Subgroups — C₂³×C₂₃×#117 #19210267

Order 5888.

14334 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#117	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
5	C₂³×C₄²×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₂₃	C₄	1	32	yes	complemented
8	C₂⁴×C₄×C₂₃	C₂²	1	4	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₂₃	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#117	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	56	yes	complemented
14	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	56	yes	—
15	C₂²×C₄²×C₂₃	C₂²	1	56	yes	complemented
16	C₂⁵×C₂₃	—	2	16	no	—
17	C₂⁵×C₂₃	C₂×C₄	1	16	yes	—
18	C₂⁵×C₂₃	C₂³	1	14	yes	—
19	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	—	2	32	no	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	16	yes	—
22	C₂³×C₄×C₂₃	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	448	yes	complemented
24	C₂³×C₄×C₂₃	C₂³	1	56	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	14	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	14	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	112	yes	—
28	C₂₃×#117	C₂³	1	512	yes	direct product factor
29	C₂×C₄⋊C₄×C₂₃	C₂³	1	224	yes	complemented
30	C₂×C₂²⋊C₄×C₂₃	C₂³	1	224	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	224	yes	complemented
32	C₂⁴×C₂₃	—	2	128	no	—
33	C₂⁴×C₂₃	—	2	112	no	—
34	C₂⁴×C₂₃	C₂²×C₄	1	112	yes	—
35	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	—
36	C₂⁴×C₂₃	#41	1	16	yes	—
37	C₂⁴×C₂₃	C₂⁴	1	28	yes	—
38	C₂⁴×C₂₃	C₂⁴	1	7	yes	—
39	C₂²×C₄×C₂₃	—	2	224	no	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	112	yes	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	112	yes	—
42	C₂²×C₄×C₂₃	C₂²×C₄	1	896	yes	complemented
43	C₂²×C₄×C₂₃	C₂⁴	1	112	yes	—
44	C₂²×C₄×C₂₃	C₂⁴	1	28	yes	—
45	C₂²×C₄×C₂₃	C₂⁴	1	28	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	64	yes	—
47	C₄⋊C₄×C₂₃	C₂⁴	1	128	yes	complemented
48	C₂²⋊C₄×C₂₃	C₂⁴	1	128	yes	—
49	C₄²×C₂₃	C₂⁴	1	128	yes	complemented
50	C₂³×#117	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₂₃	—	2	448	no	—
52	C₂³×C₂₃	—	2	112	no	—
53	C₂³×C₂₃	#117	1	64	yes	direct product factor
54	C₂³×C₂₃	C₂³×C₄	1	112	yes	—
55	C₂³×C₂₃	C₂³×C₄	1	28	yes	—
56	C₂³×C₂₃	C₂×#41	1	56	yes	—
57	C₂³×C₂₃	C₂⁵	1	8	yes	—
58	C₂³×C₂₃	C₂⁵	1	7	yes	—
59	C₂×C₄×C₂₃	—	2	224	no	—
60	C₂×C₄×C₂₃	C₂³×C₄	1	112	yes	—
61	C₂×C₄×C₂₃	C₂³×C₄	1	112	yes	—
62	C₂×C₄×C₂₃	C₂³×C₄	1	256	yes	complemented
63	C₂×C₄×C₂₃	C₂⁵	1	32	yes	—
64	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
65	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
66	C₂⁵×C₄	C₂×C₂₃	1	1	yes	characteristic
