Subgroups — C₂²×C₂₃×#488 #19208230

Order 5888.

2310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#488	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×#103	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂₃×#102	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₂₃×#125	—	2	6	no	—
11	C₂×C₂₃×#125	C₂²	1	12	yes	complemented
12	C₂×C₂₃×#124	—	2	6	no	—
13	C₂×C₂₃×#124	C₂²	1	12	yes	—
14	C₂²×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#488	C₂²	1	64	yes	direct product factor
19	C₂×C₂₃×#103	C₂²	1	24	yes	complemented
20	C₂×C₂₃×#102	C₂²	1	24	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	12	yes	complemented
22	C₂⁵×C₂₃	—	2	1	no	—
23	C₂²×Q₈×C₂₃	—	2	3	no	—
24	C₂²×Q₈×C₂₃	—	2	8	no	—
25	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
26	C₂²×D₄×C₂₃	—	2	3	no	—
27	C₂²×D₄×C₂₃	—	2	8	no	—
28	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
29	C₂³×C₄×C₂₃	—	2	1	no	—
30	C₂³×C₄×C₂₃	—	2	2	no	—
31	C₂³×C₄×C₂₃	D₄	1	2	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	—	2	8	no	—
34	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
35	C₂₃×#125	—	2	24	no	—
36	C₂₃×#125	C₂³	1	16	yes	complemented
37	C₂₃×#124	—	2	24	no	—
38	C₂₃×#124	C₂³	1	16	yes	—
39	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
43	C₂×C₄²×C₂₃	D₄	1	6	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
45	C₂₃×#103	C₂³	1	32	yes	complemented
46	C₂₃×#102	C₂³	1	32	yes	—
47	C₄×C₈×C₂₃	C₂³	1	16	yes	complemented
48	C₂⁴×C₂₃	—	2	6	no	—
49	C₂⁴×C₂₃	—	4	4	no	—
50	C₂⁴×C₂₃	—	4	8	no	—
51	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Q₈×C₂₃	—	2	6	no	—
53	C₂×Q₈×C₂₃	—	2	24	no	—
54	C₂×Q₈×C₂₃	—	4	12	no	—
55	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
56	C₂×D₄×C₂₃	—	2	6	no	—
57	C₂×D₄×C₂₃	—	2	24	no	—
58	C₂×D₄×C₂₃	—	4	12	no	—
59	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
60	C₂²×C₄×C₂₃	—	2	8	no	—
61	C₂²×C₄×C₂₃	—	2	6	no	—
62	C₂²×C₄×C₂₃	—	2	12	no	—
63	C₂²×C₄×C₂₃	—	4	4	no	—
64	C₂²×C₄×C₂₃	—	4	8	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₂₃	#41	1	8	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
69	C₂×C₈×C₂₃	—	2	24	no	—
70	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
71	C₄⋊C₄×C₂₃	—	2	32	no	—
72	C₄⋊C₄×C₂₃	—	2	12	no	—
73	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
74	C₂²⋊C₄×C₂₃	—	2	32	no	—
75	C₄²×C₂₃	C₂×D₄	1	12	yes	—
76	C₄²×C₂₃	C₂⁴	1	4	yes	—
77	C₂²×#488	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	—	2	4	no	—
79	C₂³×C₂₃	—	4	12	no	—
80	C₂³×C₂₃	—	4	24	no	—
81	C₂³×C₂₃	—	8	16	no	—
82	C₂³×C₂₃	#124	1	4	yes	—
83	C₂³×C₂₃	#135	1	8	yes	—
84	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
85	Q₈×C₂₃	—	2	8	no	—
86	Q₈×C₂₃	—	4	12	no	—
87	D₄×C₂₃	—	2	8	no	—
88	D₄×C₂₃	—	4	12	no	—
89	C₂×C₄×C₂₃	—	2	24	no	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	2	8	no	—
92	C₂×C₄×C₂₃	—	4	12	no	—
93	C₂×C₄×C₂₃	—	4	24	no	—
94	C₂×C₄×C₂₃	#124	1	12	yes	—
95	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
