Subgroups — C₂³×C₂₃×#123 #19210284

Order 5888.

15150 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#123	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₂³×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
8	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₂₃	—	2	16	no	—
10	C₂³×D₄×C₂₃	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₂₃	—	2	16	no	—
12	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₂₃	C₂²	1	28	yes	complemented
16	C₂×C₂₃×#123	C₂²	1	448	yes	direct product factor
17	C₂²×C₄⋊C₄×C₂₃	C₂²	1	56	yes	complemented
18	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	56	yes	complemented
19	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	28	yes	complemented
20	C₂⁵×C₂₃	—	2	16	no	—
21	C₂⁵×C₂₃	—	2	8	no	—
22	C₂⁵×C₂₃	—	2	16	no	—
23	C₂⁵×C₂₃	D₄	1	16	yes	—
24	C₂⁵×C₂₃	C₂³	1	14	yes	—
25	C₂⁵×C₂₃	C₂³	1	14	yes	—
26	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²×D₄×C₂₃	—	2	224	no	—
28	C₂²×D₄×C₂₃	C₂³	1	112	yes	—
29	C₂³×C₄×C₂₃	—	2	16	no	—
30	C₂³×C₄×C₂₃	—	2	224	no	—
31	C₂³×C₄×C₂₃	—	2	8	no	—
32	C₂³×C₄×C₂₃	—	2	16	no	—
33	C₂³×C₄×C₂₃	C₂³	1	28	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	14	yes	—
35	C₂³×C₄×C₂₃	C₂³	1	28	yes	—
36	C₂³×C₄×C₂₃	C₂³	1	112	yes	complemented
37	C₂₃×#123	C₂³	1	512	yes	direct product factor
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	224	yes	complemented
39	C₂×C₂²⋊C₄×C₂₃	C₂³	1	112	yes	complemented
40	C₂×C₂²⋊C₄×C₂₃	C₂³	1	224	yes	complemented
41	C₂⁴×C₂₃	—	2	128	no	—
42	C₂⁴×C₂₃	—	2	112	no	—
43	C₂⁴×C₂₃	—	2	56	no	—
44	C₂⁴×C₂₃	—	2	112	no	—
45	C₂⁴×C₂₃	—	4	64	no	—
46	C₂⁴×C₂₃	C₂×D₄	1	112	yes	—
47	C₂⁴×C₂₃	C₂×D₄	1	8	yes	—
48	C₂⁴×C₂₃	#41	1	16	yes	—
49	C₂⁴×C₂₃	C₂⁴	1	28	yes	—
50	C₂⁴×C₂₃	C₂⁴	1	28	yes	—
51	C₂⁴×C₂₃	C₂⁴	1	7	yes	—
52	C₂×D₄×C₂₃	—	2	448	no	—
53	C₂×D₄×C₂₃	C₂⁴	1	64	yes	—
54	C₂²×C₄×C₂₃	—	2	112	no	—
55	C₂²×C₄×C₂₃	—	2	448	no	—
56	C₂²×C₄×C₂₃	—	2	56	no	—
57	C₂²×C₄×C₂₃	—	2	112	no	—
58	C₂²×C₄×C₂₃	C₂⁴	1	56	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	56	yes	—
60	C₂²×C₄×C₂₃	C₂⁴	1	28	yes	—
61	C₂²×C₄×C₂₃	C₂⁴	1	64	yes	complemented
62	C₄⋊C₄×C₂₃	C₂⁴	1	128	yes	complemented
63	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	complemented
64	C₂²⋊C₄×C₂₃	C₂⁴	1	128	yes	complemented
65	C₂³×#123	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	—	2	448	no	—
67	C₂³×C₂₃	—	2	112	no	—
68	C₂³×C₂₃	—	2	56	no	—
69	C₂³×C₂₃	—	2	112	no	—
70	C₂³×C₂₃	—	4	224	no	—
71	C₂³×C₂₃	#123	1	64	yes	direct product factor
72	C₂³×C₂₃	C₂²×D₄	1	112	yes	—
73	C₂³×C₂₃	C₂²×D₄	1	28	yes	—
74	C₂³×C₂₃	C₂×#41	1	56	yes	—
75	C₂³×C₂₃	C₂⁵	1	8	yes	—
76	C₂³×C₂₃	C₂⁵	1	8	yes	—
77	C₂³×C₂₃	C₂⁵	1	7	yes	—
78	D₄×C₂₃	—	2	128	no	—
79	C₂×C₄×C₂₃	—	2	112	no	—
80	C₂×C₄×C₂₃	—	2	128	no	—
81	C₂×C₄×C₂₃	—	2	56	no	—
82	C₂×C₄×C₂₃	—	2	112	no	—
83	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
84	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
85	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
86	C₂⁴×D₄	C₂×C₂₃	1	1	yes	characteristic
87	C₂⁵×C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
88	C₂²×#123	C₂×C₂₃	1	56	yes	direct product factor
