Subgroups — C₂×C₂₃×#2216 #19181224

Order 5888.

702 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#467	C₂	1	2	yes	maximal
4	C₂₃×#2216	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#358	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
10	C₂×C₂₃×Q₃₂	—	2	4	no	—
11	C₂×C₂₃×Q₃₂	C₂²	1	4	yes	complemented
12	C₂²×C₁₆×C₂₃	—	2	4	no	—
13	C₂²×C₁₆×C₂₃	C₂²	1	4	yes	—
14	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#467	—	2	4	no	—
17	C₂₃×#467	C₂²	1	8	yes	—
18	C₂×C₂₃×#107	C₂²	1	2	yes	—
19	C₂×C₂₃×#103	—	2	2	no	—
20	C₂₃×#366	C₂²	1	4	yes	complemented
21	C₂₃×#358	C₂²	1	8	yes	complemented
22	C₂²×Q₈×C₂₃	—	2	2	no	—
23	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂×Q₁₆×C₂₃	—	2	4	no	—
25	C₂×Q₁₆×C₂₃	—	2	2	no	—
26	C₂×Q₁₆×C₂₃	—	2	4	no	—
27	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	D₄	1	2	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×C₂²⋊Q₈	—	2	8	no	—
35	C₂₃×C₂²⋊Q₈	—	4	4	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
38	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
39	C₂₃×Q₃₂	—	2	8	no	—
40	C₂×C₁₆×C₂₃	—	2	2	no	—
41	C₂×C₁₆×C₂₃	—	2	8	no	—
42	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
43	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
44	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
45	C₂₃×#107	—	2	2	no	—
46	C₂₃×#107	C₂³	1	4	yes	—
47	C₂₃×#103	—	2	8	no	—
48	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
49	C₂×Q₈×C₂₃	—	2	4	no	—
50	C₂×Q₈×C₂₃	—	4	2	no	—
51	C₂×Q₈×C₂₃	—	4	4	no	—
52	C₂×Q₈×C₂₃	—	4	4	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	4	2	no	—
56	C₂²×C₄×C₂₃	—	4	2	no	—
57	C₂²×C₄×C₂₃	D₈	1	2	yes	—
58	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₂₃	—	2	4	no	—
62	Q₁₆×C₂₃	—	2	4	no	—
63	Q₁₆×C₂₃	—	4	4	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₂₃	#41	1	2	yes	—
72	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	4	2	no	—
75	C₄⋊C₄×C₂₃	—	4	8	no	—
76	C₂²⋊C₄×C₂₃	—	4	8	no	—
77	C₁₆×C₂₃	—	2	2	no	—
78	C₁₆×C₂₃	C₂×D₄	1	4	yes	—
79	C₂×#2216	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	Q₃₂	1	4	yes	complemented
83	C₂³×C₂₃	C₂×D₈	1	2	yes	—
84	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₂₃	—	4	4	no	—
86	Q₈×C₂₃	—	4	4	no	—
87	Q₈×C₂₃	—	8	4	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	4	4	no	—
93	C₂×C₄×C₂₃	—	4	4	no	—
94	C₂×C₄×C₂₃	—	8	4	no	—
95	C₂×C₄×C₂₃	—	8	4	no	—
96	C₂×C₄×C₂₃	#121	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#135	1	2	yes	—
100	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₂₃	—	2	2	no	—
102	C₈×C₂₃	#121	1	2	yes	—
103	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
105	C₂²×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
106	C₂³×C₁₆	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#467	C₂×C₂₃	1	2	yes	—
108	#2216	C₂×C₂₃	1	8	yes	direct product factor
109	C₂×#366	C₂×C₂₃	1	1	yes	characteristic, complemented
110	C₂×#358	C₂×C₂₃	1	2	yes	complemented
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	8	no	—
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	#466	1	2	yes	—
115	C₂²×C₂₃	C₂×Q₃₂	1	2	yes	—
116	C₂²×C₂₃	C₂×Q₃₂	1	4	yes	complemented
117	C₂²×C₂₃	#508	1	2	yes	—
118	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₂₃	—	2	2	no	—
120	C₄×C₂₃	—	8	4	no	—
121	C₄×C₂₃	—	8	4	no	—
122	C₄×C₂₃	#466	1	2	yes	—
123	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
124	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
125	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
126	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₂×C₂²⋊Q₈	—	2	2	no	—
128	C₂×Q₃₂	—	2	4	no	—
129	C₂×Q₃₂	C₂²×C₂₃	1	4	yes	complemented
130	C₂²×C₁₆	—	2	4	no	—
131	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
132	C₂²×C₁₆	C₂²×C₂₃	1	4	yes	—
