Subgroups — C₂×C₈₉×#2863 #36414481

Order 22784.

2142 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	C₈₉×#2863	C₂	1	16	yes	maximal, direct product factor
2	C₂×#391×C₈₉	C₂	1	3	yes	maximal
3	C₂×#389×C₈₉	C₂	1	3	yes	maximal
4	C₂×#387×C₈₉	C₂	1	6	yes	maximal
5	C₂×#383×C₈₉	C₂	1	2	yes	maximal
6	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
9	C₂²×C₄⋊C₄×C₈₉	C₂²	1	6	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
11	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
12	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	#391×C₈₉	C₂²	1	24	yes	—
14	#389×C₈₉	C₂²	1	24	yes	—
15	#387×C₈₉	C₂²	1	48	yes	—
16	#383×C₈₉	C₂²	1	16	yes	—
17	C₂×#95×C₈₉	C₂²	1	6	yes	—
18	C₂×#95×C₈₉	C₂²	1	6	yes	—
19	C₂×#95×C₈₉	C₂²	1	6	yes	—
20	C₄³×C₈₉	C₂²	1	8	yes	—
21	C₂²×Q₈×C₈₉	—	2	12	no	—
22	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
23	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
24	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
25	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
26	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
27	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₈₉	—	2	24	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	12	no	—
30	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	—
31	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
32	C₂×C₄²×C₈₉	—	2	12	no	—
33	C₂×C₄²×C₈₉	—	2	12	no	—
34	C₂×C₄²×C₈₉	Q₈	1	8	yes	complemented
35	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
38	#95×C₈₉	C₂³	1	24	yes	—
39	#95×C₈₉	C₂³	1	24	yes	—
40	#95×C₈₉	C₂³	1	24	yes	—
41	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
42	C₂×Q₈×C₈₉	—	2	24	no	—
43	C₂×Q₈×C₈₉	—	4	16	no	—
44	C₂²×C₄×C₈₉	—	2	6	no	—
45	C₂²×C₄×C₈₉	—	2	6	no	—
46	C₂²×C₄×C₈₉	—	2	6	no	—
47	C₂²×C₄×C₈₉	—	2	6	no	—
48	C₂²×C₄×C₈₉	—	2	12	no	—
49	C₂²×C₄×C₈₉	—	2	24	no	—
50	C₂²×C₄×C₈₉	—	2	12	no	—
51	C₂²×C₄×C₈₉	C₂×Q₈	1	12	yes	—
52	C₂²×C₄×C₈₉	#41	1	12	yes	—
53	C₂²×C₄×C₈₉	#41	1	12	yes	—
54	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
55	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
56	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
57	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
58	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
59	C₄⋊C₄×C₈₉	—	2	48	no	—
60	C₄⋊C₄×C₈₉	—	2	24	no	—
61	C₄²×C₈₉	—	2	24	no	—
62	C₄²×C₈₉	—	2	24	no	—
63	C₄²×C₈₉	C₂×Q₈	1	16	yes	complemented
64	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
65	C₂³×C₈₉	C₂×#41	1	6	yes	—
66	C₂³×C₈₉	C₂×#41	1	6	yes	—
67	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
68	Q₈×C₈₉	—	4	16	no	—
69	C₂×C₄×C₈₉	—	2	24	no	—
70	C₂×C₄×C₈₉	—	2	24	no	—
71	C₂×C₄×C₈₉	—	2	6	no	—
72	C₂×C₄×C₈₉	—	2	6	no	—
73	C₂×C₄×C₈₉	—	2	6	no	—
74	C₂×C₄×C₈₉	—	2	6	no	—
75	C₂×C₄×C₈₉	—	2	12	no	—
76	C₂×C₄×C₈₉	—	2	24	no	—
77	C₂×C₄×C₈₉	—	2	12	no	—
78	C₂×C₄×C₈₉	—	4	8	no	—
79	C₂×C₄×C₈₉	—	4	24	no	—
80	C₂×C₄×C₈₉	#126	1	16	yes	—
81	C₂×C₄×C₈₉	C₂²×Q₈	1	12	yes	—
82	C₂×C₄×C₈₉	C₂×#41	1	12	yes	—
83	C₂×C₄×C₈₉	C₂×#41	1	12	yes	—
84	C₂²×C₈₉	C₂×#126	1	4	yes	—
85	C₂²×C₈₉	#529	1	12	yes	—
86	C₂²×C₈₉	#533	1	12	yes	—
87	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
88	C₂²×C₈₉	C₂²×#41	1	3	yes	—
89	C₂²×C₈₉	C₂²×#41	1	3	yes	—
90	C₄×C₈₉	—	2	12	no	—
91	C₄×C₈₉	—	2	12	no	—
92	C₄×C₈₉	—	4	4	no	—
93	C₄×C₈₉	—	4	12	no	—
94	C₄×C₈₉	C₂×#126	1	8	yes	—
95	C₂×#2863	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₈₉	#2863	1	8	yes	direct product factor
97	C₂×C₈₉	C₂²×#126	1	1	yes	characteristic
98	C₂×C₈₉	C₂×#529	1	3	yes	—
99	C₂×C₈₉	C₂×#533	1	3	yes	—
100	#2863	C₂×C₈₉	1	16	yes	direct product factor
101	C₂×#391	C₂×C₈₉	1	3	yes	—
102	C₂×#389	C₂×C₈₉	1	3	yes	—
103	C₂×#387	C₂×C₈₉	1	6	yes	—
104	C₂×#383	C₂×C₈₉	1	2	yes	—
105	C₂×C₄³	C₂×C₈₉	1	1	yes	characteristic
106	C₈₉	C₂×#2863	1	1	yes	characteristic, direct product factor, Sylow
