Subgroups — C₂²×C₂₃×#467 #19208062

Order 5888.

3006 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₂×C₂₃×#467	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂₃×#103	C₂	1	2	yes	maximal, complemented
7	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₂₃	—	2	8	no	—
10	C₂²×Q₁₆×C₂₃	C₂²	1	12	yes	complemented
11	C₂³×C₈×C₂₃	—	2	8	no	—
12	C₂³×C₈×C₂₃	C₂²	1	12	yes	—
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₂²⋊Q₈	—	2	12	no	—
16	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	24	yes	—
17	C₂²×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
20	C₂₃×#467	C₂²	1	64	yes	direct product factor
21	C₂×C₂₃×#107	C₂²	1	12	yes	complemented
22	C₂×C₂₃×#103	C₂²	1	24	yes	complemented
23	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×Q₈×C₂₃	—	2	6	no	—
25	C₂²×Q₈×C₂₃	—	2	8	no	—
26	C₂²×Q₈×C₂₃	—	2	8	no	—
27	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
28	C₂³×C₄×C₂₃	—	2	4	no	—
29	C₂³×C₄×C₂₃	—	2	4	no	—
30	C₂³×C₄×C₂₃	—	2	2	no	—
31	C₂³×C₄×C₂₃	—	2	2	no	—
32	C₂³×C₄×C₂₃	D₄	1	6	yes	—
33	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
35	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂×Q₁₆×C₂₃	—	2	48	no	—
37	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	complemented
38	C₂²×C₈×C₂₃	—	2	4	no	—
39	C₂²×C₈×C₂₃	—	2	48	no	—
40	C₂²×C₈×C₂₃	D₄	1	8	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
42	C₂²×C₈×C₂₃	C₂³	1	16	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
44	C₂₃×C₂²⋊Q₈	—	2	48	no	—
45	C₂₃×C₂²⋊Q₈	C₂³	1	32	yes	—
46	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
47	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
48	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
49	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
50	C₂₃×#107	C₂³	1	16	yes	complemented
51	C₂₃×#103	C₂³	1	32	yes	complemented
52	C₂⁴×C₂₃	—	2	4	no	—
53	C₂⁴×C₂₃	—	2	4	no	—
54	C₂⁴×C₂₃	Q₁₆	1	8	yes	complemented
55	C₂⁴×C₂₃	C₂×D₄	1	6	yes	—
56	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₂₃	—	2	12	no	—
58	C₂×Q₈×C₂₃	—	2	24	no	—
59	C₂×Q₈×C₂₃	—	2	24	no	—
60	C₂×Q₈×C₂₃	—	4	24	no	—
61	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
62	C₂²×C₄×C₂₃	—	2	4	no	—
63	C₂²×C₄×C₂₃	—	2	24	no	—
64	C₂²×C₄×C₂₃	—	2	12	no	—
65	C₂²×C₄×C₂₃	—	2	12	no	—
66	C₂²×C₄×C₂₃	—	2	12	no	—
67	C₂²×C₄×C₂₃	—	2	12	no	—
68	C₂²×C₄×C₂₃	—	4	8	no	—
69	C₂²×C₄×C₂₃	—	4	8	no	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
72	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₂₃	#41	1	4	yes	—
75	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
77	Q₁₆×C₂₃	—	2	32	no	—
78	C₂×C₈×C₂₃	—	2	12	no	—
79	C₂×C₈×C₂₃	—	2	32	no	—
80	C₂×C₈×C₂₃	C₂×D₄	1	24	yes	—
81	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
82	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
83	C₄⋊C₄×C₂₃	—	2	32	no	—
84	C₄⋊C₄×C₂₃	—	2	12	no	—
85	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
86	C₂²⋊C₄×C₂₃	—	2	32	no	—
87	C₂²×#467	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	2	12	no	—
89	C₂³×C₂₃	—	2	12	no	—
90	C₂³×C₂₃	—	2	32	no	—
91	C₂³×C₂₃	#121	1	4	yes	—
92	C₂³×C₂₃	C₂×Q₁₆	1	4	yes	—
93	C₂³×C₂₃	C₂×Q₁₆	1	24	yes	complemented
94	C₂³×C₂₃	#135	1	4	yes	—
95	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
96	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
97	Q₈×C₂₃	—	2	8	no	—
98	Q₈×C₂₃	—	2	8	no	—
99	Q₈×C₂₃	—	4	24	no	—
100	C₂×C₄×C₂₃	—	2	12	no	—
101	C₂×C₄×C₂₃	—	2	24	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	8	no	—
105	C₂×C₄×C₂₃	—	2	8	no	—
106	C₂×C₄×C₂₃	—	4	24	no	—
107	C₂×C₄×C₂₃	—	4	24	no	—
108	C₂×C₄×C₂₃	#121	1	12	yes	—
109	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
