Subgroups — C₂×C₂₃×#563 #15671147

Order 2944.

1024 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	C₂₃×#563	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#126	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#125	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#123	C₂	1	2	yes	maximal
7	C₂×C₂₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
8	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
9	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂₃×C₄⋊Q₈	C₂²	1	16	yes	complemented
13	C₂₃×#126	C₂²	1	32	yes	complemented
14	C₂₃×#125	C₂²	1	16	yes	complemented
15	C₂₃×#124	C₂²	1	8	yes	—
16	C₂₃×#123	C₂²	1	16	yes	—
17	C₂₃×C₂²⋊Q₈	C₂²	1	32	yes	complemented
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
28	C₂⁴×C₂₃	C₂³	1	2	yes	—
29	C₂×Q₈×C₂₃	—	4	4	no	—
30	C₂×Q₈×C₂₃	—	4	2	no	—
31	C₂×Q₈×C₂₃	C₂³	1	8	yes	—
32	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
33	C₂×D₄×C₂₃	—	4	2	no	—
34	C₂×D₄×C₂₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₂₃	—	4	4	no	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
42	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
43	C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
44	C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
45	C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
46	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
47	C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
48	C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
49	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
50	C₄²×C₂₃	C₂³	1	4	yes	—
51	C₄²×C₂₃	C₂³	1	8	yes	—
52	C₂³×C₂₃	—	2	4	no	—
53	C₂³×C₂₃	—	2	8	no	—
54	C₂³×C₂₃	C₂⁴	1	4	yes	—
55	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
56	Q₈×C₂₃	—	4	8	no	—
57	Q₈×C₂₃	—	4	4	no	—
58	D₄×C₂₃	—	4	4	no	—
59	C₂×C₄×C₂₃	—	2	8	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	4	no	—
62	C₂×C₄×C₂₃	—	2	4	no	—
63	C₂×C₄×C₂₃	—	2	8	no	—
64	C₂×C₄×C₂₃	—	4	8	no	—
65	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
66	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
67	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
68	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
69	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
70	C₂×#563	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₂₃	—	2	4	no	—
72	C₂²×C₂₃	—	2	8	no	—
73	C₂²×C₂₃	—	4	8	no	—
74	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
75	C₂²×C₂₃	2₋¹⁺⁴	1	4	yes	—
76	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
77	C₄×C₂₃	—	2	8	no	—
78	C₄×C₂₃	—	2	2	no	—
79	C₄×C₂₃	—	2	4	no	—
80	C₄×C₂₃	—	2	4	no	—
81	C₄×C₂₃	—	2	8	no	—
82	#563	C₂×C₂₃	1	16	yes	direct product factor
83	C₂×C₄⋊Q₈	C₂×C₂₃	1	2	yes	complemented
84	C₂×#126	C₂×C₂₃	1	4	yes	complemented
85	C₂×#125	C₂×C₂₃	1	2	yes	complemented
86	C₂×#124	C₂×C₂₃	1	1	yes	characteristic
87	C₂×#123	C₂×C₂₃	1	2	yes	—
88	C₂×C₂²⋊Q₈	C₂×C₂₃	1	4	yes	complemented
89	C₂×C₂₃	—	4	4	no	—
90	C₂×C₂₃	#563	1	4	yes	direct product factor
91	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
92	C₂×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
93	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
94	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
95	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
96	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
97	C₄⋊Q₈	C₂²×C₂₃	1	16	yes	complemented
98	#126	C₂²×C₂₃	1	32	yes	complemented
99	#125	C₂²×C₂₃	1	16	yes	complemented
100	#124	C₂²×C₂₃	1	8	yes	—
101	#123	C₂²×C₂₃	1	16	yes	—
102	C₂²⋊Q₈	C₂²×C₂₃	1	32	yes	complemented
103	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
104	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
105	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
106	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
107	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
111	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
113	C₂₃	C₂×#563	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	C₂³×C₂₃	1	2	yes	—
115	C₂×Q₈	—	4	4	no	—
116	C₂×Q₈	—	4	2	no	—
117	C₂×Q₈	C₂³×C₂₃	1	8	yes	—
118	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
119	C₂×D₄	—	4	2	no	—
120	C₂×D₄	C₂³×C₂₃	1	4	yes	—
121	C₂²×C₄	—	4	4	no	—
122	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
123	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
124	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
125	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
127	C₂²×C₄	C₂³×C₂₃	1	8	yes	—
128	C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
129	C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
130	C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
131	C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
132	C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
133	C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
134	C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
135	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
136	C₄²	C₂³×C₂₃	1	4	yes	—
137	C₄²	C₂³×C₂₃	1	8	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	—	2	8	no	—
140	C₂³	C₂⁴×C₂₃	1	4	yes	—
141	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
142	Q₈	—	4	8	no	—
143	Q₈	—	4	4	no	—
144	D₄	—	4	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
152	C₂×C₄	C₂⁴×C₂₃	1	8	yes	—
153	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
154	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
155	C₂×C₄	C₂⁴×C₂₃	1	8	yes	—
156	C₂²	—	2	4	no	—
157	C₂²	—	2	8	no	—
158	C₂²	—	4	8	no	—
159	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
160	C₂²	C₂₃×2₋¹⁺⁴	1	4	yes	—
161	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
162	C₄	—	2	8	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	2	4	no	—
166	C₄	—	2	8	no	—
167	C₂	—	4	4	no	—
168	C₂	C₂₃×#563	1	4	yes	direct product factor
169	C₂	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
170	C₂	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#563	C₂	direct (N × H)
2. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂×C₂₃×#126	C₂	semidirect
4. C₂×C₂₃×#125	C₂	semidirect
7. C₂×C₂₃×C₂²⋊Q₈	C₂	semidirect
12. C₂₃×C₄⋊Q₈	C₂²	semidirect
13. C₂₃×#126	C₂²	semidirect
14. C₂₃×#125	C₂²	semidirect
17. C₂₃×C₂²⋊Q₈	C₂²	semidirect
70. C₂×#563	C₂₃	direct (N × H)
82. #563	C₂×C₂₃	direct (N × H)
83. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
84. C₂×#126	C₂×C₂₃	semidirect
85. C₂×#125	C₂×C₂₃	semidirect
88. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
90. C₂×C₂₃	#563	direct (N × H)
97. C₄⋊Q₈	C₂²×C₂₃	semidirect
98. #126	C₂²×C₂₃	semidirect
99. #125	C₂²×C₂₃	semidirect
102. C₂²⋊Q₈	C₂²×C₂₃	semidirect
113. C₂₃	C₂×#563	direct (N × H)
168. C₂	C₂₃×#563	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×#563	2
C₂×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#126	2
C₂×C₂₃×#125	2
C₂×C₂₃×#124	2
C₂×C₂₃×#123	2
C₂×C₂₃×C₂²⋊Q₈	2
C₂×#563	23

Subgroups — C2×C23×#563 #15671147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#563 #15671147