Subgroups — C₂×C₂₃×#2982 #19207868

Order 5888.

2234 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2982	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₈×Q₈×C₂₃	C₂	1	8	yes	maximal, complemented
4	C₂×C₂₃×#431	C₂	1	6	yes	maximal
5	C₂³×Q₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₂₃	C₄	1	4	yes	—
7	C₂×C₄×Q₈×C₂₃	C₄	1	8	yes	—
8	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
9	C₂×C₄×Q₈×C₂₃	C₂²	1	8	yes	—
10	C₂²×C₄⋊C₄×C₂₃	C₄	1	3	yes	—
11	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
12	C₈×Q₈×C₂₃	C₂²	1	64	yes	complemented
13	C₂₃×#431	C₂²	1	48	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	12	yes	—
15	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	12	yes	—
17	C₂²×Q₈×C₂₃	C₈	1	16	yes	complemented
18	C₂²×Q₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	3	yes	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
24	C₄×Q₈×C₂₃	C₂×C₄	1	16	yes	—
25	C₄×Q₈×C₂₃	C₂×C₄	1	32	yes	—
26	C₄×Q₈×C₂₃	C₂³	1	16	yes	—
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	12	yes	—
28	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	24	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	6	yes	—
30	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
33	C₂₃×C₄⋊C₈	C₂³	1	48	yes	—
34	C₂₃×C₂²⋊C₈	C₂³	1	16	yes	—
35	C₄×C₈×C₂₃	C₂³	1	48	yes	—
36	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Q₈×C₂₃	C₂×C₈	1	32	yes	—
38	C₂×Q₈×C₂₃	C₂×C₈	1	64	yes	complemented
39	C₂×Q₈×C₂₃	C₂²×C₄	1	16	yes	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	—	4	6	no	—
42	C₂²×C₄×C₂₃	C₂×C₈	1	24	yes	—
43	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
46	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
49	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₂₃	—	2	16	no	—
52	C₂×C₈×C₂₃	C₂⁴	1	16	yes	—
53	C₄⋊C₄×C₂₃	C₂²×C₄	1	48	yes	—
54	C₄²×C₂₃	C₂²×C₄	1	24	yes	—
55	C₄²×C₂₃	C₂²×C₄	1	12	yes	—
56	C₄²×C₂₃	C₂⁴	1	12	yes	—
57	C₂×#2982	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	C₂²×C₈	1	4	yes	—
61	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
62	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
63	Q₈×C₂₃	C₂²×C₈	1	64	yes	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	12	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	4	12	no	—
69	C₂×C₄×C₂₃	C₂²×C₈	1	24	yes	—
70	C₂×C₄×C₂₃	C₂²×C₈	1	48	yes	—
71	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
72	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
73	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
75	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
76	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
77	C₈×C₂₃	—	2	16	no	—
78	C₂²×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
79	#2982	C₂×C₂₃	1	16	yes	direct product factor
80	C₂×C₈×Q₈	C₂×C₂₃	1	8	yes	complemented
81	C₂×#431	C₂×C₂₃	1	6	yes	—
82	C₂²×C₂₃	—	2	2	no	—
83	C₂²×C₂₃	—	2	8	no	—
84	C₂²×C₂₃	—	2	2	no	—
85	C₂²×C₂₃	#519	1	2	yes	—
86	C₂²×C₂₃	C₂³×C₈	1	4	yes	—
87	C₂²×C₂₃	C₂³×C₈	1	2	yes	—
88	C₂²×C₂₃	#568	1	2	yes	—
89	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
90	C₄×C₂₃	—	2	2	no	—
91	C₄×C₂₃	—	2	12	no	—
92	C₄×C₂₃	#519	1	2	yes	—
93	C₄×C₂₃	C₂³×C₈	1	24	yes	—
94	C₄×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
95	C₂³×Q₈	C₄×C₂₃	1	1	yes	characteristic
96	C₂×C₄×Q₈	C₄×C₂₃	1	4	yes	—
97	C₂×C₄×Q₈	C₄×C₂₃	1	8	yes	—
98	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
99	C₂×C₄×Q₈	C₂²×C₂₃	1	8	yes	—
100	C₂²×C₄⋊C₄	C₄×C₂₃	1	3	yes	—
101	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
102	C₈×Q₈	C₂²×C₂₃	1	64	yes	complemented
103	#431	C₂²×C₂₃	1	48	yes	—
104	C₂×C₄⋊C₈	C₂²×C₂₃	1	12	yes	—
105	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
106	C₂×C₄×C₈	C₂²×C₂₃	1	12	yes	—
107	C₂×C₂₃	—	2	4	no	—
108	C₂×C₂₃	#2982	1	4	yes	direct product factor
