Subgroups — C₂³×C₂₃×#121 #19210282

Order 5888.

19998 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal, complemented
2	C₂⁴×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁵×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂₃×#121	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
7	C₂⁶×C₂₃	C₂²	1	2	yes	—
8	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₂₃	—	2	32	no	—
10	C₂³×D₄×C₂₃	—	2	16	no	—
11	C₂³×D₄×C₂₃	C₂²	1	56	yes	complemented
12	C₂³×D₄×C₂₃	C₂²	1	28	yes	complemented
13	C₂⁴×C₄×C₂₃	—	2	16	no	—
14	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₂₃	C₂²	1	28	yes	complemented
18	C₂×C₂₃×#121	C₂²	1	448	yes	direct product factor
19	C₂²×C₄⋊C₄×C₂₃	C₂²	1	28	yes	complemented
20	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	56	yes	complemented
21	C₂⁵×C₂₃	—	2	8	no	—
22	C₂⁵×C₂₃	—	2	8	no	—
23	C₂⁵×C₂₃	—	2	16	no	—
24	C₂⁵×C₂₃	—	2	16	no	—
25	C₂⁵×C₂₃	—	2	16	no	—
26	C₂⁵×C₂₃	D₄	1	16	yes	complemented
27	C₂⁵×C₂₃	C₂³	1	28	yes	—
28	C₂⁵×C₂₃	C₂³	1	14	yes	—
29	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×D₄×C₂₃	—	2	448	no	—
31	C₂²×D₄×C₂₃	—	2	224	no	—
32	C₂²×D₄×C₂₃	C₂³	1	224	yes	complemented
33	C₂²×D₄×C₂₃	C₂³	1	112	yes	complemented
34	C₂³×C₄×C₂₃	—	2	8	no	—
35	C₂³×C₄×C₂₃	—	2	16	no	—
36	C₂³×C₄×C₂₃	—	2	224	no	—
37	C₂³×C₄×C₂₃	D₄	1	16	yes	complemented
38	C₂³×C₄×C₂₃	C₂³	1	14	yes	—
39	C₂³×C₄×C₂₃	C₂³	1	28	yes	—
40	C₂³×C₄×C₂₃	C₂³	1	14	yes	—
41	C₂³×C₄×C₂₃	C₂³	1	112	yes	complemented
42	C₂₃×#121	C₂³	1	512	yes	direct product factor
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	112	yes	complemented
44	C₂×C₂²⋊C₄×C₂₃	C₂³	1	224	yes	complemented
45	C₂⁴×C₂₃	—	2	128	no	—
46	C₂⁴×C₂₃	—	2	56	no	—
47	C₂⁴×C₂₃	—	2	56	no	—
48	C₂⁴×C₂₃	—	2	112	no	—
49	C₂⁴×C₂₃	—	2	112	no	—
50	C₂⁴×C₂₃	—	2	112	no	—
51	C₂⁴×C₂₃	—	4	128	no	—
52	C₂⁴×C₂₃	C₂×D₄	1	8	yes	—
53	C₂⁴×C₂₃	C₂×D₄	1	8	yes	—
54	C₂⁴×C₂₃	C₂×D₄	1	112	yes	complemented
55	C₂⁴×C₂₃	#41	1	8	yes	—
56	C₂⁴×C₂₃	C₂⁴	1	56	yes	—
57	C₂⁴×C₂₃	C₂⁴	1	28	yes	—
58	C₂⁴×C₂₃	C₂⁴	1	7	yes	—
59	C₂×D₄×C₂₃	—	2	896	no	—
60	C₂×D₄×C₂₃	—	2	448	no	—
61	C₂×D₄×C₂₃	C₂⁴	1	128	yes	complemented
62	C₂×D₄×C₂₃	C₂⁴	1	64	yes	complemented
63	C₂²×C₄×C₂₃	—	2	56	no	—
64	C₂²×C₄×C₂₃	—	2	112	no	—
65	C₂²×C₄×C₂₃	—	2	448	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	112	yes	complemented
67	C₂²×C₄×C₂₃	C₂⁴	1	28	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	56	yes	—
69	C₂²×C₄×C₂₃	C₂⁴	1	28	yes	—
70	C₂²×C₄×C₂₃	C₂⁴	1	64	yes	complemented
71	C₄⋊C₄×C₂₃	C₂⁴	1	64	yes	complemented
72	C₂²⋊C₄×C₂₃	C₂⁴	1	128	yes	complemented
73	C₂³×#121	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	—	2	448	no	—
75	C₂³×C₂₃	—	2	56	no	—
76	C₂³×C₂₃	—	2	56	no	—
77	C₂³×C₂₃	—	2	112	no	—
78	C₂³×C₂₃	—	2	112	no	—
79	C₂³×C₂₃	—	2	112	no	—
80	C₂³×C₂₃	—	4	448	no	—
