Subgroups — C₂×C₂₃×#1380 #19160181

Order 5888.

694 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#107	C₂	1	2	yes	maximal
3	C₂₃×#1380	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×C₄⋊C₄×C₂₃	—	2	2	no	—
6	C₂²×C₁₆×C₂₃	—	2	4	no	—
7	C₂²×C₁₆×C₂₃	C₄	1	4	yes	complemented
8	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×#107	—	2	2	no	—
10	C₂×C₂₃×#107	—	2	4	no	—
11	C₂×C₂₃×#107	C₄	1	8	yes	complemented
12	C₂×C₂₃×#107	C₂²	1	4	yes	—
13	C₂³×C₄×C₂₃	—	4	2	no	—
14	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	D₄	1	4	yes	—
19	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
23	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
24	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
25	C₂×C₁₆×C₂₃	—	2	8	no	—
26	C₂×C₁₆×C₂₃	—	2	4	no	—
27	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
28	C₂₃×#107	—	2	8	no	—
29	C₂₃×#107	—	2	4	no	—
30	C₂₃×#107	C₂×C₄	1	8	yes	complemented
31	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	—	4	8	no	—
33	C₂²×C₄×C₂₃	—	4	4	no	—
34	C₂²×C₄×C₂₃	—	8	4	no	—
35	C₂²×C₄×C₂₃	—	8	4	no	—
36	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
40	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₄²	1	4	yes	—
43	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
47	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
51	C₄⋊C₄×C₂₃	—	2	8	no	—
52	C₄⋊C₄×C₂₃	—	4	8	no	—
53	C₄⋊C₄×C₂₃	—	4	4	no	—
54	C₁₆×C₂₃	—	2	4	no	—
55	C₂×#1380	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₃	#102	1	4	yes	—
57	C₂³×C₂₃	#107	1	2	yes	—
58	C₂³×C₂₃	D₁₆	1	2	yes	—
59	C₂³×C₂₃	SD₃₂	1	4	yes	—
60	C₂³×C₂₃	Q₃₂	1	2	yes	—
61	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₂₃	—	4	8	no	—
63	C₂×C₄×C₂₃	—	8	16	no	—
64	C₂×C₄×C₂₃	—	8	4	no	—
65	C₂×C₄×C₂₃	—	8	4	no	—
66	C₂×C₄×C₂₃	#95	1	4	yes	—
67	C₂×C₄×C₂₃	#102	1	4	yes	—
68	C₂×C₄×C₂₃	#103	1	8	yes	—
69	C₂×C₄×C₂₃	#106	1	4	yes	—
70	C₂×C₄×C₂₃	#107	1	2	yes	—
71	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₈×C₂₃	#95	1	4	yes	—
76	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
77	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₁₆	C₂×C₂₃	1	1	yes	characteristic
80	C₂²×#107	C₂×C₂₃	1	2	yes	—
81	#1380	C₂×C₂₃	1	4	yes	direct product factor
82	C₂²×C₂₃	#340	1	4	yes	—
83	C₂²×C₂₃	#357	1	8	yes	—
84	C₂²×C₂₃	#358	1	8	yes	—
85	C₂²×C₂₃	#366	1	4	yes	—
86	C₂²×C₂₃	#367	1	4	yes	—
87	C₂²×C₂₃	C₂×#102	1	2	yes	—
88	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×D₁₆	1	1	yes	characteristic
90	C₂²×C₂₃	C₂×SD₃₂	1	2	yes	—
91	C₂²×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
92	C₄×C₂₃	—	8	8	no	—
93	C₄×C₂₃	#340	1	4	yes	—
94	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
95	C₄×C₂₃	C₂×#103	1	2	yes	—
96	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	C₂²×C₄⋊C₄	—	2	2	no	—
99	C₂²×C₁₆	—	2	4	no	—
100	C₂²×C₁₆	C₄×C₂₃	1	4	yes	complemented
101	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
102	C₂×#107	—	2	2	no	—
103	C₂×#107	—	2	4	no	—
104	C₂×#107	C₄×C₂₃	1	8	yes	complemented
105	C₂×#107	C₂²×C₂₃	1	4	yes	—
106	C₂×C₂₃	#1380	1	8	yes	direct product factor
107	C₂×C₂₃	C₂×#340	1	1	yes	characteristic
108	C₂×C₂₃	C₂×#357	1	2	yes	—
109	C₂×C₂₃	C₂×#358	1	2	yes	—
110	C₂×C₂₃	C₂×#366	1	1	yes	characteristic
111	C₂×C₂₃	C₂×#367	1	1	yes	characteristic
112	C₂³×C₄	—	4	2	no	—
113	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
115	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
116	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
117	C₂²×C₈	D₄×C₂₃	1	4	yes	—
118	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
119	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	2	8	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	4	2	no	—
123	C₂×C₄⋊C₄	—	4	4	no	—
