Subgroups — C₂²×C₂₃×#492 #19208254

Order 5888.

3950 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₂₃	C₂	1	4	yes	maximal, complemented
2	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#492	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂³×D₄×C₂₃	—	2	2	no	—
8	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂²×SD₁₆×C₂₃	—	2	32	no	—
10	C₂²×SD₁₆×C₂₃	C₂²	1	48	yes	complemented
11	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊Q₈	—	2	6	no	—
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
14	C₂×C₂₃×#127	—	2	6	no	—
15	C₂×C₂₃×#127	C₂²	1	12	yes	—
16	C₂²×C₄⋊C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#492	C₂²	1	64	yes	direct product factor
19	C₂×C₄×C₈×C₂₃	C₂²	1	12	yes	complemented
20	C₂⁵×C₂₃	—	2	2	no	—
21	C₂²×Q₈×C₂₃	—	2	6	no	—
22	C₂²×Q₈×C₂₃	—	2	16	no	—
23	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
24	C₂²×D₄×C₂₃	—	2	16	no	—
25	C₂²×D₄×C₂₃	—	2	24	no	—
26	C₂²×D₄×C₂₃	—	2	6	no	—
27	C₂²×D₄×C₂₃	—	4	16	no	—
28	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
29	C₂³×C₄×C₂₃	—	2	2	no	—
30	C₂³×C₄×C₂₃	D₄	1	2	yes	—
31	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂×SD₁₆×C₂₃	—	2	192	no	—
33	C₂×SD₁₆×C₂₃	C₂³	1	64	yes	complemented
34	C₂²×C₈×C₂₃	D₄	1	16	yes	complemented
35	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
36	C₂₃×C₄⋊Q₈	—	2	24	no	—
37	C₂₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
38	C₂₃×#127	—	2	24	no	—
39	C₂₃×#127	C₂³	1	16	yes	—
40	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
41	C₂×C₄²×C₂₃	D₄	1	6	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
43	C₄×C₈×C₂₃	C₂³	1	16	yes	complemented
44	C₂⁴×C₂₃	—	2	12	no	—
45	C₂⁴×C₂₃	—	4	8	no	—
46	C₂⁴×C₂₃	—	4	16	no	—
47	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₂₃	—	2	12	no	—
49	C₂×Q₈×C₂₃	—	2	48	no	—
50	C₂×Q₈×C₂₃	—	4	24	no	—
51	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
52	C₂×D₄×C₂₃	—	2	48	no	—
53	C₂×D₄×C₂₃	—	2	12	no	—
54	C₂×D₄×C₂₃	—	2	32	no	—
55	C₂×D₄×C₂₃	—	4	24	no	—
56	C₂×D₄×C₂₃	—	4	96	no	—
57	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₂₃	—	2	12	no	—
59	C₂²×C₄×C₂₃	—	4	8	no	—
60	C₂²×C₄×C₂₃	SD₁₆	1	16	yes	complemented
61	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
64	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
65	SD₁₆×C₂₃	—	2	128	no	—
66	C₂×C₈×C₂₃	C₂×D₄	1	48	yes	complemented
67	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
68	C₄⋊C₄×C₂₃	—	2	32	no	—
69	C₄²×C₂₃	C₂×D₄	1	12	yes	—
70	C₄²×C₂₃	C₂⁴	1	4	yes	—
71	C₂²×#492	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	—	2	8	no	—
73	C₂³×C₂₃	—	4	24	no	—
74	C₂³×C₂₃	—	4	48	no	—
75	C₂³×C₂₃	—	8	32	no	—
76	C₂³×C₂₃	#127	1	4	yes	—
77	C₂³×C₂₃	C₂×SD₁₆	1	8	yes	—
78	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
79	Q₈×C₂₃	—	2	16	no	—
80	Q₈×C₂₃	—	4	24	no	—
81	D₄×C₂₃	—	2	16	no	—
82	D₄×C₂₃	—	4	24	no	—
83	D₄×C₂₃	—	4	64	no	—
84	C₂×C₄×C₂₃	—	2	8	no	—
85	C₂×C₄×C₂₃	—	4	24	no	—
86	C₂×C₄×C₂₃	#127	1	12	yes	—
87	C₂×C₄×C₂₃	C₂×SD₁₆	1	48	yes	complemented
88	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
89	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
90	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
