Subgroups — #504×Dic₈₉ #36707071

Order 22784.

834 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	C₂×#19152015	C₂	1	1	yes	maximal, characteristic
2	#131×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26021851	C₂	1	3	yes	maximal, complemented
4	#26021848	C₂	1	3	yes	maximal, complemented
5	M₃₂×Dic₈₉	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₆×Dic₈₉	C₂	1	3	yes	maximal, complemented
7	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668019	C₄	1	1	yes	characteristic
9	#41×Dic₈₉	C₄	1	1	yes	characteristic
10	#19152015	C₄	1	8	yes	complemented
11	C₂×#15667952	C₂²	1	3	yes	—
12	C₂²×#13083549	C₂²	1	3	yes	—
13	#19151556	C₄	1	3	yes	—
14	#19151553	C₂²	1	3	yes	—
15	M₁₆×Dic₈₉	C₄	1	3	yes	—
16	C₂×C₈×Dic₈₉	C₂²	1	3	yes	—
17	#19151373	C₂²	1	6	yes	—
18	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
19	C₁₆×Dic₈₉	C₂²	1	3	yes	—
20	C₁₆×Dic₈₉	C₂²	1	3	yes	—
21	C₁₆×Dic₈₉	C₂²	1	3	yes	—
22	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
23	#504×C₈₉	C₄	1	8	yes	complemented
24	C₂×M₃₂×C₈₉	C₂²	1	3	yes	—
25	C₂²×C₁₆×C₈₉	C₂²	1	3	yes	—
26	#15668019	C₈	1	8	yes	—
27	C₂×#13083558	C₂×C₄	1	3	yes	—
28	C₂²×#10623494	C₂×C₄	1	3	yes	—
29	Q₈×Dic₈₉	C₈	1	2	yes	—
30	D₄×Dic₈₉	C₈	1	6	yes	—
31	#15667971	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₈₉	C₂×C₄	1	3	yes	—
33	#15667952	C₂×C₄	1	12	yes	complemented
34	C₂×#13083549	C₂×C₄	1	12	yes	complemented
35	C₂×#13083549	C₂³	1	1	yes	characteristic
36	C₂×#13083549	C₂³	1	3	yes	—
37	#15667922	C₂×C₄	1	6	yes	—
38	C₈×Dic₈₉	C₂×C₄	1	3	yes	—
39	C₈×Dic₈₉	C₂×C₄	1	3	yes	—
40	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
41	C₈×Dic₈₉	C₂³	1	3	yes	—
42	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
43	#131×C₈₉	C₂×C₄	1	4	yes	—
44	#131×C₈₉	C₂×C₄	1	4	yes	—
45	C₂×M₁₆×C₈₉	C₂×C₄	1	3	yes	—
46	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
47	M₃₂×C₈₉	C₂×C₄	1	12	yes	complemented
48	C₂×C₁₆×C₈₉	C₂×C₄	1	12	yes	complemented
49	C₂×C₁₆×C₈₉	C₂³	1	3	yes	—
50	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
51	C₂²×Dic₈₉	C₂×C₈	1	6	yes	—
52	#13083558	C₂×C₈	1	12	yes	—
53	C₂×#10623494	C₂×C₈	1	12	yes	—
54	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
55	C₂×#10623494	C₂²×C₄	1	3	yes	—
56	C₄⋊Dic₈₉	C₂×C₈	1	6	yes	—
57	C₂²⋊Dic₈₉	C₂×C₈	1	6	yes	—
58	C₄×Dic₈₉	C₂×C₈	1	6	yes	—
59	C₄×Dic₈₉	C₂²×C₄	1	3	yes	—
60	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
61	#13083549	C₂²×C₄	1	2	yes	—
62	#13083549	C₂²×C₄	1	6	yes	—
63	#41×C₈₉	C₄²	1	4	yes	—
64	#41×C₈₉	C₂×C₈	1	4	yes	—
65	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
66	C₂×D₄×C₈₉	C₂×C₈	1	3	yes	—