81	C₂³×C₂₃	#121	1	64	yes	direct product factor
82	C₂³×C₂₃	C₂²×D₄	1	28	yes	—
83	C₂³×C₂₃	C₂²×D₄	1	28	yes	—
84	C₂³×C₂₃	C₂²×D₄	1	112	yes	complemented
85	C₂³×C₂₃	C₂×#41	1	28	yes	—
86	C₂³×C₂₃	C₂⁵	1	16	yes	—
87	C₂³×C₂₃	C₂⁵	1	8	yes	—
88	C₂³×C₂₃	C₂⁵	1	7	yes	—
89	D₄×C₂₃	—	2	256	no	—
90	D₄×C₂₃	—	2	128	no	—
91	C₂×C₄×C₂₃	—	2	56	no	—
92	C₂×C₄×C₂₃	—	2	112	no	—
93	C₂×C₄×C₂₃	—	2	128	no	—
94	C₂×C₄×C₂₃	C₂²×D₄	1	112	yes	complemented
95	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
96	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
97	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
98	C₂⁴×D₄	C₂×C₂₃	1	2	yes	complemented
99	C₂⁴×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
100	C₂⁵×C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
101	C₂²×#121	C₂×C₂₃	1	56	yes	direct product factor
102	C₂³×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
103	C₂³×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
104	C₂²×C₂₃	—	2	8	no	—
105	C₂²×C₂₃	—	2	8	no	—
106	C₂²×C₂₃	—	2	224	no	—
107	C₂²×C₂₃	—	2	16	no	—
108	C₂²×C₂₃	—	2	16	no	—
109	C₂²×C₂₃	—	2	16	no	—
110	C₂²×C₂₃	—	4	224	no	—
111	C₂²×C₂₃	C₂×#121	1	112	yes	direct product factor
112	C₂²×C₂₃	C₂³×D₄	1	14	yes	—
113	C₂²×C₂₃	C₂³×D₄	1	14	yes	—
114	C₂²×C₂₃	C₂³×D₄	1	16	yes	complemented
115	C₂²×C₂₃	C₂²×#41	1	14	yes	—
116	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
117	C₄×C₂₃	—	2	8	no	—
118	C₄×C₂₃	—	2	16	no	—
119	C₄×C₂₃	C₂³×D₄	1	16	yes	complemented
120	C₂⁶	C₂²×C₂₃	1	2	yes	—
121	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
122	C₂³×D₄	—	2	32	no	—
123	C₂³×D₄	—	2	16	no	—
124	C₂³×D₄	C₂²×C₂₃	1	56	yes	complemented
125	C₂³×D₄	C₂²×C₂₃	1	28	yes	complemented
126	C₂⁴×C₄	—	2	16	no	—
127	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
128	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
129	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
130	C₂⁴×C₄	C₂²×C₂₃	1	28	yes	complemented
131	C₂×#121	C₂²×C₂₃	1	448	yes	direct product factor
132	C₂²×C₄⋊C₄	C₂²×C₂₃	1	28	yes	complemented
133	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	56	yes	complemented
134	C₂×C₂₃	—	2	16	no	—
135	C₂×C₂₃	—	4	16	no	—
136	C₂×C₂₃	C₂²×#121	1	28	yes	direct product factor
137	C₂×C₂₃	C₂⁴×D₄	1	1	yes	characteristic
138	C₂×C₂₃	C₂⁴×D₄	1	1	yes	characteristic
139	C₂×C₂₃	C₂³×#41	1	1	yes	characteristic
140	C₂⁵	—	2	8	no	—
141	C₂⁵	—	2	8	no	—
142	C₂⁵	—	2	16	no	—
143	C₂⁵	—	2	16	no	—
144	C₂⁵	—	2	16	no	—
145	C₂⁵	D₄×C₂₃	1	16	yes	complemented
146	C₂⁵	C₂³×C₂₃	1	28	yes	—
147	C₂⁵	C₂³×C₂₃	1	14	yes	—
148	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
149	C₂²×D₄	—	2	448	no	—
150	C₂²×D₄	—	2	224	no	—
151	C₂²×D₄	C₂³×C₂₃	1	224	yes	complemented
152	C₂²×D₄	C₂³×C₂₃	1	112	yes	complemented
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	2	16	no	—