96	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
97	C₂×C₄×C₂₃	C₂×#41	1	12	yes	—
98	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
99	C₈×C₂₃	—	2	8	no	—
100	C₂²×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
101	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic
102	C₂×#488	C₂×C₂₃	1	24	yes	direct product factor
103	C₂²×#103	C₂×C₂₃	1	2	yes	complemented
104	C₂²×#102	C₂×C₂₃	1	2	yes	—
105	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
106	C₂²×C₂₃	—	4	4	no	—
107	C₂²×C₂₃	—	4	8	no	—
108	C₂²×C₂₃	—	8	24	no	—
109	C₂²×C₂₃	#488	1	16	yes	direct product factor
110	C₂²×C₂₃	C₂×#124	1	6	yes	—
111	C₂²×C₂₃	C₂×#135	1	12	yes	—
112	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
113	C₄×C₂₃	—	2	8	no	—
114	C₄×C₂₃	—	4	4	no	—
115	C₄×C₂₃	—	4	8	no	—
116	C₄×C₂₃	C₂×#124	1	6	yes	—
117	C₄×C₂₃	C₂²×#41	1	2	yes	—
118	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
119	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
121	C₂×#125	—	2	6	no	—
122	C₂×#125	C₂²×C₂₃	1	12	yes	complemented
123	C₂×#124	—	2	6	no	—
124	C₂×#124	C₂²×C₂₃	1	12	yes	—
125	C₂²×C₄⋊C₄	—	2	2	no	—
126	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
127	C₂²×C₂²⋊C₄	—	2	2	no	—
128	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
129	#488	C₂²×C₂₃	1	64	yes	direct product factor
130	C₂×#103	C₂²×C₂₃	1	24	yes	complemented
131	C₂×#102	C₂²×C₂₃	1	24	yes	—
132	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	complemented
133	C₂×C₂₃	—	8	4	no	—
134	C₂×C₂₃	C₂×#488	1	12	yes	direct product factor
135	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
136	C₂×C₂₃	C₂²×#135	1	2	yes	—
137	C₂⁵	—	2	1	no	—
138	C₂²×Q₈	—	2	3	no	—
139	C₂²×Q₈	—	2	8	no	—
140	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	—	2	8	no	—
143	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
144	C₂³×C₄	—	2	1	no	—
145	C₂³×C₄	—	2	2	no	—
146	C₂³×C₄	D₄×C₂₃	1	2	yes	—
147	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
148	C₂²×C₈	—	2	8	no	—
149	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
150	#125	—	2	24	no	—
151	#125	C₂³×C₂₃	1	16	yes	complemented
152	#124	—	2	24	no	—
153	#124	C₂³×C₂₃	1	16	yes	—
154	C₂×C₄⋊C₄	—	2	24	no	—
155	C₂×C₄⋊C₄	—	2	6	no	—
156	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
157	C₂×C₂²⋊C₄	—	2	24	no	—
158	C₂×C₄²	D₄×C₂₃	1	6	yes	—
159	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
160	#103	C₂³×C₂₃	1	32	yes	complemented
161	#102	C₂³×C₂₃	1	32	yes	—
162	C₄×C₈	C₂³×C₂₃	1	16	yes	complemented
163	C₂₃	C₂²×#488	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴	—	2	6	no	—
165	C₂⁴	—	4	4	no	—
166	C₂⁴	—	4	8	no	—
167	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
168	C₂×Q₈	—	2	6	no	—
169	C₂×Q₈	—	2	24	no	—
170	C₂×Q₈	—	4	12	no	—
171	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
172	C₂×D₄	—	2	6	no	—
173	C₂×D₄	—	2	24	no	—
174	C₂×D₄	—	4	12	no	—
175	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	6	no	—