91	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
92	C₈×C₂₃	C₂²×D₄	1	16	yes	complemented
93	C₂³×SD₁₆	C₂×C₂₃	1	4	yes	complemented
94	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
95	C₂²×#127	C₂×C₂₃	1	1	yes	characteristic
96	C₂×#492	C₂×C₂₃	1	24	yes	direct product factor
97	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
98	C₂²×C₂₃	—	4	8	no	—
99	C₂²×C₂₃	—	4	16	no	—
100	C₂²×C₂₃	—	8	48	no	—
101	C₂²×C₂₃	#492	1	16	yes	direct product factor
102	C₂²×C₂₃	C₂×#127	1	6	yes	—
103	C₂²×C₂₃	C₂²×SD₁₆	1	12	yes	—
104	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
105	C₄×C₂₃	—	4	8	no	—
106	C₄×C₂₃	C₂×#127	1	6	yes	—
107	C₄×C₂₃	C₂²×SD₁₆	1	16	yes	complemented
108	C₄×C₂₃	C₂³×D₄	1	2	yes	—
109	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
110	C₂³×D₄	—	2	2	no	—
111	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
112	C₂²×SD₁₆	—	2	32	no	—
113	C₂²×SD₁₆	C₂²×C₂₃	1	48	yes	complemented
114	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
115	C₂×C₄⋊Q₈	—	2	6	no	—
116	C₂×C₄⋊Q₈	C₂²×C₂₃	1	12	yes	complemented
117	C₂×#127	—	2	6	no	—
118	C₂×#127	C₂²×C₂₃	1	12	yes	—
119	C₂²×C₄⋊C₄	—	2	2	no	—
120	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
121	#492	C₂²×C₂₃	1	64	yes	direct product factor
122	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	complemented
123	C₂×C₂₃	—	8	8	no	—
124	C₂×C₂₃	C₂×#492	1	12	yes	direct product factor
125	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
126	C₂×C₂₃	C₂³×SD₁₆	1	2	yes	—
127	C₂⁵	—	2	2	no	—
128	C₂²×Q₈	—	2	6	no	—
129	C₂²×Q₈	—	2	16	no	—
130	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
131	C₂²×D₄	—	2	16	no	—
132	C₂²×D₄	—	2	6	no	—
133	C₂²×D₄	—	2	24	no	—
134	C₂²×D₄	—	4	16	no	—
135	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
136	C₂³×C₄	—	2	2	no	—
137	C₂³×C₄	D₄×C₂₃	1	2	yes	—
138	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
139	C₂×SD₁₆	—	2	192	no	—
140	C₂×SD₁₆	C₂³×C₂₃	1	64	yes	complemented
141	C₂²×C₈	D₄×C₂₃	1	16	yes	complemented
142	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
143	C₄⋊Q₈	—	2	24	no	—
144	C₄⋊Q₈	C₂³×C₂₃	1	16	yes	complemented
145	#127	—	2	24	no	—
146	#127	C₂³×C₂₃	1	16	yes	—
147	C₂×C₄⋊C₄	—	2	24	no	—
148	C₂×C₄²	D₄×C₂₃	1	6	yes	—
149	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
150	C₄×C₈	C₂³×C₂₃	1	16	yes	complemented
151	C₂₃	C₂²×#492	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	—	2	12	no	—
153	C₂⁴	—	4	8	no	—
154	C₂⁴	—	4	16	no	—
155	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
156	C₂×Q₈	—	2	12	no	—
157	C₂×Q₈	—	2	48	no	—
158	C₂×Q₈	—	4	24	no	—
159	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
160	C₂×D₄	—	2	48	no	—
161	C₂×D₄	—	2	12	no	—
162	C₂×D₄	—	2	32	no	—
163	C₂×D₄	—	4	24	no	—
164	C₂×D₄	—	4	96	no	—
165	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	4	8	no	—
168	C₂²×C₄	SD₁₆×C₂₃	1	16	yes	complemented
169	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
170	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
171	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
172	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
173	SD₁₆	—	2	128	no	—