89	C₂³×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
90	C₂³×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
91	C₂³×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
92	C₂²×C₂₃	—	2	16	no	—
93	C₂²×C₂₃	—	2	224	no	—
94	C₂²×C₂₃	—	2	8	no	—
95	C₂²×C₂₃	—	2	16	no	—
96	C₂²×C₂₃	—	4	112	no	—
97	C₂²×C₂₃	C₂×#123	1	112	yes	direct product factor
98	C₂²×C₂₃	C₂³×D₄	1	16	yes	—
99	C₂²×C₂₃	C₂³×D₄	1	14	yes	—
100	C₂²×C₂₃	C₂²×#41	1	28	yes	—
101	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
102	C₄×C₂₃	—	2	16	no	—
103	C₄×C₂₃	—	2	8	no	—
104	C₄×C₂₃	—	2	16	no	—
105	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
106	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
107	C₂³×D₄	—	2	16	no	—
108	C₂³×D₄	C₂²×C₂₃	1	28	yes	—
109	C₂⁴×C₄	—	2	16	no	—
110	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
111	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
112	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
113	C₂⁴×C₄	C₂²×C₂₃	1	28	yes	complemented
114	C₂×#123	C₂²×C₂₃	1	448	yes	direct product factor
115	C₂²×C₄⋊C₄	C₂²×C₂₃	1	56	yes	complemented
116	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	28	yes	complemented
117	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	56	yes	complemented
118	C₂×C₂₃	—	2	16	no	—
119	C₂×C₂₃	—	4	8	no	—
120	C₂×C₂₃	C₂²×#123	1	28	yes	direct product factor
121	C₂×C₂₃	C₂⁴×D₄	1	1	yes	characteristic
122	C₂×C₂₃	C₂³×#41	1	2	yes	—
123	C₂⁵	—	2	16	no	—
124	C₂⁵	—	2	8	no	—
125	C₂⁵	—	2	16	no	—
126	C₂⁵	D₄×C₂₃	1	16	yes	—
127	C₂⁵	C₂³×C₂₃	1	14	yes	—
128	C₂⁵	C₂³×C₂₃	1	14	yes	—
129	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
130	C₂²×D₄	—	2	224	no	—
131	C₂²×D₄	C₂³×C₂₃	1	112	yes	—
132	C₂³×C₄	—	2	16	no	—
133	C₂³×C₄	—	2	224	no	—
134	C₂³×C₄	—	2	8	no	—
135	C₂³×C₄	—	2	16	no	—
136	C₂³×C₄	C₂³×C₂₃	1	28	yes	—
137	C₂³×C₄	C₂³×C₂₃	1	28	yes	—
138	C₂³×C₄	C₂³×C₂₃	1	14	yes	—
139	C₂³×C₄	C₂³×C₂₃	1	112	yes	complemented
140	#123	C₂³×C₂₃	1	512	yes	direct product factor
141	C₂×C₄⋊C₄	C₂³×C₂₃	1	224	yes	complemented
142	C₂×C₂²⋊C₄	C₂³×C₂₃	1	112	yes	complemented
143	C₂×C₂²⋊C₄	C₂³×C₂₃	1	224	yes	complemented
144	C₂₃	C₂³×#123	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁴	—	2	128	no	—
146	C₂⁴	—	2	112	no	—
147	C₂⁴	—	2	56	no	—
148	C₂⁴	—	2	112	no	—
149	C₂⁴	—	4	64	no	—
150	C₂⁴	C₂×D₄×C₂₃	1	112	yes	—
151	C₂⁴	C₂×D₄×C₂₃	1	8	yes	—
152	C₂⁴	#41×C₂₃	1	16	yes	—
153	C₂⁴	C₂⁴×C₂₃	1	28	yes	—
154	C₂⁴	C₂⁴×C₂₃	1	28	yes	—
155	C₂⁴	C₂⁴×C₂₃	1	7	yes	—
156	C₂×D₄	—	2	448	no	—
157	C₂×D₄	C₂⁴×C₂₃	1	64	yes	—
158	C₂²×C₄	—	2	112	no	—
159	C₂²×C₄	—	2	448	no	—
160	C₂²×C₄	—	2	56	no	—
161	C₂²×C₄	—	2	112	no	—
162	C₂²×C₄	C₂⁴×C₂₃	1	56	yes	—
163	C₂²×C₄	C₂⁴×C₂₃	1	56	yes	—
164	C₂²×C₄	C₂⁴×C₂₃	1	28	yes	—
165	C₂²×C₄	C₂⁴×C₂₃	1	64	yes	complemented
166	C₄⋊C₄	C₂⁴×C₂₃	1	128	yes	complemented
167	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	complemented
168	C₂²⋊C₄	C₂⁴×C₂₃	1	128	yes	complemented
169	C₂³	—	2	112	no	—
170	C₂³	—	2	448	no	—
171	C₂³	—	2	56	no	—
172	C₂³	—	2	112	no	—
173	C₂³	—	4	224	no	—
174	C₂³	C₂₃×#123	1	64	yes	direct product factor
