Subgroups — C₂²×C₂₃×#563 #19210316

Order 5888.

7238 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#563	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂₃×#126	C₂	1	4	yes	maximal, complemented
4	C₂²×C₂₃×#125	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₃×#123	C₂	1	2	yes	maximal
7	C₂²×C₂₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
8	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
9	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#563	C₂²	1	256	yes	direct product factor
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	48	yes	complemented
14	C₂×C₂₃×#126	C₂²	1	96	yes	complemented
15	C₂×C₂₃×#125	C₂²	1	48	yes	complemented
16	C₂×C₂₃×#124	C₂²	1	24	yes	—
17	C₂×C₂₃×#123	C₂²	1	48	yes	—
18	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	96	yes	complemented
19	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
22	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
26	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
29	C₂⁵×C₂₃	C₂³	1	2	yes	—
30	C₂²×Q₈×C₂₃	—	4	8	no	—
31	C₂²×Q₈×C₂₃	—	4	4	no	—
32	C₂²×Q₈×C₂₃	C₂³	1	24	yes	—
33	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
34	C₂²×D₄×C₂₃	—	4	4	no	—
35	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
36	C₂³×C₄×C₂₃	—	4	8	no	—
37	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
38	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
42	C₂³×C₄×C₂₃	C₂³	1	24	yes	—
43	C₂₃×C₄⋊Q₈	C₂³	1	128	yes	complemented
44	C₂₃×#126	C₂³	1	256	yes	complemented
45	C₂₃×#125	C₂³	1	128	yes	complemented
46	C₂₃×#124	C₂³	1	64	yes	—
47	C₂₃×#123	C₂³	1	128	yes	—
48	C₂₃×C₂²⋊Q₈	C₂³	1	256	yes	complemented
49	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
50	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
51	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
52	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
53	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
54	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
55	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
56	C₂×C₂²⋊C₄×C₂₃	C₂³	1	24	yes	—
57	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
58	C₂×C₄²×C₂₃	C₂³	1	24	yes	—
59	C₂⁴×C₂₃	—	2	8	no	—
60	C₂⁴×C₂₃	—	2	16	no	—
61	C₂⁴×C₂₃	C₂⁴	1	12	yes	—
62	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
63	C₂×Q₈×C₂₃	—	4	48	no	—
64	C₂×Q₈×C₂₃	—	4	24	no	—
65	C₂×Q₈×C₂₃	C₂⁴	1	32	yes	—
66	C₂×Q₈×C₂₃	C₂⁴	1	16	yes	—
67	C₂×D₄×C₂₃	—	4	24	no	—
68	C₂×D₄×C₂₃	C₂⁴	1	16	yes	—
69	C₂²×C₄×C₂₃	—	2	16	no	—
70	C₂²×C₄×C₂₃	—	2	4	no	—
71	C₂²×C₄×C₂₃	—	2	8	no	—
72	C₂²×C₄×C₂₃	—	2	8	no	—
73	C₂²×C₄×C₂₃	—	2	16	no	—
74	C₂²×C₄×C₂₃	—	4	48	no	—
75	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
76	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
77	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
78	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
79	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
80	C₂²×C₄×C₂₃	C₂⁴	1	32	yes	—
81	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
82	C₄⋊C₄×C₂₃	C₂⁴	1	64	yes	—
83	C₄⋊C₄×C₂₃	C₂⁴	1	64	yes	—
84	C₄⋊C₄×C₂₃	C₂⁴	1	64	yes	—
85	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
86	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	—
87	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	—
88	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
89	C₄²×C₂₃	C₂⁴	1	16	yes	—
90	C₄²×C₂₃	C₂⁴	1	32	yes	—
91	C₂²×#563	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	24	no	—
93	C₂³×C₂₃	—	2	48	no	—
94	C₂³×C₂₃	—	4	32	no	—
95	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
96	C₂³×C₂₃	2₋¹⁺⁴	1	8	yes	—
97	C₂³×C₂₃	C₂⁵	1	8	yes	—
98	C₂³×C₂₃	C₂⁵	1	3	yes	—
99	Q₈×C₂₃	—	4	32	no	—
100	Q₈×C₂₃	—	4	16	no	—
101	D₄×C₂₃	—	4	16	no	—
102	C₂×C₄×C₂₃	—	2	48	no	—
103	C₂×C₄×C₂₃	—	2	12	no	—