110	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
111	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
112	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
113	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
114	C₈×C₂₃	—	2	4	no	—
115	C₈×C₂₃	C₂²×D₄	1	8	yes	—
116	C₂³×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
117	C₂⁴×C₈	C₂×C₂₃	1	1	yes	characteristic
118	C₂²×C₂²⋊Q₈	C₂×C₂₃	1	2	yes	—
119	C₂×#467	C₂×C₂₃	1	24	yes	direct product factor
120	C₂²×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
121	C₂²×#103	C₂×C₂₃	1	2	yes	complemented
122	C₂²×C₂₃	—	2	4	no	—
123	C₂²×C₂₃	—	2	48	no	—
124	C₂²×C₂₃	—	2	4	no	—
125	C₂²×C₂₃	#467	1	16	yes	direct product factor
126	C₂²×C₂₃	C₂×#121	1	6	yes	—
127	C₂²×C₂₃	C₂²×Q₁₆	1	6	yes	—
128	C₂²×C₂₃	C₂²×Q₁₆	1	8	yes	complemented
129	C₂²×C₂₃	C₂×#135	1	6	yes	—
130	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	4	8	no	—
133	C₄×C₂₃	—	4	8	no	—
134	C₄×C₂₃	C₂×#121	1	6	yes	—
135	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
136	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
137	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
138	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
139	C₂²×Q₁₆	—	2	8	no	—
140	C₂²×Q₁₆	C₂²×C₂₃	1	12	yes	complemented
141	C₂³×C₈	—	2	8	no	—
142	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₂³×C₈	C₂²×C₂₃	1	12	yes	—
144	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₂²⋊Q₈	—	2	12	no	—
146	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	24	yes	—
147	C₂²×C₄⋊C₄	—	2	2	no	—
148	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
149	C₂²×C₂²⋊C₄	—	2	2	no	—
150	#467	C₂²×C₂₃	1	64	yes	direct product factor
151	C₂×#107	C₂²×C₂₃	1	12	yes	complemented
152	C₂×#103	C₂²×C₂₃	1	24	yes	complemented
153	C₂×C₂₃	—	2	8	no	—
154	C₂×C₂₃	C₂×#467	1	12	yes	direct product factor
155	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
156	C₂×C₂₃	C₂³×Q₁₆	1	1	yes	characteristic
157	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
158	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
159	C₂²×Q₈	—	2	6	no	—
160	C₂²×Q₈	—	2	8	no	—
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
163	C₂³×C₄	—	2	4	no	—
164	C₂³×C₄	—	2	4	no	—
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
168	C₂³×C₄	D₄×C₂₃	1	6	yes	—
169	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
170	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
171	C₂×Q₁₆	—	2	48	no	—
172	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	complemented
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	48	no	—
175	C₂²×C₈	D₄×C₂₃	1	8	yes	—
176	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
177	C₂²×C₈	C₂³×C₂₃	1	16	yes	—
178	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
179	C₂²⋊Q₈	—	2	48	no	—
180	C₂²⋊Q₈	C₂³×C₂₃	1	32	yes	—
181	C₂×C₄⋊C₄	—	2	24	no	—
182	C₂×C₄⋊C₄	—	2	6	no	—
183	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
184	C₂×C₂²⋊C₄	—	2	24	no	—
185	#107	C₂³×C₂₃	1	16	yes	complemented
186	#103	C₂³×C₂₃	1	32	yes	complemented
187	C₂₃	C₂²×#467	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	2	4	no	—
190	C₂⁴	Q₁₆×C₂₃	1	8	yes	complemented
191	C₂⁴	C₂×D₄×C₂₃	1	6	yes	—
192	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₂×Q₈	—	2	12	no	—
194	C₂×Q₈	—	2	24	no	—
195	C₂×Q₈	—	2	24	no	—
196	C₂×Q₈	—	4	24	no	—
197	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	24	no	—
200	C₂²×C₄	—	2	12	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	4	8	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
207	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
