Subgroups — C₈₉×#35341 #36413189

Order 22784.

2358 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#125×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2861	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2642	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2641	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2591	C₂	1	8	yes	maximal, complemented
7	C₈₉×#2562	C₂	1	8	yes	maximal, complemented
8	C₈₉×#2311	C₂	1	4	yes	maximal, complemented
9	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₈₉	C₂²	1	2	yes	—
11	#532×C₈₉	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#125×C₈₉	—	2	8	no	—
15	C₂×#125×C₈₉	C₂²	1	4	yes	—
16	C₂×#125×C₈₉	C₂²	1	2	yes	—
17	C₂×#125×C₈₉	C₂²	1	2	yes	—
18	C₂×#124×C₈₉	C₂²	1	2	yes	—
19	C₂×#121×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
21	C₂×#117×C₈₉	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
24	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	#400×C₈₉	C₂²	1	16	yes	—
26	#394×C₈₉	C₂²	1	16	yes	—
27	#391×C₈₉	C₂²	1	8	yes	—
28	#390×C₈₉	C₂²	1	8	yes	—
29	#388×C₈₉	C₂²	1	32	yes	—
30	#386×C₈₉	C₂²	1	16	yes	—
31	#380×C₈₉	C₂²	1	16	yes	—
32	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
33	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
34	#376×C₈₉	C₂²	1	2	yes	—
35	C₄³×C₈₉	C₂²	1	2	yes	—
36	C₂×#41×C₈₉	—	2	8	no	—
37	C₂×#41×C₈₉	—	2	16	no	—
38	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
39	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
40	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
41	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
42	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
43	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
44	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
45	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₈₉	—	2	4	no	—
47	#127×C₈₉	—	2	4	no	—
48	#125×C₈₉	—	2	16	no	—
49	#125×C₈₉	—	2	8	no	—
50	#125×C₈₉	—	2	8	no	—
51	#124×C₈₉	—	2	8	no	—
52	#121×C₈₉	—	2	16	no	—
53	C₄×D₄×C₈₉	—	2	16	no	—
54	#117×C₈₉	—	2	8	no	—
55	C₂×C₄⋊C₄×C₈₉	—	2	16	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
57	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
64	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
65	C₂×C₂²⋊C₄×C₈₉	C₂³	1	32	yes	—
66	C₂×C₄²×C₈₉	D₄	1	8	yes	—
67	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
68	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
69	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
72	#95×C₈₉	C₂³	1	8	yes	—
73	#95×C₈₉	C₂³	1	32	yes	—
74	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
75	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
76	#41×C₈₉	—	4	32	no	—
77	C₂×Q₈×C₈₉	—	2	4	no	—
78	C₂×Q₈×C₈₉	—	2	8	no	—
79	C₂×D₄×C₈₉	—	2	8	no	—
80	C₂×D₄×C₈₉	—	2	16	no	—
81	C₂×D₄×C₈₉	—	2	8	no	—
82	C₂×D₄×C₈₉	—	2	4	no	—
83	C₂×D₄×C₈₉	—	2	8	no	—
84	C₂×D₄×C₈₉	—	4	8	no	—
85	C₂²×C₄×C₈₉	—	2	8	no	—
86	C₂²×C₄×C₈₉	—	2	8	no	—
87	C₂²×C₄×C₈₉	—	2	16	no	—
88	C₂²×C₄×C₈₉	—	4	4	no	—
89	C₂²×C₄×C₈₉	—	4	8	no	—
90	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
91	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₈₉	#41	1	8	yes	—
93	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
94	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
95	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
96	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
97	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
98	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	—	2	16	no	—
101	C₄⋊C₄×C₈₉	—	2	16	no	—
102	C₄⋊C₄×C₈₉	—	2	8	no	—
103	C₄⋊C₄×C₈₉	—	2	8	no	—
104	C₄⋊C₄×C₈₉	—	2	8	no	—
105	C₄⋊C₄×C₈₉	—	2	8	no	—
106	C₂²⋊C₄×C₈₉	—	2	8	no	—
107	C₂²⋊C₄×C₈₉	—	2	16	no	—
108	C₂²⋊C₄×C₈₉	—	4	32	no	—
109	C₄²×C₈₉	—	2	8	no	—
110	C₄²×C₈₉	—	4	8	no	—
111	C₄²×C₈₉	C₂×D₄	1	8	yes	—
112	C₄²×C₈₉	C₂×D₄	1	4	yes	—
113	C₄²×C₈₉	C₂×D₄	1	4	yes	—
114	C₂³×C₈₉	—	2	4	no	—
115	C₂³×C₈₉	—	2	4	no	—
116	C₂³×C₈₉	—	2	8	no	—
117	C₂³×C₈₉	—	2	16	no	—
118	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
