Subgroups — C₂₃×#21728 #19169354

Order 5888.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#125	C₂	1	2	yes	maximal
3	C₂₃×#1991	C₂	1	2	yes	maximal
4	C₂₃×#1944	C₂	1	2	yes	maximal
5	C₂₃×#1943	C₂	1	2	yes	maximal
6	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1776	C₂	1	2	yes	maximal
8	C₂₃×#1776	C₂	1	2	yes	maximal
9	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×#125	C₂²	1	2	yes	—
11	C₂₃×#499	C₄	1	8	yes	complemented
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#107	C₂²	1	2	yes	—
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₂₃×#106	C₂²	1	2	yes	—
16	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#389	C₂²	1	4	yes	—
23	C₂₃×#387	—	2	4	no	—
24	C₂₃×#382	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
26	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
27	C₄³×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂₃×#340	C₂²	1	4	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#125	—	2	4	no	—
35	C₂₃×#125	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
40	C₂×C₄²×C₂₃	—	2	4	no	—
41	C₂×C₄²×C₂₃	D₄	1	4	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×#107	—	2	4	no	—
46	C₂₃×#107	C₂×C₄	1	8	yes	—
47	C₂₃×#106	—	2	4	no	—
48	C₂₃×#106	C₂×C₄	1	8	yes	—
49	C₂₃×#97	—	2	2	no	—
50	C₂₃×#97	—	2	2	no	—
51	C₂₃×#97	Q₈	1	4	yes	—
52	C₂₃×#97	Q₈	1	4	yes	—
53	C₄×C₈×C₂₃	—	2	2	no	—
54	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
55	C₂₃×#95	—	2	4	no	—
56	C₂₃×#95	—	2	4	no	—
57	C₂₃×#95	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	2	4	no	—
59	C₂²×C₄×C₂₃	—	2	4	no	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₂₃	—	2	1	no	—
65	C₂×C₈×C₂₃	—	2	1	no	—
66	C₂×C₈×C₂₃	—	2	1	no	—
67	C₂×C₈×C₂₃	—	2	1	no	—
68	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₂₃	#41	1	2	yes	—
75	C₂×C₈×C₂₃	#41	1	2	yes	—
76	C₄⋊C₄×C₂₃	—	2	16	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	4	8	no	—
81	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
82	C₄²×C₂₃	—	2	1	no	—
83	C₄²×C₂₃	—	2	1	no	—
84	C₄²×C₂₃	—	4	4	no	—
85	C₄²×C₂₃	—	4	8	no	—
86	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
88	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₂₃	#41	1	2	yes	—
90	#21728	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	2	8	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	—	4	4	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
104	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
110	C₈×C₂₃	—	2	2	no	—
111	C₈×C₂₃	—	2	2	no	—
112	C₈×C₂₃	C₄×Q₈	1	4	yes	—
113	C₈×C₂₃	C₄×Q₈	1	4	yes	—
114	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
115	C₄×#125	C₂×C₂₃	1	2	yes	—
116	#1991	C₂×C₂₃	1	2	yes	—
117	#1944	C₂×C₂₃	1	2	yes	—
118	#1943	C₂×C₂₃	1	2	yes	—
119	#1846	C₂×C₂₃	1	1	yes	characteristic
120	#1776	C₂×C₂₃	1	2	yes	—
121	#1776	C₂×C₂₃	1	2	yes	—
122	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
123	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₂₃	#532	1	1	yes	characteristic
126	C₂²×C₂₃	#575	1	4	yes	—
127	C₄×C₂₃	—	2	4	no	—
128	C₄×C₂₃	—	2	4	no	—
129	C₄×C₂₃	—	4	2	no	—
130	C₄×C₂₃	—	4	8	no	—
131	C₄×C₂₃	—	8	4	no	—
132	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₄×C₂₃	#533	1	1	yes	characteristic
135	C₄×C₂₃	#533	1	1	yes	characteristic
136	C₂×#125	C₂²×C₂₃	1	2	yes	—
137	#499	C₄×C₂₃	1	8	yes	complemented
138	C₂×#107	C₂²×C₂₃	1	2	yes	—
139	C₂×#107	C₂²×C₂₃	1	2	yes	—
140	C₂×#106	C₂²×C₂₃	1	2	yes	—
141	C₂×#106	C₂²×C₂₃	1	2	yes	—
142	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	#389	C₂²×C₂₃	1	4	yes	—
149	#387	—	2	4	no	—
150	#382	C₂²×C₂₃	1	2	yes	—
151	C₄×C₄⋊C₄	—	2	4	no	—
152	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
153	C₄³	C₂²×C₂₃	1	1	yes	characteristic
154	#340	C₂²×C₂₃	1	4	yes	—
155	#340	C₂²×C₂₃	1	4	yes	—
156	C₂×C₂₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₂₃	#2943	1	2	yes	—
158	C₂×C₂₃	#3166	1	2	yes	—
159	C₂×C₂₃	#3183	1	2	yes	—
160	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
164	#125	—	2	4	no	—
165	#125	C₂×C₄×C₂₃	1	8	yes	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
169	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
170	C₂×C₄²	—	2	4	no	—
171	C₂×C₄²	D₄×C₂₃	1	4	yes	—
172	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#107	C₂×C₄×C₂₃	1	8	yes	—
177	#106	—	2	4	no	—
178	#106	C₂×C₄×C₂₃	1	8	yes	—
179	#97	—	2	2	no	—
180	#97	—	2	2	no	—
181	#97	Q₈×C₂₃	1	4	yes	—
182	#97	Q₈×C₂₃	1	4	yes	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
185	#95	—	2	4	no	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	C₂₃	#21728	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
194	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
200	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
201	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
202	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
203	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
204	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
205	C₂×C₈	#41×C₂₃	1	2	yes	—
206	C₂×C₈	#41×C₂₃	1	2	yes	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	8	no	—
212	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	4	4	no	—
216	C₄²	—	4	8	no	—
217	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
218	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
219	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
220	C₄²	#41×C₂₃	1	2	yes	—
221	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
233	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
234	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₄×Q₈×C₂₃	1	4	yes	—
243	C₈	C₄×Q₈×C₂₃	1	4	yes	—
244	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
245	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
246	C₂²	C₂₃×#532	1	1	yes	characteristic
247	C₂²	C₂₃×#575	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	8	no	—
252	C₄	—	8	4	no	—
253	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
254	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
255	C₄	C₂₃×#533	1	1	yes	characteristic
256	C₄	C₂₃×#533	1	1	yes	characteristic
257	C₂	C₄×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
258	C₂	C₂₃×#2943	1	2	yes	—
259	C₂	C₂₃×#3166	1	2	yes	—
260	C₂	C₂₃×#3183	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
11. C₂₃×#499	C₄	semidirect
90. #21728	C₂₃	direct (N × H)
137. #499	C₄×C₂₃	semidirect
188. C₂₃	#21728	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×#499	2
C₄×C₂₃×#125	2
C₂₃×#1991	2
C₂₃×#1944	2
C₂₃×#1943	2
C₂₃×#1846	2
C₂₃×#1776	2
C₄×C₂₃×#97	2
#21728	23

Subgroups — C23×#21728 #19169354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21728 #19169354