133	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
134	#467	—	2	4	no	—
135	#467	C₂²×C₂₃	1	8	yes	—
136	C₂×#107	C₂²×C₂₃	1	2	yes	—
137	C₂×#103	—	2	2	no	—
138	#366	C₂²×C₂₃	1	4	yes	complemented
139	#358	C₂²×C₂₃	1	8	yes	complemented
140	C₂×C₂₃	—	2	4	no	—
141	C₂×C₂₃	#2216	1	4	yes	direct product factor
142	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
143	C₂×C₂₃	C₂²×Q₃₂	1	1	yes	characteristic
144	C₂×C₂₃	C₂×#508	1	1	yes	characteristic
145	C₂²×Q₈	—	2	2	no	—
146	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
147	C₂×Q₁₆	—	2	4	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	4	no	—
150	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₈	D₄×C₂₃	1	2	yes	—
155	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
156	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
157	C₂²⋊Q₈	—	2	8	no	—
158	C₂²⋊Q₈	—	4	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	4	2	no	—
161	C₂×C₂²⋊C₄	—	4	2	no	—
162	Q₃₂	—	2	8	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	2	8	no	—
165	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
166	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
167	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
168	#107	—	2	2	no	—
169	#107	C₂³×C₂₃	1	4	yes	—
170	#103	—	2	8	no	—
171	C₂₃	C₂×#2216	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	D₈×C₂₃	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	2	4	no	—
187	Q₁₆	—	4	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
195	C₂×C₈	#41×C₂₃	1	2	yes	—
196	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₁₆	—	2	2	no	—
202	C₁₆	C₂×D₄×C₂₃	1	4	yes	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	C₂₃×Q₃₂	1	4	yes	complemented
206	C₂³	C₂×D₈×C₂₃	1	2	yes	—
207	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
208	Q₈	—	4	4	no	—
209	Q₈	—	4	4	no	—
210	Q₈	—	8	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	C₂₃×#121	1	2	yes	—
220	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
221	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂₃×#135	1	2	yes	—
223	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	C₂₃×#121	1	2	yes	—
226	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
227	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
228	C₂²	—	2	8	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	C₂₃×#466	1	2	yes	—
232	C₂²	C₂×C₂₃×Q₃₂	1	2	yes	—
233	C₂²	C₂×C₂₃×Q₃₂	1	4	yes	complemented
234	C₂²	C₂₃×#508	1	2	yes	—
235	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	4	no	—
239	C₄	C₂₃×#466	1	2	yes	—
240	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
241	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	C₂₃×#2216	1	4	yes	direct product factor
244	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
245	C₂	C₂²×C₂₃×Q₃₂	1	1	yes	characteristic
246	C₂	C₂×C₂₃×#508	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×Q₃₂	C₂	semidirect
4. C₂₃×#2216	C₂	direct (N × H)
5. C₂×C₂₃×#366	C₂	semidirect
6. C₂×C₂₃×#358	C₂	semidirect
11. C₂×C₂₃×Q₃₂	C₂²	semidirect
20. C₂₃×#366	C₂²	semidirect
21. C₂₃×#358	C₂²	semidirect
79. C₂×#2216	C₂₃	direct (N × H)
82. C₂³×C₂₃	Q₃₂	semidirect
105. C₂²×Q₃₂	C₂×C₂₃	semidirect
108. #2216	C₂×C₂₃	direct (N × H)
109. C₂×#366	C₂×C₂₃	semidirect
110. C₂×#358	C₂×C₂₃	semidirect
116. C₂²×C₂₃	C₂×Q₃₂	semidirect
129. C₂×Q₃₂	C₂²×C₂₃	semidirect
138. #366	C₂²×C₂₃	semidirect
139. #358	C₂²×C₂₃	semidirect
141. C₂×C₂₃	#2216	direct (N × H)
171. C₂₃	C₂×#2216	direct (N × H)
205. C₂³	C₂₃×Q₃₂	semidirect
233. C₂²	C₂×C₂₃×Q₃₂	semidirect
243. C₂	C₂₃×#2216	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×Q₃₂	2
C₂³×C₁₆×C₂₃	2
C₂×C₂₃×#467	2
C₂₃×#2216	2
C₂×C₂₃×#366	2
C₂×C₂₃×#358	2
C₂×#2216	23

Subgroups — C2×C23×#2216 #19181224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2216 #19181224