Subgroups — C₂×C₈₉×#3166 #36438510

Order 22784.

1902 conjugacy classes of subgroups.

The subgroup lattice has 298 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	1	yes	maximal, characteristic
2	C₂²×#125×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3166	C₂	1	16	yes	maximal, direct product factor
4	C₂×#501×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₂×#499×C₈₉	C₂	1	4	yes	maximal, complemented
6	C₂×#428×C₈₉	C₂	1	4	yes	maximal
7	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
9	#533×C₈₉	—	2	4	no	—
10	#533×C₈₉	C₂²	1	8	yes	—
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
12	C₂×#125×C₈₉	—	2	2	no	—
13	C₂×#125×C₈₉	—	2	4	no	—
14	C₂×#125×C₈₉	C₂²	1	2	yes	—
15	C₂×#125×C₈₉	C₂²	1	2	yes	—
16	C₂×#125×C₈₉	C₂²	1	2	yes	—
17	C₂×#125×C₈₉	C₂²	1	8	yes	—
18	C₂×C₂²⋊Q₈×C₈₉	—	2	2	no	—
19	C₂×C₄×Q₈×C₈₉	—	2	2	no	—
20	C₂×#117×C₈₉	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₈₉	—	2	2	no	—
22	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
23	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
24	#501×C₈₉	C₂²	1	32	yes	complemented
25	#499×C₈₉	C₂²	1	32	yes	complemented
26	#428×C₈₉	C₂²	1	32	yes	—
27	C₂×#107×C₈₉	C₂²	1	8	yes	—
28	C₂×#106×C₈₉	C₂²	1	8	yes	—
29	C₄×M₁₆×C₈₉	C₂²	1	8	yes	—
30	C₂×#97×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
32	C₂²×Q₈×C₈₉	—	2	2	no	—
33	C₂³×C₄×C₈₉	—	2	2	no	—
34	C₂³×C₄×C₈₉	D₄	1	2	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₈₉	Q₈	1	16	yes	complemented
37	C₂×M₁₆×C₈₉	C₂³	1	8	yes	—
38	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₈₉	—	2	4	no	—
40	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
41	#125×C₈₉	—	2	16	no	—
42	#125×C₈₉	—	2	4	no	—
43	#125×C₈₉	—	2	4	no	—
44	#125×C₈₉	—	2	4	no	—
45	#125×C₈₉	C₂³	1	8	yes	—
46	#125×C₈₉	C₂³	1	8	yes	—
47	#125×C₈₉	C₂³	1	8	yes	—
48	C₂²⋊Q₈×C₈₉	—	2	16	no	—
49	C₄×Q₈×C₈₉	—	2	16	no	—
50	#117×C₈₉	—	2	4	no	—
51	#117×C₈₉	C₂³	1	8	yes	—
52	C₂×C₄⋊C₄×C₈₉	—	2	16	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
55	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
62	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
63	C₂×C₄²×C₈₉	—	2	2	no	—
64	C₂×C₄²×C₈₉	D₄	1	2	yes	—
65	C₂×C₄²×C₈₉	D₄	1	4	yes	—
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
69	#107×C₈₉	C₂³	1	32	yes	—
70	#106×C₈₉	C₂³	1	32	yes	—
71	#97×C₈₉	C₂³	1	8	yes	—
72	C₄×C₈×C₈₉	C₂³	1	8	yes	—
73	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×Q₈×C₈₉	—	2	8	no	—
75	C₂×Q₈×C₈₉	—	4	8	no	—
76	C₂²×C₄×C₈₉	—	2	8	no	—
77	C₂²×C₄×C₈₉	—	2	4	no	—
78	C₂²×C₄×C₈₉	—	2	4	no	—
79	C₂²×C₄×C₈₉	—	4	4	no	—
80	C₂²×C₄×C₈₉	—	4	8	no	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
87	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
88	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
89	M₁₆×C₈₉	C₂×Q₈	1	32	yes	complemented
90	C₂×C₈×C₈₉	C₂×Q₈	1	16	yes	—
91	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
92	C₄⋊C₄×C₈₉	—	2	16	no	—
93	C₄⋊C₄×C₈₉	—	2	16	no	—
94	C₄⋊C₄×C₈₉	—	2	8	no	—
95	C₄⋊C₄×C₈₉	—	2	4	no	—
96	C₄⋊C₄×C₈₉	—	2	16	no	—
97	C₄⋊C₄×C₈₉	—	2	8	no	—
98	C₄⋊C₄×C₈₉	—	2	4	no	—
99	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
100	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
101	C₂²⋊C₄×C₈₉	—	2	8	no	—
102	C₄²×C₈₉	—	2	8	no	—
103	C₄²×C₈₉	C₂×D₄	1	8	yes	—
104	C₄²×C₈₉	C₂×D₄	1	2	yes	—
105	C₄²×C₈₉	C₂×D₄	1	2	yes	—
106	C₄²×C₈₉	C₂⁴	1	2	yes	—
107	C₄²×C₈₉	C₂⁴	1	2	yes	—
