Subgroups — C₂₃×#10842 #19158468

Order 5888.

950 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#382	C₂	1	2	yes	maximal
3	C₂×C₂₃×#340	C₂	1	4	yes	maximal
4	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
5	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	—	2	4	no	—
8	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	complemented
9	C₂₃×#382	—	2	4	no	—
10	C₂₃×#382	C₄	1	8	yes	complemented
11	C₂×C₂₃×#95	—	2	4	no	—
12	C₂₃×#340	—	2	8	no	—
13	C₂³×C₄×C₂₃	—	2	4	no	—
14	C₂³×C₄×C₂₃	D₄	1	2	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	—	2	4	no	—
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₂₃	D₄	1	4	yes	—
20	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
21	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
24	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₂₃	D₄	1	4	yes	—
28	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
29	C₄×C₈×C₂₃	—	2	8	no	—
30	C₂₃×#95	—	2	16	no	—
31	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	—	2	8	no	—
33	C₂²×C₄×C₂₃	—	2	8	no	—
34	C₂²×C₄×C₂₃	—	4	4	no	—
35	C₂²×C₄×C₂₃	—	4	4	no	—
36	C₂²×C₄×C₂₃	—	4	16	no	—
37	C₂²×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
38	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₂₃	#41	1	4	yes	—
46	C₂×C₈×C₂₃	—	2	4	no	—
47	C₂×C₈×C₂₃	—	2	4	no	—
48	C₂×C₈×C₂₃	—	2	8	no	—
49	C₂×C₈×C₂₃	—	2	8	no	—
50	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
51	C₄⋊C₄×C₂₃	—	2	16	no	—
52	C₄⋊C₄×C₂₃	—	4	8	no	—
53	C₄²×C₂₃	C₄²	1	4	yes	—
54	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
56	#10842	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
58	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₂₃	#124	1	2	yes	—
60	C₂³×C₂₃	#127	1	1	yes	characteristic
61	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂³×C₂₃	#135	1	8	yes	—
63	C₂×C₄×C₂₃	—	2	16	no	—
64	C₂×C₄×C₂₃	—	2	8	no	—
65	C₂×C₄×C₂₃	—	4	8	no	—
66	C₂×C₄×C₂₃	—	4	16	no	—
67	C₂×C₄×C₂₃	—	8	8	no	—
68	C₂×C₄×C₂₃	—	8	8	no	—
69	C₂×C₄×C₂₃	#95	1	8	yes	—
70	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₃	#117	1	2	yes	—
75	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
76	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
77	C₂×C₄×C₂₃	#121	1	4	yes	—
78	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
79	C₂×C₄×C₂₃	#125	1	2	yes	—
80	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
81	C₈×C₂₃	—	2	8	no	—
82	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
83	C₂×#382	C₂×C₂₃	1	2	yes	—
84	C₂×#340	C₂×C₂₃	1	4	yes	—
85	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
86	C₂²×C₂₃	#382	1	1	yes	characteristic
87	C₂²×C₂₃	#384	1	1	yes	characteristic
88	C₂²×C₂₃	#390	1	2	yes	—
89	C₂²×C₂₃	#391	1	2	yes	—
90	C₂²×C₂₃	#416	1	8	yes	—
91	C₂²×C₂₃	#427	1	4	yes	—
92	C₂²×C₂₃	#488	1	8	yes	—
93	C₂²×C₂₃	#495	1	4	yes	—
94	C₂²×C₂₃	#499	1	4	yes	—
95	C₄×C₂₃	—	2	8	no	—
96	C₄×C₂₃	—	8	8	no	—
97	C₄×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
98	C₄×C₂₃	#380	1	2	yes	—
99	C₄×C₂₃	#389	1	4	yes	—
100	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
101	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
102	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
103	C₂×C₄×C₈	—	2	4	no	—
104	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	complemented
105	#382	—	2	4	no	—
106	#382	C₄×C₂₃	1	8	yes	complemented
107	C₂×#95	—	2	4	no	—
108	#340	—	2	8	no	—
109	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
110	C₂×C₂₃	#1735	1	2	yes	—
111	C₂×C₂₃	#1835	1	2	yes	—
112	C₂×C₂₃	#1846	1	2	yes	—
113	C₂×C₂₃	#1958	1	4	yes	—
114	C₂×C₂₃	#1991	1	4	yes	—
115	C₂³×C₄	—	2	4	no	—
116	C₂³×C₄	D₄×C₂₃	1	2	yes	—
117	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
121	C₂²×C₈	D₄×C₂₃	1	4	yes	—
122	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
123	C₂×C₄⋊C₄	—	2	8	no	—
124	C₂×C₄⋊C₄	—	2	8	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
127	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
128	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
129	C₂×C₄²	D₄×C₂₃	1	4	yes	—
130	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
131	C₄×C₈	—	2	8	no	—
132	#95	—	2	16	no	—
133	C₂₃	#10842	1	1	yes	characteristic, direct product factor, Sylow
134	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	—	4	16	no	—
140	C₂²×C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
141	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	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×D₄×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
148	C₂²×C₄	#41×C₂₃	1	4	yes	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
154	C₄⋊C₄	—	2	16	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄²	C₄²×C₂₃	1	4	yes	—
157	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
158	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
159	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
160	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
161	C₂³	C₂₃×#124	1	2	yes	—
162	C₂³	C₂₃×#127	1	1	yes	characteristic
163	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
164	C₂³	C₂₃×#135	1	8	yes	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	—	4	16	no	—
169	C₂×C₄	—	8	8	no	—
170	C₂×C₄	—	8	8	no	—
171	C₂×C₄	C₂₃×#95	1	8	yes	—
172	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
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	2	yes	—
177	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
178	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
179	C₂×C₄	C₂₃×#121	1	4	yes	—
180	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
181	C₂×C₄	C₂₃×#125	1	2	yes	—
182	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
183	C₈	—	2	8	no	—
184	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
185	C₂²	C₂₃×#382	1	1	yes	characteristic
186	C₂²	C₂₃×#384	1	1	yes	characteristic
187	C₂²	C₂₃×#390	1	2	yes	—
188	C₂²	C₂₃×#391	1	2	yes	—
189	C₂²	C₂₃×#416	1	8	yes	—
190	C₂²	C₂₃×#427	1	4	yes	—
191	C₂²	C₂₃×#488	1	8	yes	—
192	C₂²	C₂₃×#495	1	4	yes	—
193	C₂²	C₂₃×#499	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	8	8	no	—
196	C₄	C₄×C₄⋊C₄×C₂₃	1	2	yes	—
197	C₄	C₂₃×#380	1	2	yes	—
198	C₄	C₂₃×#389	1	4	yes	—
199	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
200	C₂	C₂₃×#1735	1	2	yes	—
201	C₂	C₂₃×#1835	1	2	yes	—
202	C₂	C₂₃×#1846	1	2	yes	—
203	C₂	C₂₃×#1958	1	4	yes	—
204	C₂	C₂₃×#1991	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
8. C₂×C₄×C₈×C₂₃	C₄	semidirect
10. C₂₃×#382	C₄	semidirect
56. #10842	C₂₃	direct (N × H)
104. C₂×C₄×C₈	C₄×C₂₃	semidirect
106. #382	C₄×C₂₃	semidirect
133. C₂₃	#10842	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₂₃	2
C₂×C₂₃×#382	2
C₂×C₂₃×#340	2
#10842	23

Subgroups — C23×#10842 #19158468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10842 #19158468