Subgroups — C₂₃×#17828 #19165454

Order 5888.

702 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2906	C₂	1	2	yes	maximal
3	C₂₃×#1571	C₂	1	6	yes	maximal, complemented
4	C₂₃×#1567	C₂	1	3	yes	maximal, complemented
5	C₂×C₂₃×#335	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₃×#131	—	2	2	no	—
7	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₂₃	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₂₃	C₄	1	1	yes	characteristic
10	C₈×D₄×C₂₃	C₄	1	3	yes	—
11	C₈×D₄×C₂₃	C₄	1	3	yes	—
12	C₂₃×#431	C₂²	1	3	yes	—
13	C₂₃×#424	—	2	6	no	—
14	C₂₃×#423	C₂²	1	6	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	6	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
17	C₂₃×#335	C₂²	1	1	yes	characteristic
18	C₂₃×#335	C₂²	1	3	yes	—
19	C₂₃×#335	C₂²	1	6	yes	—
20	C₂₃×#335	C₂²	1	6	yes	—
21	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
22	C₂₃×#131	—	4	8	no	—
23	C₂×M₁₆×C₂₃	—	2	6	no	—
24	C₂²×C₈×C₂₃	—	2	6	no	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
27	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Q₈×C₂₃	Q₈	1	1	yes	characteristic
29	C₄×D₄×C₂₃	C₂×C₄	1	3	yes	—
30	C₄×D₄×C₂₃	Q₈	1	3	yes	—
31	C₂₃×#117	D₄	1	3	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
33	C₂₃×C₄⋊C₈	—	2	6	no	—
34	C₂₃×C₄⋊C₈	—	2	6	no	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
36	C₂₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
37	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
38	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
39	C₂₃×C₂²⋊C₈	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₂₃	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₂₃	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂³	1	3	yes	—
44	#41×C₂₃	D₈	1	4	yes	—
45	#41×C₂₃	Q₁₆	1	4	yes	—
46	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×D₄×C₂₃	C₄⋊C₄	1	3	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
49	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₂₃	C₂×Q₈	1	3	yes	—
51	M₁₆×C₂₃	—	4	12	no	—
52	C₂×C₈×C₂₃	—	2	6	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	6	no	—
55	C₂×C₈×C₂₃	—	4	3	no	—
56	C₂×C₈×C₂₃	—	4	12	no	—
57	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
58	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
59	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	C₄⋊C₄	1	3	yes	—
62	C₄⋊C₄×C₂₃	C₄⋊C₄	1	3	yes	—
63	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	6	yes	—
64	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
65	C₄²×C₂₃	C₂×D₄	1	3	yes	—
66	C₄²×C₂₃	C₂×Q₈	1	3	yes	—
67	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
68	#17828	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
70	Q₈×C₂₃	#107	1	4	yes	—
71	D₄×C₂₃	#107	1	12	yes	—
72	C₂×C₄×C₂₃	—	2	3	no	—
73	C₂×C₄×C₂₃	—	2	3	no	—
74	C₂×C₄×C₂₃	—	4	3	no	—
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
79	C₂×C₄×C₂₃	C₂×Q₁₆	1	6	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	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₂₃	—	2	3	no	—
84	C₈×C₂₃	—	2	1	no	—
85	C₈×C₂₃	—	2	3	no	—
86	C₈×C₂₃	—	4	2	no	—
87	C₈×C₂₃	—	4	6	no	—
88	C₈×C₂₃	#131	1	2	yes	complemented
89	C₈×#41	C₂×C₂₃	1	1	yes	characteristic
90	#2906	C₂×C₂₃	1	2	yes	—
91	#1571	C₂×C₂₃	1	6	yes	complemented
92	#1567	C₂×C₂₃	1	3	yes	complemented
93	C₂×#335	C₂×C₂₃	1	3	yes	complemented
94	C₂²×C₂₃	—	2	3	no	—
95	C₂²×C₂₃	—	2	3	no	—
96	C₂²×C₂₃	C₂×#107	1	6	yes	—
97	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₂₃	—	2	3	no	—
99	C₄×C₂₃	—	2	1	no	—
100	C₄×C₂₃	—	2	3	no	—
101	C₄×C₂₃	C₂×#107	1	6	yes	—
102	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
103	C₄×C₂₃	#568	1	1	yes	characteristic
104	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
105	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
106	C₂×#131	—	2	2	no	—
107	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
108	C₈×Q₈	C₄×C₂₃	1	1	yes	characteristic
109	C₈×Q₈	C₄×C₂₃	1	1	yes	characteristic
110	C₈×D₄	C₄×C₂₃	1	3	yes	—
111	C₈×D₄	C₄×C₂₃	1	3	yes	—
