Subgroups — C₂₃×#21956 #19169582

Order 5888.

770 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	2	yes	maximal
2	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1940	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1798	C₂	1	4	yes	maximal
5	C₂×C₂₃×#117	C₂²	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#432	—	2	8	no	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	8	yes	—
10	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
11	C₂×C₂₃×#97	C₂²	1	4	yes	—
12	C₂₃×#384	C₄	1	2	yes	—
13	C₂₃×#382	C₄	1	2	yes	—
14	C₂₃×#380	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
16	C₂₃×#336	C₂²	1	8	yes	—
17	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
20	C₂₃×#117	—	4	4	no	—
21	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
22	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	D₄	1	4	yes	—
27	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
28	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
29	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
30	C₂₃×C₄⋊C₈	—	2	8	no	—
31	C₂₃×C₄⋊C₈	—	4	8	no	—
32	C₂₃×C₂²⋊C₈	—	2	8	no	—
33	C₂₃×#97	—	2	8	no	—
34	C₂₃×#95	C₂×C₄	1	8	yes	—
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	—	4	4	no	—
39	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₂₃	—	2	8	no	—
48	C₂×C₈×C₂₃	—	2	8	no	—
49	C₂×C₈×C₂₃	—	2	8	no	—
50	C₄⋊C₄×C₂₃	—	4	8	no	—
51	C₄⋊C₄×C₂₃	—	4	4	no	—
52	C₂²⋊C₄×C₂₃	—	4	4	no	—
53	C₄²×C₂₃	—	2	8	no	—
54	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
55	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
56	C₄²×C₂₃	C₂×D₄	1	4	yes	—
57	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
58	#21956	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	—	2	1	no	—
60	C₂³×C₂₃	—	2	1	no	—
61	C₂³×C₂₃	—	2	4	no	—
62	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	—	2	1	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	4	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	2	8	no	—
71	C₂×C₄×C₂₃	—	2	16	no	—
72	C₂×C₄×C₂₃	—	4	8	no	—
73	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
79	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
80	C₈×C₂₃	—	4	8	no	—
81	C₂×#432	C₂×C₂₃	1	2	yes	—
82	#2285	C₂×C₂₃	1	1	yes	characteristic
83	#1940	C₂×C₂₃	1	8	yes	complemented
84	#1798	C₂×C₂₃	1	4	yes	—
85	C₂²×C₂₃	—	2	2	no	—
86	C₂²×C₂₃	—	2	4	no	—
87	C₂²×C₂₃	—	4	2	no	—
88	C₂²×C₂₃	—	4	2	no	—
89	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₃	#518	1	1	yes	characteristic
91	C₂²×C₂₃	#519	1	1	yes	characteristic
92	C₂²×C₂₃	#568	1	4	yes	—
93	C₄×C₂₃	—	2	2	no	—
94	C₄×C₂₃	—	2	4	no	—
95	C₄×C₂₃	—	2	4	no	—
96	C₄×C₂₃	—	4	8	no	—
97	C₄×C₂₃	#537	1	2	yes	—
98	C₄×C₂₃	#544	1	2	yes	—
99	C₂×#117	C₂²×C₂₃	1	2	yes	—
100	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
101	#432	—	2	8	no	—
102	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
103	C₂×C₄⋊C₈	C₂²×C₂₃	1	8	yes	—
104	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
105	C₂×#97	C₂²×C₂₃	1	4	yes	—
106	#384	C₄×C₂₃	1	2	yes	—
107	#382	C₄×C₂₃	1	2	yes	—
108	#380	C₄×C₂₃	1	4	yes	—
109	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
110	#336	C₂²×C₂₃	1	8	yes	—
111	C₂×C₂₃	—	4	2	no	—
112	C₂×C₂₃	#2316	1	1	yes	characteristic
113	C₂×C₂₃	#2905	1	2	yes	—
114	C₂×C₂₃	#2980	1	2	yes	—
115	C₂×C₂₃	#2987	1	2	yes	—
116	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
117	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
119	#117	—	4	4	no	—
120	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
121	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
122	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
123	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂×C₄²	D₄×C₂₃	1	4	yes	—
126	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
127	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
128	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
129	C₄⋊C₈	—	2	8	no	—
130	C₄⋊C₈	—	4	8	no	—
131	C₂²⋊C₈	—	2	8	no	—
132	#97	—	2	8	no	—
133	#95	C₂×C₄×C₂₃	1	8	yes	—
134	C₂₃	#21956	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
140	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
141	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
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	C₂⁴×C₂₃	1	1	yes	characteristic
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	8	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	8	no	—
154	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
155	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
156	C₄²	C₂×D₄×C₂₃	1	4	yes	—
157	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
158	C₂³	—	2	4	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	1	no	—
161	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
162	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	4	8	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	4	yes	—
175	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
176	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
177	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
178	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
179	C₈	—	4	8	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
185	C₂²	C₂₃×#518	1	1	yes	characteristic
186	C₂²	C₂₃×#519	1	1	yes	characteristic
187	C₂²	C₂₃×#568	1	4	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	4	8	no	—
192	C₄	C₂₃×#537	1	2	yes	—
193	C₄	C₂₃×#544	1	2	yes	—
194	C₂	—	4	2	no	—
195	C₂	C₂₃×#2316	1	1	yes	characteristic
196	C₂	C₂₃×#2905	1	2	yes	—
197	C₂	C₂₃×#2980	1	2	yes	—
198	C₂	C₂₃×#2987	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1940	C₂	semidirect
58. #21956	C₂₃	direct (N × H)
83. #1940	C₂×C₂₃	semidirect
134. C₂₃	#21956	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×#432	2
C₂₃×#2285	2
C₂₃×#1940	2
C₂₃×#1798	2
#21956	23

Subgroups — C23×#21956 #19169582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21956 #19169582