Subgroups — C₂×C₂₃×#436 #15670547

Order 2944.

552 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#436	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄×C₂₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
9	C₂×M₁₆×C₂₃	—	2	8	no	—
10	C₂×M₁₆×C₂₃	C₂²	1	8	yes	complemented
11	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₂₃	C₄	1	4	yes	—
15	C₄×D₄×C₂₃	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
20	C₂₃×C₂²⋊C₈	C₂²	1	8	yes	complemented
21	C₄×C₈×C₂₃	C₂²	1	4	yes	complemented
22	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₂₃	—	2	4	no	—
24	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₃	—	2	4	no	—
26	C₂²×C₄×C₂₃	—	2	4	no	—
27	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	M₁₆×C₂₃	—	2	16	no	—
34	C₂×C₈×C₂₃	—	2	4	no	—
35	C₂×C₈×C₂₃	—	2	2	no	—
36	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
37	C₂×C₈×C₂₃	C₂³	1	4	yes	—
38	C₂×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂×C₈×C₂₃	C₂³	1	2	yes	—
40	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
42	C₄²×C₂₃	C₂×C₄	1	2	yes	—
43	C₄²×C₂₃	C₂³	1	2	yes	—
44	C₂³×C₂₃	—	2	4	no	—
45	C₂³×C₂₃	—	2	4	no	—
46	C₂³×C₂₃	—	2	4	no	—
47	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
48	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	D₄×C₂₃	—	2	8	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	8	no	—
53	C₂×C₄×C₂₃	—	2	4	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	M₁₆	1	4	yes	complemented
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₃	#41	1	2	yes	—
62	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
63	C₈×C₂₃	—	2	4	no	—
64	C₈×C₂₃	—	2	2	no	—
65	C₈×C₂₃	C₂×D₄	1	4	yes	complemented
66	C₂×#436	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₂₃	—	2	4	no	—
68	C₂²×C₂₃	—	2	4	no	—
69	C₂²×C₂₃	—	2	4	no	—
70	C₂²×C₂₃	—	4	8	no	—
71	C₂²×C₂₃	C₄×D₄	1	2	yes	—
72	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
73	C₂²×C₂₃	#131	1	2	yes	—
74	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
75	C₄×C₂₃	—	2	4	no	—
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	C₄×D₄	1	2	yes	—
78	C₄×C₂₃	C₂×M₁₆	1	4	yes	complemented
79	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
81	C₂²×M₁₆	C₂×C₂₃	1	2	yes	complemented
82	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
83	#436	C₂×C₂₃	1	8	yes	direct product factor
84	C₂×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
85	C₂×C₂²⋊C₈	C₂×C₂₃	1	2	yes	complemented
86	C₂×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
87	C₂×C₂₃	—	4	4	no	—
88	C₂×C₂₃	#436	1	4	yes	direct product factor
89	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
90	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
91	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
92	C₂²×D₄	C₄×C₂₃	1	1	yes	characteristic
93	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
94	C₂×M₁₆	—	2	8	no	—
95	C₂×M₁₆	C₂²×C₂₃	1	8	yes	complemented
96	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
97	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
99	C₄×D₄	C₄×C₂₃	1	4	yes	—
100	C₄×D₄	C₂²×C₂₃	1	4	yes	—
101	C₂×C₄⋊C₄	C₄×C₂₃	1	1	yes	characteristic
102	C₂×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
103	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
104	C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
105	C₂²⋊C₈	C₂²×C₂₃	1	8	yes	complemented
106	C₄×C₈	C₂²×C₂₃	1	4	yes	complemented
