Subgroups — C₂₃×#21676 #19169302

Order 5888.

658 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	C₂₃×#2402	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1940	C₂	1	2	yes	maximal
3	C₂₃×#1917	C₂	1	4	yes	maximal
4	C₂₃×#1832	C₂	1	1	yes	maximal, characteristic
5	C₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
6	C₈×C₄⋊C₄×C₂₃	C₂	1	4	yes	maximal
7	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₂₃×#97	C₂²	1	2	yes	—
11	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#400	C₄	1	4	yes	—
16	C₂₃×#398	C₄	1	4	yes	—
17	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
18	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂₃×#376	C₂²	1	1	yes	characteristic
20	C₂₃×#336	C₂²	1	4	yes	—
21	C₂₃×#336	C₂²	1	4	yes	—
22	C₂₃×#336	C₂²	1	4	yes	—
23	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
24	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂₃×C₄⋊C₈	—	2	8	no	—
34	C₂₃×C₄⋊C₈	—	2	4	no	—
35	C₂₃×#97	—	2	4	no	—
36	C₂₃×#97	—	2	2	no	—
37	C₄×C₈×C₂₃	—	2	4	no	—
38	C₄×C₈×C₂₃	—	2	8	no	—
39	C₄×C₈×C₂₃	Q₈	1	4	yes	—
40	C₂₃×#95	C₂×C₄	1	4	yes	—
41	C₂₃×#95	C₂×C₄	1	2	yes	—
42	C₂₃×#95	C₂×C₄	1	4	yes	—
43	C₂₃×#95	C₂×C₄	1	4	yes	—
44	C₂₃×#95	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	4	no	—
54	C₂×C₈×C₂₃	—	2	4	no	—
55	C₂×C₈×C₂₃	—	2	4	no	—
56	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₂₃	#41	1	4	yes	—
59	C₂×C₈×C₂₃	#41	1	4	yes	—
60	C₄⋊C₄×C₂₃	—	2	8	no	—
61	C₄⋊C₄×C₂₃	—	2	8	no	—
62	C₄⋊C₄×C₂₃	—	2	8	no	—
63	C₄²×C₂₃	—	2	4	no	—
64	C₄²×C₂₃	—	2	4	no	—
65	C₄²×C₂₃	—	2	2	no	—
66	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
68	C₄²×C₂₃	#41	1	2	yes	—
69	#21676	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
81	C₂×C₄×C₂₃	#131	1	8	yes	—
82	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
86	C₈×C₂₃	—	2	4	no	—
87	C₈×C₂₃	—	2	2	no	—
88	C₈×C₂₃	—	4	4	no	—
89	C₈×C₂₃	#125	1	4	yes	—
90	#2402	C₂×C₂₃	1	1	yes	characteristic
91	#1940	C₂×C₂₃	1	2	yes	—
92	#1917	C₂×C₂₃	1	4	yes	—
93	#1832	C₂×C₂₃	1	1	yes	characteristic
94	C₈×C₄⋊C₄	C₂×C₂₃	1	4	yes	—
95	C₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
96	#1726	C₂×C₂₃	1	1	yes	characteristic
97	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
98	C₂²×C₂₃	C₄×#41	1	2	yes	—
99	C₂²×C₂₃	C₂×#131	1	4	yes	—
100	C₄×C₂₃	—	2	4	no	—
101	C₄×C₂₃	—	2	8	no	—
102	C₄×C₂₃	—	4	4	no	—
103	C₄×C₂₃	—	4	4	no	—
104	C₄×C₂₃	C₂×#125	1	1	yes	characteristic
105	C₄×C₂₃	#529	1	2	yes	—
106	C₄×C₂₃	#533	1	1	yes	characteristic
107	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
108	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
109	C₂×#97	C₂²×C₂₃	1	2	yes	—
110	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
113	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
114	#400	C₄×C₂₃	1	4	yes	—
115	#398	C₄×C₂₃	1	4	yes	—
116	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
117	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
118	#376	C₂²×C₂₃	1	1	yes	characteristic
119	#336	C₂²×C₂₃	1	4	yes	—
120	#336	C₂²×C₂₃	1	4	yes	—
121	#336	C₂²×C₂₃	1	4	yes	—
122	C₂×C₂₃	C₄×#125	1	1	yes	characteristic
123	C₂×C₂₃	#2960	1	2	yes	—
124	C₂×C₂₃	#2966	1	4	yes	—
125	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
126	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
128	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
130	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
131	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
132	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
133	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
134	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
135	C₄⋊C₈	—	2	8	no	—
136	C₄⋊C₈	—	2	4	no	—
137	#97	—	2	4	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	—	2	8	no	—
141	C₄×C₈	Q₈×C₂₃	1	4	yes	—
142	#95	C₂×C₄×C₂₃	1	4	yes	—
143	#95	C₂×C₄×C₂₃	1	2	yes	—
144	#95	C₂×C₄×C₂₃	1	4	yes	—
145	#95	C₂×C₄×C₂₃	1	4	yes	—
146	#95	C₂×C₄×C₂₃	1	2	yes	—
147	C₂₃	#21676	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
149	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
150	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
151	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
152	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
160	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
161	C₂×C₈	#41×C₂₃	1	4	yes	—
162	C₂×C₈	#41×C₂₃	1	4	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
171	C₄²	#41×C₂₃	1	2	yes	—
172	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
183	C₂×C₄	C₂₃×#131	1	8	yes	—
184	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
186	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	2	2	no	—
190	C₈	—	4	4	no	—
191	C₈	C₂₃×#125	1	4	yes	—
192	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
193	C₂²	C₄×#41×C₂₃	1	2	yes	—
194	C₂²	C₂×C₂₃×#131	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	8	no	—
197	C₄	—	4	4	no	—
198	C₄	—	4	4	no	—
199	C₄	C₂×C₂₃×#125	1	1	yes	characteristic
200	C₄	C₂₃×#529	1	2	yes	—
201	C₄	C₂₃×#533	1	1	yes	characteristic
202	C₂	C₄×C₂₃×#125	1	1	yes	characteristic
203	C₂	C₂₃×#2960	1	2	yes	—
204	C₂	C₂₃×#2966	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₃	C₂³
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle, prime radical	C₂³×C₂₃	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
69. #21676	C₂₃	direct (N × H)
147. C₂₃	#21676	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×#2402	2
C₂₃×#1940	2
C₂₃×#1917	2
C₂₃×#1832	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1726	2
#21676	23

Subgroups — C23×#21676 #19169302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21676 #19169302