Subgroups — C₂²×C₂₃×#508 #19210254

Order 5888.

2118 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#135	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#508	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂₃×SD₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₂₃	—	2	2	no	—
8	C₂×C₂₃×#135	—	2	12	no	—
9	C₂×C₂₃×#135	C₂²	1	24	yes	complemented
10	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₂₃	—	2	2	no	—
12	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#508	C₂²	1	64	yes	direct product factor
15	C₂×C₂₃×Q₃₂	C₂²	1	12	yes	complemented
16	C₂×C₂₃×SD₃₂	C₂²	1	24	yes	complemented
17	C₂×C₂₃×D₁₆	C₂²	1	12	yes	complemented
18	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₂₃	C₂²	1	12	yes	complemented
21	C₂×#41×C₂₃	—	2	24	no	—
22	C₂×#41×C₂₃	—	4	12	no	—
23	C₂²×Q₈×C₂₃	—	2	2	no	—
24	C₂²×D₄×C₂₃	—	2	2	no	—
25	C₂²×D₄×C₂₃	—	4	2	no	—
26	C₂³×C₄×C₂₃	—	4	2	no	—
27	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
28	C₂₃×#135	—	2	48	no	—
29	C₂₃×#135	C₂³	1	32	yes	complemented
30	C₂×Q₁₆×C₂₃	—	2	6	no	—
31	C₂×Q₁₆×C₂₃	C₂³	1	12	yes	—
32	C₂×SD₁₆×C₂₃	—	2	24	no	—
33	C₂×D₈×C₂₃	—	2	6	no	—
34	C₂×D₈×C₂₃	C₂³	1	12	yes	—
35	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	D₄	1	6	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
39	C₂₃×Q₃₂	C₂³	1	16	yes	complemented
40	C₂₃×SD₃₂	C₂³	1	32	yes	complemented
41	C₂₃×D₁₆	C₂³	1	16	yes	complemented
42	C₂×C₁₆×C₂₃	C₂³	1	6	yes	—
43	C₂×C₁₆×C₂₃	C₂³	1	6	yes	—
44	C₂×C₁₆×C₂₃	C₂³	1	16	yes	complemented
45	C₂⁴×C₂₃	—	4	2	no	—
46	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
47	#41×C₂₃	—	2	32	no	—
48	#41×C₂₃	—	4	48	no	—
49	C₂×Q₈×C₂₃	—	2	12	no	—
50	C₂×Q₈×C₂₃	—	4	6	no	—
51	C₂×D₄×C₂₃	—	2	12	no	—
52	C₂×D₄×C₂₃	—	4	24	no	—
53	C₂×D₄×C₂₃	—	4	6	no	—
54	C₂²×C₄×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	—	4	24	no	—
56	C₂²×C₄×C₂₃	D₈	1	6	yes	—
57	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
60	Q₁₆×C₂₃	—	2	12	no	—
61	Q₁₆×C₂₃	C₂⁴	1	8	yes	—
62	SD₁₆×C₂₃	—	2	32	no	—
63	D₈×C₂₃	—	2	12	no	—
64	D₈×C₂₃	C₂⁴	1	8	yes	—
65	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
66	C₂×C₈×C₂₃	C₂×D₄	1	6	yes	—
67	C₂×C₈×C₂₃	C₂×D₄	1	12	yes	—
68	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
69	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
70	C₁₆×C₂₃	C₂⁴	1	4	yes	—
71	C₁₆×C₂₃	C₂⁴	1	4	yes	—
72	C₂²×#508	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	—	2	4	no	—
74	C₂³×C₂₃	—	4	12	no	—
75	C₂³×C₂₃	—	8	8	no	—
76	C₂³×C₂₃	C₂×D₈	1	6	yes	—
77	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₂₃	—	2	8	no	—
79	Q₈×C₂₃	—	4	12	no	—
80	D₄×C₂₃	—	2	8	no	—
81	D₄×C₂₃	—	4	32	no	—
82	D₄×C₂₃	—	4	12	no	—
83	C₂×C₄×C₂₃	—	4	12	no	—
84	C₂×C₄×C₂₃	—	4	32	no	—
85	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
86	C₂×C₄×C₂₃	C₂×D₈	1	3	yes	—
87	C₂×C₄×C₂₃	C₂×D₈	1	12	yes	—
88	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
89	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