107	C₂₃	C₂×#436	1	1	yes	characteristic, direct product factor, Sylow
108	C₂⁴	C₂×C₄×C₂₃	1	2	yes	—
109	C₂×D₄	—	2	4	no	—
110	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
115	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
116	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
117	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
118	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
119	M₁₆	—	2	16	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	D₄×C₂₃	1	4	yes	complemented
123	C₂×C₈	C₂³×C₂₃	1	2	yes	—
124	C₂×C₈	C₂³×C₂₃	1	4	yes	—
125	C₂×C₈	C₂³×C₂₃	1	2	yes	—
126	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
127	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
128	C₄²	C₂×C₄×C₂₃	1	2	yes	—
129	C₄²	C₂³×C₂₃	1	2	yes	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	4	no	—
132	C₂³	—	2	4	no	—
133	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
134	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
135	D₄	—	2	8	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	M₁₆×C₂₃	1	4	yes	complemented
142	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
143	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
144	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
147	C₂×C₄	#41×C₂₃	1	2	yes	—
148	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
149	C₈	—	2	4	no	—
150	C₈	—	2	2	no	—
151	C₈	C₂×D₄×C₂₃	1	4	yes	complemented
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	4	8	no	—
156	C₂²	C₄×D₄×C₂₃	1	2	yes	—
157	C₂²	C₂×M₁₆×C₂₃	1	2	yes	—
158	C₂²	C₂₃×#131	1	2	yes	—
159	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	4	no	—
162	C₄	C₄×D₄×C₂₃	1	2	yes	—
163	C₄	C₂×M₁₆×C₂₃	1	4	yes	complemented
164	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
165	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
166	C₂	—	4	4	no	—
167	C₂	C₂₃×#436	1	4	yes	direct product factor
168	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
169	C₂	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
170	C₂	C₂×C₂₃×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×C₄
prime radical	C₂²×D₄×C₂₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₁₆×C₂₃	C₂	semidirect
3. C₂₃×#436	C₂	direct (N × H)
4. C₂×C₂₃×C₄⋊C₈	C₂	semidirect
5. C₂×C₂₃×C₂²⋊C₈	C₂	semidirect
6. C₂×C₄×C₈×C₂₃	C₂	semidirect
10. C₂×M₁₆×C₂₃	C₂²	semidirect
19. C₂₃×C₄⋊C₈	C₂²	semidirect
20. C₂₃×C₂²⋊C₈	C₂²	semidirect
21. C₄×C₈×C₂₃	C₂²	semidirect
36. C₂×C₈×C₂₃	D₄	semidirect
55. C₂×C₄×C₂₃	M₁₆	semidirect
65. C₈×C₂₃	C₂×D₄	semidirect
66. C₂×#436	C₂₃	direct (N × H)
78. C₄×C₂₃	C₂×M₁₆	semidirect
81. C₂²×M₁₆	C₂×C₂₃	semidirect
83. #436	C₂×C₂₃	direct (N × H)
84. C₂×C₄⋊C₈	C₂×C₂₃	semidirect
85. C₂×C₂²⋊C₈	C₂×C₂₃	semidirect
86. C₂×C₄×C₈	C₂×C₂₃	semidirect
88. C₂×C₂₃	#436	direct (N × H)
95. C₂×M₁₆	C₂²×C₂₃	semidirect
104. C₄⋊C₈	C₂²×C₂₃	semidirect
105. C₂²⋊C₈	C₂²×C₂₃	semidirect
106. C₄×C₈	C₂²×C₂₃	semidirect
107. C₂₃	C₂×#436	direct (N × H)
122. C₂×C₈	D₄×C₂₃	semidirect
141. C₂×C₄	M₁₆×C₂₃	semidirect
151. C₈	C₂×D₄×C₂₃	semidirect
163. C₄	C₂×M₁₆×C₂₃	semidirect
167. C₂	C₂₃×#436	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×M₁₆×C₂₃	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#436	2
C₂×C₂₃×C₄⋊C₈	2
C₂×C₂₃×C₂²⋊C₈	2
C₂×C₄×C₈×C₂₃	2
C₂×#436	23

Subgroups — C2×C23×#436 #15670547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#436 #15670547