Subgroups — C₂₃×#21726 #19169352

Order 5888.

854 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₂₃×#1990	C₂	1	2	yes	maximal
4	C₂₃×#1943	C₂	1	4	yes	maximal
5	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1776	C₂	1	4	yes	maximal
7	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
10	C₂₃×#500	C₄	1	8	yes	complemented
11	C₂×C₂₃×#107	C₂²	1	4	yes	—
12	C₂×C₂₃×#106	C₂²	1	4	yes	—
13	C₂×C₂₃×#97	C₂²	1	2	yes	—
14	C₂×C₂₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#391	—	2	2	no	—
18	C₂₃×#389	—	2	2	no	—
19	C₂₃×#389	C₂²	1	4	yes	—
20	C₂₃×#384	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
23	C₄³×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#340	C₂²	1	8	yes	—
25	C₂²×Q₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
28	C₂₃×C₄⋊Q₈	—	2	4	no	—
29	C₂₃×C₄⋊Q₈	C₂×C₄	1	8	yes	—
30	C₄×Q₈×C₂₃	—	4	8	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₂₃	—	2	4	no	—
38	C₂×C₄²×C₂₃	D₄	1	2	yes	—
39	C₂×C₄²×C₂₃	D₄	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×#107	C₂×C₄	1	16	yes	—
44	C₂₃×#106	—	2	8	no	—
45	C₂₃×#97	—	2	4	no	—
46	C₂₃×#97	Q₈	1	8	yes	complemented
47	C₄×C₈×C₂₃	—	2	2	no	—
48	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
49	C₂₃×#95	—	2	4	no	—
50	C₂₃×#95	—	2	4	no	—
51	C₂×Q₈×C₂₃	—	2	8	no	—
52	C₂×Q₈×C₂₃	—	4	4	no	—
53	C₂×Q₈×C₂₃	—	4	4	no	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₂₃	#41	1	4	yes	—
67	C₄⋊C₄×C₂₃	—	2	8	no	—
68	C₄⋊C₄×C₂₃	—	2	4	no	—
69	C₄⋊C₄×C₂₃	—	2	8	no	—
70	C₄⋊C₄×C₂₃	—	4	8	no	—
71	C₄⋊C₄×C₂₃	—	4	4	no	—
72	C₄⋊C₄×C₂₃	—	4	4	no	—
73	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
74	C₄²×C₂₃	—	2	4	no	—
75	C₄²×C₂₃	—	2	4	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	—	4	8	no	—
78	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
80	C₄²×C₂₃	#41	1	2	yes	—
81	#21726	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₂₃	—	4	16	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	8	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	4	4	no	—
91	C₂×C₄×C₂₃	—	4	4	no	—
92	C₂×C₄×C₂₃	—	4	4	no	—
93	C₂×C₄×C₂₃	—	4	4	no	—
94	C₂×C₄×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
98	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
99	C₂×C₄×C₂₃	#137	1	4	yes	—
100	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
104	C₈×C₂₃	—	2	4	no	—
105	C₈×C₂₃	C₄×Q₈	1	8	yes	complemented
106	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
107	C₄×C₄⋊Q₈	C₂×C₂₃	1	2	yes	—
108	#1990	C₂×C₂₃	1	2	yes	—
109	#1943	C₂×C₂₃	1	4	yes	—
110	#1844	C₂×C₂₃	1	1	yes	characteristic
111	#1776	C₂×C₂₃	1	4	yes	—
112	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
113	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂²×C₂₃	#532	1	1	yes	characteristic
116	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
117	C₂²×C₂₃	C₂×#137	1	2	yes	—
118	C₄×C₂₃	—	2	4	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	—	4	8	no	—
121	C₄×C₂₃	—	8	4	no	—
122	C₄×C₂₃	#441	1	4	yes	—
123	C₄×C₂₃	#442	1	4	yes	—
124	C₄×C₂₃	C₂×C₄×Q₈	1	2	yes	—
125	C₄×C₂₃	#533	1	2	yes	—
126	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
127	C₂×C₄×Q₈	—	2	2	no	—
128	#500	C₄×C₂₃	1	8	yes	complemented
129	C₂×#107	C₂²×C₂₃	1	4	yes	—
130	C₂×#106	C₂²×C₂₃	1	4	yes	—
131	C₂×#97	C₂²×C₂₃	1	2	yes	—
132	C₂×#97	C₂²×C₂₃	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
135	#391	—	2	2	no	—