178	C₂²×C₄	—	2	12	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	—	4	8	no	—
181	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
182	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
183	C₂²×C₄	#41×C₂₃	1	8	yes	—
184	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
185	C₂×C₈	—	2	24	no	—
186	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	12	no	—
189	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
190	C₂²⋊C₄	—	2	32	no	—
191	C₄²	C₂×D₄×C₂₃	1	12	yes	—
192	C₄²	C₂⁴×C₂₃	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	4	24	no	—
195	C₂³	—	4	12	no	—
196	C₂³	—	8	16	no	—
197	C₂³	C₂₃×#124	1	4	yes	—
198	C₂³	C₂₃×#135	1	8	yes	—
199	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
200	Q₈	—	2	8	no	—
201	Q₈	—	4	12	no	—
202	D₄	—	2	8	no	—
203	D₄	—	4	12	no	—
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	4	24	no	—
208	C₂×C₄	—	4	12	no	—
209	C₂×C₄	C₂₃×#124	1	12	yes	—
210	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
211	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
212	C₂×C₄	C₂×#41×C₂₃	1	12	yes	—
213	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
214	C₈	—	2	8	no	—
215	C₂²	—	4	4	no	—
216	C₂²	—	4	8	no	—
217	C₂²	—	8	24	no	—
218	C₂²	C₂₃×#488	1	16	yes	direct product factor
219	C₂²	C₂×C₂₃×#124	1	6	yes	—
220	C₂²	C₂×C₂₃×#135	1	12	yes	—
221	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
222	C₄	—	2	8	no	—
223	C₄	—	4	4	no	—
224	C₄	—	4	8	no	—
225	C₄	C₂×C₂₃×#124	1	6	yes	—
226	C₄	C₂²×#41×C₂₃	1	2	yes	—
227	C₂	—	8	4	no	—
228	C₂	C₂×C₂₃×#488	1	12	yes	direct product factor
229	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
230	C₂	C₂²×C₂₃×#135	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
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
1. C₂²×C₂₃×#125	C₂	semidirect
3. C₂×C₂₃×#488	C₂	direct (N × H)
4. C₂²×C₂₃×#103	C₂	semidirect
6. C₂²×C₄×C₈×C₂₃	C₂	semidirect
11. C₂×C₂₃×#125	C₂²	semidirect
18. C₂₃×#488	C₂²	direct (N × H)
19. C₂×C₂₃×#103	C₂²	semidirect
21. C₂×C₄×C₈×C₂₃	C₂²	semidirect
36. C₂₃×#125	C₂³	semidirect
45. C₂₃×#103	C₂³	semidirect
47. C₄×C₈×C₂₃	C₂³	semidirect
77. C₂²×#488	C₂₃	direct (N × H)
100. C₂²×#125	C₂×C₂₃	semidirect
102. C₂×#488	C₂×C₂₃	direct (N × H)
103. C₂²×#103	C₂×C₂₃	semidirect
105. C₂²×C₄×C₈	C₂×C₂₃	semidirect
109. C₂²×C₂₃	#488	direct (N × H)
122. C₂×#125	C₂²×C₂₃	semidirect
129. #488	C₂²×C₂₃	direct (N × H)
130. C₂×#103	C₂²×C₂₃	semidirect
132. C₂×C₄×C₈	C₂²×C₂₃	semidirect
134. C₂×C₂₃	C₂×#488	direct (N × H)
151. #125	C₂³×C₂₃	semidirect
160. #103	C₂³×C₂₃	semidirect
162. C₄×C₈	C₂³×C₂₃	semidirect
163. C₂₃	C₂²×#488	direct (N × H)
218. C₂²	C₂₃×#488	direct (N × H)
228. C₂	C₂×C₂₃×#488	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#125	2
C₂²×C₂₃×#124	2
C₂×C₂₃×#488	2
C₂²×C₂₃×#103	2
C₂²×C₂₃×#102	2
C₂²×C₄×C₈×C₂₃	2
C₂²×#488	23

Subgroups — C22×C23×#488 #19208230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#488 #19208230