155	C₂³×C₄	—	2	224	no	—
156	C₂³×C₄	D₄×C₂₃	1	16	yes	complemented
157	C₂³×C₄	C₂³×C₂₃	1	14	yes	—
158	C₂³×C₄	C₂³×C₂₃	1	28	yes	—
159	C₂³×C₄	C₂³×C₂₃	1	14	yes	—
160	C₂³×C₄	C₂³×C₂₃	1	112	yes	complemented
161	#121	C₂³×C₂₃	1	512	yes	direct product factor
162	C₂×C₄⋊C₄	C₂³×C₂₃	1	112	yes	complemented
163	C₂×C₂²⋊C₄	C₂³×C₂₃	1	224	yes	complemented
164	C₂₃	C₂³×#121	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁴	—	2	128	no	—
166	C₂⁴	—	2	56	no	—
167	C₂⁴	—	2	56	no	—
168	C₂⁴	—	2	112	no	—
169	C₂⁴	—	2	112	no	—
170	C₂⁴	—	2	112	no	—
171	C₂⁴	—	4	128	no	—
172	C₂⁴	C₂×D₄×C₂₃	1	8	yes	—
173	C₂⁴	C₂×D₄×C₂₃	1	8	yes	—
174	C₂⁴	C₂×D₄×C₂₃	1	112	yes	complemented
175	C₂⁴	#41×C₂₃	1	8	yes	—
176	C₂⁴	C₂⁴×C₂₃	1	56	yes	—
177	C₂⁴	C₂⁴×C₂₃	1	28	yes	—
178	C₂⁴	C₂⁴×C₂₃	1	7	yes	—
179	C₂×D₄	—	2	896	no	—
180	C₂×D₄	—	2	448	no	—
181	C₂×D₄	C₂⁴×C₂₃	1	128	yes	complemented
182	C₂×D₄	C₂⁴×C₂₃	1	64	yes	complemented
183	C₂²×C₄	—	2	56	no	—
184	C₂²×C₄	—	2	112	no	—
185	C₂²×C₄	—	2	448	no	—
186	C₂²×C₄	C₂×D₄×C₂₃	1	112	yes	complemented
187	C₂²×C₄	C₂⁴×C₂₃	1	28	yes	—
188	C₂²×C₄	C₂⁴×C₂₃	1	56	yes	—
189	C₂²×C₄	C₂⁴×C₂₃	1	28	yes	—
190	C₂²×C₄	C₂⁴×C₂₃	1	64	yes	complemented
191	C₄⋊C₄	C₂⁴×C₂₃	1	64	yes	complemented
192	C₂²⋊C₄	C₂⁴×C₂₃	1	128	yes	complemented
193	C₂³	—	2	56	no	—
194	C₂³	—	2	56	no	—
195	C₂³	—	2	448	no	—
196	C₂³	—	2	112	no	—
197	C₂³	—	2	112	no	—
198	C₂³	—	2	112	no	—
199	C₂³	—	4	448	no	—
200	C₂³	C₂₃×#121	1	64	yes	direct product factor
201	C₂³	C₂²×D₄×C₂₃	1	28	yes	—
202	C₂³	C₂²×D₄×C₂₃	1	28	yes	—
203	C₂³	C₂²×D₄×C₂₃	1	112	yes	complemented
204	C₂³	C₂×#41×C₂₃	1	28	yes	—
205	C₂³	C₂⁵×C₂₃	1	16	yes	—
206	C₂³	C₂⁵×C₂₃	1	8	yes	—
207	C₂³	C₂⁵×C₂₃	1	7	yes	—
208	D₄	—	2	256	no	—
209	D₄	—	2	128	no	—
210	C₂×C₄	—	2	56	no	—
211	C₂×C₄	—	2	112	no	—
212	C₂×C₄	—	2	128	no	—
213	C₂×C₄	C₂²×D₄×C₂₃	1	112	yes	complemented
214	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
215	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
216	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
217	C₂²	—	2	224	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	8	no	—
220	C₂²	—	2	16	no	—
221	C₂²	—	2	16	no	—
222	C₂²	—	2	16	no	—
223	C₂²	—	4	224	no	—
224	C₂²	C₂×C₂₃×#121	1	112	yes	direct product factor
225	C₂²	C₂³×D₄×C₂₃	1	14	yes	—
226	C₂²	C₂³×D₄×C₂₃	1	14	yes	—
227	C₂²	C₂³×D₄×C₂₃	1	16	yes	complemented
228	C₂²	C₂²×#41×C₂₃	1	14	yes	—
229	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	—	2	16	no	—
232	C₄	C₂³×D₄×C₂₃	1	16	yes	complemented
233	C₂	—	2	16	no	—
234	C₂	—	4	16	no	—
235	C₂	C₂²×C₂₃×#121	1	28	yes	direct product factor
236	C₂	C₂⁴×D₄×C₂₃	1	1	yes	characteristic
237	C₂	C₂⁴×D₄×C₂₃	1	1	yes	characteristic
238	C₂	C₂³×#41×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₂₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
