Subgroups — C₂×C₈₉×#3249 #36438746

Order 22784.

1898 conjugacy classes of subgroups.

The subgroup lattice has 304 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×#3249	C₂	1	16	yes	maximal, direct product factor
3	C₂×#477×C₈₉	C₂	1	4	yes	maximal, complemented
4	C₂×#467×C₈₉	C₂	1	4	yes	maximal
5	C₂×#462×C₈₉	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₈₉	C₂²	1	4	yes	—
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	2	yes	—
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
13	C₂×#125×C₈₉	—	2	2	no	—
14	C₂×C₂²⋊Q₈×C₈₉	—	2	2	no	—
15	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
16	C₂×C₄×Q₈×C₈₉	—	2	2	no	—
17	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	—
18	C₂×#117×C₈₉	—	2	2	no	—
19	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#477×C₈₉	C₂²	1	32	yes	complemented
21	#467×C₈₉	C₂²	1	32	yes	—
22	#462×C₈₉	C₂²	1	32	yes	complemented
23	C₂×#107×C₈₉	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₈₉	C₂²	1	8	yes	—
26	C₂×#103×C₈₉	C₂²	1	8	yes	—
27	C₂²×Q₈×C₈₉	—	2	2	no	—
28	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
29	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×Q₁₆×C₈₉	—	4	8	no	—
33	C₂×Q₁₆×C₈₉	C₂³	1	16	yes	—
34	C₂²×C₈×C₈₉	—	4	4	no	—
35	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
36	C₂²×C₈×C₈₉	C₂³	1	8	yes	—
37	C₄⋊Q₈×C₈₉	—	2	4	no	—
38	C₄⋊Q₈×C₈₉	—	2	4	no	—
39	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
40	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
41	#125×C₈₉	—	2	16	no	—
42	C₂²⋊Q₈×C₈₉	—	2	16	no	—
43	C₂²⋊Q₈×C₈₉	—	2	8	no	—
44	C₂²⋊Q₈×C₈₉	C₂³	1	16	yes	—
45	C₄×Q₈×C₈₉	—	2	16	no	—
46	C₄×Q₈×C₈₉	—	2	8	no	—
47	C₄×Q₈×C₈₉	C₂³	1	16	yes	—
48	#117×C₈₉	—	2	16	no	—
49	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
54	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
55	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
56	C₂×C₄²×C₈₉	—	2	4	no	—
57	C₂×C₄²×C₈₉	D₄	1	2	yes	—
58	C₂×C₄²×C₈₉	D₄	1	4	yes	—
59	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
60	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
61	#107×C₈₉	C₂³	1	16	yes	—
62	C₄⋊C₈×C₈₉	C₂³	1	16	yes	—
63	#103×C₈₉	C₂³	1	32	yes	—
64	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₈₉	—	2	4	no	—
66	C₂×Q₈×C₈₉	—	2	8	no	—
67	C₂×Q₈×C₈₉	—	2	8	no	—
68	C₂×Q₈×C₈₉	—	2	8	no	—
69	C₂×Q₈×C₈₉	—	4	8	no	—
70	C₂×Q₈×C₈₉	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	4	no	—
74	C₂²×C₄×C₈₉	—	2	4	no	—
75	C₂²×C₄×C₈₉	—	4	2	no	—
76	C₂²×C₄×C₈₉	Q₁₆	1	8	yes	complemented
77	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
85	Q₁₆×C₈₉	—	4	16	no	—
86	C₂×C₈×C₈₉	—	2	8	no	—
87	C₂×C₈×C₈₉	—	4	8	no	—
88	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
89	C₄⋊C₄×C₈₉	—	2	8	no	—
90	C₄⋊C₄×C₈₉	—	2	8	no	—
91	C₄⋊C₄×C₈₉	—	2	8	no	—
92	C₄⋊C₄×C₈₉	—	2	16	no	—
93	C₄⋊C₄×C₈₉	—	2	16	no	—
94	C₄⋊C₄×C₈₉	—	2	4	no	—
95	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
96	C₂²⋊C₄×C₈₉	—	2	16	no	—
97	C₄²×C₈₉	—	2	16	no	—
98	C₄²×C₈₉	C₂×D₄	1	8	yes	—
99	C₄²×C₈₉	C₂×D₄	1	4	yes	—
100	C₄²×C₈₉	C₂⁴	1	4	yes	—
101	C₂³×C₈₉	—	2	2	no	—
102	C₂³×C₈₉	—	2	2	no	—
103	C₂³×C₈₉	C₂×Q₁₆	1	4	yes	—
104	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
105	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₈₉	—	2	8	no	—
107	Q₈×C₈₉	—	2	8	no	—
108	Q₈×C₈₉	—	4	16	no	—
109	Q₈×C₈₉	—	4	8	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	8	no	—
116	C₂×C₄×C₈₉	—	2	8	no	—
117	C₂×C₄×C₈₉	—	4	4	no	—
118	C₂×C₄×C₈₉	—	4	8	no	—
119	C₂×C₄×C₈₉	—	4	8	no	—
120	C₂×C₄×C₈₉	C₂×Q₁₆	1	8	yes	—
121	C₂×C₄×C₈₉	C₂×Q₁₆	1	16	yes	complemented
122	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
126	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
128	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
129	C₈×C₈₉	—	2	8	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	8	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	#537	1	2	yes	—
134	C₂²×C₈₉	C₂²×Q₁₆	1	4	yes	—
135	C₂²×C₈₉	C₂²×Q₁₆	1	2	yes	—
136	C₂²×C₈₉	#575	1	2	yes	—
137	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	—	2	4	no	—
140	C₄×C₈₉	—	4	8	no	—
141	C₄×C₈₉	—	4	8	no	—
142	C₄×C₈₉	#537	1	2	yes	—
143	C₄×C₈₉	C₂²×Q₁₆	1	8	yes	—
144	C₄×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
145	C₄×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×#3249	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₈₉	—	2	4	no	—
148	C₂×C₈₉	#3249	1	4	yes	direct product factor
149	C₂×C₈₉	C₂×#537	1	1	yes	characteristic
