Subgroups — C₂₃×#10833 #19158459

Order 5888.

950 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#336	C₂	1	4	yes	maximal
4	C₂³×C₈×C₂₃	C₂²	1	4	yes	—
5	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×C₄⋊C₈	—	2	4	no	—
8	C₂×C₂₃×C₄⋊C₈	—	2	4	no	—
9	C₂×C₂₃×C₄⋊C₈	C₄	1	8	yes	complemented
10	C₂₃×#376	—	2	4	no	—
11	C₂×C₂₃×#95	C₄	1	4	yes	—
12	C₂₃×#336	—	2	8	no	—
13	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
14	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	—	2	8	no	—
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂²×C₈×C₂₃	—	2	4	no	—
19	C₂²×C₈×C₂₃	—	2	8	no	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
21	C₂×C₄²×C₂₃	—	2	4	no	—
22	C₂×C₄²×C₂₃	—	2	4	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	D₄	1	4	yes	—
25	C₂×C₄²×C₂₃	D₄	1	2	yes	—
26	C₂×C₄²×C₂₃	D₄	1	4	yes	—
27	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
28	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
29	C₂₃×C₄⋊C₈	—	2	16	no	—
30	C₂₃×#95	—	2	8	no	—
31	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	—	2	8	no	—
33	C₂²×C₄×C₂₃	—	2	4	no	—
34	C₂²×C₄×C₂₃	—	2	4	no	—
35	C₂²×C₄×C₂₃	—	2	16	no	—
36	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
37	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₂₃	#41	1	4	yes	—
47	C₂×C₈×C₂₃	—	2	8	no	—
48	C₂×C₈×C₂₃	—	2	16	no	—
49	C₂×C₈×C₂₃	—	4	8	no	—
50	C₂×C₈×C₂₃	—	4	4	no	—
51	C₂×C₈×C₂₃	—	4	4	no	—
52	C₄²×C₂₃	—	2	8	no	—
53	C₄²×C₂₃	—	2	8	no	—
54	C₄²×C₂₃	C₂²⋊C₄	1	8	yes	—
55	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
56	#10833	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
58	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₂³×C₂₃	#117	1	2	yes	—
62	C₂³×C₂₃	#131	1	8	yes	—
63	C₂×C₄×C₂₃	—	2	8	no	—
64	C₂×C₄×C₂₃	—	2	8	no	—
65	C₂×C₄×C₂₃	—	2	8	no	—
66	C₂×C₄×C₂₃	—	2	8	no	—
67	C₂×C₄×C₂₃	—	2	16	no	—
68	C₂×C₄×C₂₃	—	4	8	no	—
69	C₂×C₄×C₂₃	—	4	8	no	—
70	C₂×C₄×C₂₃	#95	1	8	yes	—
71	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
74	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
75	C₂×C₄×C₂₃	#121	1	4	yes	—
76	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
77	C₂×C₄×C₂₃	#124	1	2	yes	—
78	C₂×C₄×C₂₃	#125	1	2	yes	—
79	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₈×C₂₃	—	4	8	no	—
83	C₂²×C₄⋊C₈	C₂×C₂₃	1	2	yes	—
84	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
85	C₂×#336	C₂×C₂₃	1	4	yes	—
86	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
87	C₂²×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
88	C₂²×C₂₃	#380	1	2	yes	—
89	C₂²×C₂₃	#382	1	1	yes	characteristic
90	C₂²×C₂₃	#384	1	1	yes	characteristic
91	C₂²×C₂₃	#405	1	4	yes	—
92	C₂²×C₂₃	#408	1	4	yes	—
93	C₂²×C₂₃	#424	1	8	yes	—
94	C₂²×C₂₃	#424	1	4	yes	—
95	C₂²×C₂₃	#433	1	8	yes	—
96	C₄×C₂₃	—	2	8	no	—
97	C₄×C₂₃	—	4	8	no	—
98	C₄×C₂₃	#389	1	4	yes	—
99	C₄×C₂₃	#390	1	2	yes	—
100	C₄×C₂₃	#391	1	2	yes	—
101	C₂³×C₈	C₂²×C₂₃	1	4	yes	—
102	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
103	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	2	4	no	—
105	C₂×C₄⋊C₈	—	2	4	no	—
106	C₂×C₄⋊C₈	C₄×C₂₃	1	8	yes	complemented
107	#376	—	2	4	no	—
108	C₂×#95	C₄×C₂₃	1	4	yes	—
109	#336	—	2	8	no	—
110	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
111	C₂×C₂₃	#1729	1	2	yes	—
112	C₂×C₂₃	#1768	1	4	yes	—
