Subgroups — C₂₃×#12051 #19159677

Order 5888.

224 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2169	C₂	1	1	yes	maximal, characteristic
3	C₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₈×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×M₃₂	—	2	1	no	—
9	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₄×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
11	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×C₄⋊C₁₆	—	2	1	no	—
13	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
16	C₂₃×#346	—	2	1	no	—
17	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#334	C₄	1	1	yes	characteristic
20	C₂₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
21	C₂₃×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×SD₁₆×C₂₃	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	—	2	1	no	—
25	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂₃×M₃₂	—	4	2	no	—
28	C₂×C₁₆×C₂₃	—	2	1	no	—
29	C₂×C₁₆×C₂₃	—	2	1	no	—
30	C₂×C₁₆×C₂₃	—	2	1	no	—
31	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
33	C₂₃×#106	C₈	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	—	2	1	no	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂₃×#103	C₈	1	1	yes	characteristic
38	C₂₃×#102	C₈	1	1	yes	characteristic
39	C₂₃×C₂²⋊C₈	—	2	1	no	—
40	C₄×C₈×C₂₃	—	2	1	no	—
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	SD₁₆×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	1	no	—
49	C₂×C₈×C₂₃	—	2	1	no	—
50	C₂×C₈×C₂₃	—	2	1	no	—
51	C₂×C₈×C₂₃	—	4	1	no	—
52	C₂×C₈×C₂₃	—	4	1	no	—
53	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
60	C₂²⋊C₄×C₂₃	—	2	1	no	—
61	C₄²×C₂₃	—	2	1	no	—
62	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₁₆×C₂₃	—	2	2	no	—
64	C₁₆×C₂₃	—	2	1	no	—
65	C₁₆×C₂₃	—	4	1	no	—
66	C₁₆×C₂₃	—	4	1	no	—
67	C₁₆×C₂₃	SD₁₆	1	2	yes	complemented
68	#12051	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	—	2	1	no	—
70	Q₈×C₂₃	—	2	1	no	—
71	Q₈×C₂₃	—	2	1	no	—
72	D₄×C₂₃	—	2	1	no	—
73	D₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	4	1	no	—
78	C₂×C₄×C₂₃	—	4	1	no	—
79	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
82	C₈×C₂₃	—	2	2	no	—
83	C₈×C₂₃	—	2	1	no	—
84	C₈×C₂₃	—	2	1	no	—
85	C₈×C₂₃	—	4	1	no	—
86	C₈×C₂₃	—	4	1	no	—
87	C₈×C₂₃	M₃₂	1	2	yes	complemented
88	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₂₃	#135	1	1	yes	characteristic
90	#2183	C₂×C₂₃	1	1	yes	characteristic, complemented
91	#2169	C₂×C₂₃	1	1	yes	characteristic
92	C₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
93	#1367	C₂×C₂₃	1	1	yes	characteristic, complemented
94	Q₈⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
95	D₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
96	C₈×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
97	C₂²×C₂₃	—	4	1	no	—
98	C₂²×C₂₃	—	4	1	no	—
99	C₂²×C₂₃	—	4	1	no	—
100	C₂²×C₂₃	C₈×D₄	1	1	yes	characteristic
101	C₄×C₂₃	—	2	1	no	—
102	C₄×C₂₃	—	4	1	no	—
103	C₄×C₂₃	—	4	1	no	—
104	C₄×C₂₃	—	4	1	no	—
105	C₄×C₂₃	C₄×SD₁₆	1	1	yes	characteristic
106	C₄×C₂₃	#443	1	1	yes	characteristic
107	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
108	C₄×C₂₃	#504	1	1	yes	characteristic
109	C₂×M₃₂	—	2	1	no	—
110	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
111	C₄×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
112	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
113	C₄⋊C₁₆	—	2	1	no	—
114	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
115	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
116	C₂²⋊C₁₆	—	2	1	no	—
117	#346	—	2	1	no	—
118	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
119	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
120	#334	C₄×C₂₃	1	1	yes	characteristic
121	Q₈⋊C₈	C₄×C₂₃	1	1	yes	characteristic
122	D₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
123	C₈²	C₂²×C₂₃	1	1	yes	characteristic
124	C₂×C₂₃	—	8	1	no	—
125	C₂×C₂₃	C₈×SD₁₆	1	1	yes	characteristic
126	C₂×C₂₃	#2169	1	1	yes	characteristic
127	C₂×C₂₃	#2170	1	1	yes	characteristic
128	C₂×SD₁₆	C₈×C₂₃	1	1	yes	characteristic
129	C₂²×C₈	—	2	1	no	—
130	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
131	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
132	M₃₂	—	4	2	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
137	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
138	#106	C₈×C₂₃	1	1	yes	characteristic
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
142	#103	C₈×C₂₃	1	1	yes	characteristic
143	#102	C₈×C₂₃	1	1	yes	characteristic
144	C₂²⋊C₈	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
147	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
148	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
149	C₂₃	#12051	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
151	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
152	C₂²×C₄	—	2	1	no	—
153	SD₁₆	—	2	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
161	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
162	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
165	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
166	C₂²⋊C₄	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
169	C₁₆	—	2	2	no	—
170	C₁₆	—	2	1	no	—
171	C₁₆	—	4	1	no	—
172	C₁₆	—	4	1	no	—
173	C₁₆	SD₁₆×C₂₃	1	2	yes	complemented
174	C₂³	—	2	1	no	—
175	Q₈	—	2	1	no	—
176	Q₈	—	2	1	no	—
177	D₄	—	2	1	no	—
178	D₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	1	no	—
191	C₈	—	4	1	no	—
192	C₈	C₂₃×M₃₂	1	2	yes	complemented
193	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
194	C₈	C₂₃×#135	1	1	yes	characteristic
195	C₂²	—	4	1	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	C₈×D₄×C₂₃	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	1	no	—
203	C₄	C₄×SD₁₆×C₂₃	1	1	yes	characteristic
204	C₄	C₂₃×#443	1	1	yes	characteristic
205	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
206	C₄	C₂₃×#504	1	1	yes	characteristic
207	C₂	—	8	1	no	—
208	C₂	C₈×SD₁₆×C₂₃	1	1	yes	characteristic
209	C₂	C₂₃×#2169	1	1	yes	characteristic
210	C₂	C₂₃×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₈×D₄
prime radical	C₂×D₄×C₂₃	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₄×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2183	C₂	semidirect
4. C₂₃×#1367	C₂	semidirect
5. C₂₃×Q₈⋊C₁₆	C₂	semidirect
7. C₈×C₁₆×C₂₃	C₂	semidirect
67. C₁₆×C₂₃	SD₁₆	semidirect
68. #12051	C₂₃	direct (N × H)
87. C₈×C₂₃	M₃₂	semidirect
90. #2183	C₂×C₂₃	semidirect
93. #1367	C₂×C₂₃	semidirect
94. Q₈⋊C₁₆	C₂×C₂₃	semidirect
96. C₈×C₁₆	C₂×C₂₃	semidirect
149. C₂₃	#12051	direct (N × H)
173. C₁₆	SD₁₆×C₂₃	semidirect
192. C₈	C₂₃×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2183	2
C₂₃×#2169	2
C₈×SD₁₆×C₂₃	2
C₂₃×#1367	2
C₂₃×Q₈⋊C₁₆	2
C₂₃×D₄⋊C₁₆	2
C₈×C₁₆×C₂₃	2
#12051	23

Subgroups — C23×#12051 #19159677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12051 #19159677