67	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
68	M₁₆×C₈₉	C₄²	1	12	yes	—
69	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
71	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
72	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
74	C₁₆×C₈₉	C₂²×C₄	1	6	yes	—
75	C₂³×C₈₉	C₂²×C₈	1	3	yes	—
76	Q₈×C₈₉	C₄×C₈	1	4	yes	—
77	D₄×C₈₉	C₄×C₈	1	12	yes	—
78	C₂×C₄×C₈₉	C₂×C₄²	1	6	yes	—
79	C₂×C₄×C₈₉	C₂²×C₈	1	6	yes	—
80	C₂×C₄×C₈₉	C₂²×C₈	1	3	yes	—
81	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
82	C₂×Dic₈₉	—	2	12	no	—
83	C₂×Dic₈₉	C₂²×C₈	1	6	yes	—
84	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
85	C₈×C₈₉	C₂×C₄²	1	6	yes	—
86	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
88	#10623494	C₂²×C₈	1	2	yes	—
89	#10623494	C₂²×C₈	1	6	yes	—
90	C₂²×C₈₉	—	2	6	no	—
91	C₂²×C₈₉	C₂×C₄×C₈	1	6	yes	—
92	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
93	C₄×C₈₉	C₂×C₄×C₈	1	6	yes	—
94	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
95	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
96	Dic₈₉	—	2	6	no	—
97	Dic₈₉	#504	1	4	yes	direct product factor
98	C₄×#504	—	89	1	no	maximal, Sylow
99	C₂×C₈₉	—	2	6	no	—
100	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
101	C₂×C₈₉	C₂×#504	1	2	yes	—
102	C₂×#504	—	89	1	no	—
103	C₂×#504	D₈₉	1	1	yes	characteristic
104	C₄×#131	—	89	1	no	—
105	#2108	—	89	3	no	—
106	#2108	—	89	3	no	—
107	C₄×M₃₂	—	89	3	no	—
108	C₂×C₄×C₁₆	—	89	3	no	—
109	C₈₉	C₄×#504	1	1	yes	characteristic, complemented, Sylow
110	C₂×#131	—	89	1	no	—
111	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
112	C₄×#41	—	89	1	no	—
113	#504	—	89	8	no	—
114	#504	Dic₈₉	1	8	yes	direct product factor
115	C₂×M₃₂	—	89	3	no	—
116	C₂×M₃₂	C₂×D₈₉	1	3	yes	—
117	C₂²×C₁₆	—	89	3	no	—
118	C₂²×C₁₆	C₂×D₈₉	1	3	yes	—
119	#405	—	89	3	no	—
120	#405	—	89	3	no	—
121	C₄×M₁₆	—	89	3	no	—
122	C₂×C₄×C₈	—	89	3	no	—
123	#346	—	89	6	no	—
124	C₄×C₁₆	—	89	3	no	—
125	C₄×C₁₆	—	89	3	no	—
126	C₄×C₁₆	—	89	1	no	—
127	C₄×C₁₆	—	89	3	no	—
128	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
129	#131	—	89	8	no	—
130	#131	C₄×D₈₉	1	4	yes	—
131	#131	C₂×Dic₈₉	1	4	yes	—
132	C₂×M₁₆	—	89	3	no	—
133	C₂×M₁₆	C₄×D₈₉	1	3	yes	—
134	C₂²×C₈	—	89	3	no	—
135	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
136	C₄×Q₈	—	89	2	no	—
137	C₄×D₄	—	89	6	no	—
138	#117	—	89	3	no	—
139	C₂×C₄²	—	89	3	no	—
140	M₃₂	—	89	12	no	—
141	M₃₂	C₂×Dic₈₉	1	12	yes	complemented
142	C₂×C₁₆	—	89	12	no	—
143	C₂×C₁₆	—	89	1	no	—
144	C₂×C₁₆	—	89	3	no	—
145	C₂×C₁₆	C₂×Dic₈₉	1	12	yes	complemented
146	C₂×C₁₆	C₂²×D₈₉	1	3	yes	—
147	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
148	#97	—	89	6	no	—
