Subgroups — C₂×C₈₉×#3160 #36438503

Order 22784.

1942 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₈₉	C₂	1	2	yes	maximal
2	C₈₉×#3160	C₂	1	16	yes	maximal, direct product factor
3	C₂×#500×C₈₉	C₂	1	2	yes	maximal, complemented
4	C₂×#499×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₂×#498×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#427×C₈₉	C₂	1	4	yes	maximal
7	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
9	#533×C₈₉	—	2	8	no	—
10	#533×C₈₉	C₂²	1	16	yes	—
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	4	yes	—
12	C₂×#125×C₈₉	C₂²	1	4	yes	—
13	C₂×C₂²⋊Q₈×C₈₉	—	2	4	no	—
14	C₂×C₄×Q₈×C₈₉	—	2	4	no	—
15	C₂×#117×C₈₉	C₂²	1	4	yes	—
16	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	#500×C₈₉	C₂²	1	16	yes	complemented
18	#499×C₈₉	C₂²	1	32	yes	complemented
19	#498×C₈₉	C₂²	1	16	yes	complemented
20	#427×C₈₉	C₂²	1	32	yes	—
21	C₂×#107×C₈₉	C₂²	1	8	yes	—
22	C₂×#106×C₈₉	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	8	yes	—
27	C₂²×Q₈×C₈₉	—	2	4	no	—
28	C₂³×C₄×C₈₉	D₄	1	2	yes	—
29	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	Q₈	1	16	yes	complemented
31	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
32	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₈₉	C₂³	1	8	yes	—
34	C₄⋊Q₈×C₈₉	—	2	8	no	—
35	C₄⋊Q₈×C₈₉	C₂³	1	16	yes	—
36	#125×C₈₉	—	2	8	no	—
37	#125×C₈₉	C₂³	1	16	yes	—
38	C₂²⋊Q₈×C₈₉	—	2	32	no	—
39	C₄×Q₈×C₈₉	—	2	32	no	—
40	#117×C₈₉	—	2	8	no	—
41	#117×C₈₉	C₂³	1	16	yes	—
42	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
45	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
46	C₂×C₄²×C₈₉	—	2	4	no	—
47	C₂×C₄²×C₈₉	D₄	1	4	yes	—
48	C₂×C₄²×C₈₉	D₄	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
52	#107×C₈₉	C₂³	1	32	yes	—
53	#106×C₈₉	C₂³	1	32	yes	—
54	C₄×C₈×C₈₉	C₂³	1	4	yes	—
55	C₄×C₈×C₈₉	C₂³	1	8	yes	—
56	C₄×C₈×C₈₉	C₂³	1	4	yes	—
57	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₈₉	—	2	16	no	—
59	C₂×Q₈×C₈₉	—	4	16	no	—
60	C₂²×C₄×C₈₉	—	2	8	no	—
61	C₂²×C₄×C₈₉	—	2	8	no	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
64	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
68	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
70	C₂×C₈×C₈₉	C₂×Q₈	1	8	yes	—
71	C₂×C₈×C₈₉	C₂×Q₈	1	8	yes	—
72	C₂×C₈×C₈₉	C₂×Q₈	1	32	yes	complemented
73	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
74	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
75	C₄⋊C₄×C₈₉	—	2	32	no	—
76	C₄⋊C₄×C₈₉	—	2	16	no	—
77	C₄⋊C₄×C₈₉	—	2	8	no	—
78	C₄⋊C₄×C₈₉	C₂⁴	1	16	yes	—
79	C₂²⋊C₄×C₈₉	—	2	16	no	—
80	C₄²×C₈₉	—	2	16	no	—
81	C₄²×C₈₉	C₂×D₄	1	8	yes	—
82	C₄²×C₈₉	C₂×D₄	1	2	yes	—
83	C₄²×C₈₉	C₂×D₄	1	2	yes	—
84	C₄²×C₈₉	C₂⁴	1	2	yes	—
85	C₄²×C₈₉	C₂⁴	1	2	yes	—
86	C₂³×C₈₉	—	2	4	no	—
87	C₂³×C₈₉	C₄⋊Q₈	1	4	yes	—
88	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
89	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₈₉	—	4	32	no	—
91	C₂×C₄×C₈₉	—	2	4	no	—
92	C₂×C₄×C₈₉	—	2	16	no	—
93	C₂×C₄×C₈₉	—	2	16	no	—
94	C₂×C₄×C₈₉	—	4	16	no	—
95	C₂×C₄×C₈₉	C₄⋊Q₈	1	8	yes	—
96	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄×C₈₉	#135	1	8	yes	—