90	C₈×C₂₃	C₂²×D₄	1	3	yes	—
91	C₈×C₂₃	C₂²×D₄	1	4	yes	—
92	C₈×C₂₃	C₂⁵	1	1	yes	characteristic
93	C₂²×#135	C₂×C₂₃	1	2	yes	complemented
94	C₂×#508	C₂×C₂₃	1	24	yes	direct product factor
95	C₂²×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
96	C₂²×SD₃₂	C₂×C₂₃	1	2	yes	complemented
97	C₂²×D₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
98	C₂³×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
99	C₂²×C₂₃	—	2	12	no	—
100	C₂²×C₂₃	—	4	8	no	—
101	C₂²×C₂₃	—	8	24	no	—
102	C₂²×C₂₃	#508	1	4	yes	direct product factor
103	C₂²×C₂₃	C₂²×D₈	1	4	yes	—
104	C₂²×C₂₃	C₂²×D₈	1	3	yes	—
105	C₄×C₂₃	—	4	8	no	—
106	C₄×C₂₃	C₂²×D₈	1	4	yes	—
107	C₄×C₂₃	C₂²×D₈	1	3	yes	—
108	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
109	C₂²×#41	—	2	2	no	—
110	C₂×#135	—	2	12	no	—
111	C₂×#135	C₂²×C₂₃	1	24	yes	complemented
112	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
113	C₂²×SD₁₆	—	2	2	no	—
114	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
115	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
116	#508	C₂²×C₂₃	1	64	yes	direct product factor
117	C₂×Q₃₂	C₂²×C₂₃	1	12	yes	complemented
118	C₂×SD₃₂	C₂²×C₂₃	1	24	yes	complemented
119	C₂×D₁₆	C₂²×C₂₃	1	12	yes	complemented
120	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
121	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂²×C₂₃	1	12	yes	complemented
123	C₂×C₂₃	—	2	4	no	—
124	C₂×C₂₃	—	8	8	no	—
125	C₂×C₂₃	C₂×#508	1	6	yes	direct product factor
126	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
127	C₂×#41	—	2	24	no	—
128	C₂×#41	—	4	12	no	—
129	C₂²×Q₈	—	2	2	no	—
130	C₂²×D₄	—	2	2	no	—
131	C₂²×D₄	—	4	2	no	—
132	C₂³×C₄	—	4	2	no	—
133	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
134	#135	—	2	48	no	—
135	#135	C₂³×C₂₃	1	32	yes	complemented
136	C₂×Q₁₆	—	2	6	no	—
137	C₂×Q₁₆	C₂³×C₂₃	1	12	yes	—
138	C₂×SD₁₆	—	2	24	no	—
139	C₂×D₈	—	2	6	no	—
140	C₂×D₈	C₂³×C₂₃	1	12	yes	—
141	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
142	C₂²×C₈	D₄×C₂₃	1	6	yes	—
143	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
145	Q₃₂	C₂³×C₂₃	1	16	yes	complemented
146	SD₃₂	C₂³×C₂₃	1	32	yes	complemented
147	D₁₆	C₂³×C₂₃	1	16	yes	complemented
148	C₂×C₁₆	C₂³×C₂₃	1	6	yes	—
149	C₂×C₁₆	C₂³×C₂₃	1	6	yes	—
150	C₂×C₁₆	C₂³×C₂₃	1	16	yes	complemented
151	C₂₃	C₂²×#508	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	—	4	2	no	—
153	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
154	#41	—	2	32	no	—
155	#41	—	4	48	no	—
156	C₂×Q₈	—	2	12	no	—
157	C₂×Q₈	—	4	6	no	—
158	C₂×D₄	—	2	12	no	—
159	C₂×D₄	—	4	24	no	—
160	C₂×D₄	—	4	6	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	—	4	24	no	—
163	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	D₈×C₂₃	1	6	yes	—
165	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
167	Q₁₆	—	2	12	no	—
168	Q₁₆	C₂⁴×C₂₃	1	8	yes	—
169	SD₁₆	—	2	32	no	—
170	D₈	—	2	12	no	—
171	D₈	C₂⁴×C₂₃	1	8	yes	—
