Subgroups — C₂×C₂₃×#1845 #19168421

Order 5888.

1358 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#107	C₂	1	4	yes	maximal
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#384	C₂	1	2	yes	maximal
4	C₂₃×#1845	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₂₃	—	2	4	no	—
7	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#107	—	2	16	no	—
10	C₂×C₂₃×#107	C₂²	1	16	yes	—
11	C₂×C₄×C₈×C₂₃	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
13	C₂₃×#384	—	2	4	no	—
14	C₂₃×#384	C₂²	1	8	yes	—
15	C₂³×C₄×C₂₃	—	2	4	no	—
16	C₂³×C₄×C₂₃	D₄	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₂₃	D₄	1	8	yes	—
20	C₂²×C₈×C₂₃	Q₈	1	8	yes	—
21	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
22	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
25	C₂×C₄⋊C₄×C₂₃	—	4	16	no	—
26	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
31	C₂₃×#107	—	2	32	no	—
32	C₄×C₈×C₂₃	C₂×C₄	1	16	yes	complemented
33	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	—	2	8	no	—
35	C₂²×C₄×C₂₃	—	4	8	no	—
36	C₂²×C₄×C₂₃	—	4	16	no	—
37	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₂₃	D₈	1	8	yes	—
39	C₂²×C₄×C₂₃	Q₁₆	1	8	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
44	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₄⋊C₄	1	32	yes	complemented
48	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
50	C₂×C₈×C₂₃	C₂×Q₈	1	8	yes	—
51	C₄⋊C₄×C₂₃	—	2	16	no	—
52	C₄⋊C₄×C₂₃	—	4	8	no	—
53	C₄⋊C₄×C₂₃	—	4	32	no	—
54	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
55	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
56	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
57	C₂×#1845	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
59	C₂³×C₂₃	#127	1	2	yes	—
60	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
61	C₂³×C₂₃	C₂×D₈	1	4	yes	—
62	C₂³×C₂₃	C₂×Q₁₆	1	4	yes	—
63	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	—	4	8	no	—
65	C₂×C₄×C₂₃	—	4	16	no	—
66	C₂×C₄×C₂₃	—	8	16	no	—
67	C₂×C₄×C₂₃	#107	1	32	yes	complemented
68	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
69	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
71	C₂×C₄×C₂₃	#127	1	2	yes	—
72	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
73	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₂₃	C₂×D₈	1	8	yes	—
75	C₂×C₄×C₂₃	C₂×Q₁₆	1	8	yes	—
76	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
78	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
79	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
80	C₈×C₂₃	C₂×C₄⋊C₄	1	16	yes	complemented
81	C₂²×#107	C₂×C₂₃	1	4	yes	—
82	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
83	C₂×#384	C₂×C₂₃	1	2	yes	—
84	#1845	C₂×C₂₃	1	8	yes	direct product factor
85	C₂²×C₂₃	#384	1	4	yes	—
86	C₂²×C₂₃	C₂×#107	1	8	yes	—
87	C₂²×C₂₃	#493	1	4	yes	—
88	C₂²×C₂₃	#494	1	4	yes	—
89	C₂²×C₂₃	#500	1	8	yes	—
90	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
92	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂²×C₂₃	C₂²×D₈	1	2	yes	—
94	C₂²×C₂₃	C₂²×Q₁₆	1	2	yes	—
95	C₄×C₂₃	—	8	8	no	—
96	C₄×C₂₃	#384	1	4	yes	—
97	C₄×C₂₃	C₂×#107	1	16	yes	complemented
98	C₄×C₂₃	C₂²×C₄⋊C₄	1	2	yes	—
99	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
100	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
101	C₂²×C₄⋊C₄	—	2	4	no	—
102	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
103	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
104	C₂×#107	—	2	16	no	—
105	C₂×#107	C₂²×C₂₃	1	16	yes	—
106	C₂×C₄×C₈	C₄×C₂₃	1	8	yes	complemented
107	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
108	#384	—	2	4	no	—
109	#384	C₂²×C₂₃	1	8	yes	—
110	C₂×C₂₃	#1845	1	8	yes	direct product factor
111	C₂×C₂₃	C₂×#384	1	1	yes	characteristic
112	C₂×C₂₃	C₂²×#107	1	2	yes	—
113	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
114	C₂×C₂₃	C₂×#494	1	1	yes	characteristic
115	C₂×C₂₃	C₂×#500	1	2	yes	—
116	C₂³×C₄	—	2	4	no	—
117	C₂³×C₄	D₄×C₂₃	1	2	yes	—
118	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
120	C₂²×C₈	D₄×C₂₃	1	8	yes	—