67	C₂²×#117	C₂×C₂₃	1	56	yes	direct product factor
68	C₂³×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
69	C₂³×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
70	C₂³×C₄²	C₂×C₂₃	1	2	yes	complemented
71	C₂²×C₂₃	—	2	16	no	—
72	C₂²×C₂₃	—	2	224	no	—
73	C₂²×C₂₃	C₂×#117	1	112	yes	direct product factor
74	C₂²×C₂₃	C₂⁴×C₄	1	16	yes	—
75	C₂²×C₂₃	C₂⁴×C₄	1	14	yes	—
76	C₂²×C₂₃	C₂²×#41	1	28	yes	—
77	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
78	C₄×C₂₃	—	2	32	no	—
79	C₄×C₂₃	C₂⁴×C₄	1	16	yes	—
80	C₄×C₂₃	C₂⁴×C₄	1	16	yes	—
81	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
82	C₂⁴×C₄	C₄×C₂₃	1	32	yes	complemented
83	C₂⁴×C₄	C₂²×C₂₃	1	4	yes	—
84	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
85	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
86	C₂⁴×C₄	C₂²×C₂₃	1	28	yes	—
87	C₂×#117	C₂²×C₂₃	1	448	yes	direct product factor
88	C₂²×C₄⋊C₄	C₂²×C₂₃	1	56	yes	complemented
89	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	56	yes	—
90	C₂²×C₄²	C₂²×C₂₃	1	56	yes	complemented
91	C₂×C₂₃	—	2	16	no	—
92	C₂×C₂₃	C₂²×#117	1	28	yes	direct product factor
93	C₂×C₂₃	C₂⁵×C₄	1	1	yes	characteristic
94	C₂×C₂₃	C₂³×#41	1	2	yes	—
95	C₂⁵	—	2	16	no	—
96	C₂⁵	C₂×C₄×C₂₃	1	16	yes	—
97	C₂⁵	C₂³×C₂₃	1	14	yes	—
98	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
99	C₂³×C₄	—	2	32	no	—
100	C₂³×C₄	C₂×C₄×C₂₃	1	16	yes	—
101	C₂³×C₄	C₂×C₄×C₂₃	1	16	yes	—
102	C₂³×C₄	C₂×C₄×C₂₃	1	448	yes	complemented
103	C₂³×C₄	C₂³×C₂₃	1	56	yes	—
104	C₂³×C₄	C₂³×C₂₃	1	14	yes	—
105	C₂³×C₄	C₂³×C₂₃	1	14	yes	—
106	C₂³×C₄	C₂³×C₂₃	1	112	yes	—
107	#117	C₂³×C₂₃	1	512	yes	direct product factor
108	C₂×C₄⋊C₄	C₂³×C₂₃	1	224	yes	complemented
109	C₂×C₂²⋊C₄	C₂³×C₂₃	1	224	yes	—
110	C₂×C₄²	C₂³×C₂₃	1	224	yes	complemented
111	C₂₃	C₂³×#117	1	1	yes	characteristic, direct product factor, Sylow
112	C₂⁴	—	2	128	no	—
113	C₂⁴	—	2	112	no	—
114	C₂⁴	C₂²×C₄×C₂₃	1	112	yes	—
115	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	—
116	C₂⁴	#41×C₂₃	1	16	yes	—
117	C₂⁴	C₂⁴×C₂₃	1	28	yes	—
118	C₂⁴	C₂⁴×C₂₃	1	7	yes	—
119	C₂²×C₄	—	2	224	no	—
120	C₂²×C₄	C₂²×C₄×C₂₃	1	112	yes	—
121	C₂²×C₄	C₂²×C₄×C₂₃	1	112	yes	—
122	C₂²×C₄	C₂²×C₄×C₂₃	1	896	yes	complemented
123	C₂²×C₄	C₂⁴×C₂₃	1	112	yes	—
124	C₂²×C₄	C₂⁴×C₂₃	1	28	yes	—
125	C₂²×C₄	C₂⁴×C₂₃	1	28	yes	—
126	C₂²×C₄	C₂⁴×C₂₃	1	64	yes	—
127	C₄⋊C₄	C₂⁴×C₂₃	1	128	yes	complemented
128	C₂²⋊C₄	C₂⁴×C₂₃	1	128	yes	—
129	C₄²	C₂⁴×C₂₃	1	128	yes	complemented
130	C₂³	—	2	112	no	—
131	C₂³	—	2	448	no	—
132	C₂³	C₂₃×#117	1	64	yes	direct product factor
133	C₂³	C₂³×C₄×C₂₃	1	112	yes	—
134	C₂³	C₂³×C₄×C₂₃	1	28	yes	—
135	C₂³	C₂×#41×C₂₃	1	56	yes	—
