Subgroups — C₈₉×#29545 #36407393

Order 22784.

746 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2319	C₂	1	2	yes	maximal, complemented
2	C₈₉×#2086	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2065	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2054	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1925	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1919	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1879	C₂	1	2	yes	maximal
8	C₈₉×#1864	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1832	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
12	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#106×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#102×C₈₉	C₂²	1	2	yes	—
17	C₂×#102×C₈₉	C₂²	1	2	yes	—
18	C₂×#102×C₈₉	C₂²	1	2	yes	—
19	#390×C₈₉	C₂²	1	2	yes	—
20	#390×C₈₉	C₂²	1	2	yes	—
21	#389×C₈₉	C₂²	1	2	yes	—
22	#389×C₈₉	C₂²	1	2	yes	—
23	#388×C₈₉	—	2	2	no	—
24	#388×C₈₉	—	2	2	no	—
25	#381×C₈₉	—	2	2	no	—
26	#380×C₈₉	—	2	2	no	—
27	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
28	#376×C₈₉	C₂²	1	1	yes	characteristic
29	#340×C₈₉	C₂²	1	1	yes	characteristic
30	#340×C₈₉	C₂²	1	2	yes	—
31	#340×C₈₉	C₂²	1	1	yes	characteristic
32	#340×C₈₉	C₂²	1	2	yes	—
33	#336×C₈₉	C₂²	1	2	yes	—
34	#336×C₈₉	C₂²	1	2	yes	—
35	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
36	C₂³×C₄×C₈₉	—	2	2	no	—
37	C₂×D₈×C₈₉	—	2	4	no	—
38	C₂×D₈×C₈₉	—	4	1	no	—
39	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₄×D₄×C₈₉	—	2	4	no	—
43	C₄×D₄×C₈₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
54	C₂×C₄²×C₈₉	—	2	2	no	—
55	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	#107×C₈₉	—	4	1	no	—
58	#106×C₈₉	—	2	4	no	—
59	C₄⋊C₈×C₈₉	—	4	1	no	—
60	C₄⋊C₈×C₈₉	—	4	1	no	—
61	#102×C₈₉	—	2	4	no	—
62	#102×C₈₉	—	2	4	no	—
63	#102×C₈₉	—	4	2	no	—
64	#95×C₈₉	—	2	2	no	—
65	#95×C₈₉	—	2	2	no	—
66	#95×C₈₉	—	2	2	no	—
67	#95×C₈₉	D₄	1	2	yes	—
68	#95×C₈₉	D₄	1	1	yes	characteristic
69	#95×C₈₉	D₄	1	1	yes	characteristic
70	C₂⁴×C₈₉	—	2	2	no	—
71	C₂×D₄×C₈₉	—	2	2	no	—
72	C₂×D₄×C₈₉	—	2	2	no	—
73	C₂×D₄×C₈₉	—	2	2	no	—
74	C₂×D₄×C₈₉	—	2	2	no	—
75	C₂×D₄×C₈₉	—	2	2	no	—
76	C₂×D₄×C₈₉	#41	1	4	yes	—
77	C₂²×C₄×C₈₉	—	2	2	no	—
78	C₂²×C₄×C₈₉	—	2	8	no	—
79	C₂²×C₄×C₈₉	—	2	2	no	—
80	C₂²×C₄×C₈₉	—	2	2	no	—
81	C₂²×C₄×C₈₉	—	8	2	no	—
82	C₂²×C₄×C₈₉	—	8	2	no	—
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
88	D₈×C₈₉	—	4	4	no	—
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	2	2	no	—
99	C₄⋊C₄×C₈₉	—	2	2	no	—
100	C₄⋊C₄×C₈₉	—	2	2	no	—
101	C₄⋊C₄×C₈₉	—	8	2	no	—
102	C₄⋊C₄×C₈₉	—	8	2	no	—
103	C₄⋊C₄×C₈₉	#41	1	4	yes	—
104	C₂²⋊C₄×C₈₉	—	2	8	no	—
105	C₂²⋊C₄×C₈₉	—	8	2	no	—
106	C₂²⋊C₄×C₈₉	—	8	2	no	—
107	C₂²⋊C₄×C₈₉	—	8	2	no	—
108	C₂²⋊C₄×C₈₉	—	8	2	no	—
109	C₄²×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	—	2	1	no	—
113	C₄²×C₈₉	—	8	2	no	—
114	C₂³×C₈₉	—	2	4	no	—
115	C₂³×C₈₉	—	4	2	no	—
116	C₂³×C₈₉	—	4	2	no	—
117	C₂³×C₈₉	—	4	2	no	—
118	C₂³×C₈₉	—	4	2	no	—
119	C₂³×C₈₉	—	4	2	no	—