98	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
99	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
102	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
103	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
104	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
105	C₈×C₈₉	C₂²×Q₈	1	8	yes	—
106	C₈×C₈₉	C₂²×Q₈	1	8	yes	—
107	C₂²×C₈₉	—	2	8	no	—
108	C₂²×C₈₉	—	2	4	no	—
109	C₂²×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
110	C₂²×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
111	C₂²×C₈₉	C₂×#135	1	4	yes	—
112	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
113	C₄×C₈₉	—	2	4	no	—
114	C₄×C₈₉	—	4	16	no	—
115	C₄×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
116	C₄×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
117	C₄×C₈₉	C₂×#135	1	8	yes	—
118	C₄×C₈₉	C₂³×Q₈	1	2	yes	—
119	C₂×#3160	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₈₉	—	2	4	no	—
121	C₂×C₈₉	#3160	1	4	yes	direct product factor
122	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₈₉	C₂²×#135	1	2	yes	—
124	C₂×#533	C₂×C₈₉	1	2	yes	—
125	#3160	C₂×C₈₉	1	16	yes	direct product factor
126	C₂×#500	C₂×C₈₉	1	2	yes	complemented
127	C₂×#499	C₂×C₈₉	1	4	yes	complemented
128	C₂×#498	C₂×C₈₉	1	2	yes	complemented
129	C₂×#427	C₂×C₈₉	1	4	yes	—
130	C₂²×C₄×C₈	C₂×C₈₉	1	1	yes	characteristic
131	C₈₉	C₂×#3160	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₈	C₂²×C₈₉	1	2	yes	—
133	#533	—	2	8	no	—
134	#533	C₂²×C₈₉	1	16	yes	—
135	C₂×C₄⋊Q₈	C₂²×C₈₉	1	4	yes	—
136	C₂×#125	C₂²×C₈₉	1	4	yes	—
137	C₂×C₂²⋊Q₈	—	2	4	no	—
138	C₂×C₄×Q₈	—	2	4	no	—
139	C₂×#117	C₂²×C₈₉	1	4	yes	—
140	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
141	#500	C₂²×C₈₉	1	16	yes	complemented
142	#499	C₂²×C₈₉	1	32	yes	complemented
143	#498	C₂²×C₈₉	1	16	yes	complemented
144	#427	C₂²×C₈₉	1	32	yes	—
145	C₂×#107	C₂²×C₈₉	1	8	yes	—
146	C₂×#106	C₂²×C₈₉	1	8	yes	—
147	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₈₉	1	8	yes	—
151	C₂²×Q₈	—	2	4	no	—
152	C₂³×C₄	D₄×C₈₉	1	2	yes	—
153	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
154	C₂²×C₈	Q₈×C₈₉	1	16	yes	complemented
155	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
156	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
157	C₂²×C₈	C₂³×C₈₉	1	8	yes	—
158	C₄⋊Q₈	—	2	8	no	—
159	C₄⋊Q₈	C₂³×C₈₉	1	16	yes	—
160	#125	—	2	8	no	—
161	#125	C₂³×C₈₉	1	16	yes	—
162	C₂²⋊Q₈	—	2	32	no	—
163	C₄×Q₈	—	2	32	no	—
164	#117	—	2	8	no	—
165	#117	C₂³×C₈₉	1	16	yes	—
166	C₂×C₄⋊C₄	—	2	8	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
169	C₂×C₂²⋊C₄	—	2	4	no	—
170	C₂×C₄²	—	2	4	no	—
171	C₂×C₄²	D₄×C₈₉	1	2	yes	—
172	C₂×C₄²	D₄×C₈₉	1	4	yes	—
173	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
176	#107	C₂³×C₈₉	1	32	yes	—
177	#106	C₂³×C₈₉	1	32	yes	—
178	C₄×C₈	C₂³×C₈₉	1	4	yes	—
179	C₄×C₈	C₂³×C₈₉	1	8	yes	—
180	C₄×C₈	C₂³×C₈₉	1	4	yes	—
181	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
182	C₂×Q₈	—	2	16	no	—
183	C₂×Q₈	—	4	16	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
188	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
