Subgroups — C₂₃×#20825 #19168451

Order 5888.

842 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2283	C₂	1	2	yes	maximal
3	C₂₃×#1846	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1835	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1809	C₂	1	4	yes	maximal
6	C₂₃×#1776	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#107	C₂²	1	4	yes	—
10	C₂×C₂₃×#106	C₂²	1	4	yes	—
11	C₄×M₁₆×C₂₃	C₄	1	8	yes	complemented
12	C₂×C₂₃×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#383	—	2	2	no	—
16	C₂₃×#382	C₂²	1	4	yes	—
17	C₂₃×#381	—	2	2	no	—
18	C₂₃×#380	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
20	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
21	C₂₃×#376	—	2	2	no	—
22	C₂₃×#340	C₂²	1	8	yes	—
23	C₂³×C₄×C₂₃	D₄	1	2	yes	—
24	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₂₃	—	2	4	no	—
26	C₂×M₁₆×C₂₃	—	2	4	no	—
27	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
30	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
31	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
32	C₂×C₄²×C₂₃	—	2	4	no	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
36	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×#107	—	2	8	no	—
39	C₂₃×#106	—	2	8	no	—
40	C₂₃×#97	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
43	C₂₃×#95	—	2	4	no	—
44	C₂₃×#95	—	2	4	no	—
45	C₂₃×#95	—	2	4	no	—
46	C₂₃×#95	—	2	4	no	—
47	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	8	no	—
51	C₂²×C₄×C₂₃	—	4	4	no	—
52	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₂₃	—	2	16	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₂₃	#41	1	4	yes	—
65	C₂×C₈×C₂₃	#41	1	4	yes	—
66	C₄⋊C₄×C₂₃	—	2	8	no	—
67	C₄⋊C₄×C₂₃	—	2	8	no	—
68	C₂²⋊C₄×C₂₃	—	4	8	no	—
69	C₄²×C₂₃	—	4	8	no	—
70	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
71	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
74	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
75	#20825	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	—	2	2	no	—
77	C₂³×C₂₃	—	2	4	no	—
78	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	4	8	no	—
87	C₂×C₄×C₂₃	—	4	8	no	—
88	C₂×C₄×C₂₃	—	4	8	no	—
89	C₂×C₄×C₂₃	—	4	8	no	—
90	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
97	C₈×C₂₃	—	2	4	no	—
98	C₈×C₂₃	#117	1	8	yes	—
99	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
100	#2283	C₂×C₂₃	1	2	yes	—
101	#1846	C₂×C₂₃	1	2	yes	complemented
102	#1835	C₂×C₂₃	1	2	yes	complemented
103	#1809	C₂×C₂₃	1	4	yes	—
104	#1776	C₂×C₂₃	1	4	yes	complemented
105	C₂²×C₂₃	—	2	4	no	—
106	C₂²×C₂₃	—	2	2	no	—
107	C₂²×C₂₃	—	4	4	no	—
108	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₂₃	#532	1	2	yes	—
110	C₂²×C₂₃	#575	1	4	yes	—
111	C₄×C₂₃	—	2	4	no	—
112	C₄×C₂₃	—	2	4	no	—
113	C₄×C₂₃	—	2	2	no	—
114	C₄×C₂₃	—	8	8	no	—
115	C₄×C₂₃	C₂×#117	1	2	yes	—
116	C₄×C₂₃	#533	1	1	yes	characteristic
117	C₄×C₂₃	#533	1	1	yes	characteristic
118	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
119	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×#107	C₂²×C₂₃	1	4	yes	—
121	C₂×#106	C₂²×C₂₃	1	4	yes	—
122	C₄×M₁₆	C₄×C₂₃	1	8	yes	complemented
123	C₂×#97	C₂²×C₂₃	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
126	#383	—	2	2	no	—
127	#382	C₂²×C₂₃	1	4	yes	—
128	#381	—	2	2	no	—
129	#380	C₂²×C₂₃	1	4	yes	—
130	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
131	C₄×C₂²⋊C₄	—	2	2	no	—
132	#376	—	2	2	no	—
133	#340	C₂²×C₂₃	1	8	yes	—
134	C₂×C₂₃	—	4	2	no	—
135	C₂×C₂₃	#2285	1	1	yes	characteristic
136	C₂×C₂₃	#2911	1	2	yes	—
137	C₂×C₂₃	#3183	1	4	yes	—
138	C₂³×C₄	D₄×C₂₃	1	2	yes	—
139	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
143	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
144	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
146	C₂×C₂²⋊C₄	—	2	4	no	—
147	C₂×C₄²	—	2	4	no	—
148	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
149	C₂×C₄²	D₄×C₂₃	1	2	yes	—
150	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
151	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
153	#107	—	2	8	no	—
154	#106	—	2	8	no	—
155	#97	C₂×C₄×C₂₃	1	8	yes	—
156	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
157	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
158	#95	—	2	4	no	—
159	#95	—	2	4	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	C₂₃	#20825	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
169	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
172	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
173	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
175	M₁₆	—	2	16	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
179	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
180	C₂×C₈	#41×C₂₃	1	4	yes	—
181	C₂×C₈	#41×C₂₃	1	4	yes	—
182	C₄⋊C₄	—	2	8	no	—
183	C₄⋊C₄	—	2	8	no	—
184	C₂²⋊C₄	—	4	8	no	—
185	C₄²	—	4	8	no	—
186	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
187	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
188	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
189	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
190	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	2	no	—
193	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
194	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
208	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
211	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
212	C₈	—	2	4	no	—
213	C₈	C₂₃×#117	1	8	yes	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	4	4	no	—
217	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
218	C₂²	C₂₃×#532	1	2	yes	—
219	C₂²	C₂₃×#575	1	4	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	4	no	—
222	C₄	—	2	2	no	—
223	C₄	—	8	8	no	—
224	C₄	C₂×C₂₃×#117	1	2	yes	—
225	C₄	C₂₃×#533	1	1	yes	characteristic
226	C₄	C₂₃×#533	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	C₂₃×#2285	1	1	yes	characteristic
229	C₂	C₂₃×#2911	1	2	yes	—
230	C₂	C₂₃×#3183	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1846	C₂	semidirect
4. C₂₃×#1835	C₂	semidirect
6. C₂₃×#1776	C₂	semidirect
11. C₄×M₁₆×C₂₃	C₄	semidirect
75. #20825	C₂₃	direct (N × H)
101. #1846	C₂×C₂₃	semidirect
102. #1835	C₂×C₂₃	semidirect
104. #1776	C₂×C₂₃	semidirect
122. C₄×M₁₆	C₄×C₂₃	semidirect
162. C₂₃	#20825	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×C₄×M₁₆×C₂₃	2
C₂₃×#2283	2
C₂₃×#1846	2
C₂₃×#1835	2
C₂₃×#1809	2
C₂₃×#1776	2
#20825	23

Subgroups — C23×#20825 #19168451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20825 #19168451