120	C₂³×C₈₉	—	4	2	no	—
121	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	D₄×C₈₉	—	2	4	no	—
123	D₄×C₈₉	—	2	4	no	—
124	D₄×C₈₉	—	4	2	no	—
125	D₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	2	4	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	—	4	2	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	—	8	8	no	—
147	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
148	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
149	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
151	C₈×C₈₉	—	4	1	no	—
152	C₈×C₈₉	—	4	1	no	—
153	C₈×C₈₉	—	4	2	no	—
154	C₂²×C₈₉	—	4	4	no	—
155	C₂²×C₈₉	—	4	4	no	—
156	C₂²×C₈₉	—	4	4	no	—
157	C₂²×C₈₉	—	8	2	no	—
158	C₂²×C₈₉	—	8	2	no	—
159	C₂²×C₈₉	—	8	2	no	—
160	C₂²×C₈₉	—	8	2	no	—
161	C₂²×C₈₉	#525	1	2	yes	—
162	C₂²×C₈₉	#537	1	1	yes	characteristic
163	C₂²×C₈₉	#576	1	1	yes	characteristic
164	C₂²×C₈₉	#576	1	1	yes	characteristic
165	C₂²×C₈₉	#577	1	2	yes	—
166	C₄×C₈₉	—	4	2	no	—
167	C₄×C₈₉	—	4	4	no	—
168	C₄×C₈₉	—	4	2	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	—	8	2	no	—
172	C₄×C₈₉	—	8	2	no	—
173	C₄×C₈₉	#551	1	1	yes	characteristic
174	C₄×C₈₉	#552	1	2	yes	—
175	C₄×C₈₉	#555	1	1	yes	characteristic
176	#29545	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂×C₈₉	—	8	4	no	—
178	C₂×C₈₉	#2609	1	1	yes	characteristic
179	C₂×C₈₉	#3118	1	2	yes	—
180	C₂×C₈₉	#3119	1	2	yes	—
181	C₂×C₈₉	#3259	1	1	yes	characteristic
182	C₂×C₈₉	#3263	1	1	yes	characteristic
183	#2319	C₂×C₈₉	1	2	yes	complemented
184	#2086	C₂×C₈₉	1	1	yes	characteristic, complemented
185	#2065	C₂×C₈₉	1	2	yes	complemented
186	#2054	C₂×C₈₉	1	1	yes	characteristic
187	#1925	C₂×C₈₉	1	2	yes	complemented
188	#1919	C₂×C₈₉	1	2	yes	complemented
189	#1879	C₂×C₈₉	1	2	yes	—
190	#1864	C₂×C₈₉	1	2	yes	complemented
191	#1832	C₂×C₈₉	1	1	yes	characteristic, complemented
192	C₈₉	#29545	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
194	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
195	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
196	C₂×#106	C₂²×C₈₉	1	2	yes	—
197	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
198	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
199	C₂×#102	C₂²×C₈₉	1	2	yes	—
200	C₂×#102	C₂²×C₈₉	1	2	yes	—
201	C₂×#102	C₂²×C₈₉	1	2	yes	—
202	#390	C₂²×C₈₉	1	2	yes	—
203	#390	C₂²×C₈₉	1	2	yes	—
204	#389	C₂²×C₈₉	1	2	yes	—
205	#389	C₂²×C₈₉	1	2	yes	—
206	#388	—	2	2	no	—
207	#388	—	2	2	no	—
208	#381	—	2	2	no	—
209	#380	—	2	2	no	—
210	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
211	#376	C₂²×C₈₉	1	1	yes	characteristic
212	#340	C₂²×C₈₉	1	1	yes	characteristic
213	#340	C₂²×C₈₉	1	2	yes	—
214	#340	C₂²×C₈₉	1	1	yes	characteristic
215	#340	C₂²×C₈₉	1	2	yes	—
216	#336	C₂²×C₈₉	1	2	yes	—
217	#336	C₂²×C₈₉	1	2	yes	—
218	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
219	C₂³×C₄	—	2	2	no	—
220	C₂×D₈	—	2	4	no	—
221	C₂×D₈	—	4	1	no	—
222	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
223	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
