Subgroups — C₂³×C₂₃×#124 #19210286

Order 5888.

14334 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×#124	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₂²⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁶×C₂₃	C₂²	1	2	yes	—
7	C₂³×Q₈×C₂₃	—	2	16	no	—
8	C₂³×Q₈×C₂₃	C₂²	1	28	yes	complemented
9	C₂³×D₄×C₂₃	—	2	16	no	—
10	C₂³×D₄×C₂₃	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#124	C₂²	1	448	yes	direct product factor
15	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	112	yes	complemented
16	C₂²×C₄²×C₂₃	C₂²	1	28	yes	complemented
17	C₂⁵×C₂₃	—	2	16	no	—
18	C₂⁵×C₂₃	—	2	32	no	—
19	C₂⁵×C₂₃	C₂³	1	28	yes	—
20	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₂₃	—	2	224	no	—
22	C₂²×Q₈×C₂₃	C₂³	1	112	yes	complemented
23	C₂²×D₄×C₂₃	—	2	224	no	—
24	C₂²×D₄×C₂₃	C₂³	1	112	yes	—
25	C₂³×C₄×C₂₃	—	2	16	no	—
26	C₂³×C₄×C₂₃	—	2	16	no	—
27	C₂³×C₄×C₂₃	D₄	1	16	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	28	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	28	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	14	yes	—
31	C₂₃×#124	C₂³	1	512	yes	direct product factor
32	C₂×C₂²⋊C₄×C₂₃	C₂³	1	448	yes	complemented
33	C₂×C₄²×C₂₃	C₂³	1	112	yes	complemented
34	C₂⁴×C₂₃	—	2	112	no	—
35	C₂⁴×C₂₃	—	2	224	no	—
36	C₂⁴×C₂₃	—	4	128	no	—
37	C₂⁴×C₂₃	C₂×D₄	1	8	yes	—
38	C₂⁴×C₂₃	#41	1	16	yes	—
39	C₂⁴×C₂₃	C₂⁴	1	56	yes	—
40	C₂⁴×C₂₃	C₂⁴	1	7	yes	—
41	C₂×Q₈×C₂₃	—	2	448	no	—
42	C₂×Q₈×C₂₃	C₂⁴	1	64	yes	complemented
43	C₂×D₄×C₂₃	—	2	448	no	—
44	C₂×D₄×C₂₃	C₂⁴	1	64	yes	—
45	C₂²×C₄×C₂₃	—	2	112	no	—
46	C₂²×C₄×C₂₃	—	2	112	no	—
47	C₂²×C₄×C₂₃	C₂×D₄	1	112	yes	—
48	C₂²×C₄×C₂₃	C₂⁴	1	56	yes	—
49	C₂²×C₄×C₂₃	C₂⁴	1	56	yes	—
50	C₂²×C₄×C₂₃	C₂⁴	1	28	yes	—
51	C₂²⋊C₄×C₂₃	C₂⁴	1	256	yes	complemented
52	C₄²×C₂₃	C₂⁴	1	64	yes	complemented
53	C₂³×#124	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₂₃	—	2	112	no	—
55	C₂³×C₂₃	—	2	224	no	—
56	C₂³×C₂₃	—	4	448	no	—
57	C₂³×C₂₃	#124	1	64	yes	direct product factor
58	C₂³×C₂₃	C₂²×D₄	1	28	yes	—
59	C₂³×C₂₃	C₂×#41	1	56	yes	—
60	C₂³×C₂₃	C₂⁵	1	16	yes	—
61	C₂³×C₂₃	C₂⁵	1	7	yes	—
62	Q₈×C₂₃	—	2	128	no	—
63	D₄×C₂₃	—	2	128	no	—
64	C₂×C₄×C₂₃	—	2	112	no	—
65	C₂×C₄×C₂₃	—	2	112	no	—
66	C₂×C₄×C₂₃	C₂²×D₄	1	112	yes	—
67	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
68	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
69	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
70	C₂⁴×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
71	C₂⁴×D₄	C₂×C₂₃	1	1	yes	characteristic
