Subgroups — C₂²×C₂₃×#494 #19208256

Order 5888.

3262 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₂₃	C₂	1	4	yes	maximal
2	C₂²×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₂×C₂₃×#494	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₂₃	C₂²	1	4	yes	—
6	C₂²×Q₁₆×C₂₃	—	2	32	no	—
7	C₂²×Q₁₆×C₂₃	C₂²	1	48	yes	—
8	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊Q₈	—	2	12	no	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	24	yes	—
11	C₂²×C₄⋊C₄×C₂₃	—	2	4	no	—
12	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#494	C₂²	1	64	yes	direct product factor
14	C₂×C₄×C₈×C₂₃	C₂²	1	12	yes	—
15	C₂²×Q₈×C₂₃	—	2	12	no	—
16	C₂²×Q₈×C₂₃	—	2	32	no	—
17	C₂²×Q₈×C₂₃	C₂³	1	24	yes	—
18	C₂³×C₄×C₂₃	—	2	4	no	—
19	C₂³×C₄×C₂₃	D₄	1	2	yes	—
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×Q₁₆×C₂₃	—	2	192	no	—
22	C₂×Q₁₆×C₂₃	C₂³	1	64	yes	—
23	C₂²×C₈×C₂₃	D₄	1	16	yes	—
24	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
25	C₂₃×C₄⋊Q₈	—	2	48	no	—
26	C₂₃×C₄⋊Q₈	C₂³	1	32	yes	—
27	C₂×C₄⋊C₄×C₂₃	—	2	48	no	—
28	C₂×C₄²×C₂₃	D₄	1	6	yes	—
29	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
30	C₄×C₈×C₂₃	C₂³	1	16	yes	—
31	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×Q₈×C₂₃	—	2	24	no	—
33	C₂×Q₈×C₂₃	—	2	96	no	—
34	C₂×Q₈×C₂₃	—	4	48	no	—
35	C₂×Q₈×C₂₃	C₂⁴	1	16	yes	—
36	C₂²×C₄×C₂₃	—	2	24	no	—
37	C₂²×C₄×C₂₃	—	4	16	no	—
38	C₂²×C₄×C₂₃	Q₁₆	1	16	yes	complemented
39	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
42	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
43	Q₁₆×C₂₃	—	2	128	no	—
44	C₂×C₈×C₂₃	C₂×D₄	1	48	yes	—
45	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
46	C₄⋊C₄×C₂₃	—	2	64	no	—
47	C₄²×C₂₃	C₂×D₄	1	12	yes	—
48	C₄²×C₂₃	C₂⁴	1	4	yes	—
49	C₂²×#494	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₂₃	#127	1	4	yes	—
51	C₂³×C₂₃	C₂×Q₁₆	1	8	yes	—
52	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
53	Q₈×C₂₃	—	2	32	no	—
54	Q₈×C₂₃	—	4	48	no	—
55	C₂×C₄×C₂₃	—	2	16	no	—
56	C₂×C₄×C₂₃	—	4	48	no	—
57	C₂×C₄×C₂₃	#127	1	12	yes	—
58	C₂×C₄×C₂₃	C₂×Q₁₆	1	48	yes	complemented
59	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
60	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
61	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
62	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
63	C₈×C₂₃	C₂²×D₄	1	16	yes	—
64	C₂³×Q₁₆	C₂×C₂₃	1	4	yes	—
65	C₂²×C₄⋊Q₈	C₂×C₂₃	1	2	yes	—
66	C₂×#494	C₂×C₂₃	1	24	yes	direct product factor
67	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
68	C₂²×C₂₃	#494	1	16	yes	direct product factor
69	C₂²×C₂₃	C₂×#127	1	6	yes	—
70	C₂²×C₂₃	C₂²×Q₁₆	1	12	yes	—
71	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
72	C₄×C₂₃	—	4	16	no	—
73	C₄×C₂₃	C₂×#127	1	6	yes	—
74	C₄×C₂₃	C₂²×Q₁₆	1	16	yes	complemented
75	C₄×C₂₃	C₂³×D₄	1	2	yes	—
76	C₂³×Q₈	C₂²×C₂₃	1	4	yes	—
77	C₂²×Q₁₆	—	2	32	no	—
78	C₂²×Q₁₆	C₂²×C₂₃	1	48	yes	—
79	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
80	C₂×C₄⋊Q₈	—	2	12	no	—
81	C₂×C₄⋊Q₈	C₂²×C₂₃	1	24	yes	—
82	C₂²×C₄⋊C₄	—	2	4	no	—
83	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
84	#494	C₂²×C₂₃	1	64	yes	direct product factor
85	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	—