224	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
225	C₄×D₄	—	2	4	no	—
226	C₄×D₄	—	2	4	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
239	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
240	#107	—	4	1	no	—
241	#106	—	2	4	no	—
242	C₄⋊C₈	—	4	1	no	—
243	C₄⋊C₈	—	4	1	no	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#102	—	4	2	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	#95	D₄×C₈₉	1	2	yes	—
251	#95	D₄×C₈₉	1	1	yes	characteristic
252	#95	D₄×C₈₉	1	1	yes	characteristic
253	C₂⁴	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	#41×C₈₉	1	4	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	8	2	no	—
265	C₂²×C₄	—	8	2	no	—
266	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
270	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
271	D₈	—	4	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	8	2	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	#41×C₈₉	1	4	yes	—
287	C₂²⋊C₄	—	2	8	no	—
288	C₂²⋊C₄	—	8	2	no	—
289	C₂²⋊C₄	—	8	2	no	—
290	C₂²⋊C₄	—	8	2	no	—
291	C₂²⋊C₄	—	8	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	8	2	no	—
297	C₂³	—	2	4	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
305	D₄	—	2	4	no	—
306	D₄	—	2	4	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	8	no	—
330	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
331	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
332	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
333	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	#525×C₈₉	1	2	yes	—
345	C₂²	#537×C₈₉	1	1	yes	characteristic
346	C₂²	#576×C₈₉	1	1	yes	characteristic
347	C₂²	#576×C₈₉	1	1	yes	characteristic
348	C₂²	#577×C₈₉	1	2	yes	—
349	C₄	—	4	2	no	—
350	C₄	—	4	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	#551×C₈₉	1	1	yes	characteristic
357	C₄	#552×C₈₉	1	2	yes	—
358	C₄	#555×C₈₉	1	1	yes	characteristic
359	C₂	—	8	4	no	—
360	C₂	C₈₉×#2609	1	1	yes	characteristic
361	C₂	C₈₉×#3118	1	2	yes	—
362	C₂	C₈₉×#3119	1	2	yes	—
363	C₂	C₈₉×#3259	1	1	yes	characteristic
364	C₂	C₈₉×#3263	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂²×D₈×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ #376×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#2319	C₂	semidirect
2. C₈₉×#2086	C₂	semidirect
3. C₈₉×#2065	C₂	semidirect
5. C₈₉×#1925	C₂	semidirect
6. C₈₉×#1919	C₂	semidirect
8. C₈₉×#1864	C₂	semidirect
9. C₈₉×#1832	C₂	semidirect
176. #29545	C₈₉	direct (N × H)
183. #2319	C₂×C₈₉	semidirect
184. #2086	C₂×C₈₉	semidirect
185. #2065	C₂×C₈₉	semidirect
187. #1925	C₂×C₈₉	semidirect
188. #1919	C₂×C₈₉	semidirect
190. #1864	C₂×C₈₉	semidirect
191. #1832	C₂×C₈₉	semidirect
192. C₈₉	#29545	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×#2319	2
C₈₉×#2086	2
C₈₉×#2065	2
C₈₉×#2054	2
C₈₉×#1925	2
C₈₉×#1919	2
C₈₉×#1879	2
C₈₉×#1864	2
C₈₉×#1832	2
#29545	89

Subgroups — C89×#29545 #36407393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29545 #36407393