149	C₄×C₈	—	89	3	no	—
150	C₄×C₈	—	89	3	no	—
151	C₄×C₈	—	89	1	no	—
152	C₄×C₈	—	89	3	no	—
153	#41	C₄×Dic₈₉	1	4	yes	—
154	#41	C₈×D₈₉	1	4	yes	—
155	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
156	C₂×D₄	C₈×D₈₉	1	3	yes	—
157	C₂²×C₄	—	89	6	no	—
158	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
159	M₁₆	—	89	12	no	—
160	M₁₆	C₄×Dic₈₉	1	12	yes	—
161	C₂×C₈	—	89	12	no	—
162	C₂×C₈	—	89	1	no	—
163	C₂×C₈	—	89	3	no	—
164	C₂×C₈	C₂×C₄×D₈₉	1	6	yes	—
165	C₂×C₈	C₂×C₄×D₈₉	1	3	yes	—
166	C₂×C₈	C₂²×Dic₈₉	1	6	yes	—
167	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
168	C₄⋊C₄	—	89	6	no	—
169	C₂²⋊C₄	—	89	6	no	—
170	C₄²	—	89	6	no	—
171	C₄²	—	89	1	no	—
172	C₄²	—	89	3	no	—
173	C₁₆	—	89	2	no	—
174	C₁₆	—	89	6	no	—
175	C₁₆	C₂²×Dic₈₉	1	2	yes	—
176	C₁₆	C₂²×Dic₈₉	1	6	yes	—
177	C₂³	C₂×C₈×D₈₉	1	3	yes	—
178	Q₈	C₈×Dic₈₉	1	4	yes	—
179	D₄	C₈×Dic₈₉	1	12	yes	—
180	C₂×C₄	—	89	2	no	—
181	C₂×C₄	—	89	6	no	—
182	C₂×C₄	—	178	12	no	—
183	C₂×C₄	C₂×C₄×Dic₈₉	1	6	yes	—
184	C₂×C₄	C₂×C₈×D₈₉	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₈₉	1	6	yes	—
186	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
187	C₈	—	89	2	no	—
188	C₈	—	89	6	no	—
189	C₈	C₂×C₄×Dic₈₉	1	6	yes	—
190	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
191	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
192	C₂²	—	2	6	no	—
193	C₂²	C₂×C₈×Dic₈₉	1	6	yes	—
194	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
195	C₄	—	89	4	no	—
196	C₄	—	178	6	no	—
197	C₄	C₂×C₈×Dic₈₉	1	6	yes	—
198	C₄	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
199	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
200	C₂	—	2	6	no	—
201	C₂	C₂²×C₈×Dic₈₉	1	1	yes	characteristic
202	C₂	#504×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#504×C₈₉	C₂
socle	C₂²×C₈₉	C₂³×C₈
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	C₄×#504
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
3. #26021851	C₂	semidirect
4. #26021848	C₂	semidirect
5. M₃₂×Dic₈₉	C₂	semidirect
6. C₂×C₁₆×Dic₈₉	C₂	semidirect
10. #19152015	C₄	semidirect
23. #504×C₈₉	C₄	semidirect
33. #15667952	C₂×C₄	semidirect
34. C₂×#13083549	C₂×C₄	semidirect
47. M₃₂×C₈₉	C₂×C₄	semidirect
48. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
97. Dic₈₉	#504	direct (N × H)
109. C₈₉	C₄×#504	semidirect
114. #504	Dic₈₉	direct (N × H)
141. M₃₂	C₂×Dic₈₉	semidirect
145. C₂×C₁₆	C₂×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152015	2
#131×Dic₈₉	2
#26021851	2
#26021848	2
M₃₂×Dic₈₉	2
C₂×C₁₆×Dic₈₉	2
C₂×#504×C₈₉	2
C₄×#504	89

Subgroups — #504×Dic89 #36707071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #504×Dic₈₉ #36707071