Subgroups — C₈₉×#35346 #36413194

Order 22784.

7198 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#120×C₈₉	C₂	1	3	yes	maximal, complemented
3	C₈₉×#2404	C₂	1	12	yes	maximal, complemented
4	C₈₉×#2403	C₂	1	12	yes	maximal, complemented
5	C₈₉×#2308	C₂	1	3	yes	maximal, complemented
6	C₂⁶×C₈₉	C₂²	1	1	yes	characteristic, complemented
7	C₂×H(5,𝔽₂)×C₈₉	—	2	24	no	—
8	C₂²×#41×C₈₉	C₂²	1	6	yes	complemented
9	C₂³×D₄×C₈₉	C₂²	1	3	yes	—
10	C₂³×D₄×C₈₉	C₂²	1	6	yes	complemented
11	C₂³×D₄×C₈₉	C₂²	1	6	yes	complemented
12	C₂×#120×C₈₉	—	2	24	no	—
13	C₂×#120×C₈₉	C₂²	1	24	yes	complemented
14	C₂×#120×C₈₉	C₂²	1	8	yes	complemented
15	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	complemented
16	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	6	yes	complemented
17	#393×C₈₉	C₂²	1	8	yes	complemented
18	#392×C₈₉	C₂²	1	24	yes	complemented
19	#386×C₈₉	C₂²	1	24	yes	complemented
20	#386×C₈₉	C₂²	1	24	yes	complemented
21	#381×C₈₉	C₂²	1	6	yes	complemented
22	#379×C₈₉	C₂²	1	6	yes	complemented
23	C₂⁵×C₈₉	—	2	12	no	—
24	C₂⁵×C₈₉	—	4	8	no	—
25	C₂⁵×C₈₉	C₂³	1	6	yes	—
26	C₂⁵×C₈₉	C₂³	1	3	yes	—
27	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
28	C₂⁵×C₈₉	C₂³	1	2	yes	complemented
29	H(5,𝔽₂)×C₈₉	—	2	128	no	—
30	C₂×#41×C₈₉	—	2	48	no	—
31	C₂×#41×C₈₉	—	2	24	no	—
32	C₂²×Q₈×C₈₉	C₂³	1	2	yes	complemented
33	C₂²×D₄×C₈₉	—	2	24	no	—
34	C₂²×D₄×C₈₉	—	2	24	no	—
35	C₂²×D₄×C₈₉	—	2	48	no	—
36	C₂²×D₄×C₈₉	—	2	24	no	—
37	C₂²×D₄×C₈₉	—	4	24	no	—
38	C₂²×D₄×C₈₉	D₄	1	24	yes	complemented
39	C₂²×D₄×C₈₉	C₂³	1	24	yes	—
40	C₂²×D₄×C₈₉	C₂³	1	12	yes	—
41	C₂²×D₄×C₈₉	C₂³	1	6	yes	complemented
42	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
43	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
44	C₂³×C₄×C₈₉	C₂³	1	6	yes	complemented
45	#120×C₈₉	—	2	96	no	—
46	#120×C₈₉	—	4	16	no	—
47	C₂×C₂²⋊C₄×C₈₉	—	2	12	no	—
48	C₂×C₂²⋊C₄×C₈₉	—	2	24	no	—
49	C₂×C₂²⋊C₄×C₈₉	C₂³	1	12	yes	—
50	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
51	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	complemented
52	#95×C₈₉	C₂³	1	24	yes	complemented
53	C₂⁴×C₈₉	—	2	12	no	—
54	C₂⁴×C₈₉	—	2	24	no	—
55	C₂⁴×C₈₉	—	2	12	no	—
56	C₂⁴×C₈₉	—	2	12	no	—
57	C₂⁴×C₈₉	—	2	24	no	—
58	C₂⁴×C₈₉	—	4	4	no	—
59	C₂⁴×C₈₉	—	4	12	no	—
