Subgroups — C₈₉×#7657 #36385505

Order 22784.

370 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2864	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1271	C₂	1	2	yes	maximal
5	C₈₉×#1271	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₈₉	—	2	1	no	—
7	#389×C₈₉	C₄	1	1	yes	characteristic
8	#389×C₈₉	C₄	1	1	yes	characteristic
9	#389×C₈₉	C₂²	1	1	yes	characteristic
10	#389×C₈₉	C₂²	1	1	yes	characteristic
11	#387×C₈₉	—	2	1	no	—
12	#387×C₈₉	—	2	1	no	—
13	#387×C₈₉	—	2	1	no	—
14	#387×C₈₉	—	2	1	no	—
15	#382×C₈₉	C₄	1	1	yes	characteristic
16	#382×C₈₉	C₄	1	1	yes	characteristic
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×C₈×C₈₉	—	2	2	no	—
24	C₂²×C₈×C₈₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₈₉	—	2	4	no	—
41	#95×C₈₉	—	2	1	no	—
42	#95×C₈₉	—	2	1	no	—
43	#95×C₈₉	—	2	1	no	—
44	#95×C₈₉	—	2	1	no	—
45	#95×C₈₉	—	2	1	no	—
46	#95×C₈₉	—	2	1	no	—
47	#95×C₈₉	—	2	1	no	—
48	#95×C₈₉	—	2	1	no	—
49	C₂²×C₄×C₈₉	—	2	1	no	—
50	C₂²×C₄×C₈₉	—	2	1	no	—
51	C₂²×C₄×C₈₉	—	2	1	no	—
52	C₂²×C₄×C₈₉	—	2	1	no	—
53	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	—	4	2	no	—
62	C₂×C₈×C₈₉	—	4	2	no	—
63	C₂×C₈×C₈₉	—	4	2	no	—
64	C₂×C₈×C₈₉	—	4	2	no	—
65	C₂×C₈×C₈₉	—	4	2	no	—
66	C₄⋊C₄×C₈₉	—	2	2	no	—
67	C₄⋊C₄×C₈₉	—	2	2	no	—
68	C₄⋊C₄×C₈₉	—	4	2	no	—
69	C₄⋊C₄×C₈₉	—	4	1	no	—
70	C₄⋊C₄×C₈₉	—	4	1	no	—
71	C₄⋊C₄×C₈₉	—	4	2	no	—
72	C₄⋊C₄×C₈₉	—	4	2	no	—
73	C₄⋊C₄×C₈₉	—	4	2	no	—
74	C₄²×C₈₉	—	2	2	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	2	2	no	—
79	C₄²×C₈₉	—	4	1	no	—
80	C₄²×C₈₉	—	4	1	no	—
81	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	4	1	no	—
99	C₂×C₄×C₈₉	—	4	1	no	—
100	C₂×C₄×C₈₉	—	4	1	no	—
101	C₂×C₄×C₈₉	—	4	1	no	—
102	C₂×C₄×C₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	—	4	1	no	—
104	C₂×C₄×C₈₉	—	4	1	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	—	4	1	no	—
107	C₂×C₄×C₈₉	—	4	1	no	—
108	C₂×C₄×C₈₉	—	4	1	no	—
109	C₂×C₄×C₈₉	—	4	1	no	—
110	C₂×C₄×C₈₉	#101	1	2	yes	—
111	C₂×C₄×C₈₉	#104	1	2	yes	—
112	C₂×C₄×C₈₉	#131	1	2	yes	—
113	C₂×C₄×C₈₉	#135	1	2	yes	—
114	C₂×C₄×C₈₉	#135	1	2	yes	—
115	C₈×C₈₉	—	4	4	no	—
116	C₈×C₈₉	—	8	2	no	—
117	C₂²×C₈₉	#408	1	1	yes	characteristic
118	C₂²×C₈₉	#410	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
120	C₂²×C₈₉	#416	1	1	yes	characteristic
121	C₂²×C₈₉	#416	1	1	yes	characteristic
122	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
123	C₂²×C₈₉	#421	1	1	yes	characteristic
124	C₄×C₈₉	—	2	2	no	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	—	2	2	no	—
129	C₄×C₈₉	—	4	1	no	—
130	C₄×C₈₉	—	4	1	no	—
131	C₄×C₈₉	—	8	1	no	—
132	C₄×C₈₉	—	8	1	no	—
133	C₄×C₈₉	—	8	1	no	—
134	C₄×C₈₉	—	8	1	no	—