172	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
173	C₂×C₈	C₂×D₄×C₂₃	1	6	yes	—
174	C₂×C₈	C₂×D₄×C₂₃	1	12	yes	—
175	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
176	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
177	C₁₆	C₂⁴×C₂₃	1	4	yes	—
178	C₁₆	C₂⁴×C₂₃	1	4	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	4	12	no	—
181	C₂³	—	8	8	no	—
182	C₂³	C₂×D₈×C₂₃	1	6	yes	—
183	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	Q₈	—	4	12	no	—
186	D₄	—	2	8	no	—
187	D₄	—	4	32	no	—
188	D₄	—	4	12	no	—
189	C₂×C₄	—	4	12	no	—
190	C₂×C₄	—	4	32	no	—
191	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
192	C₂×C₄	C₂×D₈×C₂₃	1	3	yes	—
193	C₂×C₄	C₂×D₈×C₂₃	1	12	yes	—
194	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
195	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
196	C₈	C₂²×D₄×C₂₃	1	3	yes	—
197	C₈	C₂²×D₄×C₂₃	1	4	yes	—
198	C₈	C₂⁵×C₂₃	1	1	yes	characteristic
199	C₂²	—	2	12	no	—
200	C₂²	—	4	8	no	—
201	C₂²	—	8	24	no	—
202	C₂²	C₂₃×#508	1	4	yes	direct product factor
203	C₂²	C₂²×D₈×C₂₃	1	4	yes	—
204	C₂²	C₂²×D₈×C₂₃	1	3	yes	—
205	C₄	—	4	8	no	—
206	C₄	C₂²×D₈×C₂₃	1	4	yes	—
207	C₄	C₂²×D₈×C₂₃	1	3	yes	—
208	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
209	C₂	—	2	4	no	—
210	C₂	—	8	8	no	—
211	C₂	C₂×C₂₃×#508	1	6	yes	direct product factor
212	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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×#135	C₂	semidirect
2. C₂×C₂₃×#508	C₂	direct (N × H)
3. C₂²×C₂₃×Q₃₂	C₂	semidirect
4. C₂²×C₂₃×SD₃₂	C₂	semidirect
5. C₂²×C₂₃×D₁₆	C₂	semidirect
6. C₂³×C₁₆×C₂₃	C₂	semidirect
9. C₂×C₂₃×#135	C₂²	semidirect
14. C₂₃×#508	C₂²	direct (N × H)
15. C₂×C₂₃×Q₃₂	C₂²	semidirect
16. C₂×C₂₃×SD₃₂	C₂²	semidirect
17. C₂×C₂₃×D₁₆	C₂²	semidirect
20. C₂²×C₁₆×C₂₃	C₂²	semidirect
29. C₂₃×#135	C₂³	semidirect
39. C₂₃×Q₃₂	C₂³	semidirect
40. C₂₃×SD₃₂	C₂³	semidirect
41. C₂₃×D₁₆	C₂³	semidirect
44. C₂×C₁₆×C₂₃	C₂³	semidirect
72. C₂²×#508	C₂₃	direct (N × H)
93. C₂²×#135	C₂×C₂₃	semidirect
94. C₂×#508	C₂×C₂₃	direct (N × H)
95. C₂²×Q₃₂	C₂×C₂₃	semidirect
96. C₂²×SD₃₂	C₂×C₂₃	semidirect
97. C₂²×D₁₆	C₂×C₂₃	semidirect
98. C₂³×C₁₆	C₂×C₂₃	semidirect
102. C₂²×C₂₃	#508	direct (N × H)
111. C₂×#135	C₂²×C₂₃	semidirect
116. #508	C₂²×C₂₃	direct (N × H)
117. C₂×Q₃₂	C₂²×C₂₃	semidirect
118. C₂×SD₃₂	C₂²×C₂₃	semidirect
119. C₂×D₁₆	C₂²×C₂₃	semidirect
122. C₂²×C₁₆	C₂²×C₂₃	semidirect
125. C₂×C₂₃	C₂×#508	direct (N × H)
135. #135	C₂³×C₂₃	semidirect
145. Q₃₂	C₂³×C₂₃	semidirect
146. SD₃₂	C₂³×C₂₃	semidirect
147. D₁₆	C₂³×C₂₃	semidirect
150. C₂×C₁₆	C₂³×C₂₃	semidirect
151. C₂₃	C₂²×#508	direct (N × H)
202. C₂²	C₂₃×#508	direct (N × H)
211. C₂	C₂×C₂₃×#508	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×#135	2
C₂×C₂₃×#508	2
C₂²×C₂₃×Q₃₂	2
C₂²×C₂₃×SD₃₂	2
C₂²×C₂₃×D₁₆	2
C₂³×C₁₆×C₂₃	2
C₂²×#508	23

Subgroups — C22×C23×#508 #19210254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#508 #19210254