136	C₂³	C₂⁵×C₂₃	1	8	yes	—
137	C₂³	C₂⁵×C₂₃	1	7	yes	—
138	C₂×C₄	—	2	224	no	—
139	C₂×C₄	C₂³×C₄×C₂₃	1	112	yes	—
140	C₂×C₄	C₂³×C₄×C₂₃	1	112	yes	—
141	C₂×C₄	C₂³×C₄×C₂₃	1	256	yes	complemented
142	C₂×C₄	C₂⁵×C₂₃	1	32	yes	—
143	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
144	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
145	C₂²	—	2	224	no	—
146	C₂²	—	2	16	no	—
147	C₂²	C₂×C₂₃×#117	1	112	yes	direct product factor
148	C₂²	C₂⁴×C₄×C₂₃	1	16	yes	—
149	C₂²	C₂⁴×C₄×C₂₃	1	14	yes	—
150	C₂²	C₂²×#41×C₂₃	1	28	yes	—
151	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
152	C₄	—	2	32	no	—
153	C₄	C₂⁴×C₄×C₂₃	1	16	yes	—
154	C₄	C₂⁴×C₄×C₂₃	1	16	yes	—
155	C₂	—	2	16	no	—
156	C₂	C₂²×C₂₃×#117	1	28	yes	direct product factor
157	C₂	C₂⁵×C₄×C₂₃	1	1	yes	characteristic
158	C₂	C₂³×#41×C₂₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄×C₂₃	C₂²
commutator	C₂	C₂⁵×C₄×C₂₃
Frattini, cosocle	C₂²	C₂⁶×C₂₃
socle	C₂⁵×C₂₃	C₂³
prime radical	C₂⁶×C₂₃	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×#117	C₂	direct (N × H)
3. C₂³×C₄⋊C₄×C₂₃	C₂	semidirect
5. C₂³×C₄²×C₂₃	C₂	semidirect
7. C₂⁴×C₄×C₂₃	C₄	semidirect
12. C₂×C₂₃×#117	C₂²	direct (N × H)
13. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
15. C₂²×C₄²×C₂₃	C₂²	semidirect
23. C₂³×C₄×C₂₃	C₂×C₄	semidirect
28. C₂₃×#117	C₂³	direct (N × H)
29. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
31. C₂×C₄²×C₂₃	C₂³	semidirect
42. C₂²×C₄×C₂₃	C₂²×C₄	semidirect
47. C₄⋊C₄×C₂₃	C₂⁴	semidirect
49. C₄²×C₂₃	C₂⁴	semidirect
50. C₂³×#117	C₂₃	direct (N × H)
53. C₂³×C₂₃	#117	direct (N × H)
62. C₂×C₄×C₂₃	C₂³×C₄	semidirect
67. C₂²×#117	C₂×C₂₃	direct (N × H)
68. C₂³×C₄⋊C₄	C₂×C₂₃	semidirect
70. C₂³×C₄²	C₂×C₂₃	semidirect
73. C₂²×C₂₃	C₂×#117	direct (N × H)
82. C₂⁴×C₄	C₄×C₂₃	semidirect
87. C₂×#117	C₂²×C₂₃	direct (N × H)
88. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
90. C₂²×C₄²	C₂²×C₂₃	semidirect
92. C₂×C₂₃	C₂²×#117	direct (N × H)
102. C₂³×C₄	C₂×C₄×C₂₃	semidirect
107. #117	C₂³×C₂₃	direct (N × H)
108. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
110. C₂×C₄²	C₂³×C₂₃	semidirect
111. C₂₃	C₂³×#117	direct (N × H)
122. C₂²×C₄	C₂²×C₄×C₂₃	semidirect
127. C₄⋊C₄	C₂⁴×C₂₃	semidirect
129. C₄²	C₂⁴×C₂₃	semidirect
132. C₂³	C₂₃×#117	direct (N × H)
141. C₂×C₄	C₂³×C₄×C₂₃	semidirect
147. C₂²	C₂×C₂₃×#117	direct (N × H)
156. C₂	C₂²×C₂₃×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₂₃	2
C₂²×C₂₃×#117	2
C₂³×C₄⋊C₄×C₂₃	2
C₂³×C₂²⋊C₄×C₂₃	2
C₂³×C₄²×C₂₃	2
C₂³×#117	23

Subgroups — C23×C23×#117 #19210267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#117 #19210267