150	C₂×C₈₉	C₂³×Q₁₆	1	1	yes	characteristic
151	C₂×C₈₉	C₂×#575	1	1	yes	characteristic
152	C₂×#533	C₂×C₈₉	1	2	yes	—
153	#3249	C₂×C₈₉	1	16	yes	direct product factor
154	C₂×#477	C₂×C₈₉	1	4	yes	complemented
155	C₂×#467	C₂×C₈₉	1	4	yes	—
156	C₂×#462	C₂×C₈₉	1	4	yes	complemented
157	C₂²×C₄⋊C₈	C₂×C₈₉	1	1	yes	characteristic
158	C₈₉	C₂×#3249	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×Q₁₆	C₂²×C₈₉	1	4	yes	—
160	C₂³×C₈	C₂²×C₈₉	1	2	yes	—
161	#533	—	2	8	no	—
162	#533	C₂²×C₈₉	1	16	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
164	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
165	C₂×#125	—	2	2	no	—
166	C₂×C₂²⋊Q₈	—	2	2	no	—
167	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
168	C₂×C₄×Q₈	—	2	2	no	—
169	C₂×C₄×Q₈	C₂²×C₈₉	1	4	yes	—
170	C₂×#117	—	2	2	no	—
171	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
172	#477	C₂²×C₈₉	1	32	yes	complemented
173	#467	C₂²×C₈₉	1	32	yes	—
174	#462	C₂²×C₈₉	1	32	yes	complemented
175	C₂×#107	C₂²×C₈₉	1	4	yes	—
176	C₂×C₄⋊C₈	C₂²×C₈₉	1	4	yes	—
177	C₂×C₄⋊C₈	C₂²×C₈₉	1	8	yes	—
178	C₂×#103	C₂²×C₈₉	1	8	yes	—
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
181	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
182	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
183	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
184	C₂×Q₁₆	—	4	8	no	—
185	C₂×Q₁₆	C₂³×C₈₉	1	16	yes	—
186	C₂²×C₈	—	4	4	no	—
187	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
188	C₂²×C₈	C₂³×C₈₉	1	8	yes	—
189	C₄⋊Q₈	—	2	4	no	—
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
192	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
193	#125	—	2	16	no	—
194	C₂²⋊Q₈	—	2	16	no	—
195	C₂²⋊Q₈	—	2	8	no	—
196	C₂²⋊Q₈	C₂³×C₈₉	1	16	yes	—
197	C₄×Q₈	—	2	16	no	—
198	C₄×Q₈	—	2	8	no	—
199	C₄×Q₈	C₂³×C₈₉	1	16	yes	—
200	#117	—	2	16	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
207	C₂×C₂²⋊C₄	—	2	4	no	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	D₄×C₈₉	1	2	yes	—
210	C₂×C₄²	D₄×C₈₉	1	4	yes	—
211	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
212	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
213	#107	C₂³×C₈₉	1	16	yes	—
214	C₄⋊C₈	C₂³×C₈₉	1	16	yes	—
215	#103	C₂³×C₈₉	1	32	yes	—
216	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	—	2	8	no	—
221	C₂×Q₈	—	4	8	no	—
222	C₂×Q₈	C₂⁴×C₈₉	1	8	yes	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	Q₁₆×C₈₉	1	8	yes	complemented
229	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
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	4	yes	—
233	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
236	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
237	Q₁₆	—	4	16	no	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	4	8	no	—
240	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
241	C₄⋊C₄	—	2	8	no	—
242	C₄⋊C₄	—	2	8	no	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	16	no	—
245	C₄⋊C₄	—	2	16	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
248	C₂²⋊C₄	—	2	16	no	—
249	C₄²	—	2	16	no	—
250	C₄²	C₂×D₄×C₈₉	1	8	yes	—
251	C₄²	C₂×D₄×C₈₉	1	4	yes	—
252	C₄²	C₂⁴×C₈₉	1	4	yes	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	C₂×Q₁₆×C₈₉	1	4	yes	—
256	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
257	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
258	Q₈	—	2	8	no	—
259	Q₈	—	2	8	no	—
260	Q₈	—	4	16	no	—
261	Q₈	—	4	8	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	C₂×Q₁₆×C₈₉	1	8	yes	—
273	C₂×C₄	C₂×Q₁₆×C₈₉	1	16	yes	complemented
274	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
276	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
278	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
279	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
280	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
281	C₈	—	2	8	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	8	no	—
285	C₂²	#537×C₈₉	1	2	yes	—
286	C₂²	C₂²×Q₁₆×C₈₉	1	4	yes	—
287	C₂²	C₂²×Q₁₆×C₈₉	1	2	yes	—
288	C₂²	#575×C₈₉	1	2	yes	—
289	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	8	no	—
293	C₄	—	4	8	no	—
294	C₄	#537×C₈₉	1	2	yes	—
295	C₄	C₂²×Q₁₆×C₈₉	1	8	yes	—
296	C₄	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
297	C₄	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
298	C₂	—	2	4	no	—
299	C₂	C₈₉×#3249	1	4	yes	direct product factor
300	C₂	C₂×#537×C₈₉	1	1	yes	characteristic
301	C₂	C₂³×Q₁₆×C₈₉	1	1	yes	characteristic
302	C₂	C₂×#575×C₈₉	1	1	yes	characteristic