72	C₂²×#124	C₂×C₂₃	1	56	yes	direct product factor
73	C₂³×C₂²⋊C₄	C₂×C₂₃	1	4	yes	complemented
74	C₂³×C₄²	C₂×C₂₃	1	1	yes	characteristic, complemented
75	C₂²×C₂₃	—	2	16	no	—
76	C₂²×C₂₃	—	2	32	no	—
77	C₂²×C₂₃	—	4	224	no	—
78	C₂²×C₂₃	C₂×#124	1	112	yes	direct product factor
79	C₂²×C₂₃	C₂³×D₄	1	14	yes	—
80	C₂²×C₂₃	C₂²×#41	1	28	yes	—
81	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
82	C₄×C₂₃	—	2	16	no	—
83	C₄×C₂₃	—	2	16	no	—
84	C₄×C₂₃	C₂³×D₄	1	16	yes	—
85	C₂⁶	C₂²×C₂₃	1	2	yes	—
86	C₂³×Q₈	—	2	16	no	—
87	C₂³×Q₈	C₂²×C₂₃	1	28	yes	complemented
88	C₂³×D₄	—	2	16	no	—
89	C₂³×D₄	C₂²×C₂₃	1	28	yes	—
90	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
91	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
92	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
93	C₂×#124	C₂²×C₂₃	1	448	yes	direct product factor
94	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	112	yes	complemented
95	C₂²×C₄²	C₂²×C₂₃	1	28	yes	complemented
96	C₂×C₂₃	—	4	16	no	—
97	C₂×C₂₃	C₂²×#124	1	28	yes	direct product factor
98	C₂×C₂₃	C₂⁴×D₄	1	1	yes	characteristic
99	C₂×C₂₃	C₂³×#41	1	2	yes	—
100	C₂⁵	—	2	16	no	—
101	C₂⁵	—	2	32	no	—
102	C₂⁵	C₂³×C₂₃	1	28	yes	—
103	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
104	C₂²×Q₈	—	2	224	no	—
105	C₂²×Q₈	C₂³×C₂₃	1	112	yes	complemented
106	C₂²×D₄	—	2	224	no	—
107	C₂²×D₄	C₂³×C₂₃	1	112	yes	—
108	C₂³×C₄	—	2	16	no	—
109	C₂³×C₄	—	2	16	no	—
110	C₂³×C₄	D₄×C₂₃	1	16	yes	—
111	C₂³×C₄	C₂³×C₂₃	1	28	yes	—
112	C₂³×C₄	C₂³×C₂₃	1	28	yes	—
113	C₂³×C₄	C₂³×C₂₃	1	14	yes	—
114	#124	C₂³×C₂₃	1	512	yes	direct product factor
115	C₂×C₂²⋊C₄	C₂³×C₂₃	1	448	yes	complemented
116	C₂×C₄²	C₂³×C₂₃	1	112	yes	complemented
117	C₂₃	C₂³×#124	1	1	yes	characteristic, direct product factor, Sylow
118	C₂⁴	—	2	112	no	—
119	C₂⁴	—	2	224	no	—
120	C₂⁴	—	4	128	no	—
121	C₂⁴	C₂×D₄×C₂₃	1	8	yes	—
122	C₂⁴	#41×C₂₃	1	16	yes	—
123	C₂⁴	C₂⁴×C₂₃	1	56	yes	—
124	C₂⁴	C₂⁴×C₂₃	1	7	yes	—
125	C₂×Q₈	—	2	448	no	—
126	C₂×Q₈	C₂⁴×C₂₃	1	64	yes	complemented
127	C₂×D₄	—	2	448	no	—
128	C₂×D₄	C₂⁴×C₂₃	1	64	yes	—
129	C₂²×C₄	—	2	112	no	—
130	C₂²×C₄	—	2	112	no	—
131	C₂²×C₄	C₂×D₄×C₂₃	1	112	yes	—
132	C₂²×C₄	C₂⁴×C₂₃	1	56	yes	—
133	C₂²×C₄	C₂⁴×C₂₃	1	56	yes	—
134	C₂²×C₄	C₂⁴×C₂₃	1	28	yes	—
135	C₂²⋊C₄	C₂⁴×C₂₃	1	256	yes	complemented
136	C₄²	C₂⁴×C₂₃	1	64	yes	complemented
137	C₂³	—	2	112	no	—
138	C₂³	—	2	224	no	—
139	C₂³	—	4	448	no	—
140	C₂³	C₂₃×#124	1	64	yes	direct product factor
141	C₂³	C₂²×D₄×C₂₃	1	28	yes	—
142	C₂³	C₂×#41×C₂₃	1	56	yes	—
143	C₂³	C₂⁵×C₂₃	1	16	yes	—