175	C₂³	C₂²×D₄×C₂₃	1	112	yes	—
176	C₂³	C₂²×D₄×C₂₃	1	28	yes	—
177	C₂³	C₂×#41×C₂₃	1	56	yes	—
178	C₂³	C₂⁵×C₂₃	1	8	yes	—
179	C₂³	C₂⁵×C₂₃	1	8	yes	—
180	C₂³	C₂⁵×C₂₃	1	7	yes	—
181	D₄	—	2	128	no	—
182	C₂×C₄	—	2	112	no	—
183	C₂×C₄	—	2	128	no	—
184	C₂×C₄	—	2	56	no	—
185	C₂×C₄	—	2	112	no	—
186	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
187	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
188	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
189	C₂²	—	2	224	no	—
190	C₂²	—	2	16	no	—
191	C₂²	—	2	8	no	—
192	C₂²	—	2	16	no	—
193	C₂²	—	4	112	no	—
194	C₂²	C₂×C₂₃×#123	1	112	yes	direct product factor
195	C₂²	C₂³×D₄×C₂₃	1	16	yes	—
196	C₂²	C₂³×D₄×C₂₃	1	14	yes	—
197	C₂²	C₂²×#41×C₂₃	1	28	yes	—
198	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
199	C₄	—	2	16	no	—
200	C₄	—	2	8	no	—
201	C₄	—	2	16	no	—
202	C₂	—	2	16	no	—
203	C₂	—	4	8	no	—
204	C₂	C₂²×C₂₃×#123	1	28	yes	direct product factor
205	C₂	C₂⁴×D₄×C₂₃	1	1	yes	characteristic
206	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
2. C₂⁵×C₄×C₂₃	C₂	semidirect
3. C₂²×C₂₃×#123	C₂	direct (N × H)
4. C₂³×C₄⋊C₄×C₂₃	C₂	semidirect
5. C₂³×C₂²⋊C₄×C₂₃	C₂	semidirect
6. C₂³×C₂²⋊C₄×C₂₃	C₂	semidirect
15. C₂⁴×C₄×C₂₃	C₂²	semidirect
16. C₂×C₂₃×#123	C₂²	direct (N × H)
17. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
18. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
19. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
36. C₂³×C₄×C₂₃	C₂³	semidirect
37. C₂₃×#123	C₂³	direct (N × H)
38. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
39. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
40. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
61. C₂²×C₄×C₂₃	C₂⁴	semidirect
62. C₄⋊C₄×C₂₃	C₂⁴	semidirect
63. C₂²⋊C₄×C₂₃	C₂⁴	semidirect
64. C₂²⋊C₄×C₂₃	C₂⁴	semidirect
65. C₂³×#123	C₂₃	direct (N × H)
71. C₂³×C₂₃	#123	direct (N × H)
87. C₂⁵×C₄	C₂×C₂₃	semidirect
88. C₂²×#123	C₂×C₂₃	direct (N × H)
89. C₂³×C₄⋊C₄	C₂×C₂₃	semidirect
90. C₂³×C₂²⋊C₄	C₂×C₂₃	semidirect
91. C₂³×C₂²⋊C₄	C₂×C₂₃	semidirect
97. C₂²×C₂₃	C₂×#123	direct (N × H)
113. C₂⁴×C₄	C₂²×C₂₃	semidirect
114. C₂×#123	C₂²×C₂₃	direct (N × H)
115. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
116. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
117. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
120. C₂×C₂₃	C₂²×#123	direct (N × H)
139. C₂³×C₄	C₂³×C₂₃	semidirect
140. #123	C₂³×C₂₃	direct (N × H)
141. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
142. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
143. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
144. C₂₃	C₂³×#123	direct (N × H)
165. C₂²×C₄	C₂⁴×C₂₃	semidirect
166. C₄⋊C₄	C₂⁴×C₂₃	semidirect
167. C₂²⋊C₄	C₂⁴×C₂₃	semidirect
168. C₂²⋊C₄	C₂⁴×C₂₃	semidirect
174. C₂³	C₂₃×#123	direct (N × H)
194. C₂²	C₂×C₂₃×#123	direct (N × H)
204. C₂	C₂²×C₂₃×#123	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×D₄×C₂₃	2
C₂⁵×C₄×C₂₃	2
C₂²×C₂₃×#123	2
C₂³×C₄⋊C₄×C₂₃	2
C₂³×C₂²⋊C₄×C₂₃	2
C₂³×#123	23

Subgroups — C23×C23×#123 #19210284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#123 #19210284