Subgroups — C₈₉×#7656 #36385504

Order 22784.

432 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2861	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1272	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1270	C₂	1	2	yes	maximal
6	C₂×#125×C₈₉	C₄	1	1	yes	characteristic
7	C₂×#124×C₈₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₈₉	—	2	1	no	—
9	#391×C₈₉	C₄	1	1	yes	characteristic
10	#391×C₈₉	C₂²	1	1	yes	characteristic
11	#390×C₈₉	C₄	1	1	yes	characteristic
12	#390×C₈₉	C₂²	1	1	yes	characteristic
13	#388×C₈₉	—	2	1	no	—
14	#388×C₈₉	—	2	1	no	—
15	#388×C₈₉	—	2	1	no	—
16	#388×C₈₉	—	2	1	no	—
17	C₄³×C₈₉	C₂²	1	1	yes	characteristic
18	#336×C₈₉	—	2	1	no	—
19	#336×C₈₉	—	2	1	no	—
20	#336×C₈₉	—	2	1	no	—
21	#336×C₈₉	C₂²	1	2	yes	—
22	#336×C₈₉	C₂²	1	2	yes	—
23	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	—	2	2	no	—
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	#125×C₈₉	—	4	2	no	—
28	#124×C₈₉	—	4	2	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₈₉	—	2	4	no	—
47	#95×C₈₉	—	2	1	no	—
48	#95×C₈₉	—	2	1	no	—
49	#95×C₈₉	—	2	1	no	—
50	#95×C₈₉	—	2	1	no	—
51	C₂⁴×C₈₉	—	2	1	no	—
52	C₂×Q₈×C₈₉	—	2	2	no	—
53	C₂×Q₈×C₈₉	—	4	1	no	—
54	C₂×Q₈×C₈₉	—	4	1	no	—
55	C₂×D₄×C₈₉	—	2	2	no	—
56	C₂×D₄×C₈₉	—	4	1	no	—
57	C₂×D₄×C₈₉	—	4	1	no	—
58	C₂²×C₄×C₈₉	—	2	1	no	—
59	C₂²×C₄×C₈₉	—	2	1	no	—
60	C₂²×C₄×C₈₉	—	2	1	no	—
61	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	—	2	4	no	—
69	C₂×C₈×C₈₉	—	4	2	no	—
70	C₂×C₈×C₈₉	—	4	2	no	—
71	C₂×C₈×C₈₉	—	4	2	no	—
72	C₂×C₈×C₈₉	—	4	2	no	—
73	C₂×C₈×C₈₉	—	4	2	no	—
74	C₄⋊C₄×C₈₉	—	4	2	no	—
75	C₄⋊C₄×C₈₉	—	4	1	no	—
76	C₄⋊C₄×C₈₉	—	4	1	no	—
77	C₄⋊C₄×C₈₉	—	4	2	no	—
78	C₂²⋊C₄×C₈₉	—	4	2	no	—
79	C₂²⋊C₄×C₈₉	—	4	2	no	—
80	C₂²⋊C₄×C₈₉	—	4	2	no	—
81	C₂²⋊C₄×C₈₉	—	4	2	no	—
82	C₂²⋊C₄×C₈₉	—	4	2	no	—
83	C₂²⋊C₄×C₈₉	—	4	2	no	—
84	C₄²×C₈₉	—	2	2	no	—
85	C₄²×C₈₉	—	2	2	no	—
86	C₄²×C₈₉	—	2	2	no	—
87	C₄²×C₈₉	—	2	2	no	—
88	C₄²×C₈₉	—	2	2	no	—
89	C₄²×C₈₉	—	4	1	no	—
90	C₄²×C₈₉	—	4	1	no	—
91	C₂³×C₈₉	—	4	1	no	—
92	C₂³×C₈₉	—	4	1	no	—
93	C₂³×C₈₉	—	4	1	no	—
94	C₂³×C₈₉	—	4	1	no	—
95	C₂³×C₈₉	—	4	1	no	—
96	C₂³×C₈₉	—	4	1	no	—
97	C₂³×C₈₉	—	4	1	no	—
98	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	Q₈×C₈₉	—	8	2	no	—
100	D₄×C₈₉	—	8	2	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	1	no	—
124	C₂×C₄×C₈₉	—	4	1	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	#100	1	2	yes	—
127	C₂×C₄×C₈₉	#104	1	2	yes	—
128	C₂×C₄×C₈₉	#131	1	2	yes	—
129	C₂×C₄×C₈₉	#135	1	2	yes	—
130	C₂×C₄×C₈₉	#135	1	2	yes	—
131	C₈×C₈₉	—	4	4	no	—
132	C₈×C₈₉	—	8	2	no	—
133	C₂²×C₈₉	—	8	1	no	—
134	C₂²×C₈₉	—	8	1	no	—
135	C₂²×C₈₉	—	8	1	no	—
136	C₂²×C₈₉	—	8	1	no	—
137	C₂²×C₈₉	—	8	1	no	—
138	C₂²×C₈₉	—	8	1	no	—
139	C₂²×C₈₉	—	8	1	no	—
140	C₂²×C₈₉	#408	1	1	yes	characteristic
141	C₂²×C₈₉	#410	1	1	yes	characteristic
142	C₂²×C₈₉	C₂×#100	1	1	yes	characteristic
143	C₂²×C₈₉	#416	1	1	yes	characteristic
144	C₂²×C₈₉	#416	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
146	C₂²×C₈₉	#421	1	1	yes	characteristic
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	2	2	no	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	—	8	1	no	—
155	C₄×C₈₉	—	8	1	no	—
156	C₄×C₈₉	—	8	1	no	—
157	#7656	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₈₉	—	16	1	no	—
159	C₂×C₈₉	#1469	1	1	yes	characteristic