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×D₄×C₂₃	C₂	semidirect
2. C₂⁴×D₄×C₂₃	C₂	semidirect
3. C₂⁵×C₄×C₂₃	C₂	semidirect
4. C₂²×C₂₃×#121	C₂	direct (N × H)
5. C₂³×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂³×C₂²⋊C₄×C₂₃	C₂	semidirect
11. C₂³×D₄×C₂₃	C₂²	semidirect
12. C₂³×D₄×C₂₃	C₂²	semidirect
17. C₂⁴×C₄×C₂₃	C₂²	semidirect
18. C₂×C₂₃×#121	C₂²	direct (N × H)
19. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
20. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
26. C₂⁵×C₂₃	D₄	semidirect
32. C₂²×D₄×C₂₃	C₂³	semidirect
33. C₂²×D₄×C₂₃	C₂³	semidirect
37. C₂³×C₄×C₂₃	D₄	semidirect
41. C₂³×C₄×C₂₃	C₂³	semidirect
42. C₂₃×#121	C₂³	direct (N × H)
43. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
44. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
54. C₂⁴×C₂₃	C₂×D₄	semidirect
61. C₂×D₄×C₂₃	C₂⁴	semidirect
62. C₂×D₄×C₂₃	C₂⁴	semidirect
66. C₂²×C₄×C₂₃	C₂×D₄	semidirect
70. C₂²×C₄×C₂₃	C₂⁴	semidirect
71. C₄⋊C₄×C₂₃	C₂⁴	semidirect
72. C₂²⋊C₄×C₂₃	C₂⁴	semidirect
73. C₂³×#121	C₂₃	direct (N × H)
81. C₂³×C₂₃	#121	direct (N × H)
84. C₂³×C₂₃	C₂²×D₄	semidirect
94. C₂×C₄×C₂₃	C₂²×D₄	semidirect
98. C₂⁴×D₄	C₂×C₂₃	semidirect
99. C₂⁴×D₄	C₂×C₂₃	semidirect
100. C₂⁵×C₄	C₂×C₂₃	semidirect
101. C₂²×#121	C₂×C₂₃	direct (N × H)
102. C₂³×C₄⋊C₄	C₂×C₂₃	semidirect
103. C₂³×C₂²⋊C₄	C₂×C₂₃	semidirect
111. C₂²×C₂₃	C₂×#121	direct (N × H)
114. C₂²×C₂₃	C₂³×D₄	semidirect
119. C₄×C₂₃	C₂³×D₄	semidirect
124. C₂³×D₄	C₂²×C₂₃	semidirect
125. C₂³×D₄	C₂²×C₂₃	semidirect
130. C₂⁴×C₄	C₂²×C₂₃	semidirect
131. C₂×#121	C₂²×C₂₃	direct (N × H)
132. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
133. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
136. C₂×C₂₃	C₂²×#121	direct (N × H)
145. C₂⁵	D₄×C₂₃	semidirect
151. C₂²×D₄	C₂³×C₂₃	semidirect
152. C₂²×D₄	C₂³×C₂₃	semidirect
156. C₂³×C₄	D₄×C₂₃	semidirect
160. C₂³×C₄	C₂³×C₂₃	semidirect
161. #121	C₂³×C₂₃	direct (N × H)
162. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
163. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
164. C₂₃	C₂³×#121	direct (N × H)
174. C₂⁴	C₂×D₄×C₂₃	semidirect
181. C₂×D₄	C₂⁴×C₂₃	semidirect
182. C₂×D₄	C₂⁴×C₂₃	semidirect
186. C₂²×C₄	C₂×D₄×C₂₃	semidirect
190. C₂²×C₄	C₂⁴×C₂₃	semidirect
191. C₄⋊C₄	C₂⁴×C₂₃	semidirect
192. C₂²⋊C₄	C₂⁴×C₂₃	semidirect
200. C₂³	C₂₃×#121	direct (N × H)
203. C₂³	C₂²×D₄×C₂₃	semidirect
213. C₂×C₄	C₂²×D₄×C₂₃	semidirect
224. C₂²	C₂×C₂₃×#121	direct (N × H)
227. C₂²	C₂³×D₄×C₂₃	semidirect
232. C₄	C₂³×D₄×C₂₃	semidirect
235. C₂	C₂²×C₂₃×#121	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×#121	2
C₂³×C₄⋊C₄×C₂₃	2
C₂³×C₂²⋊C₄×C₂₃	2
C₂³×#121	23

Subgroups — C23×C23×#121 #19210282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#121 #19210282