	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
2. C₈₉×#3249	C₂	direct (N × H)
3. C₂×#477×C₈₉	C₂	semidirect
5. C₂×#462×C₈₉	C₂	semidirect
20. #477×C₈₉	C₂²	semidirect
22. #462×C₈₉	C₂²	semidirect
76. C₂²×C₄×C₈₉	Q₁₆	semidirect
121. C₂×C₄×C₈₉	C₂×Q₁₆	semidirect
146. C₂×#3249	C₈₉	direct (N × H)
148. C₂×C₈₉	#3249	direct (N × H)
153. #3249	C₂×C₈₉	direct (N × H)
154. C₂×#477	C₂×C₈₉	semidirect
156. C₂×#462	C₂×C₈₉	semidirect
158. C₈₉	C₂×#3249	direct (N × H)
172. #477	C₂²×C₈₉	semidirect
174. #462	C₂²×C₈₉	semidirect
228. C₂²×C₄	Q₁₆×C₈₉	semidirect
273. C₂×C₄	C₂×Q₁₆×C₈₉	semidirect
299. C₂	C₈₉×#3249	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×#3249	2
C₂×#477×C₈₉	2
C₂×#467×C₈₉	2
C₂×#462×C₈₉	2
C₂²×C₄⋊C₈×C₈₉	2
C₂×#3249	89

Subgroups — C2×C89×#3249 #36438746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3249 #36438746