Subgroups — C₂×C₈₉×#1690 #36396993

Order 22784.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#475×C₈₉	C₂	1	2	yes	maximal
3	C₂×#446×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1690	C₂	1	8	yes	maximal, direct product factor
5	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#332×C₈₉	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₈₉	—	2	1	no	—
8	C₂²×Q₁₆×C₈₉	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	#494×C₈₉	—	2	2	no	—
12	#494×C₈₉	C₂²	1	4	yes	—
13	#475×C₈₉	—	2	4	no	—
14	#475×C₈₉	C₂²	1	8	yes	—
15	#446×C₈₉	—	2	2	no	—
16	#446×C₈₉	C₂²	1	4	yes	—
17	C₂×#106×C₈₉	—	2	2	no	—
18	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
19	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#103×C₈₉	—	2	2	no	—
22	C₂×#97×C₈₉	—	2	1	no	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#334×C₈₉	C₂²	1	4	yes	—
25	#332×C₈₉	C₂²	1	8	yes	—
26	C₂²×Q₈×C₈₉	—	2	2	no	—
27	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂×Q₁₆×C₈₉	—	2	4	no	—
29	C₂×Q₁₆×C₈₉	—	2	4	no	—
30	C₂×Q₁₆×C₈₉	—	4	4	no	—
31	C₂×Q₁₆×C₈₉	—	4	4	no	—
32	C₂²×C₈×C₈₉	—	2	1	no	—
33	C₂²×C₈×C₈₉	—	2	2	no	—
34	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₈₉	—	2	2	no	—
37	C₄⋊Q₈×C₈₉	—	2	2	no	—
38	C₄⋊Q₈×C₈₉	—	2	2	no	—
39	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
40	C₄×Q₈×C₈₉	—	2	1	no	—
41	C₄×Q₈×C₈₉	—	2	1	no	—
42	C₄×Q₈×C₈₉	D₄	1	2	yes	—
43	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	—	2	1	no	—
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	#106×C₈₉	—	2	8	no	—
51	C₄⋊C₈×C₈₉	—	2	4	no	—
52	C₄⋊C₈×C₈₉	—	2	1	no	—
53	C₄⋊C₈×C₈₉	—	2	2	no	—
54	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
55	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
56	#103×C₈₉	—	2	8	no	—
57	#97×C₈₉	—	2	4	no	—
58	C₄×C₈×C₈₉	D₄	1	2	yes	—
59	C₄×C₈×C₈₉	C₂³	1	2	yes	—
60	C₂×Q₈×C₈₉	—	2	1	no	—
61	C₂×Q₈×C₈₉	—	2	4	no	—
62	C₂×Q₈×C₈₉	—	2	4	no	—
63	C₂×Q₈×C₈₉	—	4	2	no	—
64	C₂×Q₈×C₈₉	—	4	4	no	—
65	C₂×Q₈×C₈₉	—	4	4	no	—
66	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₈₉	—	2	1	no	—
68	C₂²×C₄×C₈₉	—	2	1	no	—
69	C₂²×C₄×C₈₉	—	4	2	no	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
73	Q₁₆×C₈₉	—	4	8	no	—
74	Q₁₆×C₈₉	—	4	8	no	—
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	—	2	2	no	—
77	C₂×C₈×C₈₉	—	2	4	no	—
78	C₂×C₈×C₈₉	—	4	2	no	—
79	C₂×C₈×C₈₉	—	4	4	no	—
80	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
81	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
83	C₄⋊C₄×C₈₉	—	2	2	no	—
84	C₄⋊C₄×C₈₉	—	2	1	no	—
85	C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₄⋊C₄×C₈₉	—	2	4	no	—
87	C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₄⋊C₄×C₈₉	—	4	2	no	—
89	C₄⋊C₄×C₈₉	—	4	2	no	—
90	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
91	C₄²×C₈₉	—	2	2	no	—
92	C₄²×C₈₉	—	2	2	no	—
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₈₉	#41	1	1	yes	characteristic
96	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₂³×C₈₉	#121	1	1	yes	characteristic
98	Q₈×C₈₉	—	2	4	no	—
99	Q₈×C₈₉	—	4	2	no	—
100	Q₈×C₈₉	—	4	4	no	—
101	Q₈×C₈₉	—	4	4	no	—
102	Q₈×C₈₉	—	8	4	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	—	4	2	no	—
108	C₂×C₄×C₈₉	—	4	4	no	—
109	C₂×C₄×C₈₉	—	8	4	no	—
110	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₄×C₈₉	#137	1	4	yes	—
116	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
119	C₈×C₈₉	—	2	2	no	—
120	C₈×C₈₉	—	4	2	no	—
121	C₈×C₈₉	—	4	4	no	—
122	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
123	C₂²×C₈₉	#460	1	2	yes	—
124	C₂²×C₈₉	#470	1	2	yes	—
125	C₂²×C₈₉	#473	1	2	yes	—
126	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	4	2	no	—
130	C₄×C₈₉	—	4	2	no	—
131	C₄×C₈₉	—	8	4	no	—
132	C₄×C₈₉	#460	1	2	yes	—
133	C₄×C₈₉	#470	1	2	yes	—
