Subgroups — C₂₃×#17692 #19165318

Order 5888.

872 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3504	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2971	C₂	1	2	yes	maximal
3	C₂₃×#1537	C₂	1	3	yes	maximal, complemented
4	C₂₃×#1528	C₂	1	6	yes	maximal, complemented
5	C₂₃×#1526	C₂	1	3	yes	maximal, complemented
6	C₂₃×#557	—	2	2	no	—
7	C₂₃×#554	—	2	6	no	—
8	C₂₃×#544	C₄	1	6	yes	—
9	C₂₃×#533	C₂²	1	3	yes	—
10	C₂×C₂₃×#125	C₂²	1	3	yes	—
11	C₂₃×#520	C₄	1	2	yes	—
12	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₂₃	—	2	2	no	—
14	C₂₃×#436	—	2	6	no	—
15	C₂₃×#432	C₂²	1	6	yes	—
16	C₄×M₁₆×C₂₃	C₂²	1	6	yes	—
17	C₂₃×#330	C₂²	1	1	yes	characteristic
18	C₂₃×#330	C₂²	1	6	yes	—
19	C₂₃×#330	C₂²	1	3	yes	—
20	C₂₃×#330	C₂²	1	6	yes	—
21	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	—	2	6	no	—
23	C₂₃×C₄⋊Q₈	—	2	6	no	—
24	C₂₃×C₄⋊Q₈	C₂×C₄	1	6	yes	—
25	C₂₃×#125	—	2	6	no	—
26	C₂₃×#125	—	2	6	no	—
27	C₂₃×#125	C₂×C₄	1	6	yes	—
28	C₂₃×#125	C₂³	1	3	yes	—
29	C₂₃×#125	C₂³	1	1	yes	characteristic
30	C₂₃×C₂²⋊Q₈	—	2	6	no	—
31	C₂₃×C₂²⋊Q₈	—	2	12	no	—
32	C₄×Q₈×C₂₃	—	2	2	no	—
33	C₄×Q₈×C₂₃	—	2	6	no	—
34	C₄×Q₈×C₂₃	D₄	1	2	yes	—
35	C₄×D₄×C₂₃	—	2	6	no	—
36	C₄×D₄×C₂₃	D₄	1	6	yes	—
37	C₂₃×#117	C₂×C₄	1	6	yes	—
38	C₂₃×#117	C₂×C₄	1	3	yes	—
39	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
42	C₂₃×C₄⋊C₈	—	2	6	no	—
43	C₂₃×C₂²⋊C₈	—	2	6	no	—
44	C₂₃×#97	C₂³	1	2	yes	—
45	C₂₃×#97	C₂³	1	6	yes	—
46	C₄×C₈×C₂₃	—	2	6	no	—
47	#41×C₂₃	—	2	4	no	—
48	C₂×Q₈×C₂₃	—	2	6	no	—
49	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₄×C₂₃	C₂²⋊C₄	1	3	yes	—
51	C₂²×C₄×C₂₃	—	2	6	no	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
54	M₁₆×C₂₃	—	4	12	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	2	6	no	—
57	C₄⋊C₄×C₂₃	—	2	12	no	—
58	C₄⋊C₄×C₂₃	—	2	6	no	—
59	C₄⋊C₄×C₂₃	—	2	6	no	—
60	C₄⋊C₄×C₂₃	—	2	6	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
63	C₄⋊C₄×C₂₃	C₂²×C₄	1	6	yes	—
64	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂²⋊C₄×C₂₃	—	2	6	no	—
66	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
67	C₄²×C₂₃	—	2	6	no	—
68	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
69	C₄²×C₂₃	C₂×D₄	1	6	yes	—
70	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
71	#17692	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
73	Q₈×C₂₃	—	8	6	no	—
74	Q₈×C₂₃	#101	1	4	yes	—
75	D₄×C₂₃	—	2	6	no	—
76	C₂×C₄×C₂₃	—	2	6	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	6	no	—
79	C₂×C₄×C₂₃	—	4	6	no	—
80	C₂×C₄×C₂₃	—	8	6	no	—
81	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
82	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
83	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
85	C₈×C₂₃	—	2	4	no	—
86	C₈×C₂₃	—	4	6	no	—
87	#3504	C₂×C₂₃	1	1	yes	characteristic
88	#2971	C₂×C₂₃	1	2	yes	—
89	#1537	C₂×C₂₃	1	3	yes	complemented
90	#1528	C₂×C₂₃	1	6	yes	complemented
91	#1526	C₂×C₂₃	1	3	yes	complemented
92	C₂²×C₂₃	—	2	3	no	—
93	C₂²×C₂₃	—	2	6	no	—
94	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₂₃	—	2	1	no	—
96	C₄×C₂₃	—	2	2	no	—
97	C₄×C₂₃	—	2	6	no	—
98	C₄×C₂₃	—	4	6	no	—
99	C₄×C₂₃	—	4	2	no	—
100	C₄×C₂₃	C₂×#101	1	6	yes	—
101	#557	—	2	2	no	—
102	#554	—	2	6	no	—
103	#544	C₄×C₂₃	1	6	yes	—
104	#533	C₂²×C₂₃	1	3	yes	—
105	C₂×#125	C₂²×C₂₃	1	3	yes	—