108	C₂³×C₈₉	—	2	4	no	—
109	C₂³×C₈₉	C₄⋊Q₈	1	4	yes	—
110	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
111	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₈₉	—	4	16	no	—
113	C₂×C₄×C₈₉	—	2	4	no	—
114	C₂×C₄×C₈₉	—	2	4	no	—
115	C₂×C₄×C₈₉	—	2	8	no	—
116	C₂×C₄×C₈₉	—	2	8	no	—
117	C₂×C₄×C₈₉	—	4	8	no	—
118	C₂×C₄×C₈₉	—	4	16	no	—
119	C₂×C₄×C₈₉	—	4	8	no	—
120	C₂×C₄×C₈₉	—	4	8	no	—
121	C₂×C₄×C₈₉	C₄⋊Q₈	1	8	yes	—
122	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
123	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
124	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
125	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
128	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
129	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
130	C₈×C₈₉	C₂²×Q₈	1	16	yes	—
131	C₂²×C₈₉	—	2	8	no	—
132	C₂²×C₈₉	—	2	4	no	—
133	C₂²×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
134	C₂²×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
135	C₂²×C₈₉	#575	1	4	yes	—
136	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
137	C₄×C₈₉	—	2	4	no	—
138	C₄×C₈₉	—	2	4	no	—
139	C₄×C₈₉	—	4	8	no	—
140	C₄×C₈₉	—	4	8	no	—
141	C₄×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
142	C₄×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
143	C₄×C₈₉	C₂³×Q₈	1	2	yes	—
144	C₂×#3166	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₈₉	—	2	4	no	—
146	C₂×C₈₉	#3166	1	4	yes	direct product factor
147	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₈₉	C₂×#575	1	2	yes	—
149	C₂×#533	C₂×C₈₉	1	1	yes	characteristic
150	C₂²×#125	C₂×C₈₉	1	1	yes	characteristic
151	#3166	C₂×C₈₉	1	16	yes	direct product factor
152	C₂×#501	C₂×C₈₉	1	4	yes	complemented
153	C₂×#499	C₂×C₈₉	1	4	yes	complemented
154	C₂×#428	C₂×C₈₉	1	4	yes	—
155	C₂×C₄×M₁₆	C₂×C₈₉	1	1	yes	characteristic
156	C₈₉	C₂×#3166	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
158	#533	—	2	4	no	—
159	#533	C₂²×C₈₉	1	8	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
161	C₂×#125	—	2	2	no	—
162	C₂×#125	—	2	4	no	—
163	C₂×#125	C₂²×C₈₉	1	2	yes	—
164	C₂×#125	C₂²×C₈₉	1	2	yes	—
165	C₂×#125	C₂²×C₈₉	1	2	yes	—
166	C₂×#125	C₂²×C₈₉	1	8	yes	—
167	C₂×C₂²⋊Q₈	—	2	2	no	—
168	C₂×C₄×Q₈	—	2	2	no	—
169	C₂×#117	C₂²×C₈₉	1	2	yes	—
170	C₂²×C₄⋊C₄	—	2	2	no	—
171	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
172	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
173	#501	C₂²×C₈₉	1	32	yes	complemented
174	#499	C₂²×C₈₉	1	32	yes	complemented
175	#428	C₂²×C₈₉	1	32	yes	—
176	C₂×#107	C₂²×C₈₉	1	8	yes	—
177	C₂×#106	C₂²×C₈₉	1	8	yes	—
178	C₄×M₁₆	C₂²×C₈₉	1	8	yes	—
179	C₂×#97	C₂²×C₈₉	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
181	C₂²×Q₈	—	2	2	no	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	D₄×C₈₉	1	2	yes	—
184	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
185	C₂×M₁₆	Q₈×C₈₉	1	16	yes	complemented
186	C₂×M₁₆	C₂³×C₈₉	1	8	yes	—
187	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
188	C₄⋊Q₈	—	2	4	no	—
189	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
190	#125	—	2	16	no	—
191	#125	—	2	4	no	—
192	#125	—	2	4	no	—
193	#125	—	2	4	no	—
194	#125	C₂³×C₈₉	1	8	yes	—
195	#125	C₂³×C₈₉	1	8	yes	—
196	#125	C₂³×C₈₉	1	8	yes	—
197	C₂²⋊Q₈	—	2	16	no	—
198	C₄×Q₈	—	2	16	no	—
199	#117	—	2	4	no	—
200	#117	C₂³×C₈₉	1	8	yes	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	16	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
