Subgroups — C₂₃×#2606 #15670225

Order 2944.

380 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#398	C₂	1	2	yes	maximal
2	C₂₃×#398	C₂	1	2	yes	maximal
3	C₂₃×#389	C₂	1	2	yes	maximal
4	C₂₃×#389	C₂	1	2	yes	maximal
5	C₂₃×#387	C₂	1	2	yes	maximal
6	C₂₃×#387	C₂	1	2	yes	maximal
7	C₂₃×#383	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#376	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#95	C₂²	1	2	yes	—
22	C₂₃×#95	C₂²	1	1	yes	characteristic
23	C₂₃×#95	C₂²	1	1	yes	characteristic
24	C₂₃×#95	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	1	yes	characteristic
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	1	yes	characteristic
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₄⋊C₄×C₂₃	—	2	4	no	—
41	C₄⋊C₄×C₂₃	—	2	4	no	—
42	C₄⋊C₄×C₂₃	—	2	4	no	—
43	C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₄⋊C₄×C₂₃	—	2	4	no	—
45	C₄⋊C₄×C₂₃	—	4	2	no	—
46	C₄⋊C₄×C₂₃	—	4	2	no	—
47	C₄²×C₂₃	—	2	4	no	—
48	C₄²×C₂₃	—	4	1	no	—
49	C₄²×C₂₃	—	4	1	no	—
50	C₄²×C₂₃	Q₈	1	4	yes	—
51	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	—	2	2	no	—
56	C₂×C₄×C₂₃	—	2	1	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₂₃	#41	1	4	yes	—
76	C₂×C₄×C₂₃	#41	1	4	yes	—
77	#2606	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₂₃	C₂×#41	1	2	yes	—
80	C₂²×C₂₃	C₂×#41	1	2	yes	—
81	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
83	C₄×C₂₃	—	2	2	no	—
84	C₄×C₂₃	—	2	2	no	—
85	C₄×C₂₃	—	2	2	no	—
86	C₄×C₂₃	—	2	4	no	—
87	C₄×C₂₃	—	2	4	no	—
88	C₄×C₂₃	—	4	2	no	—
89	C₄×C₂₃	—	4	2	no	—
90	C₄×C₂₃	—	4	2	no	—
91	C₄×C₂₃	—	4	2	no	—
92	#398	C₂×C₂₃	1	2	yes	—
93	#398	C₂×C₂₃	1	2	yes	—
94	#389	C₂×C₂₃	1	2	yes	—
95	#389	C₂×C₂₃	1	2	yes	—
96	#387	C₂×C₂₃	1	2	yes	—
97	#387	C₂×C₂₃	1	2	yes	—
98	#383	C₂×C₂₃	1	1	yes	characteristic
99	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
100	#376	C₂×C₂₃	1	1	yes	characteristic
101	C₂×C₂₃	#533	1	1	yes	characteristic
102	C₂×C₂₃	#533	1	1	yes	characteristic
103	C₂×C₂₃	#544	1	1	yes	characteristic
104	C₂×C₂₃	#552	1	2	yes	—
105	C₂×C₂₃	#553	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
108	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
109	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
113	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
114	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
115	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
117	#95	C₂²×C₂₃	1	2	yes	—
118	#95	C₂²×C₂₃	1	1	yes	characteristic
119	#95	C₂²×C₂₃	1	1	yes	characteristic
120	#95	C₂²×C₂₃	1	2	yes	—
121	#95	C₂²×C₂₃	1	1	yes	characteristic
122	#95	C₂²×C₂₃	1	2	yes	—
123	#95	C₂²×C₂₃	1	1	yes	characteristic
124	#95	C₂²×C₂₃	1	2	yes	—
125	#95	C₂²×C₂₃	1	2	yes	—
126	#95	C₂²×C₂₃	1	2	yes	—
127	C₂₃	#2606	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
129	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
130	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
132	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	4	1	no	—
146	C₄²	—	4	1	no	—
147	C₄²	Q₈×C₂₃	1	4	yes	—
148	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
170	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
171	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
172	C₂×C₄	#41×C₂₃	1	4	yes	—
173	C₂×C₄	#41×C₂₃	1	4	yes	—
174	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
175	C₂²	C₂×#41×C₂₃	1	2	yes	—
176	C₂²	C₂×#41×C₂₃	1	2	yes	—
177	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
178	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₂	C₂₃×#533	1	1	yes	characteristic
189	C₂	C₂₃×#533	1	1	yes	characteristic
190	C₂	C₂₃×#544	1	1	yes	characteristic
191	C₂	C₂₃×#552	1	2	yes	—
192	C₂	C₂₃×#553	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
77. #2606	C₂₃	direct (N × H)
127. C₂₃	#2606	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×#398	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#383	2
C₄×C₄⋊C₄×C₂₃	2
C₂₃×#376	2
#2606	23

Subgroups — C23×#2606 #15670225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2606 #15670225