106	#520	C₄×C₂₃	1	2	yes	—
107	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
108	C₈×Q₈	—	2	2	no	—
109	#436	—	2	6	no	—
110	#432	C₂²×C₂₃	1	6	yes	—
111	C₄×M₁₆	C₂²×C₂₃	1	6	yes	—
112	#330	C₂²×C₂₃	1	3	yes	—
113	#330	C₂²×C₂₃	1	6	yes	—
114	#330	C₂²×C₂₃	1	1	yes	characteristic
115	#330	C₂²×C₂₃	1	6	yes	—
116	C₂×C₂₃	—	4	3	no	—
117	C₂×C₂₃	C₂²×#101	1	1	yes	characteristic
118	C₂×C₂₃	#2901	1	2	yes	—
119	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
120	C₂×M₁₆	—	2	6	no	—
121	C₄⋊Q₈	—	2	6	no	—
122	C₄⋊Q₈	C₂×C₄×C₂₃	1	6	yes	—
123	#125	—	2	6	no	—
124	#125	—	2	6	no	—
125	#125	C₂×C₄×C₂₃	1	6	yes	—
126	#125	C₂³×C₂₃	1	3	yes	—
127	#125	C₂³×C₂₃	1	1	yes	characteristic
128	C₂²⋊Q₈	—	2	6	no	—
129	C₂²⋊Q₈	—	2	12	no	—
130	C₄×Q₈	—	2	2	no	—
131	C₄×Q₈	—	2	6	no	—
132	C₄×Q₈	D₄×C₂₃	1	2	yes	—
133	C₄×D₄	—	2	6	no	—
134	C₄×D₄	D₄×C₂₃	1	6	yes	—
135	#117	C₂×C₄×C₂₃	1	6	yes	—
136	#117	C₂×C₄×C₂₃	1	3	yes	—
137	C₂×C₄⋊C₄	—	2	6	no	—
138	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	6	yes	—
139	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
140	C₄⋊C₈	—	2	6	no	—
141	C₂²⋊C₈	—	2	6	no	—
142	#97	C₂³×C₂₃	1	2	yes	—
143	#97	C₂³×C₂₃	1	6	yes	—
144	C₄×C₈	—	2	6	no	—
145	C₂₃	#17692	1	1	yes	characteristic, direct product factor, Sylow
146	#41	—	2	4	no	—
147	C₂×Q₈	—	2	6	no	—
148	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
149	C₂×D₄	C₂²⋊C₄×C₂₃	1	3	yes	—
150	C₂²×C₄	—	2	6	no	—
151	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
152	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
153	M₁₆	—	4	12	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	6	no	—
156	C₄⋊C₄	—	2	12	no	—
157	C₄⋊C₄	—	2	6	no	—
158	C₄⋊C₄	—	2	6	no	—
159	C₄⋊C₄	—	2	6	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
162	C₄⋊C₄	C₂²×C₄×C₂₃	1	6	yes	—
163	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
164	C₂²⋊C₄	—	2	6	no	—
165	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
166	C₄²	—	2	6	no	—
167	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
168	C₄²	C₂×D₄×C₂₃	1	6	yes	—
169	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
170	C₂³	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
171	Q₈	—	8	6	no	—
172	Q₈	C₂₃×#101	1	4	yes	—
173	D₄	—	2	6	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	4	6	no	—
178	C₂×C₄	—	8	6	no	—
179	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
180	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
181	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	4	6	no	—
185	C₂²	—	2	3	no	—
186	C₂²	—	2	6	no	—
187	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	6	no	—
191	C₄	—	4	6	no	—
192	C₄	—	4	2	no	—
193	C₄	C₂×C₂₃×#101	1	6	yes	—
194	C₂	—	4	3	no	—
195	C₂	C₂²×C₂₃×#101	1	1	yes	characteristic
196	C₂	C₂₃×#2901	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1537	C₂	semidirect
4. C₂₃×#1528	C₂	semidirect
5. C₂₃×#1526	C₂	semidirect
71. #17692	C₂₃	direct (N × H)
89. #1537	C₂×C₂₃	semidirect
90. #1528	C₂×C₂₃	semidirect
91. #1526	C₂×C₂₃	semidirect
145. C₂₃	#17692	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3504	2
C₂₃×#2971	2
C₂₃×#1537	2
C₂₃×#1528	2
C₂₃×#1526	2
#17692	23

Subgroups — C23×#17692 #19165318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17692 #19165318