135	#7657	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₈₉	#1470	1	1	yes	characteristic
137	C₂×C₈₉	#1482	1	1	yes	characteristic
138	C₂×C₈₉	#1494	1	1	yes	characteristic
139	C₂×C₈₉	#1500	1	1	yes	characteristic
140	C₂×C₈₉	#1521	1	1	yes	characteristic
141	C₂×C₈₉	#1531	1	1	yes	characteristic
142	C₂×C₈₉	#1544	1	1	yes	characteristic
143	#2864	C₂×C₈₉	1	1	yes	characteristic
144	#1839	C₂×C₈₉	1	1	yes	characteristic
145	#1744	C₂×C₈₉	1	1	yes	characteristic
146	#1271	C₂×C₈₉	1	2	yes	—
147	#1271	C₂×C₈₉	1	2	yes	—
148	C₈₉	#7657	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×C₄×C₈	—	2	1	no	—
150	#389	C₄×C₈₉	1	1	yes	characteristic
151	#389	C₄×C₈₉	1	1	yes	characteristic
152	#389	C₂²×C₈₉	1	1	yes	characteristic
153	#389	C₂²×C₈₉	1	1	yes	characteristic
154	#387	—	2	1	no	—
155	#387	—	2	1	no	—
156	#387	—	2	1	no	—
157	#387	—	2	1	no	—
158	#382	C₄×C₈₉	1	1	yes	characteristic
159	#382	C₄×C₈₉	1	1	yes	characteristic
160	C₄³	C₂²×C₈₉	1	1	yes	characteristic
161	#336	—	2	1	no	—
162	#336	—	2	1	no	—
163	#336	—	2	1	no	—
164	#336	C₂²×C₈₉	1	2	yes	—
165	#336	C₂²×C₈₉	1	2	yes	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
183	C₄×C₈	—	2	4	no	—
184	#95	—	2	1	no	—
185	#95	—	2	1	no	—
186	#95	—	2	1	no	—
187	#95	—	2	1	no	—
188	#95	—	2	1	no	—
189	#95	—	2	1	no	—
190	#95	—	2	1	no	—
191	#95	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₂³	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	#101×C₈₉	1	2	yes	—
254	C₂×C₄	#104×C₈₉	1	2	yes	—
255	C₂×C₄	#131×C₈₉	1	2	yes	—
256	C₂×C₄	#135×C₈₉	1	2	yes	—
257	C₂×C₄	#135×C₈₉	1	2	yes	—
258	C₈	—	4	4	no	—
259	C₈	—	8	2	no	—
260	C₂²	#408×C₈₉	1	1	yes	characteristic
261	C₂²	#410×C₈₉	1	1	yes	characteristic
262	C₂²	C₂×#101×C₈₉	1	1	yes	characteristic
263	C₂²	#416×C₈₉	1	1	yes	characteristic
264	C₂²	#416×C₈₉	1	1	yes	characteristic
265	C₂²	C₂×#104×C₈₉	1	1	yes	characteristic
266	C₂²	#421×C₈₉	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	8	1	no	—
275	C₄	—	8	1	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	1	no	—
278	C₂	C₈₉×#1470	1	1	yes	characteristic
279	C₂	C₈₉×#1482	1	1	yes	characteristic
280	C₂	C₈₉×#1494	1	1	yes	characteristic
281	C₂	C₈₉×#1500	1	1	yes	characteristic
282	C₂	C₈₉×#1521	1	1	yes	characteristic
283	C₂	C₈₉×#1531	1	1	yes	characteristic
284	C₂	C₈₉×#1544	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
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
135. #7657	C₈₉	direct (N × H)
148. C₈₉	#7657	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×#2864	2
C₈₉×#1839	2
C₈₉×#1744	2
C₈₉×#1271	2
#7657	89

Subgroups — C89×#7657 #36385505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7657 #36385505