104	C₂×C₄×C₂₃	—	2	24	no	—
105	C₂×C₄×C₂₃	—	2	24	no	—
106	C₂×C₄×C₂₃	—	2	48	no	—
107	C₂×C₄×C₂₃	—	4	32	no	—
108	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
109	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
110	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
111	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
112	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
113	C₂×#563	C₂×C₂₃	1	48	yes	direct product factor
114	C₂²×C₄⋊Q₈	C₂×C₂₃	1	2	yes	complemented
115	C₂²×#126	C₂×C₂₃	1	4	yes	complemented
116	C₂²×#125	C₂×C₂₃	1	2	yes	complemented
117	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic
118	C₂²×#123	C₂×C₂₃	1	2	yes	—
119	C₂²×C₂²⋊Q₈	C₂×C₂₃	1	4	yes	complemented
120	C₂²×C₂₃	—	2	8	no	—
121	C₂²×C₂₃	—	2	16	no	—
122	C₂²×C₂₃	—	4	48	no	—
123	C₂²×C₂₃	#563	1	16	yes	direct product factor
124	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	6	yes	—
125	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	12	yes	—
126	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
127	C₄×C₂₃	—	2	16	no	—
128	C₄×C₂₃	—	2	4	no	—
129	C₄×C₂₃	—	2	8	no	—
130	C₄×C₂₃	—	2	8	no	—
131	C₄×C₂₃	—	2	16	no	—
132	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
133	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
135	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
136	#563	C₂²×C₂₃	1	256	yes	direct product factor
137	C₂×C₄⋊Q₈	C₂²×C₂₃	1	48	yes	complemented
138	C₂×#126	C₂²×C₂₃	1	96	yes	complemented
139	C₂×#125	C₂²×C₂₃	1	48	yes	complemented
140	C₂×#124	C₂²×C₂₃	1	24	yes	—
141	C₂×#123	C₂²×C₂₃	1	48	yes	—
142	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	96	yes	complemented
143	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
144	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
146	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
147	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
148	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
149	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
150	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
152	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
153	C₂×C₂₃	—	4	8	no	—
154	C₂×C₂₃	C₂×#563	1	12	yes	direct product factor
155	C₂×C₂₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	2	yes	—
157	C₂⁵	C₂³×C₂₃	1	2	yes	—
158	C₂²×Q₈	—	4	8	no	—
159	C₂²×Q₈	—	4	4	no	—
160	C₂²×Q₈	C₂³×C₂₃	1	24	yes	—
161	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
162	C₂²×D₄	—	4	4	no	—
163	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
164	C₂³×C₄	—	4	8	no	—
165	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
166	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
167	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
168	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
169	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₂₃	1	24	yes	—
171	C₄⋊Q₈	C₂³×C₂₃	1	128	yes	complemented
172	#126	C₂³×C₂₃	1	256	yes	complemented
173	#125	C₂³×C₂₃	1	128	yes	complemented
174	#124	C₂³×C₂₃	1	64	yes	—
175	#123	C₂³×C₂₃	1	128	yes	—
176	C₂²⋊Q₈	C₂³×C₂₃	1	256	yes	complemented
177	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
178	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
179	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
180	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
181	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
182	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
184	C₂×C₂²⋊C₄	C₂³×C₂₃	1	24	yes	—
185	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
186	C₂×C₄²	C₂³×C₂₃	1	24	yes	—
187	C₂₃	C₂²×#563	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	—	2	8	no	—
189	C₂⁴	—	2	16	no	—
190	C₂⁴	C₂⁴×C₂₃	1	12	yes	—
191	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