136	#389	—	2	2	no	—
137	#389	C₂²×C₂₃	1	4	yes	—
138	#384	C₂²×C₂₃	1	2	yes	—
139	C₄×C₄⋊C₄	—	2	2	no	—
140	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
141	C₄³	C₂²×C₂₃	1	1	yes	characteristic
142	#340	C₂²×C₂₃	1	8	yes	—
143	C₂×C₂₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
144	C₂×C₂₃	C₂×#441	1	1	yes	characteristic
145	C₂×C₂₃	C₂×#442	1	1	yes	characteristic
146	C₂×C₂₃	#3165	1	2	yes	—
147	C₂×C₂₃	#3182	1	2	yes	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
150	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
151	C₄⋊Q₈	—	2	4	no	—
152	C₄⋊Q₈	C₂×C₄×C₂₃	1	8	yes	—
153	C₄×Q₈	—	4	8	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
159	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
160	C₂×C₄²	—	2	4	no	—
161	C₂×C₄²	D₄×C₂₃	1	2	yes	—
162	C₂×C₄²	D₄×C₂₃	1	2	yes	—
163	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
166	#107	C₂×C₄×C₂₃	1	16	yes	—
167	#106	—	2	8	no	—
168	#97	—	2	4	no	—
169	#97	Q₈×C₂₃	1	8	yes	complemented
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
172	#95	—	2	4	no	—
173	#95	—	2	4	no	—
174	C₂₃	#21726	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
188	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
189	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
190	C₂×C₈	#41×C₂₃	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	8	no	—
202	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
203	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
204	C₄²	#41×C₂₃	1	2	yes	—
205	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
206	Q₈	—	4	16	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
219	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
220	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
221	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
222	C₂×C₄	C₂₃×#137	1	4	yes	—
223	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
224	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
225	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
226	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
227	C₈	—	2	4	no	—
228	C₈	C₄×Q₈×C₂₃	1	8	yes	complemented
229	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
230	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
231	C₂²	C₂₃×#532	1	1	yes	characteristic
232	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
233	C₂²	C₂×C₂₃×#137	1	2	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	8	no	—
237	C₄	—	8	4	no	—
238	C₄	C₂₃×#441	1	4	yes	—
239	C₄	C₂₃×#442	1	4	yes	—
240	C₄	C₂×C₄×Q₈×C₂₃	1	2	yes	—
241	C₄	C₂₃×#533	1	2	yes	—
242	C₂	C₄×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
243	C₂	C₂×C₂₃×#441	1	1	yes	characteristic
244	C₂	C₂×C₂₃×#442	1	1	yes	characteristic
245	C₂	C₂₃×#3165	1	2	yes	—
246	C₂	C₂₃×#3182	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂₃×#500	C₄	semidirect
46. C₂₃×#97	Q₈	semidirect
81. #21726	C₂₃	direct (N × H)
105. C₈×C₂₃	C₄×Q₈	semidirect
128. #500	C₄×C₂₃	semidirect
169. #97	Q₈×C₂₃	semidirect
174. C₂₃	#21726	direct (N × H)
228. C₈	C₄×Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#500	2
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#1990	2
C₂₃×#1943	2
C₂₃×#1844	2
C₂₃×#1776	2
C₄×C₂₃×#97	2
#21726	23

Subgroups — C23×#21726 #19169352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21726 #19169352