Subgroups — #1381×D₈₉ #36469313

Order 22784.

574 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₈₉	C₂	1	2	yes	maximal
3	#26018346	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018345	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×D₈₉	—	2	2	no	—
7	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×D₈₉	—	2	4	no	—
10	C₂×C₁₆×D₈₉	C₄	1	2	yes	—
11	C₂×C₁₆×D₈₉	C₄	1	2	yes	—
12	C₂×#15667948	C₂²	1	2	yes	—
13	C₂×#15667947	—	2	2	no	—
14	#108×D₈₉	—	2	4	no	—
15	#108×D₈₉	C₄	1	8	yes	—
16	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#108×C₈₉	C₂²	1	2	yes	—
18	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083557	—	4	2	no	—
21	M₁₆×D₈₉	—	2	8	no	—
22	M₁₆×D₈₉	—	4	4	no	—
23	C₂²×#10623494	D₄	1	1	yes	characteristic
24	C₂×C₈×D₈₉	—	4	2	no	—
25	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₈₉	D₄	1	4	yes	—
29	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
30	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
31	C₂×#13083549	—	2	2	no	—
32	C₂×#13083549	C₂×C₄	1	2	yes	—
33	C₁₆×D₈₉	—	2	4	no	—
34	C₁₆×D₈₉	—	2	4	no	—
35	#15667948	—	2	2	no	—
36	#15667948	C₂×C₄	1	4	yes	—
37	#15667947	—	2	8	no	—
38	C₂×M₁₆×C₈₉	—	2	2	no	—
39	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₁₆×C₈₉	—	2	2	no	—
41	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
42	#108×C₈₉	—	2	2	no	—
43	#108×C₈₉	C₂×C₄	1	4	yes	—
44	C₂³×D₈₉	Q₁₆	1	1	yes	characteristic
45	C₂²×Dic₈₉	Q₁₆	1	1	yes	characteristic
46	C₂×C₄×D₈₉	C₂²⋊C₄	1	4	yes	—
47	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
50	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
51	C₂×C₄×D₈₉	SD₁₆	1	4	yes	—
52	#13083558	—	2	4	no	—
53	#13083558	—	4	2	no	—
54	#13083557	—	4	8	no	—
55	C₂×#10623494	—	4	2	no	—
56	C₂×#10623494	C₂²⋊C₄	1	4	yes	—
57	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
59	C₈×D₈₉	—	4	8	no	—
60	C₈×D₈₉	C₄²	1	2	yes	—
61	C₈×D₈₉	C₄²	1	2	yes	—
62	C₈×D₈₉	C₂²⋊C₄	1	2	yes	—
63	C₈×D₈₉	C₂²⋊C₄	1	2	yes	—
64	C₈×D₈₉	C₄⋊C₄	1	4	yes	—
65	C₈×D₈₉	C₄⋊C₄	1	4	yes	—
66	#13083549	—	2	2	no	—
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	M₁₆×C₈₉	—	2	4	no	—
69	M₁₆×C₈₉	—	4	2	no	—
70	C₂×C₈×C₈₉	—	4	2	no	—
71	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
75	C₁₆×C₈₉	—	2	2	no	—
76	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₂²×D₈₉	—	2	4	no	—
78	C₂²×D₈₉	#103	1	4	yes	—
79	C₂²×D₈₉	#107	1	1	yes	characteristic
80	C₂²×D₈₉	#107	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
85	C₂×Dic₈₉	#103	1	4	yes	—
86	C₂×Dic₈₉	#107	1	1	yes	characteristic
87	C₂×Dic₈₉	#107	1	1	yes	characteristic
88	C₄×D₈₉	#95	1	2	yes	—
89	C₄×D₈₉	#95	1	2	yes	—
90	C₄×D₈₉	#102	1	4	yes	—
91	C₄×D₈₉	#102	1	4	yes	—
92	C₄×D₈₉	#106	1	2	yes	—
93	C₄×D₈₉	#106	1	2	yes	—
94	C₈×C₈₉	—	4	4	no	—
95	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
96	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
98	#10623494	—	4	4	no	—
99	#10623494	#95	1	2	yes	—
100	#10623494	#95	1	2	yes	—
101	C₂²×C₈₉	—	2	2	no	—
102	C₂²×C₈₉	C₂×#103	1	2	yes	—
103	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
104	C₂×D₈₉	—	2	2	no	—
105	C₂×D₈₉	—	2	8	no	—
106	C₂×D₈₉	#340	1	2	yes	—
107	C₂×D₈₉	#340	1	2	yes	—
108	C₂×D₈₉	#368	1	4	yes	—
109	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
110	C₄×C₈₉	C₂×#102	1	2	yes	—
111	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
112	Dic₈₉	#340	1	2	yes	—
113	Dic₈₉	#340	1	2	yes	—
114	C₂×#1381	—	89	1	no	maximal, Sylow
115	C₂×C₈₉	—	2	2	no	—
116	C₂×C₈₉	C₂×#340	1	1	yes	characteristic
117	C₂×C₈₉	C₂×#368	1	2	yes	—
118	D₈₉	—	2	2	no	—
119	D₈₉	#1381	1	4	yes	direct product factor
120	C₂³×C₁₆	—	89	1	no	—
121	C₂²×#108	—	89	2	no	—
122	#1381	—	89	1	no	—
123	#1381	—	89	2	no	—
124	#1381	D₈₉	1	1	yes	characteristic, direct product factor
125	C₈₉	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