191	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	—
192	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
194	C₂×C₈	C₂×Q₈×C₈₉	1	8	yes	—
195	C₂×C₈	C₂×Q₈×C₈₉	1	8	yes	—
196	C₂×C₈	C₂×Q₈×C₈₉	1	32	yes	complemented
197	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
198	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
199	C₄⋊C₄	—	2	32	no	—
200	C₄⋊C₄	—	2	16	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	C₂⁴×C₈₉	1	16	yes	—
203	C₂²⋊C₄	—	2	16	no	—
204	C₄²	—	2	16	no	—
205	C₄²	C₂×D₄×C₈₉	1	8	yes	—
206	C₄²	C₂×D₄×C₈₉	1	2	yes	—
207	C₄²	C₂×D₄×C₈₉	1	2	yes	—
208	C₄²	C₂⁴×C₈₉	1	2	yes	—
209	C₄²	C₂⁴×C₈₉	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	C₄⋊Q₈×C₈₉	1	4	yes	—
212	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
213	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
214	Q₈	—	4	32	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	2	16	no	—
218	C₂×C₄	—	4	16	no	—
219	C₂×C₄	C₄⋊Q₈×C₈₉	1	8	yes	—
220	C₂×C₄	C₄⋊Q₈×C₈₉	1	4	yes	—
221	C₂×C₄	#135×C₈₉	1	8	yes	—
222	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
223	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
224	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
226	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
227	C₂×C₄	C₂²×Q₈×C₈₉	1	8	yes	—
228	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
229	C₈	C₂²×Q₈×C₈₉	1	8	yes	—
230	C₈	C₂²×Q₈×C₈₉	1	8	yes	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	8	no	—
233	C₂²	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
234	C₂²	C₂×C₄⋊Q₈×C₈₉	1	2	yes	—
235	C₂²	C₂×#135×C₈₉	1	4	yes	—
236	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	4	16	no	—
239	C₄	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
240	C₄	C₂×C₄⋊Q₈×C₈₉	1	2	yes	—
241	C₄	C₂×#135×C₈₉	1	8	yes	—
242	C₄	C₂³×Q₈×C₈₉	1	2	yes	—
243	C₂	—	2	4	no	—
244	C₂	C₈₉×#3160	1	4	yes	direct product factor
245	C₂	C₂²×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
246	C₂	C₂²×#135×C₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₈₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂³×C₈₉	C₂²×D₄
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#3160	C₂	direct (N × H)
3. C₂×#500×C₈₉	C₂	semidirect
4. C₂×#499×C₈₉	C₂	semidirect
5. C₂×#498×C₈₉	C₂	semidirect
17. #500×C₈₉	C₂²	semidirect
18. #499×C₈₉	C₂²	semidirect
19. #498×C₈₉	C₂²	semidirect
30. C₂²×C₈×C₈₉	Q₈	semidirect
72. C₂×C₈×C₈₉	C₂×Q₈	semidirect
119. C₂×#3160	C₈₉	direct (N × H)
121. C₂×C₈₉	#3160	direct (N × H)
125. #3160	C₂×C₈₉	direct (N × H)
126. C₂×#500	C₂×C₈₉	semidirect
127. C₂×#499	C₂×C₈₉	semidirect
128. C₂×#498	C₂×C₈₉	semidirect
131. C₈₉	C₂×#3160	direct (N × H)
141. #500	C₂²×C₈₉	semidirect
142. #499	C₂²×C₈₉	semidirect
143. #498	C₂²×C₈₉	semidirect
154. C₂²×C₈	Q₈×C₈₉	semidirect
196. C₂×C₈	C₂×Q₈×C₈₉	semidirect
244. C₂	C₈₉×#3160	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×#533×C₈₉	2
C₈₉×#3160	2
C₂×#500×C₈₉	2
C₂×#499×C₈₉	2
C₂×#498×C₈₉	2
C₂×#427×C₈₉	2
C₂²×C₄×C₈×C₈₉	2
C₂×#3160	89

Subgroups — C2×C89×#3160 #36438503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3160 #36438503