174	C₂×C₈	C₂×D₄×C₂₃	1	48	yes	complemented
175	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
176	C₄⋊C₄	—	2	32	no	—
177	C₄²	C₂×D₄×C₂₃	1	12	yes	—
178	C₄²	C₂⁴×C₂₃	1	4	yes	—
179	C₂³	—	2	8	no	—
180	C₂³	—	4	48	no	—
181	C₂³	—	4	24	no	—
182	C₂³	—	8	32	no	—
183	C₂³	C₂₃×#127	1	4	yes	—
184	C₂³	C₂×SD₁₆×C₂₃	1	8	yes	—
185	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
186	Q₈	—	2	16	no	—
187	Q₈	—	4	24	no	—
188	D₄	—	2	16	no	—
189	D₄	—	4	64	no	—
190	D₄	—	4	24	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	24	no	—
193	C₂×C₄	C₂₃×#127	1	12	yes	—
194	C₂×C₄	C₂×SD₁₆×C₂₃	1	48	yes	complemented
195	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
196	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
197	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
198	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
199	C₈	C₂²×D₄×C₂₃	1	16	yes	complemented
200	C₂²	—	4	8	no	—
201	C₂²	—	4	16	no	—
202	C₂²	—	8	48	no	—
203	C₂²	C₂₃×#492	1	16	yes	direct product factor
204	C₂²	C₂×C₂₃×#127	1	6	yes	—
205	C₂²	C₂²×SD₁₆×C₂₃	1	12	yes	—
206	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
207	C₄	—	4	8	no	—
208	C₄	C₂×C₂₃×#127	1	6	yes	—
209	C₄	C₂²×SD₁₆×C₂₃	1	16	yes	complemented
210	C₄	C₂³×D₄×C₂₃	1	2	yes	—
211	C₂	—	8	8	no	—
212	C₂	C₂×C₂₃×#492	1	12	yes	direct product factor
213	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
214	C₂	C₂³×SD₁₆×C₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂²×C₂₃×#127	C₂
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. C₂³×SD₁₆×C₂₃	C₂	semidirect
2. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
4. C₂×C₂₃×#492	C₂	direct (N × H)
5. C₂²×C₄×C₈×C₂₃	C₂	semidirect
10. C₂²×SD₁₆×C₂₃	C₂²	semidirect
13. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
18. C₂₃×#492	C₂²	direct (N × H)
19. C₂×C₄×C₈×C₂₃	C₂²	semidirect
33. C₂×SD₁₆×C₂₃	C₂³	semidirect
34. C₂²×C₈×C₂₃	D₄	semidirect
37. C₂₃×C₄⋊Q₈	C₂³	semidirect
43. C₄×C₈×C₂₃	C₂³	semidirect
60. C₂²×C₄×C₂₃	SD₁₆	semidirect
66. C₂×C₈×C₂₃	C₂×D₄	semidirect
71. C₂²×#492	C₂₃	direct (N × H)
87. C₂×C₄×C₂₃	C₂×SD₁₆	semidirect
92. C₈×C₂₃	C₂²×D₄	semidirect
93. C₂³×SD₁₆	C₂×C₂₃	semidirect
94. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
96. C₂×#492	C₂×C₂₃	direct (N × H)
97. C₂²×C₄×C₈	C₂×C₂₃	semidirect
101. C₂²×C₂₃	#492	direct (N × H)
107. C₄×C₂₃	C₂²×SD₁₆	semidirect
113. C₂²×SD₁₆	C₂²×C₂₃	semidirect
116. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
121. #492	C₂²×C₂₃	direct (N × H)
122. C₂×C₄×C₈	C₂²×C₂₃	semidirect
124. C₂×C₂₃	C₂×#492	direct (N × H)
140. C₂×SD₁₆	C₂³×C₂₃	semidirect
141. C₂²×C₈	D₄×C₂₃	semidirect
144. C₄⋊Q₈	C₂³×C₂₃	semidirect
150. C₄×C₈	C₂³×C₂₃	semidirect
151. C₂₃	C₂²×#492	direct (N × H)
168. C₂²×C₄	SD₁₆×C₂₃	semidirect
174. C₂×C₈	C₂×D₄×C₂₃	semidirect
194. C₂×C₄	C₂×SD₁₆×C₂₃	semidirect
199. C₈	C₂²×D₄×C₂₃	semidirect
203. C₂²	C₂₃×#492	direct (N × H)
209. C₄	C₂²×SD₁₆×C₂₃	semidirect
212. C₂	C₂×C₂₃×#492	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×SD₁₆×C₂₃	2
C₂²×C₂₃×C₄⋊Q₈	2
C₂²×C₂₃×#127	2
C₂×C₂₃×#492	2
C₂²×C₄×C₈×C₂₃	2
C₂²×#492	23

Subgroups — C22×C23×#492 #19208254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#492 #19208254