119	C₂³×C₈₉	C₂×#41	1	4	yes	—
120	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
121	Q₈×C₈₉	—	4	8	no	—
122	D₄×C₈₉	—	4	16	no	—
123	D₄×C₈₉	—	4	16	no	—
124	D₄×C₈₉	—	4	8	no	—
125	C₂×C₄×C₈₉	—	2	4	no	—
126	C₂×C₄×C₈₉	—	2	4	no	—
127	C₂×C₄×C₈₉	—	2	4	no	—
128	C₂×C₄×C₈₉	—	2	8	no	—
129	C₂×C₄×C₈₉	—	2	8	no	—
130	C₂×C₄×C₈₉	—	2	8	no	—
131	C₂×C₄×C₈₉	—	2	16	no	—
132	C₂×C₄×C₈₉	—	2	16	no	—
133	C₂×C₄×C₈₉	—	4	8	no	—
134	C₂×C₄×C₈₉	—	4	16	no	—
135	C₂×C₄×C₈₉	—	4	16	no	—
136	C₂×C₄×C₈₉	#124	1	16	yes	—
137	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
138	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
139	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
140	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
142	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
143	C₂²×C₈₉	—	2	4	no	—
144	C₂²×C₈₉	—	2	8	no	—
145	C₂²×C₈₉	—	4	4	no	—
146	C₂²×C₈₉	—	4	8	no	—
147	C₂²×C₈₉	—	4	16	no	—
148	C₂²×C₈₉	C₂×#124	1	4	yes	—
149	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
150	C₂²×C₈₉	C₂²×#41	1	2	yes	—
151	C₂²×C₈₉	#585	1	4	yes	—
152	C₄×C₈₉	—	2	4	no	—
153	C₄×C₈₉	—	2	4	no	—
154	C₄×C₈₉	—	4	4	no	—
155	C₄×C₈₉	—	4	8	no	—
156	C₄×C₈₉	—	4	8	no	—
157	C₄×C₈₉	C₂×#124	1	4	yes	—
158	C₄×C₈₉	C₂×#124	1	4	yes	—
159	#35341	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂×C₈₉	—	2	4	no	—
161	C₂×C₈₉	—	8	4	no	—
162	C₂×C₈₉	C₂²×#124	1	1	yes	characteristic
163	C₂×C₈₉	#3449	1	2	yes	—
164	C₂×C₈₉	#3512	1	4	yes	—
165	C₂×#532	C₂×C₈₉	1	1	yes	characteristic
166	C₂²×#125	C₂×C₈₉	1	1	yes	characteristic, complemented
167	#2861	C₂×C₈₉	1	4	yes	complemented
168	#2642	C₂×C₈₉	1	2	yes	complemented
169	#2641	C₂×C₈₉	1	2	yes	complemented
170	#2591	C₂×C₈₉	1	8	yes	complemented
171	#2562	C₂×C₈₉	1	8	yes	complemented
172	#2311	C₂×C₈₉	1	4	yes	complemented
173	C₄×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
174	C₈₉	#35341	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×#41	C₂²×C₈₉	1	2	yes	—
176	#532	—	2	8	no	—
177	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
178	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
179	C₂×#125	—	2	8	no	—
180	C₂×#125	C₂²×C₈₉	1	4	yes	—
181	C₂×#125	C₂²×C₈₉	1	2	yes	—
182	C₂×#125	C₂²×C₈₉	1	2	yes	—
183	C₂×#124	C₂²×C₈₉	1	2	yes	—
184	C₂×#121	C₂²×C₈₉	1	4	yes	—
185	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
186	C₂×#117	C₂²×C₈₉	1	2	yes	—
187	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
188	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
189	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
190	#400	C₂²×C₈₉	1	16	yes	—
191	#394	C₂²×C₈₉	1	16	yes	—
192	#391	C₂²×C₈₉	1	8	yes	—
193	#390	C₂²×C₈₉	1	8	yes	—
194	#388	C₂²×C₈₉	1	32	yes	—
195	#386	C₂²×C₈₉	1	16	yes	—
196	#380	C₂²×C₈₉	1	16	yes	—
197	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
198	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
199	#376	C₂²×C₈₉	1	2	yes	—
200	C₄³	C₂²×C₈₉	1	2	yes	—
201	C₂×#41	—	2	8	no	—
202	C₂×#41	—	2	16	no	—
203	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
204	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
205	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
206	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
207	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
208	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
209	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
210	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	4	no	—
212	#127	—	2	4	no	—
213	#125	—	2	16	no	—
214	#125	—	2	8	no	—
215	#125	—	2	8	no	—
216	#124	—	2	8	no	—
217	#121	—	2	16	no	—
218	C₄×D₄	—	2	16	no	—
219	#117	—	2	8	no	—
220	C₂×C₄⋊C₄	—	2	16	no	—
221	C₂×C₄⋊C₄	—	2	8	no	—
222	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
223	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
224	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
225	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
226	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
227	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
