Subgroups — C₂₃×#12869 #19160495

Order 5888.

404 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1810	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1382	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃×#117	—	2	1	no	—
7	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂₃×#108	C₂²	1	2	yes	—
9	C₂₃×#427	—	2	1	no	—
10	C₂₃×#427	—	2	1	no	—
11	C₂₃×#427	C₄	1	2	yes	—
12	C₂₃×#427	C₂²	1	2	yes	—
13	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
14	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
15	C₂×C₂₃×#106	—	2	1	no	—
16	C₂₃×#407	—	2	1	no	—
17	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
19	C₂₃×#341	—	2	1	no	—
20	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	—	2	4	no	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×#117	—	2	2	no	—
30	C₂₃×#117	—	2	2	no	—
31	C₂₃×#117	—	4	1	no	—
32	C₂₃×#117	—	4	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
36	C₂×C₄²×C₂₃	—	4	1	no	—
37	C₂×C₁₆×C₂₃	—	2	2	no	—
38	C₂×C₁₆×C₂₃	—	2	4	no	—
39	C₂₃×#108	—	2	4	no	—
40	C₂₃×#107	—	2	1	no	—
41	C₂₃×#107	—	2	1	no	—
42	C₂₃×#107	C₂×C₄	1	2	yes	—
43	C₂₃×#107	C₂×C₄	1	2	yes	—
44	C₂₃×#106	—	2	2	no	—
45	C₂₃×#106	—	2	2	no	—
46	C₂₃×C₂²⋊C₈	—	4	1	no	—
47	C₂₃×C₂²⋊C₈	—	4	1	no	—
48	C₂⁴×C₂₃	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	4	1	no	—
52	C₂²×C₄×C₂₃	—	4	1	no	—
53	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	M₁₆×C₂₃	—	4	2	no	—
60	M₁₆×C₂₃	—	4	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₂₃	#41	1	4	yes	—
72	C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₄⋊C₄×C₂₃	—	4	1	no	—
75	C₄⋊C₄×C₂₃	—	4	1	no	—
76	C₂²⋊C₄×C₂₃	—	4	2	no	—
77	C₂²⋊C₄×C₂₃	—	4	2	no	—
78	C₄²×C₂₃	—	4	2	no	—
79	C₄²×C₂₃	—	4	2	no	—
80	C₁₆×C₂₃	—	4	2	no	—
81	#12869	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	2	2	no	—
84	C₂³×C₂₃	—	4	1	no	—
85	C₂³×C₂₃	#102	1	1	yes	characteristic
86	C₂³×C₂₃	#102	1	1	yes	characteristic
87	C₂³×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	8	2	no	—
98	C₂×C₄×C₂₃	—	8	2	no	—
99	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	#123	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#137	1	2	yes	—
105	C₈×C₂₃	—	2	1	no	—
106	C₈×C₂₃	—	2	1	no	—
107	C₈×C₂₃	—	8	2	no	—
108	C₈×C₂₃	#117	1	2	yes	—
109	C₈×C₂₃	#123	1	2	yes	—
110	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
111	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
112	#1810	C₂×C₂₃	1	1	yes	characteristic
113	#1382	C₂×C₂₃	1	4	yes	complemented
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	4	4	no	—
122	C₂²×C₂₃	#359	1	2	yes	—
123	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
124	C₂²×C₂₃	#483	1	2	yes	—
125	C₄×C₂₃	—	2	1	no	—
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	8	2	no	—
128	C₄×C₂₃	—	8	2	no	—
129	C₄×C₂₃	#381	1	1	yes	characteristic
130	C₄×C₂₃	#419	1	1	yes	characteristic
131	C₄×C₂₃	#480	1	2	yes	—
132	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×#117	—	2	1	no	—
134	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
135	C₂×#108	C₂²×C₂₃	1	2	yes	—
136	#427	—	2	1	no	—
137	#427	—	2	1	no	—
138	#427	C₄×C₂₃	1	2	yes	—
139	#427	C₂²×C₂₃	1	2	yes	—
140	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
141	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
142	C₂×#106	—	2	1	no	—
143	#407	—	2	1	no	—
144	C₂²⋊C₁₆	—	2	1	no	—
145	C₂²⋊C₁₆	—	2	1	no	—
146	#341	—	2	1	no	—
147	C₂×C₂₃	—	2	2	no	—
148	C₂×C₂₃	—	2	2	no	—
149	C₂×C₂₃	—	4	1	no	—
150	C₂×C₂₃	#1823	1	1	yes	characteristic
151	C₂×C₂₃	C₂×#359	1	1	yes	characteristic
152	C₂×C₂₃	#2143	1	1	yes	characteristic
153	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
154	C₂×M₁₆	—	2	2	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
162	#117	—	2	2	no	—
163	#117	—	2	2	no	—
164	#117	—	4	1	no	—
165	#117	—	4	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₂²⋊C₄	—	4	1	no	—
169	C₂×C₄²	—	4	1	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	4	no	—
172	#108	—	2	4	no	—
173	#107	—	2	1	no	—
174	#107	—	2	1	no	—
175	#107	C₂×C₄×C₂₃	1	2	yes	—
176	#107	C₂×C₄×C₂₃	1	2	yes	—
177	#106	—	2	2	no	—
178	#106	—	2	2	no	—
179	C₂²⋊C₈	—	4	1	no	—
180	C₂²⋊C₈	—	4	1	no	—
181	C₂₃	#12869	1	1	yes	characteristic, direct product factor, Sylow
182	C₂⁴	SD₁₆×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
190	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
203	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
205	C₂×C₈	#41×C₂₃	1	4	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₁₆	—	4	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	1	no	—
218	C₂³	C₂₃×#102	1	1	yes	characteristic
219	C₂³	C₂₃×#102	1	1	yes	characteristic
220	C₂³	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
233	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	C₂₃×#123	1	2	yes	—
236	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂₃×#137	1	2	yes	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	8	2	no	—
241	C₈	C₂₃×#117	1	2	yes	—
242	C₈	C₂₃×#123	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	C₂₃×#359	1	2	yes	—
252	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
253	C₂²	C₂₃×#483	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	C₂₃×#381	1	1	yes	characteristic
259	C₄	C₂₃×#419	1	1	yes	characteristic
260	C₄	C₂₃×#480	1	2	yes	—
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	4	1	no	—
264	C₂	C₂₃×#1823	1	1	yes	characteristic
265	C₂	C₂×C₂₃×#359	1	1	yes	characteristic
266	C₂	C₂₃×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#102
prime radical	C₂⁴×C₂₃	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1382	C₂	semidirect
81. #12869	C₂₃	direct (N × H)
113. #1382	C₂×C₂₃	semidirect
181. C₂₃	#12869	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×#427	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂₃×#1810	2
C₂₃×#1382	2
#12869	23

Subgroups — C23×#12869 #19160495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12869 #19160495