Subgroups — C₂₃×#13480 #19161106

Order 5888.

378 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×#357	C₂	1	2	yes	maximal
2	C₂₃×#1970	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1380	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂₃×#107	—	2	1	no	—
9	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#107	C₂²	1	2	yes	—
11	C₂×C₂₃×#102	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#390	—	2	1	no	—
14	C₂₃×#384	—	2	1	no	—
15	C₂₃×#357	—	2	4	no	—
16	C₂²×D₄×C₂₃	—	2	1	no	—
17	C₂×D₈×C₂₃	—	2	2	no	—
18	C₂×D₈×C₂₃	—	2	2	no	—
19	C₂×D₈×C₂₃	—	2	1	no	—
20	C₂×D₈×C₂₃	D₄	1	2	yes	—
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
25	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
26	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
27	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
28	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
29	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₂₃	—	2	2	no	—
31	C₂×C₁₆×C₂₃	—	2	2	no	—
32	C₂×C₁₆×C₂₃	—	2	2	no	—
33	C₂₃×#107	—	2	2	no	—
34	C₂₃×#107	—	2	2	no	—
35	C₂₃×#107	—	2	2	no	—
36	C₂₃×#107	—	4	2	no	—
37	C₂₃×#102	—	4	2	no	—
38	C₄×C₈×C₂₃	—	2	2	no	—
39	C₂⁴×C₂₃	—	4	1	no	—
40	C₂×D₄×C₂₃	—	2	2	no	—
41	C₂×D₄×C₂₃	—	4	1	no	—
42	C₂×D₄×C₂₃	—	4	1	no	—
43	C₂×D₄×C₂₃	—	4	1	no	—
44	C₂×D₄×C₂₃	—	4	1	no	—
45	C₂×D₄×C₂₃	—	4	1	no	—
46	C₂²×C₄×C₂₃	—	4	1	no	—
47	C₂²×C₄×C₂₃	—	4	2	no	—
48	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	D₈×C₂₃	—	2	4	no	—
52	D₈×C₂₃	—	4	2	no	—
53	C₂×C₈×C₂₃	—	2	1	no	—
54	C₂×C₈×C₂₃	—	2	1	no	—
55	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	#41	1	2	yes	—
58	C₂×C₈×C₂₃	#41	1	2	yes	—
59	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	—	4	2	no	—
61	C₄⋊C₄×C₂₃	—	4	1	no	—
62	C₄⋊C₄×C₂₃	—	4	1	no	—
63	C₄⋊C₄×C₂₃	—	4	2	no	—
64	C₄⋊C₄×C₂₃	—	8	2	no	—
65	C₄⋊C₄×C₂₃	—	8	2	no	—
66	C₂²⋊C₄×C₂₃	—	8	2	no	—
67	C₂²⋊C₄×C₂₃	—	8	2	no	—
68	C₄²×C₂₃	—	2	1	no	—
69	C₄²×C₂₃	—	2	1	no	—
70	C₁₆×C₂₃	—	4	2	no	—
71	#13480	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	—	4	2	no	—
73	C₂³×C₂₃	—	8	1	no	—
74	C₂³×C₂₃	—	8	1	no	—
75	C₂³×C₂₃	—	8	1	no	—
76	C₂³×C₂₃	—	8	1	no	—
77	C₂³×C₂₃	—	8	1	no	—
78	C₂³×C₂₃	—	8	1	no	—
79	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
80	D₄×C₂₃	—	4	2	no	—
81	D₄×C₂₃	—	4	2	no	—
82	D₄×C₂₃	—	8	1	no	—
83	D₄×C₂₃	—	8	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	8	2	no	—
89	C₂×C₄×C₂₃	—	8	1	no	—
90	C₂×C₄×C₂₃	—	8	1	no	—
91	C₂×C₄×C₂₃	—	8	2	no	—
92	C₂×C₄×C₂₃	—	8	1	no	—
93	C₂×C₄×C₂₃	—	8	2	no	—
94	C₂×C₄×C₂₃	D₁₆	1	2	yes	—
95	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
96	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
102	C₈×C₂₃	—	2	2	no	—
103	C₈×C₂₃	#123	1	2	yes	—
104	C₈×C₂₃	#126	1	2	yes	—
105	C₂×#357	C₂×C₂₃	1	2	yes	—
106	#1970	C₂×C₂₃	1	1	yes	characteristic
107	#1845	C₂×C₂₃	1	1	yes	characteristic, complemented
108	#1380	C₂×C₂₃	1	2	yes	complemented
109	#1374	C₂×C₂₃	1	1	yes	characteristic, complemented
110	C₂²×C₂₃	—	8	2	no	—
111	C₂²×C₂₃	—	8	2	no	—
112	C₂²×C₂₃	—	8	2	no	—
113	C₂²×C₂₃	—	16	1	no	—
114	C₂²×C₂₃	—	16	1	no	—
115	C₂²×C₂₃	—	16	1	no	—
116	C₂²×C₂₃	—	16	1	no	—
117	C₂²×C₂₃	#459	1	1	yes	characteristic
118	C₂²×C₂₃	#480	1	1	yes	characteristic
119	C₂²×C₂₃	#486	1	1	yes	characteristic
120	C₂²×C₂₃	C₂×D₁₆	1	1	yes	characteristic
121	C₂²×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
122	C₂²×C₂₃	#509	1	1	yes	characteristic
123	C₂²×C₂₃	#510	1	1	yes	characteristic
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	16	2	no	—
127	C₄×C₂₃	—	16	1	no	—
128	C₄×C₂₃	#397	1	1	yes	characteristic
129	C₄×C₂₃	#461	1	1	yes	characteristic
130	C₄×C₂₃	#481	1	1	yes	characteristic
131	C₄×C₂₃	#489	1	1	yes	characteristic
132	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
134	C₂×#107	—	2	1	no	—
