Subgroups — C₂×C₂₃×#1369 #19159627

Order 5888.

222 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	2	yes	maximal
2	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1369	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₂₃×M₃₂	—	2	2	no	—
5	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#364	—	2	2	no	—
9	C₂₃×#364	C₂²	1	4	yes	—
10	C₂₃×#346	C₄	1	2	yes	—
11	C₂₃×#346	C₂²	1	2	yes	—
12	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
15	C₂₃×M₃₂	—	2	4	no	—
16	C₂₃×M₃₂	—	4	2	no	—
17	C₂×C₁₆×C₂₃	—	2	1	no	—
18	C₂×C₁₆×C₂₃	—	2	1	no	—
19	C₂×C₁₆×C₂₃	—	4	2	no	—
20	C₂×C₁₆×C₂₃	C₈	1	2	yes	—
21	C₂×C₁₆×C₂₃	C₈	1	2	yes	—
22	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	—	2	1	no	—
29	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
35	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₂₃	—	2	1	no	—
41	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
43	C₁₆×C₂₃	—	2	2	no	—
44	C₁₆×C₂₃	—	4	4	no	—
45	C₁₆×C₂₃	C₂×C₈	1	4	yes	—
46	C₂×#1369	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₂₃	#108	1	1	yes	characteristic
48	C₂×C₄×C₂₃	—	2	1	no	—
49	C₂×C₄×C₂₃	—	2	1	no	—
50	C₂×C₄×C₂₃	—	4	2	no	—
51	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
55	C₂×C₄×C₂₃	#108	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₈×C₂₃	C₄⋊C₈	1	1	yes	characteristic
58	C₈×C₂₃	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₂₃	#106	1	2	yes	—
60	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
61	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
63	C₂×#364	C₂×C₂₃	1	2	yes	—
64	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
65	#1369	C₂×C₂₃	1	4	yes	direct product factor
66	C₂²×C₂₃	—	2	2	no	—
67	C₂²×C₂₃	#334	1	1	yes	characteristic
68	C₂²×C₂₃	#334	1	1	yes	characteristic
69	C₂²×C₂₃	C₂×#108	1	1	yes	characteristic
70	C₄×C₂₃	—	4	1	no	—
71	C₄×C₂₃	—	4	1	no	—
72	C₄×C₂₃	#334	1	1	yes	characteristic
73	C₄×C₂₃	#334	1	1	yes	characteristic
74	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
76	C₂×M₃₂	—	2	2	no	—
77	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
78	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
80	#364	—	2	2	no	—
81	#364	C₂²×C₂₃	1	4	yes	—
82	#346	C₄×C₂₃	1	2	yes	—
83	#346	C₂²×C₂₃	1	2	yes	—
84	C₂×C₂₃	—	2	1	no	—
85	C₂×C₂₃	—	2	1	no	—
86	C₂×C₂₃	#1369	1	2	yes	direct product factor
87	C₂×C₂₃	C₂×#334	1	1	yes	characteristic
88	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
89	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
90	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
91	M₃₂	—	2	4	no	—
92	M₃₂	—	4	2	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	4	2	no	—
96	C₂×C₁₆	C₈×C₂₃	1	2	yes	—
97	C₂×C₁₆	C₈×C₂₃	1	2	yes	—
98	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
99	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
100	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
101	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
102	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
103	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
104	C₂₃	C₂×#1369	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
107	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
108	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
109	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
110	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
111	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
112	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
113	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
114	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
115	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
116	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
119	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	4	4	no	—
122	C₁₆	C₂×C₈×C₂₃	1	4	yes	—
123	C₂³	C₂₃×#108	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
129	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
130	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
131	C₂×C₄	C₂₃×#108	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
133	C₈	C₂₃×C₄⋊C₈	1	1	yes	characteristic
134	C₈	C₂₃×C₄⋊C₈	1	1	yes	characteristic
135	C₈	C₂₃×#106	1	2	yes	—
136	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
137	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
138	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₂₃×#334	1	1	yes	characteristic
141	C₂²	C₂₃×#334	1	1	yes	characteristic
142	C₂²	C₂×C₂₃×#108	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	C₂₃×#334	1	1	yes	characteristic
146	C₄	C₂₃×#334	1	1	yes	characteristic
147	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
148	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	C₂₃×#1369	1	2	yes	direct product factor
152	C₂	C₂×C₂₃×#334	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#106
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	#334
prime radical	C₂³×C₂₃	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1369	C₂	direct (N × H)
46. C₂×#1369	C₂₃	direct (N × H)
65. #1369	C₂×C₂₃	direct (N × H)
86. C₂×C₂₃	#1369	direct (N × H)
104. C₂₃	C₂×#1369	direct (N × H)
151. C₂	C₂₃×#1369	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#364	2
C₂×C₂₃×#346	2
C₂₃×#1369	2
C₂×#1369	23

Subgroups — C2×C23×#1369 #19159627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1369 #19159627