60	C₂⁴×C₈₉	—	4	12	no	—
61	C₂⁴×C₈₉	—	4	8	no	—
62	C₂⁴×C₈₉	—	4	96	no	—
63	C₂⁴×C₈₉	—	4	16	no	—
64	C₂⁴×C₈₉	C₂×D₄	1	12	yes	—
65	C₂⁴×C₈₉	C₂×D₄	1	12	yes	complemented
66	C₂⁴×C₈₉	C₂⁴	1	6	yes	—
67	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
68	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
69	#41×C₈₉	—	2	192	no	—
70	C₂×Q₈×C₈₉	—	2	12	no	—
71	C₂×D₄×C₈₉	—	2	192	no	—
72	C₂×D₄×C₈₉	—	2	96	no	—
73	C₂×D₄×C₈₉	—	2	24	no	—
74	C₂×D₄×C₈₉	—	2	24	no	—
75	C₂×D₄×C₈₉	—	2	24	no	—
76	C₂×D₄×C₈₉	—	2	48	no	—
77	C₂×D₄×C₈₉	—	2	48	no	—
78	C₂×D₄×C₈₉	—	2	12	no	—
79	C₂×D₄×C₈₉	—	4	24	no	—
80	C₂×D₄×C₈₉	—	4	96	no	—
81	C₂×D₄×C₈₉	C₂×D₄	1	48	yes	complemented
82	C₂²×C₄×C₈₉	—	2	12	no	—
83	C₂²×C₄×C₈₉	—	2	12	no	—
84	C₂²×C₄×C₈₉	—	2	24	no	—
85	C₂²×C₄×C₈₉	—	2	12	no	—
86	C₂²×C₄×C₈₉	—	4	12	no	—
87	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	complemented
88	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
89	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
90	C₂²⋊C₄×C₈₉	—	2	24	no	—
91	C₂²⋊C₄×C₈₉	—	2	48	no	—
92	C₂²⋊C₄×C₈₉	—	4	24	no	—
93	C₂³×C₈₉	—	2	12	no	—
94	C₂³×C₈₉	—	2	48	no	—
95	C₂³×C₈₉	—	2	16	no	—
96	C₂³×C₈₉	—	2	12	no	—
97	C₂³×C₈₉	—	2	12	no	—
98	C₂³×C₈₉	—	2	24	no	—
99	C₂³×C₈₉	—	2	12	no	—
100	C₂³×C₈₉	—	2	24	no	—
101	C₂³×C₈₉	—	4	24	no	—
102	C₂³×C₈₉	—	4	24	no	—
103	C₂³×C₈₉	—	4	96	no	—
104	C₂³×C₈₉	—	4	24	no	—
105	C₂³×C₈₉	—	4	128	no	—
106	C₂³×C₈₉	—	4	24	no	—
107	C₂³×C₈₉	—	8	24	no	—
108	C₂³×C₈₉	—	8	24	no	—
109	C₂³×C₈₉	#120	1	16	yes	complemented
110	C₂³×C₈₉	C₂²×D₄	1	12	yes	—
111	C₂³×C₈₉	C₂²×D₄	1	12	yes	—
112	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
113	C₂³×C₈₉	H(5,𝔽₂)	1	8	yes	complemented
114	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
115	Q₈×C₈₉	—	2	16	no	—
116	D₄×C₈₉	—	2	48	no	—
117	D₄×C₈₉	—	2	96	no	—
118	D₄×C₈₉	—	4	96	no	—
119	C₂×C₄×C₈₉	—	2	48	no	—
120	C₂×C₄×C₈₉	—	2	24	no	—
121	C₂×C₄×C₈₉	—	2	12	no	—
122	C₂×C₄×C₈₉	—	2	12	no	—
123	C₂×C₄×C₈₉	—	2	24	no	—
124	C₂×C₄×C₈₉	—	4	24	no	—
125	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
126	C₂²×C₈₉	—	2	12	no	—
127	C₂²×C₈₉	—	2	24	no	—
128	C₂²×C₈₉	—	2	12	no	—
129	C₂²×C₈₉	—	2	24	no	—
130	C₂²×C₈₉	—	2	24	no	—
131	C₂²×C₈₉	—	4	16	no	—
132	C₂²×C₈₉	—	4	96	no	—
133	C₂²×C₈₉	—	4	4	no	—
134	C₂²×C₈₉	—	4	12	no	—