134	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
135	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₄×C₈₉	C₂×#137	1	2	yes	—
137	C₂×#1690	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₈₉	#1690	1	4	yes	direct product factor
139	C₂×C₈₉	C₂×#460	1	1	yes	characteristic
140	C₂×C₈₉	C₂×#470	1	1	yes	characteristic
141	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
142	C₂×#494	C₂×C₈₉	1	1	yes	characteristic
143	C₂×#475	C₂×C₈₉	1	2	yes	—
144	C₂×#446	C₂×C₈₉	1	1	yes	characteristic
145	#1690	C₂×C₈₉	1	8	yes	direct product factor
146	C₂×#334	C₂×C₈₉	1	1	yes	characteristic
147	C₂×#332	C₂×C₈₉	1	2	yes	—
148	C₈₉	C₂×#1690	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×Q₁₆	—	2	1	no	—
150	C₂²×Q₁₆	—	2	1	no	—
151	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
152	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
153	#494	—	2	2	no	—
154	#494	C₂²×C₈₉	1	4	yes	—
155	#475	—	2	4	no	—
156	#475	C₂²×C₈₉	1	8	yes	—
157	#446	—	2	2	no	—
158	#446	C₂²×C₈₉	1	4	yes	—
159	C₂×#106	—	2	2	no	—
160	C₂×C₄⋊C₈	—	2	1	no	—
161	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
163	C₂×#103	—	2	2	no	—
164	C₂×#97	—	2	1	no	—
165	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
166	#334	C₂²×C₈₉	1	4	yes	—
167	#332	C₂²×C₈₉	1	8	yes	—
168	C₂²×Q₈	—	2	2	no	—
169	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
170	C₂×Q₁₆	—	2	4	no	—
171	C₂×Q₁₆	—	2	4	no	—
172	C₂×Q₁₆	—	4	4	no	—
173	C₂×Q₁₆	—	4	4	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	D₄×C₈₉	1	2	yes	—
185	C₄×Q₈	C₂³×C₈₉	1	2	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
192	#106	—	2	8	no	—
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
197	C₄⋊C₈	C₂³×C₈₉	1	4	yes	—
198	#103	—	2	8	no	—
199	#97	—	2	4	no	—
200	C₄×C₈	D₄×C₈₉	1	2	yes	—
201	C₄×C₈	C₂³×C₈₉	1	2	yes	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	4	4	no	—
208	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
214	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
215	Q₁₆	—	4	8	no	—
216	Q₁₆	—	4	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	SD₁₆×C₈₉	1	4	yes	—
223	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
236	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
237	C₄²	#41×C₈₉	1	1	yes	characteristic
238	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
239	C₂³	#121×C₈₉	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
253	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
254	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
256	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
257	C₂×C₄	#137×C₈₉	1	4	yes	—
258	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	4	no	—
264	C₈	C₂×SD₁₆×C₈₉	1	4	yes	—
265	C₂²	#460×C₈₉	1	2	yes	—
266	C₂²	#470×C₈₉	1	2	yes	—
267	C₂²	#473×C₈₉	1	2	yes	—
268	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	8	4	no	—
274	C₄	#460×C₈₉	1	2	yes	—
275	C₄	#470×C₈₉	1	2	yes	—
276	C₄	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
277	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
278	C₄	C₂×#137×C₈₉	1	2	yes	—
279	C₂	C₈₉×#1690	1	4	yes	direct product factor
280	C₂	C₂×#460×C₈₉	1	1	yes	characteristic
281	C₂	C₂×#470×C₈₉	1	1	yes	characteristic
282	C₂	C₂×#473×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈₉×#1690	C₂	direct (N × H)
137. C₂×#1690	C₈₉	direct (N × H)
138. C₂×C₈₉	#1690	direct (N × H)
145. #1690	C₂×C₈₉	direct (N × H)
148. C₈₉	C₂×#1690	direct (N × H)
279. C₂	C₈₉×#1690	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#494×C₈₉	2
C₂×#475×C₈₉	2
C₂×#446×C₈₉	2
C₈₉×#1690	2
C₂×#334×C₈₉	2
C₂×#332×C₈₉	2
C₂×#1690	89

Subgroups — C2×C89×#1690 #36396993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#1690 #36396993