124	C₂×C₁₆	—	2	8	no	—
125	C₂×C₁₆	—	2	4	no	—
126	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
127	#107	—	2	8	no	—
128	#107	—	2	4	no	—
129	#107	C₂×C₄×C₂₃	1	8	yes	complemented
130	C₂₃	C₂×#1380	1	1	yes	characteristic, direct product factor, Sylow
131	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
132	C₂²×C₄	—	4	8	no	—
133	C₂²×C₄	—	4	4	no	—
134	C₂²×C₄	—	8	4	no	—
135	C₂²×C₄	—	8	4	no	—
136	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
137	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
138	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
139	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
140	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
141	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
142	C₂×C₈	C₄²×C₂₃	1	4	yes	—
143	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
144	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
145	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
146	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
147	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
148	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
149	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
150	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₁₆	—	2	4	no	—
155	C₂³	C₂₃×#102	1	4	yes	—
156	C₂³	C₂₃×#107	1	2	yes	—
157	C₂³	C₂₃×D₁₆	1	2	yes	—
158	C₂³	C₂₃×SD₃₂	1	4	yes	—
159	C₂³	C₂₃×Q₃₂	1	2	yes	—
160	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	8	16	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	C₂₃×#95	1	4	yes	—
166	C₂×C₄	C₂₃×#102	1	4	yes	—
167	C₂×C₄	C₂₃×#103	1	8	yes	—
168	C₂×C₄	C₂₃×#106	1	4	yes	—
169	C₂×C₄	C₂₃×#107	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
171	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
172	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
173	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
174	C₈	C₂₃×#95	1	4	yes	—
175	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
176	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
177	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
178	C₂²	C₂₃×#340	1	4	yes	—
179	C₂²	C₂₃×#357	1	8	yes	—
180	C₂²	C₂₃×#358	1	8	yes	—
181	C₂²	C₂₃×#366	1	4	yes	—
182	C₂²	C₂₃×#367	1	4	yes	—
183	C₂²	C₂×C₂₃×#102	1	2	yes	—
184	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
185	C₂²	C₂×C₂₃×D₁₆	1	1	yes	characteristic
186	C₂²	C₂×C₂₃×SD₃₂	1	2	yes	—
187	C₂²	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
188	C₄	—	8	8	no	—
189	C₄	C₂₃×#340	1	4	yes	—
190	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
191	C₄	C₂×C₂₃×#103	1	2	yes	—
192	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
193	C₂	C₂₃×#1380	1	8	yes	direct product factor
194	C₂	C₂×C₂₃×#340	1	1	yes	characteristic
195	C₂	C₂×C₂₃×#357	1	2	yes	—
196	C₂	C₂×C₂₃×#358	1	2	yes	—
197	C₂	C₂×C₂₃×#366	1	1	yes	characteristic
198	C₂	C₂×C₂₃×#367	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	D₈
commutator	C₈	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1380	C₂	direct (N × H)
7. C₂²×C₁₆×C₂₃	C₄	semidirect
11. C₂×C₂₃×#107	C₄	semidirect
27. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
30. C₂₃×#107	C₂×C₄	semidirect
55. C₂×#1380	C₂₃	direct (N × H)
81. #1380	C₂×C₂₃	direct (N × H)
100. C₂²×C₁₆	C₄×C₂₃	semidirect
104. C₂×#107	C₄×C₂₃	semidirect
106. C₂×C₂₃	#1380	direct (N × H)
126. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
129. #107	C₂×C₄×C₂₃	semidirect
130. C₂₃	C₂×#1380	direct (N × H)
193. C₂	C₂₃×#1380	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×C₁₆×C₂₃	2
C₂²×C₂₃×#107	2
C₂₃×#1380	2
C₂×#1380	23

Subgroups — C2×C23×#1380 #19160181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1380 #19160181