144	C₂³	C₂⁵×C₂₃	1	7	yes	—
145	Q₈	—	2	128	no	—
146	D₄	—	2	128	no	—
147	C₂×C₄	—	2	112	no	—
148	C₂×C₄	—	2	112	no	—
149	C₂×C₄	C₂²×D₄×C₂₃	1	112	yes	—
150	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
151	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
152	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
153	C₂²	—	2	16	no	—
154	C₂²	—	2	32	no	—
155	C₂²	—	4	224	no	—
156	C₂²	C₂×C₂₃×#124	1	112	yes	direct product factor
157	C₂²	C₂³×D₄×C₂₃	1	14	yes	—
158	C₂²	C₂²×#41×C₂₃	1	28	yes	—
159	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
160	C₄	—	2	16	no	—
161	C₄	—	2	16	no	—
162	C₄	C₂³×D₄×C₂₃	1	16	yes	—
163	C₂	—	4	16	no	—
164	C₂	C₂²×C₂₃×#124	1	28	yes	direct product factor
165	C₂	C₂⁴×D₄×C₂₃	1	1	yes	characteristic
166	C₂	C₂³×#41×C₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₂₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
prime radical	C₂⁴×D₄×C₂₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₂₃	C₂	semidirect
3. C₂²×C₂₃×#124	C₂	direct (N × H)
4. C₂³×C₂²⋊C₄×C₂₃	C₂	semidirect
5. C₂³×C₄²×C₂₃	C₂	semidirect
8. C₂³×Q₈×C₂₃	C₂²	semidirect
14. C₂×C₂₃×#124	C₂²	direct (N × H)
15. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
16. C₂²×C₄²×C₂₃	C₂²	semidirect
22. C₂²×Q₈×C₂₃	C₂³	semidirect
31. C₂₃×#124	C₂³	direct (N × H)
32. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
33. C₂×C₄²×C₂₃	C₂³	semidirect
42. C₂×Q₈×C₂₃	C₂⁴	semidirect
51. C₂²⋊C₄×C₂₃	C₂⁴	semidirect
52. C₄²×C₂₃	C₂⁴	semidirect
53. C₂³×#124	C₂₃	direct (N × H)
57. C₂³×C₂₃	#124	direct (N × H)
70. C₂⁴×Q₈	C₂×C₂₃	semidirect
72. C₂²×#124	C₂×C₂₃	direct (N × H)
73. C₂³×C₂²⋊C₄	C₂×C₂₃	semidirect
74. C₂³×C₄²	C₂×C₂₃	semidirect
78. C₂²×C₂₃	C₂×#124	direct (N × H)
87. C₂³×Q₈	C₂²×C₂₃	semidirect
93. C₂×#124	C₂²×C₂₃	direct (N × H)
94. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
95. C₂²×C₄²	C₂²×C₂₃	semidirect
97. C₂×C₂₃	C₂²×#124	direct (N × H)
105. C₂²×Q₈	C₂³×C₂₃	semidirect
114. #124	C₂³×C₂₃	direct (N × H)
115. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
116. C₂×C₄²	C₂³×C₂₃	semidirect
117. C₂₃	C₂³×#124	direct (N × H)
126. C₂×Q₈	C₂⁴×C₂₃	semidirect
135. C₂²⋊C₄	C₂⁴×C₂₃	semidirect
136. C₄²	C₂⁴×C₂₃	semidirect
140. C₂³	C₂₃×#124	direct (N × H)
156. C₂²	C₂×C₂₃×#124	direct (N × H)
164. C₂	C₂²×C₂₃×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₂₃	2
C₂⁴×D₄×C₂₃	2
C₂²×C₂₃×#124	2
C₂³×C₂²⋊C₄×C₂₃	2
C₂³×C₄²×C₂₃	2
C₂³×#124	23

Subgroups — C23×C23×#124 #19210286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#124 #19210286