192	C₂×Q₈	—	4	48	no	—
193	C₂×Q₈	—	4	24	no	—
194	C₂×Q₈	C₂⁴×C₂₃	1	32	yes	—
195	C₂×Q₈	C₂⁴×C₂₃	1	16	yes	—
196	C₂×D₄	—	4	24	no	—
197	C₂×D₄	C₂⁴×C₂₃	1	16	yes	—
198	C₂²×C₄	—	2	16	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	16	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	4	48	no	—
204	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
205	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
206	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
207	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
208	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
209	C₂²×C₄	C₂⁴×C₂₃	1	32	yes	—
210	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
211	C₄⋊C₄	C₂⁴×C₂₃	1	64	yes	—
212	C₄⋊C₄	C₂⁴×C₂₃	1	64	yes	—
213	C₄⋊C₄	C₂⁴×C₂₃	1	64	yes	—
214	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
215	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	—
216	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	—
217	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
218	C₄²	C₂⁴×C₂₃	1	16	yes	—
219	C₄²	C₂⁴×C₂₃	1	32	yes	—
220	C₂³	—	2	24	no	—
221	C₂³	—	2	48	no	—
222	C₂³	—	4	32	no	—
223	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
224	C₂³	C₂₃×2₋¹⁺⁴	1	8	yes	—
225	C₂³	C₂⁵×C₂₃	1	8	yes	—
226	C₂³	C₂⁵×C₂₃	1	3	yes	—
227	Q₈	—	4	32	no	—
228	Q₈	—	4	16	no	—
229	D₄	—	4	16	no	—
230	C₂×C₄	—	2	48	no	—
231	C₂×C₄	—	2	12	no	—
232	C₂×C₄	—	2	24	no	—
233	C₂×C₄	—	2	24	no	—
234	C₂×C₄	—	2	48	no	—
235	C₂×C₄	—	4	32	no	—
236	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
237	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
238	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
239	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
240	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	16	no	—
243	C₂²	—	4	48	no	—
244	C₂²	C₂₃×#563	1	16	yes	direct product factor
245	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	6	yes	—
246	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	12	yes	—
247	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
248	C₄	—	2	16	no	—
249	C₄	—	2	4	no	—
250	C₄	—	2	8	no	—
251	C₄	—	2	8	no	—
252	C₄	—	2	16	no	—
253	C₂	—	4	8	no	—
254	C₂	C₂×C₂₃×#563	1	12	yes	direct product factor
255	C₂	C₂²×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#563	C₂	direct (N × H)
2. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₂₃×#126	C₂	semidirect
4. C₂²×C₂₃×#125	C₂	semidirect
7. C₂²×C₂₃×C₂²⋊Q₈	C₂	semidirect
12. C₂₃×#563	C₂²	direct (N × H)
13. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
14. C₂×C₂₃×#126	C₂²	semidirect
15. C₂×C₂₃×#125	C₂²	semidirect
18. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
43. C₂₃×C₄⋊Q₈	C₂³	semidirect
44. C₂₃×#126	C₂³	semidirect
45. C₂₃×#125	C₂³	semidirect
48. C₂₃×C₂²⋊Q₈	C₂³	semidirect
91. C₂²×#563	C₂₃	direct (N × H)
113. C₂×#563	C₂×C₂₃	direct (N × H)
114. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
115. C₂²×#126	C₂×C₂₃	semidirect
116. C₂²×#125	C₂×C₂₃	semidirect
119. C₂²×C₂²⋊Q₈	C₂×C₂₃	semidirect
123. C₂²×C₂₃	#563	direct (N × H)
136. #563	C₂²×C₂₃	direct (N × H)
137. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
138. C₂×#126	C₂²×C₂₃	semidirect
139. C₂×#125	C₂²×C₂₃	semidirect
142. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
154. C₂×C₂₃	C₂×#563	direct (N × H)
171. C₄⋊Q₈	C₂³×C₂₃	semidirect
172. #126	C₂³×C₂₃	semidirect
173. #125	C₂³×C₂₃	semidirect
176. C₂²⋊Q₈	C₂³×C₂₃	semidirect
187. C₂₃	C₂²×#563	direct (N × H)
244. C₂²	C₂₃×#563	direct (N × H)
254. C₂	C₂×C₂₃×#563	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#563	2
C₂²×C₂₃×C₄⋊Q₈	2
C₂²×C₂₃×#126	2
C₂²×C₂₃×#125	2
C₂²×C₂₃×#124	2
C₂²×C₂₃×#123	2
C₂²×C₂₃×C₂²⋊Q₈	2
C₂²×#563	23

Subgroups — C22×C23×#563 #19210316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#563 #19210316