107	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
108	C₂²×C₄⋊C₄	C₂²×C₈₉	1	6	yes	—
109	C₂²×C₄⋊C₄	C₂²×C₈₉	1	3	yes	—
110	C₂²×C₄²	C₂²×C₈₉	1	3	yes	—
111	C₂²×C₄²	C₂²×C₈₉	1	3	yes	—
112	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
113	#391	C₂²×C₈₉	1	24	yes	—
114	#389	C₂²×C₈₉	1	24	yes	—
115	#387	C₂²×C₈₉	1	48	yes	—
116	#383	C₂²×C₈₉	1	16	yes	—
117	C₂×#95	C₂²×C₈₉	1	6	yes	—
118	C₂×#95	C₂²×C₈₉	1	6	yes	—
119	C₂×#95	C₂²×C₈₉	1	6	yes	—
120	C₄³	C₂²×C₈₉	1	8	yes	—
121	C₂²×Q₈	—	2	12	no	—
122	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
123	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
124	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
125	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
126	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
127	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	24	no	—
129	C₂×C₄⋊C₄	—	2	12	no	—
130	C₂×C₄⋊C₄	C₂³×C₈₉	1	24	yes	—
131	C₂×C₄⋊C₄	C₂³×C₈₉	1	12	yes	—
132	C₂×C₄²	—	2	12	no	—
133	C₂×C₄²	—	2	12	no	—
134	C₂×C₄²	Q₈×C₈₉	1	8	yes	complemented
135	C₂×C₄²	C₂³×C₈₉	1	12	yes	—
136	C₂×C₄²	C₂³×C₈₉	1	12	yes	—
137	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
138	#95	C₂³×C₈₉	1	24	yes	—
139	#95	C₂³×C₈₉	1	24	yes	—
140	#95	C₂³×C₈₉	1	24	yes	—
141	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
142	C₂×Q₈	—	2	24	no	—
143	C₂×Q₈	—	4	16	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	—	2	6	no	—
146	C₂²×C₄	—	2	6	no	—
147	C₂²×C₄	—	2	6	no	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	2	24	no	—
150	C₂²×C₄	—	2	12	no	—
151	C₂²×C₄	C₂×Q₈×C₈₉	1	12	yes	—
152	C₂²×C₄	#41×C₈₉	1	12	yes	—
153	C₂²×C₄	#41×C₈₉	1	12	yes	—
154	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
155	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
156	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
157	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
158	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
159	C₄⋊C₄	—	2	48	no	—
160	C₄⋊C₄	—	2	24	no	—
161	C₄²	—	2	24	no	—
162	C₄²	—	2	24	no	—
163	C₄²	C₂×Q₈×C₈₉	1	16	yes	complemented
164	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
165	C₂³	C₂×#41×C₈₉	1	6	yes	—
166	C₂³	C₂×#41×C₈₉	1	6	yes	—
167	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
168	Q₈	—	4	16	no	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	2	24	no	—
171	C₂×C₄	—	2	6	no	—
172	C₂×C₄	—	2	6	no	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	—	2	12	no	—
176	C₂×C₄	—	2	24	no	—
177	C₂×C₄	—	2	12	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	24	no	—
180	C₂×C₄	#126×C₈₉	1	16	yes	—
181	C₂×C₄	C₂²×Q₈×C₈₉	1	12	yes	—
182	C₂×C₄	C₂×#41×C₈₉	1	12	yes	—
183	C₂×C₄	C₂×#41×C₈₉	1	12	yes	—
184	C₂²	C₂×#126×C₈₉	1	4	yes	—
185	C₂²	#529×C₈₉	1	12	yes	—
186	C₂²	#533×C₈₉	1	12	yes	—
187	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
188	C₂²	C₂²×#41×C₈₉	1	3	yes	—
189	C₂²	C₂²×#41×C₈₉	1	3	yes	—
190	C₄	—	2	12	no	—
191	C₄	—	2	12	no	—
192	C₄	—	4	4	no	—
193	C₄	—	4	12	no	—
194	C₄	C₂×#126×C₈₉	1	8	yes	—
195	C₂	C₈₉×#2863	1	8	yes	direct product factor
196	C₂	C₂²×#126×C₈₉	1	1	yes	characteristic
197	C₂	C₂×#529×C₈₉	1	3	yes	—
198	C₂	C₂×#533×C₈₉	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#2863	C₂	direct (N × H)
34. C₂×C₄²×C₈₉	Q₈	semidirect
63. C₄²×C₈₉	C₂×Q₈	semidirect
95. C₂×#2863	C₈₉	direct (N × H)
96. C₂×C₈₉	#2863	direct (N × H)
100. #2863	C₂×C₈₉	direct (N × H)
106. C₈₉	C₂×#2863	direct (N × H)
134. C₂×C₄²	Q₈×C₈₉	semidirect
163. C₄²	C₂×Q₈×C₈₉	semidirect
195. C₂	C₈₉×#2863	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈₉×#2863	2
C₂×#391×C₈₉	2
C₂×#389×C₈₉	2
C₂×#387×C₈₉	2
C₂×#383×C₈₉	2
C₂×C₄³×C₈₉	2
C₂×#2863	89

Subgroups — C2×C89×#2863 #36414481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2863 #36414481