Subgroups — C₈₉×#12832 #36390680

Order 22784.

358 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	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2155	C₂	1	2	yes	maximal, complemented
3	C₈₉×#1813	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1812	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
7	#504×C₈₉	—	2	4	no	—
8	C₂×M₃₂×C₈₉	C₄	1	2	yes	complemented
9	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₈₉	C₄	1	2	yes	complemented
11	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×#108×C₈₉	C₂²	1	2	yes	—
13	#427×C₈₉	C₂²	1	2	yes	—
14	#413×C₈₉	—	2	1	no	—
15	#410×C₈₉	—	2	1	no	—
16	#365×C₈₉	—	2	4	no	—
17	#338×C₈₉	—	2	1	no	—
18	H(3,ℤ/4)×C₈₉	—	2	1	no	—
19	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
20	#131×C₈₉	—	2	2	no	—
21	#131×C₈₉	—	2	2	no	—
22	C₂×M₁₆×C₈₉	—	2	2	no	—
23	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₈₉	Q₈	1	2	yes	—
25	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	#117×C₈₉	—	2	2	no	—
28	M₃₂×C₈₉	—	2	4	no	—
29	M₃₂×C₈₉	D₄	1	4	yes	—
30	C₂×C₁₆×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	—	2	4	no	—
32	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
35	#108×C₈₉	—	2	2	no	—
36	#108×C₈₉	—	2	2	no	—
37	#107×C₈₉	—	2	2	no	—
38	#106×C₈₉	—	2	2	no	—
39	#101×C₈₉	—	4	1	no	—
40	#100×C₈₉	—	4	1	no	—
41	#99×C₈₉	—	4	2	no	—
42	#41×C₈₉	—	2	2	no	—
43	#41×C₈₉	—	2	2	no	—
44	C₂×Q₈×C₈₉	SD₁₆	1	1	yes	characteristic
45	C₂×D₄×C₈₉	SD₁₆	1	2	yes	—
46	C₂×D₄×C₈₉	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₈₉	—	2	2	no	—
50	M₁₆×C₈₉	—	2	4	no	—
51	M₁₆×C₈₉	—	4	2	no	—
52	M₁₆×C₈₉	—	4	2	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	4	2	no	—
56	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₉	#41	1	2	yes	—
61	C₄⋊C₄×C₈₉	—	4	2	no	—
62	C₂²⋊C₄×C₈₉	—	4	2	no	—
63	C₄²×C₈₉	—	4	2	no	—
64	C₁₆×C₈₉	—	2	1	no	—
65	C₁₆×C₈₉	—	2	1	no	—
66	C₁₆×C₈₉	C₂²⋊C₄	1	4	yes	—
67	C₂³×C₈₉	#106	1	2	yes	—
68	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈×C₈₉	—	2	2	no	—
70	D₄×C₈₉	—	2	4	no	—
71	D₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	2	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	8	2	no	—
75	C₂×C₄×C₈₉	#106	1	2	yes	—
76	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#121	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#135	1	2	yes	—
80	C₈×C₈₉	—	2	2	no	—
81	C₈×C₈₉	—	8	2	no	—
82	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈×C₈₉	#117	1	1	yes	characteristic
84	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
85	C₂²×C₈₉	—	2	2	no	—
86	C₂²×C₈₉	—	4	1	no	—
87	C₂²×C₈₉	—	4	2	no	—
88	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
89	C₂²×C₈₉	#465	1	2	yes	—
90	C₄×C₈₉	—	2	2	no	—
91	C₄×C₈₉	—	16	2	no	—
92	C₄×C₈₉	#380	1	1	yes	characteristic
93	C₄×C₈₉	#427	1	1	yes	characteristic
94	C₄×C₈₉	#465	1	2	yes	—
95	#12832	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₈₉	—	2	2	no	—
97	C₂×C₈₉	—	2	1	no	—
98	C₂×C₈₉	—	4	2	no	—
99	C₂×C₈₉	#1807	1	1	yes	characteristic
100	C₂×#504	C₂×C₈₉	1	1	yes	characteristic
101	#2155	C₂×C₈₉	1	2	yes	complemented
102	#1813	C₂×C₈₉	1	1	yes	characteristic
103	#1812	C₂×C₈₉	1	1	yes	characteristic
104	#1382	C₂×C₈₉	1	2	yes	complemented
105	C₈₉	#12832	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
