Subgroups — C₂×C₂₃×#431 #15670536

Order 2944.

528 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#431	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₃	C₂²	1	4	yes	—
10	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₂₃	C₄	1	4	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊C₈	C₂²	1	8	yes	complemented
17	C₂₃×C₂²⋊C₈	C₂²	1	8	yes	—
18	C₄×C₈×C₂₃	C₂²	1	8	yes	complemented
19	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	—	2	4	no	—
21	C₂²×C₄×C₂₃	—	2	4	no	—
22	C₂²×C₄×C₂₃	C₈	1	8	yes	complemented
23	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₂₃	—	2	8	no	—
34	C₂×C₈×C₂₃	C₂³	1	8	yes	—
35	C₄²×C₂₃	C₂×C₄	1	4	yes	—
36	C₄²×C₂₃	C₂×C₄	1	4	yes	—
37	C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₄²×C₂₃	C₂×C₄	1	2	yes	—
39	C₄²×C₂₃	C₂³	1	2	yes	—
40	C₄²×C₂₃	C₂³	1	2	yes	—
41	C₂³×C₂₃	—	2	2	no	—
42	C₂³×C₂₃	—	2	2	no	—
43	C₂³×C₂₃	C₂×C₈	1	4	yes	—
44	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
45	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	2	8	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
52	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
53	C₂×C₄×C₂₃	C₂×C₈	1	16	yes	complemented
54	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₃	#41	1	4	yes	—
62	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
63	C₈×C₂₃	—	2	8	no	—
64	C₂×#431	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₂₃	—	2	2	no	—
66	C₂²×C₂₃	—	2	2	no	—
67	C₂²×C₂₃	—	2	8	no	—
68	C₂²×C₂₃	#117	1	2	yes	—
69	C₂²×C₂₃	C₂²×C₈	1	4	yes	—
70	C₂²×C₂₃	C₂²×C₈	1	2	yes	—
71	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
72	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₂₃	—	2	2	no	—
74	C₄×C₂₃	—	2	2	no	—
75	C₄×C₂₃	#117	1	2	yes	—
76	C₄×C₂₃	C₂²×C₈	1	4	yes	—
77	C₄×C₂₃	C₂²×C₈	1	4	yes	—
78	C₄×C₂₃	C₂×M₁₆	1	4	yes	—
79	C₄×C₂₃	C₂×#41	1	2	yes	—
80	C₂²×C₄²	C₂×C₂₃	1	1	yes	characteristic
81	#431	C₂×C₂₃	1	8	yes	direct product factor
82	C₂×C₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
83	C₂×C₂²⋊C₈	C₂×C₂₃	1	2	yes	—
84	C₂×C₄×C₈	C₂×C₂₃	1	2	yes	complemented
85	C₂×C₂₃	—	2	4	no	—
86	C₂×C₂₃	#431	1	4	yes	direct product factor
87	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
88	C₂×C₂₃	C₂³×C₈	1	1	yes	characteristic
89	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
90	C₂³×C₄	C₄×C₂₃	1	1	yes	characteristic
91	C₂³×C₄	C₄×C₂₃	1	1	yes	characteristic
92	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₂₃	1	4	yes	—
94	C₂×C₄²	C₄×C₂₃	1	1	yes	characteristic
95	C₂×C₄²	C₄×C₂₃	1	4	yes	—
96	C₂×C₄²	C₄×C₂₃	1	1	yes	characteristic
97	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
98	C₂×C₄²	C₂²×C₂₃	1	4	yes	—
99	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
100	C₄⋊C₈	C₂²×C₂₃	1	8	yes	complemented
101	C₂²⋊C₈	C₂²×C₂₃	1	8	yes	—
102	C₄×C₈	C₂²×C₂₃	1	8	yes	complemented
103	C₂₃	C₂×#431	1	1	yes	characteristic, direct product factor, Sylow
104	C₂⁴	C₂×C₄×C₂₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	C₈×C₂₃	1	8	yes	complemented
108	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
110	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
112	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
115	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
116	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
117	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
118	C₂×C₈	—	2	8	no	—
119	C₂×C₈	C₂³×C₂₃	1	8	yes	—
120	C₄²	C₂×C₄×C₂₃	1	4	yes	—
121	C₄²	C₂×C₄×C₂₃	1	4	yes	—
122	C₄²	C₂×C₄×C₂₃	1	2	yes	—
123	C₄²	C₂×C₄×C₂₃	1	2	yes	—
124	C₄²	C₂³×C₂₃	1	2	yes	—
125	C₄²	C₂³×C₂₃	1	2	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	C₂×C₈×C₂₃	1	4	yes	—
129	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
130	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	C₂×C₈×C₂₃	1	4	yes	—
137	C₂×C₄	C₂×C₈×C₂₃	1	4	yes	—
138	C₂×C₄	C₂×C₈×C₂₃	1	16	yes	complemented
139	C₂×C₄	M₁₆×C₂₃	1	4	yes	—
140	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
141	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
142	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
144	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
145	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
146	C₂×C₄	#41×C₂₃	1	4	yes	—
147	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
148	C₈	—	2	8	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	8	no	—
151	C₂²	—	2	2	no	—
152	C₂²	C₂₃×#117	1	2	yes	—
153	C₂²	C₂²×C₈×C₂₃	1	4	yes	—
154	C₂²	C₂²×C₈×C₂₃	1	2	yes	—
155	C₂²	C₂×M₁₆×C₂₃	1	2	yes	—
156	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	C₂₃×#117	1	2	yes	—
160	C₄	C₂²×C₈×C₂₃	1	4	yes	—
161	C₄	C₂²×C₈×C₂₃	1	4	yes	—
162	C₄	C₂×M₁₆×C₂₃	1	4	yes	—
163	C₄	C₂×#41×C₂₃	1	2	yes	—
164	C₂	—	2	4	no	—
165	C₂	C₂₃×#431	1	4	yes	direct product factor
166	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
167	C₂	C₂³×C₈×C₂₃	1	1	yes	characteristic
168	C₂	C₂²×M₁₆×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₂₃	C₂²
commutator	C₂	C₂³×C₈×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#431	C₂	direct (N × H)
3. C₂×C₂₃×C₄⋊C₈	C₂	semidirect
5. C₂×C₄×C₈×C₂₃	C₂	semidirect
16. C₂₃×C₄⋊C₈	C₂²	semidirect
18. C₄×C₈×C₂₃	C₂²	semidirect
22. C₂²×C₄×C₂₃	C₈	semidirect
53. C₂×C₄×C₂₃	C₂×C₈	semidirect
64. C₂×#431	C₂₃	direct (N × H)
81. #431	C₂×C₂₃	direct (N × H)
82. C₂×C₄⋊C₈	C₂×C₂₃	semidirect
84. C₂×C₄×C₈	C₂×C₂₃	semidirect
86. C₂×C₂₃	#431	direct (N × H)
100. C₄⋊C₈	C₂²×C₂₃	semidirect
102. C₄×C₈	C₂²×C₂₃	semidirect
103. C₂₃	C₂×#431	direct (N × H)
107. C₂²×C₄	C₈×C₂₃	semidirect
138. C₂×C₄	C₂×C₈×C₂₃	semidirect
165. C₂	C₂₃×#431	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×C₄²×C₂₃	2
C₂₃×#431	2
C₂×C₂₃×C₄⋊C₈	2
C₂×C₂₃×C₂²⋊C₈	2
C₂×C₄×C₈×C₂₃	2
C₂×#431	23

Subgroups — C2×C23×#431 #15670536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#431 #15670536