121	C₂²×C₈	Q₈×C₂₃	1	8	yes	—
122	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
123	C₂×C₄⋊C₄	—	2	16	no	—
124	C₂×C₄⋊C₄	—	2	16	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	4	16	no	—
127	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
128	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
129	C₂×C₄²	D₄×C₂₃	1	2	yes	—
130	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
131	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
132	#107	—	2	32	no	—
133	C₄×C₈	C₂×C₄×C₂₃	1	16	yes	complemented
134	C₂₃	C₂×#1845	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	4	8	no	—
138	C₂²×C₄	—	4	16	no	—
139	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
140	C₂²×C₄	D₈×C₂₃	1	8	yes	—
141	C₂²×C₄	Q₁₆×C₂₃	1	8	yes	—
142	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
145	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
146	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
147	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
149	C₂×C₈	C₄⋊C₄×C₂₃	1	32	yes	complemented
150	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
151	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
152	C₂×C₈	C₂×Q₈×C₂₃	1	8	yes	—
153	C₄⋊C₄	—	2	16	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	32	no	—
156	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
157	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
158	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
159	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
160	C₂³	C₂₃×#127	1	2	yes	—
161	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
162	C₂³	C₂×D₈×C₂₃	1	4	yes	—
163	C₂³	C₂×Q₁₆×C₂₃	1	4	yes	—
164	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
165	C₂×C₄	—	4	16	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	—	8	16	no	—
168	C₂×C₄	C₂₃×#107	1	32	yes	complemented
169	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
172	C₂×C₄	C₂₃×#127	1	2	yes	—
173	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
174	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
175	C₂×C₄	C₂×D₈×C₂₃	1	8	yes	—
176	C₂×C₄	C₂×Q₁₆×C₂₃	1	8	yes	—
177	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
179	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
180	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
181	C₈	C₂×C₄⋊C₄×C₂₃	1	16	yes	complemented
182	C₂²	C₂₃×#384	1	4	yes	—
183	C₂²	C₂×C₂₃×#107	1	8	yes	—
184	C₂²	C₂₃×#493	1	4	yes	—
185	C₂²	C₂₃×#494	1	4	yes	—
186	C₂²	C₂₃×#500	1	8	yes	—
187	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
189	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
190	C₂²	C₂²×D₈×C₂₃	1	2	yes	—
191	C₂²	C₂²×Q₁₆×C₂₃	1	2	yes	—
192	C₄	—	8	8	no	—
193	C₄	C₂₃×#384	1	4	yes	—
194	C₄	C₂×C₂₃×#107	1	16	yes	complemented
195	C₄	C₂²×C₄⋊C₄×C₂₃	1	2	yes	—
196	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
197	C₂	C₂₃×#1845	1	8	yes	direct product factor
198	C₂	C₂×C₂₃×#384	1	1	yes	characteristic
199	C₂	C₂²×C₂₃×#107	1	2	yes	—
200	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
201	C₂	C₂×C₂₃×#494	1	1	yes	characteristic
202	C₂	C₂×C₂₃×#500	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
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
4. C₂₃×#1845	C₂	direct (N × H)
11. C₂×C₄×C₈×C₂₃	C₄	semidirect
32. C₄×C₈×C₂₃	C₂×C₄	semidirect
47. C₂×C₈×C₂₃	C₄⋊C₄	semidirect
57. C₂×#1845	C₂₃	direct (N × H)
67. C₂×C₄×C₂₃	#107	semidirect
80. C₈×C₂₃	C₂×C₄⋊C₄	semidirect
84. #1845	C₂×C₂₃	direct (N × H)
97. C₄×C₂₃	C₂×#107	semidirect
106. C₂×C₄×C₈	C₄×C₂₃	semidirect
110. C₂×C₂₃	#1845	direct (N × H)
133. C₄×C₈	C₂×C₄×C₂₃	semidirect
134. C₂₃	C₂×#1845	direct (N × H)
149. C₂×C₈	C₄⋊C₄×C₂₃	semidirect
168. C₂×C₄	C₂₃×#107	semidirect
181. C₈	C₂×C₄⋊C₄×C₂₃	semidirect
194. C₄	C₂×C₂₃×#107	semidirect
197. C₂	C₂₃×#1845	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×#107	2
C₂²×C₄×C₈×C₂₃	2
C₂×C₂₃×#384	2
C₂₃×#1845	2
C₂×#1845	23

Subgroups — C2×C23×#1845 #19168421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1845 #19168421