107	#504	—	2	4	no	—
108	C₂×M₃₂	C₄×C₈₉	1	2	yes	complemented
109	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
110	C₂²×C₁₆	C₄×C₈₉	1	2	yes	complemented
111	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
112	C₂×#108	C₂²×C₈₉	1	2	yes	—
113	#427	C₂²×C₈₉	1	2	yes	—
114	#413	—	2	1	no	—
115	#410	—	2	1	no	—
116	#365	—	2	4	no	—
117	#338	—	2	1	no	—
118	H(3,ℤ/4)	—	2	1	no	—
119	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
124	C₂×M₁₆	Q₈×C₈₉	1	2	yes	—
125	C₂²×C₈	C₂×C₄×C₈₉	1	2	yes	—
126	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
127	#117	—	2	2	no	—
128	M₃₂	—	2	4	no	—
129	M₃₂	D₄×C₈₉	1	4	yes	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic
133	C₂×C₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic
134	C₂×C₁₆	C₂×C₄×C₈₉	1	2	yes	—
135	#108	—	2	2	no	—
136	#108	—	2	2	no	—
137	#107	—	2	2	no	—
138	#106	—	2	2	no	—
139	#101	—	4	1	no	—
140	#100	—	4	1	no	—
141	#99	—	4	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	SD₁₆×C₈₉	1	1	yes	characteristic
145	C₂×D₄	SD₁₆×C₈₉	1	2	yes	—
146	C₂×D₄	SD₁₆×C₈₉	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊C₄×C₈₉	1	2	yes	—
148	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₄⋊C₄×C₈₉	1	2	yes	—
157	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
158	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
159	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
160	C₂×C₈	#41×C₈₉	1	2	yes	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₄²	—	4	2	no	—
164	C₁₆	—	2	1	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	C₂²⋊C₄×C₈₉	1	4	yes	—
167	C₂³	#106×C₈₉	1	2	yes	—
168	C₂³	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	#106×C₈₉	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
177	C₂×C₄	#121×C₈₉	1	2	yes	—
178	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
179	C₂×C₄	#135×C₈₉	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	8	2	no	—
182	C₈	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
183	C₈	#117×C₈₉	1	1	yes	characteristic
184	C₈	C₂²⋊Q₈×C₈₉	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#106×C₈₉	1	1	yes	characteristic
189	C₂²	#465×C₈₉	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	16	2	no	—
192	C₄	#380×C₈₉	1	1	yes	characteristic
193	C₄	#427×C₈₉	1	1	yes	characteristic
194	C₄	#465×C₈₉	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	C₈₉×#1807	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	#427
commutator	C₂×C₈	C₂²×C₄×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
socle	C₂×C₈₉	#1807
prime radical	C₂×#41×C₈₉	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂×M₁₆×C₈₉ ◁ C₂×#131×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2155	C₂	semidirect
5. C₈₉×#1382	C₂	semidirect
8. C₂×M₃₂×C₈₉	C₄	semidirect
10. C₂²×C₁₆×C₈₉	C₄	semidirect
95. #12832	C₈₉	direct (N × H)
101. #2155	C₂×C₈₉	semidirect
104. #1382	C₂×C₈₉	semidirect
105. C₈₉	#12832	direct (N × H)
108. C₂×M₃₂	C₄×C₈₉	semidirect
110. C₂²×C₁₆	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504×C₈₉	2
C₈₉×#2155	2
C₈₉×#1813	2
C₈₉×#1812	2
C₈₉×#1382	2
#12832	89

Subgroups — C89×#12832 #36390680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#12832 #36390680