109	C₂×C₂₃	C₂×#519	1	1	yes	characteristic
110	C₂×C₂₃	C₂⁴×C₈	1	1	yes	characteristic
111	C₂×C₂₃	C₂×#568	1	1	yes	characteristic
112	C₂²×Q₈	C₈×C₂₃	1	16	yes	complemented
113	C₂²×Q₈	C₂×C₄×C₂₃	1	4	yes	—
114	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	—
115	C₂³×C₄	C₂×C₄×C₂₃	1	3	yes	—
116	C₂³×C₄	C₂×C₄×C₂₃	1	3	yes	—
117	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
119	C₄×Q₈	C₂×C₄×C₂₃	1	16	yes	—
120	C₄×Q₈	C₂×C₄×C₂₃	1	32	yes	—
121	C₄×Q₈	C₂³×C₂₃	1	16	yes	—
122	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	12	yes	—
123	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	24	yes	—
124	C₂×C₄²	C₂×C₄×C₂₃	1	6	yes	—
125	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
126	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
127	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
128	C₄⋊C₈	C₂³×C₂₃	1	48	yes	—
129	C₂²⋊C₈	C₂³×C₂₃	1	16	yes	—
130	C₄×C₈	C₂³×C₂₃	1	48	yes	—
131	C₂₃	C₂×#2982	1	1	yes	characteristic, direct product factor, Sylow
132	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
133	C₂×Q₈	C₂×C₈×C₂₃	1	32	yes	—
134	C₂×Q₈	C₂×C₈×C₂₃	1	64	yes	complemented
135	C₂×Q₈	C₂²×C₄×C₂₃	1	16	yes	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	6	no	—
138	C₂²×C₄	C₂×C₈×C₂₃	1	24	yes	—
139	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
142	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
143	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
144	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
145	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
147	C₂×C₈	—	2	16	no	—
148	C₂×C₈	C₂⁴×C₂₃	1	16	yes	—
149	C₄⋊C₄	C₂²×C₄×C₂₃	1	48	yes	—
150	C₄²	C₂²×C₄×C₂₃	1	24	yes	—
151	C₄²	C₂²×C₄×C₂₃	1	12	yes	—
152	C₄²	C₂⁴×C₂₃	1	12	yes	—
153	C₂³	—	2	2	no	—
154	C₂³	—	2	2	no	—
155	C₂³	C₂²×C₈×C₂₃	1	4	yes	—
156	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
157	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
158	Q₈	C₂²×C₈×C₂₃	1	64	yes	—
159	C₂×C₄	—	2	12	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	12	no	—
164	C₂×C₄	C₂²×C₈×C₂₃	1	24	yes	—
165	C₂×C₄	C₂²×C₈×C₂₃	1	48	yes	—
166	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
167	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
168	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
169	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
170	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
171	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
172	C₈	—	2	16	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	8	no	—
175	C₂²	—	2	2	no	—
176	C₂²	C₂₃×#519	1	2	yes	—
177	C₂²	C₂³×C₈×C₂₃	1	4	yes	—
178	C₂²	C₂³×C₈×C₂₃	1	2	yes	—
179	C₂²	C₂₃×#568	1	2	yes	—
180	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	12	no	—
183	C₄	C₂₃×#519	1	2	yes	—
184	C₄	C₂³×C₈×C₂₃	1	24	yes	—
185	C₄	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	C₂₃×#2982	1	4	yes	direct product factor
188	C₂	C₂×C₂₃×#519	1	1	yes	characteristic
189	C₂	C₂⁴×C₈×C₂₃	1	1	yes	characteristic
190	C₂	C₂×C₂₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂	C₂⁴×C₈×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2982	C₂	direct (N × H)
3. C₂×C₈×Q₈×C₂₃	C₂	semidirect
12. C₈×Q₈×C₂₃	C₂²	semidirect
17. C₂²×Q₈×C₂₃	C₈	semidirect
38. C₂×Q₈×C₂₃	C₂×C₈	semidirect
57. C₂×#2982	C₂₃	direct (N × H)
79. #2982	C₂×C₂₃	direct (N × H)
80. C₂×C₈×Q₈	C₂×C₂₃	semidirect
102. C₈×Q₈	C₂²×C₂₃	semidirect
108. C₂×C₂₃	#2982	direct (N × H)
112. C₂²×Q₈	C₈×C₂₃	semidirect
131. C₂₃	C₂×#2982	direct (N × H)
134. C₂×Q₈	C₂×C₈×C₂₃	semidirect
187. C₂	C₂₃×#2982	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×C₄×Q₈×C₂₃	2
C₂₃×#2982	2
C₂×C₈×Q₈×C₂₃	2
C₂×C₂₃×#431	2
C₂×#2982	23

Subgroups — C2×C23×#2982 #19207868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2982 #19207868