209	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
210	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	D₄×C₈₉	1	2	yes	—
214	C₂×C₄²	D₄×C₈₉	1	4	yes	—
215	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
218	#107	C₂³×C₈₉	1	32	yes	—
219	#106	C₂³×C₈₉	1	32	yes	—
220	#97	C₂³×C₈₉	1	8	yes	—
221	C₄×C₈	C₂³×C₈₉	1	8	yes	—
222	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	8	no	—
230	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
233	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
235	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	—
236	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
237	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
238	M₁₆	C₂×Q₈×C₈₉	1	32	yes	complemented
239	C₂×C₈	C₂×Q₈×C₈₉	1	16	yes	—
240	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	2	16	no	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	16	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
249	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
250	C₂²⋊C₄	—	2	8	no	—
251	C₄²	—	2	8	no	—
252	C₄²	C₂×D₄×C₈₉	1	8	yes	—
253	C₄²	C₂×D₄×C₈₉	1	2	yes	—
254	C₄²	C₂×D₄×C₈₉	1	2	yes	—
255	C₄²	C₂⁴×C₈₉	1	2	yes	—
256	C₄²	C₂⁴×C₈₉	1	2	yes	—
257	C₂³	—	2	4	no	—
258	C₂³	C₄⋊Q₈×C₈₉	1	4	yes	—
259	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
260	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
261	Q₈	—	4	16	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	4	16	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	C₄⋊Q₈×C₈₉	1	8	yes	—
271	C₂×C₄	C₄⋊Q₈×C₈₉	1	4	yes	—
272	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
273	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
274	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
276	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
277	C₂×C₄	C₂²×Q₈×C₈₉	1	8	yes	—
278	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
279	C₈	C₂²×Q₈×C₈₉	1	16	yes	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	8	no	—
282	C₂²	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
283	C₂²	C₂×C₄⋊Q₈×C₈₉	1	2	yes	—
284	C₂²	#575×C₈₉	1	4	yes	—
285	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	8	no	—
289	C₄	—	4	8	no	—
290	C₄	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
291	C₄	C₂×C₄⋊Q₈×C₈₉	1	2	yes	—
292	C₄	C₂³×Q₈×C₈₉	1	2	yes	—
293	C₂	—	2	4	no	—
294	C₂	C₈₉×#3166	1	4	yes	direct product factor
295	C₂	C₂²×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
296	C₂	C₂×#575×C₈₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₈₉×#3166	C₂	direct (N × H)
4. C₂×#501×C₈₉	C₂	semidirect
5. C₂×#499×C₈₉	C₂	semidirect
24. #501×C₈₉	C₂²	semidirect
25. #499×C₈₉	C₂²	semidirect
36. C₂×M₁₆×C₈₉	Q₈	semidirect
89. M₁₆×C₈₉	C₂×Q₈	semidirect
144. C₂×#3166	C₈₉	direct (N × H)
146. C₂×C₈₉	#3166	direct (N × H)
151. #3166	C₂×C₈₉	direct (N × H)
152. C₂×#501	C₂×C₈₉	semidirect
153. C₂×#499	C₂×C₈₉	semidirect
156. C₈₉	C₂×#3166	direct (N × H)
173. #501	C₂²×C₈₉	semidirect
174. #499	C₂²×C₈₉	semidirect
185. C₂×M₁₆	Q₈×C₈₉	semidirect
238. M₁₆	C₂×Q₈×C₈₉	semidirect
294. C₂	C₈₉×#3166	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
C₂²×#125×C₈₉	2
C₈₉×#3166	2
C₂×#501×C₈₉	2
C₂×#499×C₈₉	2
C₂×#428×C₈₉	2
C₂×C₄×M₁₆×C₈₉	2
C₂×#3166	89

Subgroups — C2×C89×#3166 #36438510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3166 #36438510