄×C₂₃	—	8	4	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	8	4	no	—
108	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#127	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
111	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	8	yes	—
112	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
114	C₈×C₂₃	—	2	8	no	—
115	C₈×C₂₃	C₂×SD₁₆	1	8	yes	complemented
116	C₂×#498	C₂×C₂₃	1	1	yes	characteristic, complemented
117	C₂×#493	C₂×C₂₃	1	1	yes	characteristic
118	C₂×#492	C₂×C₂₃	1	2	yes	complemented
119	C₂×#486	C₂×C₂₃	1	2	yes	complemented
120	#1715	C₂×C₂₃	1	8	yes	direct product factor
121	C₂×#322	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂²×C₂₃	—	8	2	no	—
123	C₂²×C₂₃	—	8	2	no	—
124	C₂²×C₂₃	—	8	4	no	—
125	C₂²×C₂₃	—	8	4	no	—
126	C₂²×C₂₃	—	16	4	no	—
127	C₂²×C₂₃	—	16	4	no	—
128	C₂²×C₂₃	#492	1	2	yes	—
129	C₂²×C₂₃	#496	1	4	yes	—
130	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
131	C₄×C₂₃	—	8	4	no	—
132	C₄×C₂₃	#492	1	2	yes	—
133	C₄×C₂₃	#496	1	4	yes	—
134	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
135	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
136	C₂²×SD₁₆	—	2	4	no	—
137	C₂²×D₈	—	2	2	no	—
138	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
139	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
141	#498	—	2	2	no	—
142	#498	C₂²×C₂₃	1	4	yes	complemented
143	#493	—	2	2	no	—
144	#493	C₂²×C₂₃	1	4	yes	—
145	#492	—	2	4	no	—
146	#492	C₂²×C₂₃	1	8	yes	complemented
147	#486	—	2	4	no	—
148	#486	C₂²×C₂₃	1	8	yes	complemented
149	C₂×#106	—	2	2	no	—
150	C₂×#102	—	2	4	no	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	#322	C₂²×C₂₃	1	4	yes	complemented
154	C₂×C₂₃	—	16	2	no	—
155	C₂×C₂₃	—	16	2	no	—
156	C₂×C₂₃	#1715	1	4	yes	direct product factor
157	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
158	C₂×C₂₃	C₂×#496	1	2	yes	—
159	C₂²×Q₈	—	2	2	no	—
160	C₂²×D₄	—	2	1	no	—
161	C₂²×D₄	—	2	1	no	—
162	C₂²×D₄	—	2	2	no	—
163	C₂²×D₄	—	2	2	no	—
164	C₂×SD₁₆	—	2	16	no	—
165	C₂×SD₁₆	—	4	16	no	—
166	C₂×D₈	—	2	8	no	—
167	C₂×D₈	—	4	8	no	—
168	C₂²×C₈	D₄×C₂₃	1	4	yes	—
169	C₂²×C₈	D₄×C₂₃	1	2	yes	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
174	#127	—	2	2	no	—
175	#127	—	2	2	no	—
176	#127	—	2	1	no	—
177	#127	—	2	1	no	—
178	#127	C₂³×C₂₃	1	2	yes	—
179	#127	C₂³×C₂₃	1	2	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
183	#106	—	2	8	no	—
184	#102	—	2	16	no	—
185	C₄×C₈	D₄×C₂₃	1	4	yes	—
186	C₄×C₈	D₄×C₂₃	1	2	yes	—
187	C₄×C₈	C₂³×C₂₃	1	4	yes	—
188	C₄×C₈	C₂³×C₂₃	1	2	yes	—
189	C₂₃	C₂×#1715	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	—	4	1	no	—
191	C₂⁴	—	4	1	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	8	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	—	4	8	no	—
206	C₂×D₄	—	4	8	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
210	SD₁₆	—	4	32	no	—
211	D₈	—	4	16	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	SD₁₆×C₂₃	1	8	yes	complemented
214	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
215	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄²	C₂×D₄×C₂₃	1	2	yes	—
221	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	8	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	4	no	—
228	C₂³	—	8	4	no	—
229	C₂³	C₂₃×#127	1	1	yes	characteristic
230	Q₈	—	4	8	no	—
231	Q₈	—	8	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	D₄	—	4	8	no	—
235	D₄	—	4	8	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	4	no	—
239	D₄	—	8	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
243	C₂×C₄	C₂₃×#127	1	2	yes	—
244	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
245	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	8	yes	—
246	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
248	C₈	—	2	8	no	—
249	C₈	C₂×SD₁₆×C₂₃	1	8	yes	complemented
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	16	4	no	—
255	C₂²	—	16	4	no	—
256	C₂²	C₂₃×#492	1	2	yes	—
257	C₂²	C₂₃×#496	1	4	yes	—
258	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
259	C₄	—	8	4	no	—
260	C₄	C₂₃×#492	1	2	yes	—
261	C₄	C₂₃×#496	1	4	yes	—
262	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
263	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
264	C₂	—	16	2	no	—
265	C₂	—	16	2	no	—
266	C₂	C₂₃×#1715	1	4	yes	direct product factor
267	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
268	C₂	C₂×C₂₃×#496	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#498	C₂	semidirect
3. C₂×C₂₃×#492	C₂	semidirect
4. C₂×C₂₃×#486	C₂	semidirect
5. C₂₃×#1715	C₂	direct (N × H)
6. C₂×C₂₃×#322	C₂	semidirect
13. C₂₃×#498	C₂²	semidirect
17. C₂₃×#492	C₂²	semidirect
19. C₂₃×#486	C₂²	semidirect
24. C₂₃×#322	C₂²	semidirect
78. C₂×C₈×C₂₃	SD₁₆	semidirect
88. C₂×#1715	C₂₃	direct (N × H)
115. C₈×C₂₃	C₂×SD₁₆	semidirect
116. C₂×#498	C₂×C₂₃	semidirect
118. C₂×#492	C₂×C₂₃	semidirect
119. C₂×#486	C₂×C₂₃	semidirect
120. #1715	C₂×C₂₃	direct (N × H)
121. C₂×#322	C₂×C₂₃	semidirect
142. #498	C₂²×C₂₃	semidirect
146. #492	C₂²×C₂₃	semidirect
148. #486	C₂²×C₂₃	semidirect
153. #322	C₂²×C₂₃	semidirect
156. C₂×C₂₃	#1715	direct (N × H)
189. C₂₃	C₂×#1715	direct (N × H)
213. C₂×C₈	SD₁₆×C₂₃	semidirect
249. C₈	C₂×SD₁₆×C₂₃	semidirect
266. C₂	C₂₃×#1715	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×#498	2
C₂×C₂₃×#493	2
C₂×C₂₃×#492	2
C₂×C₂₃×#486	2
C₂₃×#1715	2
C₂×C₂₃×#322	2
C₂×#1715	23

Subgroups — C2×C23×#1715 #19167232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1715 #19167232