135	C₂×#107	C₂²×C₂₃	1	2	yes	—
136	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×#102	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
139	#390	—	2	1	no	—
140	#384	—	2	1	no	—
141	#357	—	2	4	no	—
142	C₂×C₂₃	—	16	2	no	—
143	C₂×C₂₃	#2071	1	1	yes	characteristic
144	C₂×C₂₃	#2206	1	1	yes	characteristic
145	C₂×C₂₃	#2208	1	1	yes	characteristic
146	C₂×C₂₃	#2232	1	1	yes	characteristic
147	C₂×C₂₃	#2233	1	1	yes	characteristic
148	C₂×C₂₃	#2240	1	1	yes	characteristic
149	C₂×C₂₃	#2243	1	1	yes	characteristic
150	C₂²×D₄	—	2	1	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	D₄×C₂₃	1	2	yes	—
155	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	4	1	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₂²⋊C₄	—	4	1	no	—
162	C₂×C₂²⋊C₄	—	4	1	no	—
163	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	#107	—	2	2	no	—
168	#107	—	2	2	no	—
169	#107	—	2	2	no	—
170	#107	—	4	2	no	—
171	#102	—	4	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₂₃	#13480	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁴	—	4	1	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	1	no	—
179	C₂×D₄	—	4	1	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
186	D₈	—	2	4	no	—
187	D₈	—	4	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
191	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₈	#41×C₂₃	1	2	yes	—
193	C₂×C₈	#41×C₂₃	1	2	yes	—
194	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	8	2	no	—
200	C₄⋊C₄	—	8	2	no	—
201	C₂²⋊C₄	—	8	2	no	—
202	C₂²⋊C₄	—	8	2	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₁₆	—	4	2	no	—
206	C₂³	—	4	2	no	—
207	C₂³	—	8	1	no	—
208	C₂³	—	8	1	no	—
209	C₂³	—	8	1	no	—
210	C₂³	—	8	1	no	—
211	C₂³	—	8	1	no	—
212	C₂³	—	8	1	no	—
213	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	8	1	no	—
217	D₄	—	8	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	C₂₃×D₁₆	1	2	yes	—
229	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
230	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
231	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
232	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
233	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
234	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	C₂₃×#123	1	2	yes	—
238	C₈	C₂₃×#126	1	2	yes	—
239	C₂²	—	8	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	16	1	no	—
243	C₂²	—	16	1	no	—
244	C₂²	—	16	1	no	—
245	C₂²	—	16	1	no	—
246	C₂²	C₂₃×#459	1	1	yes	characteristic
247	C₂²	C₂₃×#480	1	1	yes	characteristic
248	C₂²	C₂₃×#486	1	1	yes	characteristic
249	C₂²	C₂×C₂₃×D₁₆	1	1	yes	characteristic
250	C₂²	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
251	C₂²	C₂₃×#509	1	1	yes	characteristic
252	C₂²	C₂₃×#510	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	16	2	no	—
256	C₄	—	16	1	no	—
257	C₄	C₂₃×#397	1	1	yes	characteristic
258	C₄	C₂₃×#461	1	1	yes	characteristic
259	C₄	C₂₃×#481	1	1	yes	characteristic
260	C₄	C₂₃×#489	1	1	yes	characteristic
261	C₂	—	16	2	no	—
262	C₂	C₂₃×#2071	1	1	yes	characteristic
263	C₂	C₂₃×#2206	1	1	yes	characteristic
264	C₂	C₂₃×#2208	1	1	yes	characteristic
265	C₂	C₂₃×#2232	1	1	yes	characteristic
266	C₂	C₂₃×#2233	1	1	yes	characteristic
267	C₂	C₂₃×#2240	1	1	yes	characteristic
268	C₂	C₂₃×#2243	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1845	C₂	semidirect
4. C₂₃×#1380	C₂	semidirect
5. C₂₃×#1374	C₂	semidirect
71. #13480	C₂₃	direct (N × H)
107. #1845	C₂×C₂₃	semidirect
108. #1380	C₂×C₂₃	semidirect
109. #1374	C₂×C₂₃	semidirect
173. C₂₃	#13480	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×#357	2
C₂₃×#1970	2
C₂₃×#1845	2
C₂₃×#1380	2
C₂₃×#1374	2
#13480	23

Subgroups — C23×#13480 #19161106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13480 #19161106