126	C₂²×M₁₆	—	178	2	no	—
127	C₂³×C₈	—	89	1	no	—
128	C₂²×C₁₆	—	89	1	no	—
129	C₂²×C₁₆	—	89	2	no	—
130	C₂²×C₁₆	—	89	2	no	—
131	C₂²×C₁₆	—	178	4	no	—
132	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
133	C₂×#108	—	89	2	no	—
134	C₂×#108	—	89	8	no	—
135	C₂×#108	—	178	4	no	—
136	C₂×#108	—	178	2	no	—
137	C₂×#108	C₂×D₈₉	1	2	yes	—
138	C₂³×C₄	—	89	1	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	356	2	no	—
141	C₂×M₁₆	—	356	4	no	—
142	C₂×M₁₆	—	178	8	no	—
143	C₂×M₁₆	—	178	2	no	—
144	C₂²×C₈	—	89	1	no	—
145	C₂²×C₈	—	89	4	no	—
146	C₂²×C₈	—	89	1	no	—
147	C₂²×C₈	—	89	1	no	—
148	C₂²×C₈	—	89	4	no	—
149	C₂²×C₈	—	89	1	no	—
150	C₂²×C₈	—	89	1	no	—
151	C₂²×C₈	—	356	2	no	—
152	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	89	2	no	—
155	C₂×C₁₆	—	178	4	no	—
156	C₂×C₁₆	—	178	4	no	—
157	C₂×C₁₆	—	178	2	no	—
158	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
159	#108	—	2	2	no	—
160	#108	—	89	4	no	—
161	#108	—	178	8	no	—
162	#108	—	178	2	no	—
163	#108	C₄×D₈₉	1	4	yes	—
164	C₂⁴	—	89	1	no	—
165	C₂²×C₄	—	89	1	no	—
166	C₂²×C₄	—	89	1	no	—
167	C₂²×C₄	—	89	1	no	—
168	C₂²×C₄	—	89	4	no	—
169	C₂²×C₄	—	89	4	no	—
170	C₂²×C₄	—	89	1	no	—
171	C₂²×C₄	—	89	1	no	—
172	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
173	M₁₆	—	2	4	no	—
174	M₁₆	—	4	2	no	—
175	M₁₆	—	356	8	no	—
176	M₁₆	—	356	2	no	—
177	M₁₆	—	178	4	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	89	4	no	—
180	C₂×C₈	—	89	1	no	—
181	C₂×C₈	—	89	1	no	—
182	C₂×C₈	—	89	2	no	—
183	C₂×C₈	—	89	2	no	—
184	C₂×C₈	—	89	2	no	—
185	C₂×C₈	—	89	2	no	—
186	C₂×C₈	—	89	4	no	—
187	C₂×C₈	—	89	4	no	—
188	C₂×C₈	—	356	8	no	—
189	C₂×C₈	—	356	2	no	—
190	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
192	C₂×C₈	D₄×D₈₉	1	2	yes	—
193	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
194	C₁₆	—	2	2	no	—
195	C₁₆	—	178	2	no	—
196	C₂³	—	89	1	no	—
197	C₂³	—	89	1	no	—
198	C₂³	—	89	4	no	—
199	C₂³	—	178	4	no	—
200	C₂³	Q₁₆×D₈₉	1	1	yes	characteristic
201	C₂×C₄	—	89	1	no	—
202	C₂×C₄	—	89	1	no	—
203	C₂×C₄	—	89	4	no	—
204	C₂×C₄	—	89	2	no	—
205	C₂×C₄	—	89	2	no	—
206	C₂×C₄	—	89	2	no	—
207	C₂×C₄	—	89	2	no	—
208	C₂×C₄	—	89	4	no	—
209	C₂×C₄	—	89	4	no	—
210	C₂×C₄	C₂²⋊C₄×D₈₉	1	2	yes	—
211	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
212	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
213	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
214	C₈	—	4	4	no	—
215	C₈	—	89	2	no	—
216	C₈	—	89	2	no	—
217	C₈	—	356	4	no	—
218	C₈	C₄²×D₈₉	1	1	yes	characteristic
219	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
220	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	89	2	no	—
223	C₂²	—	89	4	no	—
224	C₂²	—	89	2	no	—
225	C₂²	—	178	8	no	—
226	C₂²	—	178	2	no	—
227	C₂²	#103×D₈₉	1	2	yes	—
228	C₂²	#107×D₈₉	1	1	yes	characteristic
229	C₄	—	89	2	no	—
230	C₄	—	89	2	no	—
231	C₄	#95×D₈₉	1	1	yes	characteristic
232	C₄	#102×D₈₉	1	2	yes	—
233	C₄	#106×D₈₉	1	1	yes	characteristic
234	C₂	—	2	2	no	—
235	C₂	—	89	4	no	—
236	C₂	—	178	2	no	—
237	C₂	#340×D₈₉	1	1	yes	characteristic
238	C₂	#368×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1381	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×D₈₉	Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1381	D₈₉
hypocenter	C₈₉	C₂×#1381
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

Complements

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

N	H ≅ G/N	type
3. #26018346	C₂	semidirect
4. #26018345	C₂	semidirect
5. C₈₉×#1381	C₂	semidirect
119. D₈₉	#1381	direct (N × H)
124. #1381	D₈₉	direct (N × H)
125. C₈₉	C₂×#1381	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₈₉	2
C₂×#108×D₈₉	2
#26018346	2
#26018345	2
C₈₉×#1381	2
C₂×#1381	89

Subgroups — #1381×D89 #36469313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1381×D₈₉ #36469313