Subgroups — C₂₃×#8041 #19155667

Order 5888.

698 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#423	C₂	1	2	yes	maximal
2	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1296	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₂₃	—	2	2	no	—
5	C₂×C₂₃×#117	C₄	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#423	—	2	4	no	—
8	C₂₃×#423	C₄	1	8	yes	complemented
9	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂₃×#384	C₂²	1	2	yes	—
12	C₂₃×#382	C₄	1	2	yes	—
13	C₂₃×#380	—	2	2	no	—
14	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	D₄	1	2	yes	—
17	C₂×M₁₆×C₂₃	—	2	8	no	—
18	C₂×M₁₆×C₂₃	—	4	8	no	—
19	C₂²×C₈×C₂₃	—	2	4	no	—
20	C₂₃×#117	—	4	4	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
24	C₂×C₄²×C₂₃	—	2	2	no	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₂₃	D₄	1	2	yes	—
28	C₂×C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	—	2	8	no	—
32	C₂₃×C₄⋊C₈	—	2	4	no	—
33	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
34	C₂₃×#95	—	2	4	no	—
35	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	2	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	—	2	4	no	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
49	M₁₆×C₂₃	—	4	16	no	—
50	C₂×C₈×C₂₃	—	2	8	no	—
51	C₂×C₈×C₂₃	—	4	8	no	—
52	C₄⋊C₄×C₂₃	—	4	4	no	—
53	C₄⋊C₄×C₂₃	—	4	4	no	—
54	C₄⋊C₄×C₂₃	—	4	4	no	—
55	C₂²⋊C₄×C₂₃	—	4	4	no	—
56	C₄²×C₂₃	—	4	4	no	—
57	C₄²×C₂₃	C₄²	1	2	yes	—
58	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
59	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
60	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
61	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
64	#8041	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	—	2	1	no	—
67	C₂³×C₂₃	—	2	4	no	—
68	C₂³×C₂₃	#95	1	2	yes	—
69	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	8	no	—
78	C₂×C₄×C₂₃	—	4	4	no	—
79	C₂×C₄×C₂₃	—	4	8	no	—
80	C₂×C₄×C₂₃	#95	1	4	yes	—
81	C₂×C₄×C₂₃	#95	1	2	yes	—
82	C₂×C₄×C₂₃	#100	1	2	yes	—
83	C₂×C₄×C₂₃	#101	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
89	C₂×C₄×C₂₃	#137	1	4	yes	—
90	C₈×C₂₃	—	4	8	no	—
91	C₂×#423	C₂×C₂₃	1	2	yes	—
92	#2285	C₂×C₂₃	1	1	yes	characteristic
93	#1296	C₂×C₂₃	1	4	yes	complemented
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
99	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
100	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
101	C₂²×C₂₃	#417	1	4	yes	—
102	C₄×C₂₃	—	2	4	no	—
103	C₄×C₂₃	—	2	1	no	—
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	8	4	no	—
106	C₄×C₂₃	#343	1	4	yes	—
107	C₄×C₂₃	#418	1	2	yes	—
108	C₄×C₂₃	#419	1	2	yes	—
109	C₄×C₂₃	#428	1	4	yes	—
110	C₂²×M₁₆	—	2	2	no	—
111	C₂×#117	C₄×C₂₃	1	2	yes	—
112	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
113	#423	—	2	4	no	—
114	#423	C₄×C₂₃	1	8	yes	complemented
115	C₂×C₄⋊C₈	—	2	2	no	—
116	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
117	#384	C₂²×C₂₃	1	2	yes	—
118	#382	C₄×C₂₃	1	2	yes	—
119	#380	—	2	2	no	—
120	C₄×C₄⋊C₄	—	2	2	no	—
121	C₂×C₂₃	—	4	2	no	—
122	C₂×C₂₃	#1560	1	2	yes	—
123	C₂×C₂₃	#1561	1	2	yes	—
124	C₂×C₂₃	#1736	1	2	yes	—
125	C₂×C₂₃	C₂×#343	1	1	yes	characteristic
126	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
127	C₂³×C₄	D₄×C₂₃	1	2	yes	—
128	C₂×M₁₆	—	2	8	no	—
129	C₂×M₁₆	—	4	8	no	—
130	C₂²×C₈	—	2	4	no	—
131	#117	—	4	4	no	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
134	C₂×C₂²⋊C₄	—	2	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
137	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
138	C₂×C₄²	D₄×C₂₃	1	2	yes	—
139	C₂×C₄²	D₄×C₂₃	1	2	yes	—
140	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
141	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
142	C₄⋊C₈	—	2	8	no	—
143	C₄⋊C₈	—	2	4	no	—
144	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
145	#95	—	2	4	no	—
146	C₂₃	#8041	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
154	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
155	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
156	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
158	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
159	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
161	M₁₆	—	4	16	no	—
162	C₂×C₈	—	2	8	no	—
163	C₂×C₈	—	4	8	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₂²⋊C₄	—	4	4	no	—
168	C₄²	—	4	4	no	—
169	C₄²	C₄²×C₂₃	1	2	yes	—
170	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
171	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
172	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
173	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
174	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
175	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	4	no	—
179	C₂³	C₂₃×#95	1	2	yes	—
180	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	C₂₃×#95	1	4	yes	—
192	C₂×C₄	C₂₃×#95	1	2	yes	—
193	C₂×C₄	C₂₃×#100	1	2	yes	—
194	C₂×C₄	C₂₃×#101	1	2	yes	—
195	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
197	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
199	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
200	C₂×C₄	C₂₃×#137	1	4	yes	—
201	C₈	—	4	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
207	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
208	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
209	C₂²	C₂₃×#417	1	4	yes	—
210	C₄	—	2	4	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	8	4	no	—
214	C₄	C₂₃×#343	1	4	yes	—
215	C₄	C₂₃×#418	1	2	yes	—
216	C₄	C₂₃×#419	1	2	yes	—
217	C₄	C₂₃×#428	1	4	yes	—
218	C₂	—	4	2	no	—
219	C₂	C₂₃×#1560	1	2	yes	—
220	C₂	C₂₃×#1561	1	2	yes	—
221	C₂	C₂₃×#1736	1	2	yes	—
222	C₂	C₂×C₂₃×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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×#1296	C₂	semidirect
8. C₂₃×#423	C₄	semidirect
64. #8041	C₂₃	direct (N × H)
93. #1296	C₂×C₂₃	semidirect
114. #423	C₄×C₂₃	semidirect
146. C₂₃	#8041	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×#423	2
C₂₃×#2285	2
C₂₃×#1296	2
#8041	23

Subgroups — C23×#8041 #19155667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8041 #19155667