112	#431	C₂²×C₂₃	1	3	yes	—
113	#424	—	2	6	no	—
114	#423	C₂²×C₂₃	1	6	yes	—
115	C₂×C₄⋊C₈	C₂²×C₂₃	1	6	yes	—
116	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
117	#335	C₂²×C₂₃	1	1	yes	characteristic
118	#335	C₂²×C₂₃	1	3	yes	—
119	#335	C₂²×C₂₃	1	6	yes	—
120	#335	C₂²×C₂₃	1	6	yes	—
121	C₂×C₂₃	—	2	6	no	—
122	C₂×C₂₃	#2906	1	1	yes	characteristic
123	C₂×C₂₃	C₂²×#107	1	1	yes	characteristic
124	C₂×C₂₃	#2916	1	1	yes	characteristic
125	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
126	#131	—	4	8	no	—
127	C₂×M₁₆	—	2	6	no	—
128	C₂²×C₈	—	2	6	no	—
129	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
130	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
131	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
132	C₄×Q₈	Q₈×C₂₃	1	1	yes	characteristic
133	C₄×D₄	C₂×C₄×C₂₃	1	3	yes	—
134	C₄×D₄	Q₈×C₂₃	1	3	yes	—
135	#117	D₄×C₂₃	1	3	yes	—
136	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
137	C₄⋊C₈	—	2	6	no	—
138	C₄⋊C₈	—	2	6	no	—
139	C₄⋊C₈	C₂×C₄×C₂₃	1	3	yes	—
140	C₄⋊C₈	C₂×C₄×C₂₃	1	3	yes	—
141	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
142	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
143	C₂²⋊C₈	C₂×C₄×C₂₃	1	6	yes	—
144	C₄×C₈	C₂×C₄×C₂₃	1	3	yes	—
145	C₄×C₈	C₂×C₄×C₂₃	1	3	yes	—
146	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
147	C₄×C₈	C₂³×C₂₃	1	3	yes	—
148	C₂₃	#17828	1	1	yes	characteristic, direct product factor, Sylow
149	#41	D₈×C₂₃	1	4	yes	—
150	#41	Q₁₆×C₂₃	1	4	yes	—
151	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
152	C₂×D₄	C₄⋊C₄×C₂₃	1	3	yes	—
153	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
154	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
155	C₂²×C₄	C₂×Q₈×C₂₃	1	3	yes	—
156	M₁₆	—	4	12	no	—
157	C₂×C₈	—	2	6	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	6	no	—
160	C₂×C₈	—	4	3	no	—
161	C₂×C₈	—	4	12	no	—
162	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
164	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
165	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
166	C₄⋊C₄	C₄⋊C₄×C₂₃	1	3	yes	—
167	C₄⋊C₄	C₄⋊C₄×C₂₃	1	3	yes	—
168	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	6	yes	—
169	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
170	C₄²	C₂×D₄×C₂₃	1	3	yes	—
171	C₄²	C₂×Q₈×C₂₃	1	3	yes	—
172	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
173	C₂³	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
174	Q₈	C₂₃×#107	1	4	yes	—
175	D₄	C₂₃×#107	1	12	yes	—
176	C₂×C₄	—	2	3	no	—
177	C₂×C₄	—	2	3	no	—
178	C₂×C₄	—	4	3	no	—
179	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
180	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
182	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
183	C₂×C₄	C₂×Q₁₆×C₂₃	1	6	yes	—
184	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	4	2	no	—
191	C₈	—	4	6	no	—
192	C₈	C₂₃×#131	1	2	yes	complemented
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂×C₂₃×#107	1	6	yes	—
196	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	3	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×C₂₃×#107	1	6	yes	—
201	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
202	C₄	C₂₃×#568	1	1	yes	characteristic
203	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
204	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	C₂₃×#2906	1	1	yes	characteristic
207	C₂	C₂²×C₂₃×#107	1	1	yes	characteristic
208	C₂	C₂₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1571	C₂	semidirect
4. C₂₃×#1567	C₂	semidirect
5. C₂×C₂₃×#335	C₂	semidirect
68. #17828	C₂₃	direct (N × H)
88. C₈×C₂₃	#131	semidirect
91. #1571	C₂×C₂₃	semidirect
92. #1567	C₂×C₂₃	semidirect
93. C₂×#335	C₂×C₂₃	semidirect
148. C₂₃	#17828	direct (N × H)
192. C₈	C₂₃×#131	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈×#41×C₂₃	2
C₂₃×#2906	2
C₂₃×#1571	2
C₂₃×#1567	2
C₂×C₂₃×#335	2
#17828	23

Subgroups — C23×#17828 #19165454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17828 #19165454