160	C₂×C₈₉	#1482	1	1	yes	characteristic
161	C₂×C₈₉	#1494	1	1	yes	characteristic
162	C₂×C₈₉	#1500	1	1	yes	characteristic
163	C₂×C₈₉	#1521	1	1	yes	characteristic
164	C₂×C₈₉	#1530	1	1	yes	characteristic
165	C₂×C₈₉	#1543	1	1	yes	characteristic
166	#2861	C₂×C₈₉	1	1	yes	characteristic
167	#1839	C₂×C₈₉	1	1	yes	characteristic, complemented
168	#1744	C₂×C₈₉	1	1	yes	characteristic, complemented
169	#1272	C₂×C₈₉	1	2	yes	complemented
170	#1270	C₂×C₈₉	1	2	yes	—
171	C₈₉	#7656	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#125	C₄×C₈₉	1	1	yes	characteristic
173	C₂×#124	C₄×C₈₉	1	1	yes	characteristic
174	C₂×C₄×C₈	—	2	1	no	—
175	#391	C₄×C₈₉	1	1	yes	characteristic
176	#391	C₂²×C₈₉	1	1	yes	characteristic
177	#390	C₄×C₈₉	1	1	yes	characteristic
178	#390	C₂²×C₈₉	1	1	yes	characteristic
179	#388	—	2	1	no	—
180	#388	—	2	1	no	—
181	#388	—	2	1	no	—
182	#388	—	2	1	no	—
183	C₄³	C₂²×C₈₉	1	1	yes	characteristic
184	#336	—	2	1	no	—
185	#336	—	2	1	no	—
186	#336	—	2	1	no	—
187	#336	C₂²×C₈₉	1	2	yes	—
188	#336	C₂²×C₈₉	1	2	yes	—
189	C₂²×Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
190	C₂²×D₄	C₂×C₄×C₈₉	1	1	yes	characteristic
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	#125	—	4	2	no	—
194	#124	—	4	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
198	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₂²⋊C₄	—	2	1	no	—
201	C₂×C₂²⋊C₄	—	2	1	no	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
206	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
208	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
212	C₄×C₈	—	2	4	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	C₂⁴	—	2	1	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	1	no	—
261	C₂³	—	4	1	no	—
262	C₂³	—	4	1	no	—
263	C₂³	—	4	1	no	—
264	C₂³	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
265	Q₈	—	8	2	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	#100×C₈₉	1	2	yes	—
293	C₂×C₄	#104×C₈₉	1	2	yes	—
294	C₂×C₄	#131×C₈₉	1	2	yes	—
295	C₂×C₄	#135×C₈₉	1	2	yes	—
296	C₂×C₄	#135×C₈₉	1	2	yes	—
297	C₈	—	4	4	no	—
298	C₈	—	8	2	no	—
299	C₂²	—	8	1	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	—	8	1	no	—
305	C₂²	—	8	1	no	—
306	C₂²	#408×C₈₉	1	1	yes	characteristic
307	C₂²	#410×C₈₉	1	1	yes	characteristic
308	C₂²	C₂×#100×C₈₉	1	1	yes	characteristic
309	C₂²	#416×C₈₉	1	1	yes	characteristic
310	C₂²	#416×C₈₉	1	1	yes	characteristic
311	C₂²	C₂×#104×C₈₉	1	1	yes	characteristic
312	C₂²	#421×C₈₉	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	1	no	—
323	C₂	—	16	1	no	—
324	C₂	C₈₉×#1469	1	1	yes	characteristic
325	C₂	C₈₉×#1482	1	1	yes	characteristic
326	C₂	C₈₉×#1494	1	1	yes	characteristic
327	C₂	C₈₉×#1500	1	1	yes	characteristic
328	C₂	C₈₉×#1521	1	1	yes	characteristic
329	C₂	C₈₉×#1530	1	1	yes	characteristic
330	C₂	C₈₉×#1543	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#1839	C₂	semidirect
3. C₈₉×#1744	C₂	semidirect
4. C₈₉×#1272	C₂	semidirect
157. #7656	C₈₉	direct (N × H)
167. #1839	C₂×C₈₉	semidirect
168. #1744	C₂×C₈₉	semidirect
169. #1272	C₂×C₈₉	semidirect
171. C₈₉	#7656	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2861	2
C₈₉×#1839	2
C₈₉×#1744	2
C₈₉×#1272	2
C₈₉×#1270	2
#7656	89

Subgroups — C89×#7656 #36385504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7656 #36385504