Subgroups — C₂₃×#20769 #19168395

Order 5888.

818 conjugacy classes of subgroups.

The subgroup lattice has 202 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	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1838	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1837	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1738	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#423	—	2	4	no	—
10	C₂₃×#423	C₂²	1	4	yes	—
11	C₂₃×#423	C₂²	1	4	yes	—
12	C₂×C₂₃×C₄⋊C₈	—	2	4	no	—
13	C₄×M₁₆×C₂₃	C₄	1	4	yes	—
14	C₄×M₁₆×C₂₃	C₄	1	4	yes	—
15	C₂×C₂₃×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#341	C₂²	1	16	yes	—
18	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
19	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	—	2	8	no	—
21	C₂×M₁₆×C₂₃	—	2	4	no	—
22	C₂×M₁₆×C₂₃	—	2	16	no	—
23	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
24	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
25	C₂²×C₈×C₂₃	—	2	4	no	—
26	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	D₄	1	4	yes	—
30	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂₃×C₄⋊C₈	—	2	8	no	—
34	C₂₃×C₄⋊C₈	—	2	8	no	—
35	C₂₃×#97	C₂×C₄	1	8	yes	—
36	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
38	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	—	4	4	no	—
40	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	#41	1	8	yes	—
48	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
49	M₁₆×C₂₃	—	2	16	no	—
50	M₁₆×C₂₃	—	2	16	no	—
51	M₁₆×C₂₃	—	4	8	no	—
52	C₂×C₈×C₂₃	—	2	8	no	—
53	C₂×C₈×C₂₃	—	2	8	no	—
54	C₂×C₈×C₂₃	—	4	8	no	—
55	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
56	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
57	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
58	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₂₃	C₂×D₄	1	2	yes	—
62	C₄²×C₂₃	C₂×D₄	1	2	yes	—
63	#20769	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	—	4	2	no	—
65	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
66	C₂³×C₂₃	#125	1	4	yes	—
67	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	—	2	8	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	4	2	no	—
72	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₂₃	#117	1	8	yes	—
77	C₂×C₄×C₂₃	#124	1	4	yes	—
78	C₂×C₄×C₂₃	#124	1	4	yes	—
79	C₂×C₄×C₂₃	#125	1	4	yes	—
80	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
83	C₈×C₂₃	—	2	8	no	—
84	C₈×C₂₃	—	4	8	no	—
85	C₂×#423	C₂×C₂₃	1	2	yes	—
86	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
87	#1838	C₂×C₂₃	1	4	yes	complemented
88	#1837	C₂×C₂₃	1	4	yes	complemented
89	#1738	C₂×C₂₃	1	4	yes	—
90	C₂²×C₂₃	—	2	4	no	—
91	C₂²×C₂₃	—	2	2	no	—
92	C₂²×C₂₃	—	4	4	no	—
93	C₂²×C₂₃	#382	1	4	yes	—
94	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₂₃	C₂×#125	1	2	yes	—
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	#382	1	4	yes	—
99	C₄×C₂₃	#430	1	4	yes	—
100	C₄×C₂₃	C₂×#117	1	2	yes	—
101	C₄×C₂₃	C₂×#124	1	2	yes	—
102	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
103	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
104	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
105	#423	—	2	4	no	—
106	#423	C₂²×C₂₃	1	4	yes	—
107	#423	C₂²×C₂₃	1	4	yes	—
108	C₂×C₄⋊C₈	—	2	4	no	—
109	C₄×M₁₆	C₄×C₂₃	1	4	yes	—
110	C₄×M₁₆	C₄×C₂₃	1	4	yes	—
111	C₂×#97	C₂²×C₂₃	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
113	#341	C₂²×C₂₃	1	16	yes	—
114	C₂×C₂₃	—	2	2	no	—
115	C₂×C₂₃	—	2	4	no	—
116	C₂×C₂₃	C₂×#382	1	1	yes	characteristic
117	C₂×C₂₃	C₂×#430	1	2	yes	—
118	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
119	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
120	C₂×M₁₆	—	2	8	no	—
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	—	2	16	no	—
123	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
124	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
127	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
128	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
129	C₂×C₄²	D₄×C₂₃	1	4	yes	—
130	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
131	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
132	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
133	C₄⋊C₈	—	2	8	no	—
134	C₄⋊C₈	—	2	8	no	—
135	#97	C₂×C₄×C₂₃	1	8	yes	—
136	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
137	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
138	C₂₃	#20769	1	1	yes	characteristic, direct product factor, Sylow
139	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
142	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
143	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
145	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	#41×C₂₃	1	8	yes	—
149	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
150	M₁₆	—	2	16	no	—
151	M₁₆	—	2	16	no	—
152	M₁₆	—	4	8	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	4	8	no	—
156	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
157	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
158	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
159	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
160	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
161	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
162	C₄²	C₂×D₄×C₂₃	1	2	yes	—
163	C₄²	C₂×D₄×C₂₃	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
166	C₂³	C₂₃×#125	1	4	yes	—
167	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
173	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
176	C₂×C₄	C₂₃×#117	1	8	yes	—
177	C₂×C₄	C₂₃×#124	1	4	yes	—
178	C₂×C₄	C₂₃×#124	1	4	yes	—
179	C₂×C₄	C₂₃×#125	1	4	yes	—
180	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
183	C₈	—	2	8	no	—
184	C₈	—	4	8	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	4	no	—
188	C₂²	C₂₃×#382	1	4	yes	—
189	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
190	C₂²	C₂×C₂₃×#125	1	2	yes	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	C₂₃×#382	1	4	yes	—
194	C₄	C₂₃×#430	1	4	yes	—
195	C₄	C₂×C₂₃×#117	1	2	yes	—
196	C₄	C₂×C₂₃×#124	1	2	yes	—
197	C₂	—	2	4	no	—
198	C₂	—	2	2	no	—
199	C₂	C₂×C₂₃×#382	1	1	yes	characteristic
200	C₂	C₂×C₂₃×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1838	C₂	semidirect
4. C₂₃×#1837	C₂	semidirect
63. #20769	C₂₃	direct (N × H)
87. #1838	C₂×C₂₃	semidirect
88. #1837	C₂×C₂₃	semidirect
138. C₂₃	#20769	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
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#1838	2
C₂₃×#1837	2
C₂₃×#1738	2
#20769	23

Subgroups — C23×#20769 #19168395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20769 #19168395