86	C₂×C₂₃	C₂×#494	1	12	yes	direct product factor
87	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
88	C₂×C₂₃	C₂³×Q₁₆	1	2	yes	—
89	C₂²×Q₈	—	2	12	no	—
90	C₂²×Q₈	—	2	32	no	—
91	C₂²×Q₈	C₂³×C₂₃	1	24	yes	—
92	C₂³×C₄	—	2	4	no	—
93	C₂³×C₄	D₄×C₂₃	1	2	yes	—
94	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
95	C₂×Q₁₆	—	2	192	no	—
96	C₂×Q₁₆	C₂³×C₂₃	1	64	yes	—
97	C₂²×C₈	D₄×C₂₃	1	16	yes	—
98	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
99	C₄⋊Q₈	—	2	48	no	—
100	C₄⋊Q₈	C₂³×C₂₃	1	32	yes	—
101	C₂×C₄⋊C₄	—	2	48	no	—
102	C₂×C₄²	D₄×C₂₃	1	6	yes	—
103	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
104	C₄×C₈	C₂³×C₂₃	1	16	yes	—
105	C₂₃	C₂²×#494	1	1	yes	characteristic, direct product factor, Sylow
106	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
107	C₂×Q₈	—	2	24	no	—
108	C₂×Q₈	—	2	96	no	—
109	C₂×Q₈	—	4	48	no	—
110	C₂×Q₈	C₂⁴×C₂₃	1	16	yes	—
111	C₂²×C₄	—	2	24	no	—
112	C₂²×C₄	—	4	16	no	—
113	C₂²×C₄	Q₁₆×C₂₃	1	16	yes	complemented
114	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
115	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
116	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
117	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
118	Q₁₆	—	2	128	no	—
119	C₂×C₈	C₂×D₄×C₂₃	1	48	yes	—
120	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
121	C₄⋊C₄	—	2	64	no	—
122	C₄²	C₂×D₄×C₂₃	1	12	yes	—
123	C₄²	C₂⁴×C₂₃	1	4	yes	—
124	C₂³	C₂₃×#127	1	4	yes	—
125	C₂³	C₂×Q₁₆×C₂₃	1	8	yes	—
126	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
127	Q₈	—	2	32	no	—
128	Q₈	—	4	48	no	—
129	C₂×C₄	—	2	16	no	—
130	C₂×C₄	—	4	48	no	—
131	C₂×C₄	C₂₃×#127	1	12	yes	—
132	C₂×C₄	C₂×Q₁₆×C₂₃	1	48	yes	complemented
133	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
134	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
135	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
136	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
137	C₈	C₂²×D₄×C₂₃	1	16	yes	—
138	C₂²	C₂₃×#494	1	16	yes	direct product factor
139	C₂²	C₂×C₂₃×#127	1	6	yes	—
140	C₂²	C₂²×Q₁₆×C₂₃	1	12	yes	—
141	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
142	C₄	—	4	16	no	—
143	C₄	C₂×C₂₃×#127	1	6	yes	—
144	C₄	C₂²×Q₁₆×C₂₃	1	16	yes	complemented
145	C₄	C₂³×D₄×C₂₃	1	2	yes	—
146	C₂	C₂×C₂₃×#494	1	12	yes	direct product factor
147	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
148	C₂	C₂³×Q₁₆×C₂₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#494	C₂	direct (N × H)
13. C₂₃×#494	C₂²	direct (N × H)
38. C₂²×C₄×C₂₃	Q₁₆	semidirect
49. C₂²×#494	C₂₃	direct (N × H)
58. C₂×C₄×C₂₃	C₂×Q₁₆	semidirect
66. C₂×#494	C₂×C₂₃	direct (N × H)
68. C₂²×C₂₃	#494	direct (N × H)
74. C₄×C₂₃	C₂²×Q₁₆	semidirect
84. #494	C₂²×C₂₃	direct (N × H)
86. C₂×C₂₃	C₂×#494	direct (N × H)
105. C₂₃	C₂²×#494	direct (N × H)
113. C₂²×C₄	Q₁₆×C₂₃	semidirect
132. C₂×C₄	C₂×Q₁₆×C₂₃	semidirect
138. C₂²	C₂₃×#494	direct (N × H)
144. C₄	C₂²×Q₁₆×C₂₃	semidirect
146. C₂	C₂×C₂₃×#494	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×Q₁₆×C₂₃	2
C₂²×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#494	2
C₂²×C₄×C₈×C₂₃	2
C₂²×#494	23

Subgroups — C22×C23×#494 #19208256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#494 #19208256