113	C₂×C₂₃	#1798	1	4	yes	—
114	C₂×C₂₃	#1832	1	2	yes	—
115	C₂×C₂₃	#1843	1	2	yes	—
116	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
117	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
118	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
119	C₂²×C₈	—	2	8	no	—
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	—	2	8	no	—
123	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
127	C₂×C₄²	D₄×C₂₃	1	4	yes	—
128	C₂×C₄²	D₄×C₂₃	1	2	yes	—
129	C₂×C₄²	D₄×C₂₃	1	4	yes	—
130	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
131	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
132	C₄⋊C₈	—	2	16	no	—
133	#95	—	2	8	no	—
134	C₂₃	#10833	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	16	no	—
140	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
141	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
142	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
143	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
145	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
150	C₂²×C₄	#41×C₂₃	1	4	yes	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	16	no	—
153	C₂×C₈	—	4	8	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	4	4	no	—
156	C₄²	—	2	8	no	—
157	C₄²	—	2	8	no	—
158	C₄²	C₂²⋊C₄×C₂₃	1	8	yes	—
159	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
160	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
161	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
162	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
163	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
164	C₂³	C₂₃×#117	1	2	yes	—
165	C₂³	C₂₃×#131	1	8	yes	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	C₂₃×#95	1	8	yes	—
174	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
176	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
177	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
178	C₂×C₄	C₂₃×#121	1	4	yes	—
179	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
180	C₂×C₄	C₂₃×#124	1	2	yes	—
181	C₂×C₄	C₂₃×#125	1	2	yes	—
182	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
183	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
184	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
185	C₈	—	4	8	no	—
186	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
187	C₂²	C₄×C₄⋊C₄×C₂₃	1	2	yes	—
188	C₂²	C₂₃×#380	1	2	yes	—
189	C₂²	C₂₃×#382	1	1	yes	characteristic
190	C₂²	C₂₃×#384	1	1	yes	characteristic
191	C₂²	C₂₃×#405	1	4	yes	—
192	C₂²	C₂₃×#408	1	4	yes	—
193	C₂²	C₂₃×#424	1	4	yes	—
194	C₂²	C₂₃×#424	1	8	yes	—
195	C₂²	C₂₃×#433	1	8	yes	—
196	C₄	—	2	8	no	—
197	C₄	—	4	8	no	—
198	C₄	C₂₃×#389	1	4	yes	—
199	C₄	C₂₃×#390	1	2	yes	—
200	C₄	C₂₃×#391	1	2	yes	—
201	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
202	C₂	C₂₃×#1729	1	2	yes	—
203	C₂	C₂₃×#1768	1	4	yes	—
204	C₂	C₂₃×#1798	1	4	yes	—
205	C₂	C₂₃×#1832	1	2	yes	—
206	C₂	C₂₃×#1843	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₂×C₂₃×C₄⋊C₈	C₄	semidirect
56. #10833	C₂₃	direct (N × H)
106. C₂×C₄⋊C₈	C₄×C₂₃	semidirect
134. C₂₃	#10833	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×C₂₃×C₄⋊C₈	2
C₂×C₂₃×#376	2
C₂×C₂₃×#336	2
#10833	23

Subgroups — C23×#10833 #19158459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10833 #19158459