228	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
229	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
230	C₂×C₂²⋊C₄	C₂³×C₈₉	1	32	yes	—
231	C₂×C₄²	D₄×C₈₉	1	8	yes	—
232	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
233	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
234	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
235	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
237	#95	C₂³×C₈₉	1	8	yes	—
238	#95	C₂³×C₈₉	1	32	yes	—
239	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
240	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
241	#41	—	4	32	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	8	no	—
244	C₂×D₄	—	2	8	no	—
245	C₂×D₄	—	2	16	no	—
246	C₂×D₄	—	2	8	no	—
247	C₂×D₄	—	2	4	no	—
248	C₂×D₄	—	2	8	no	—
249	C₂×D₄	—	4	8	no	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	—	2	8	no	—
252	C₂²×C₄	—	2	16	no	—
253	C₂²×C₄	—	4	4	no	—
254	C₂²×C₄	—	4	8	no	—
255	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
256	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
257	C₂²×C₄	#41×C₈₉	1	8	yes	—
258	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
259	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
260	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
261	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
262	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
263	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	16	no	—
266	C₄⋊C₄	—	2	16	no	—
267	C₄⋊C₄	—	2	8	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₂²⋊C₄	—	2	8	no	—
272	C₂²⋊C₄	—	2	16	no	—
273	C₂²⋊C₄	—	4	32	no	—
274	C₄²	—	2	8	no	—
275	C₄²	—	4	8	no	—
276	C₄²	C₂×D₄×C₈₉	1	8	yes	—
277	C₄²	C₂×D₄×C₈₉	1	4	yes	—
278	C₄²	C₂×D₄×C₈₉	1	4	yes	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	4	no	—
281	C₂³	—	2	8	no	—
282	C₂³	—	2	16	no	—
283	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
284	C₂³	C₂×#41×C₈₉	1	4	yes	—
285	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
286	Q₈	—	4	8	no	—
287	D₄	—	4	16	no	—
288	D₄	—	4	16	no	—
289	D₄	—	4	8	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	16	no	—
297	C₂×C₄	—	2	16	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	4	16	no	—
300	C₂×C₄	—	4	16	no	—
301	C₂×C₄	#124×C₈₉	1	16	yes	—
302	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
303	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
304	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
306	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
307	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	8	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	8	no	—
312	C₂²	—	4	16	no	—
313	C₂²	C₂×#124×C₈₉	1	4	yes	—
314	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
315	C₂²	C₂²×#41×C₈₉	1	2	yes	—
316	C₂²	#585×C₈₉	1	4	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	2	4	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	8	no	—
321	C₄	—	4	8	no	—
322	C₄	C₂×#124×C₈₉	1	4	yes	—
323	C₄	C₂×#124×C₈₉	1	4	yes	—
324	C₂	—	2	4	no	—
325	C₂	—	8	4	no	—
326	C₂	C₂²×#124×C₈₉	1	1	yes	characteristic
327	C₂	C₈₉×#3449	1	2	yes	—
328	C₂	C₈₉×#3512	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#532×C₈₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#125×C₈₉	C₂	semidirect
3. C₈₉×#2861	C₂	semidirect
4. C₈₉×#2642	C₂	semidirect
5. C₈₉×#2641	C₂	semidirect
6. C₈₉×#2591	C₂	semidirect
7. C₈₉×#2562	C₂	semidirect
8. C₈₉×#2311	C₂	semidirect
9. C₄×#117×C₈₉	C₂	semidirect
159. #35341	C₈₉	direct (N × H)
166. C₂²×#125	C₂×C₈₉	semidirect
167. #2861	C₂×C₈₉	semidirect
168. #2642	C₂×C₈₉	semidirect
169. #2641	C₂×C₈₉	semidirect
170. #2591	C₂×C₈₉	semidirect
171. #2562	C₂×C₈₉	semidirect
172. #2311	C₂×C₈₉	semidirect
173. C₄×#117	C₂×C₈₉	semidirect
174. C₈₉	#35341	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×#532×C₈₉	2
C₂²×#125×C₈₉	2
C₈₉×#2861	2
C₈₉×#2642	2
C₈₉×#2641	2
C₈₉×#2591	2
C₈₉×#2562	2
C₈₉×#2311	2
C₄×#117×C₈₉	2
#35341	89

Subgroups — C89×#35341 #36413189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35341 #36413189