208	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
209	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
210	C₂²×C₄	#41×C₂₃	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
212	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
213	Q₁₆	—	2	32	no	—
214	C₂×C₈	—	2	12	no	—
215	C₂×C₈	—	2	32	no	—
216	C₂×C₈	C₂×D₄×C₂₃	1	24	yes	—
217	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
218	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
219	C₄⋊C₄	—	2	32	no	—
220	C₄⋊C₄	—	2	12	no	—
221	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
222	C₂²⋊C₄	—	2	32	no	—
223	C₂³	—	2	12	no	—
224	C₂³	—	2	32	no	—
225	C₂³	—	2	12	no	—
226	C₂³	C₂₃×#121	1	4	yes	—
227	C₂³	C₂×Q₁₆×C₂₃	1	4	yes	—
228	C₂³	C₂×Q₁₆×C₂₃	1	24	yes	complemented
229	C₂³	C₂₃×#135	1	4	yes	—
230	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
231	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
232	Q₈	—	2	8	no	—
233	Q₈	—	2	8	no	—
234	Q₈	—	4	24	no	—
235	C₂×C₄	—	2	12	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	24	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	4	24	no	—
242	C₂×C₄	—	4	24	no	—
243	C₂×C₄	C₂₃×#121	1	12	yes	—
244	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
245	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
246	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
247	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
248	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	C₂²×D₄×C₂₃	1	8	yes	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	48	no	—
253	C₂²	—	2	4	no	—
254	C₂²	C₂₃×#467	1	16	yes	direct product factor
255	C₂²	C₂×C₂₃×#121	1	6	yes	—
256	C₂²	C₂²×Q₁₆×C₂₃	1	6	yes	—
257	C₂²	C₂²×Q₁₆×C₂₃	1	8	yes	complemented
258	C₂²	C₂×C₂₃×#135	1	6	yes	—
259	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	—	4	8	no	—
263	C₄	C₂×C₂₃×#121	1	6	yes	—
264	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
265	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
266	C₂	—	2	8	no	—
267	C₂	C₂×C₂₃×#467	1	12	yes	direct product factor
268	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
269	C₂	C₂³×Q₁₆×C₂₃	1	1	yes	characteristic
270	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂⁵×C₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₂₃	C₂	semidirect
4. C₂×C₂₃×#467	C₂	direct (N × H)
5. C₂²×C₂₃×#107	C₂	semidirect
6. C₂²×C₂₃×#103	C₂	semidirect
10. C₂²×Q₁₆×C₂₃	C₂²	semidirect
20. C₂₃×#467	C₂²	direct (N × H)
21. C₂×C₂₃×#107	C₂²	semidirect
22. C₂×C₂₃×#103	C₂²	semidirect
37. C₂×Q₁₆×C₂₃	C₂³	semidirect
50. C₂₃×#107	C₂³	semidirect
51. C₂₃×#103	C₂³	semidirect
54. C₂⁴×C₂₃	Q₁₆	semidirect
87. C₂²×#467	C₂₃	direct (N × H)
93. C₂³×C₂₃	C₂×Q₁₆	semidirect
116. C₂³×Q₁₆	C₂×C₂₃	semidirect
119. C₂×#467	C₂×C₂₃	direct (N × H)
120. C₂²×#107	C₂×C₂₃	semidirect
121. C₂²×#103	C₂×C₂₃	semidirect
125. C₂²×C₂₃	#467	direct (N × H)
128. C₂²×C₂₃	C₂²×Q₁₆	semidirect
140. C₂²×Q₁₆	C₂²×C₂₃	semidirect
150. #467	C₂²×C₂₃	direct (N × H)
151. C₂×#107	C₂²×C₂₃	semidirect
152. C₂×#103	C₂²×C₂₃	semidirect
154. C₂×C₂₃	C₂×#467	direct (N × H)
172. C₂×Q₁₆	C₂³×C₂₃	semidirect
185. #107	C₂³×C₂₃	semidirect
186. #103	C₂³×C₂₃	semidirect
187. C₂₃	C₂²×#467	direct (N × H)
190. C₂⁴	Q₁₆×C₂₃	semidirect
228. C₂³	C₂×Q₁₆×C₂₃	semidirect
254. C₂²	C₂₃×#467	direct (N × H)
257. C₂²	C₂²×Q₁₆×C₂₃	semidirect
267. C₂	C₂×C₂₃×#467	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₂₃	2
C₂⁴×C₈×C₂₃	2
C₂²×C₂₃×C₂²⋊Q₈	2
C₂×C₂₃×#467	2
C₂²×C₂₃×#107	2
C₂²×C₂₃×#103	2
C₂²×#467	23

Subgroups — C22×C23×#467 #19208062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#467 #19208062