135	C₂²×C₈₉	—	4	24	no	—
136	C₂²×C₈₉	—	8	24	no	—
137	C₂²×C₈₉	—	8	16	no	—
138	C₂²×C₈₉	C₂×#120	1	12	yes	—
139	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
140	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	4	yes	—
141	C₄×C₈₉	—	2	12	no	—
142	C₄×C₈₉	—	4	12	no	—
143	#35346	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₈₉	—	2	12	no	—
145	C₂×C₈₉	—	4	8	no	—
146	C₂×C₈₉	—	4	12	no	—
147	C₂×C₈₉	C₂²×#120	1	1	yes	characteristic
148	C₂×C₈₉	#3484	1	6	yes	—
149	C₂²×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic, complemented
150	C₂²×#120	C₂×C₈₉	1	3	yes	complemented
151	#2404	C₂×C₈₉	1	12	yes	complemented
152	#2403	C₂×C₈₉	1	12	yes	complemented
153	#2308	C₂×C₈₉	1	3	yes	complemented
154	C₈₉	#35346	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁶	C₂²×C₈₉	1	1	yes	characteristic, complemented
156	C₂×H(5,𝔽₂)	—	2	24	no	—
157	C₂²×#41	C₂²×C₈₉	1	6	yes	complemented
158	C₂³×D₄	C₂²×C₈₉	1	3	yes	—
159	C₂³×D₄	C₂²×C₈₉	1	6	yes	complemented
160	C₂³×D₄	C₂²×C₈₉	1	6	yes	complemented
161	C₂×#120	—	2	24	no	—
162	C₂×#120	C₂²×C₈₉	1	24	yes	complemented
163	C₂×#120	C₂²×C₈₉	1	8	yes	complemented
164	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	complemented
165	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	6	yes	complemented
166	#393	C₂²×C₈₉	1	8	yes	complemented
167	#392	C₂²×C₈₉	1	24	yes	complemented
168	#386	C₂²×C₈₉	1	24	yes	complemented
169	#386	C₂²×C₈₉	1	24	yes	complemented
170	#381	C₂²×C₈₉	1	6	yes	complemented
171	#379	C₂²×C₈₉	1	6	yes	complemented
172	C₂⁵	—	2	12	no	—
173	C₂⁵	—	4	8	no	—
174	C₂⁵	C₂³×C₈₉	1	6	yes	—
175	C₂⁵	C₂³×C₈₉	1	3	yes	—
176	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
177	C₂⁵	C₂³×C₈₉	1	2	yes	complemented
178	H(5,𝔽₂)	—	2	128	no	—
179	C₂×#41	—	2	48	no	—
180	C₂×#41	—	2	24	no	—
181	C₂²×Q₈	C₂³×C₈₉	1	2	yes	complemented
182	C₂²×D₄	—	2	24	no	—
183	C₂²×D₄	—	2	24	no	—
184	C₂²×D₄	—	2	48	no	—
185	C₂²×D₄	—	2	24	no	—
186	C₂²×D₄	—	4	24	no	—
187	C₂²×D₄	D₄×C₈₉	1	24	yes	complemented
188	C₂²×D₄	C₂³×C₈₉	1	24	yes	—
189	C₂²×D₄	C₂³×C₈₉	1	12	yes	—
190	C₂²×D₄	C₂³×C₈₉	1	6	yes	complemented
191	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
192	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
193	C₂³×C₄	C₂³×C₈₉	1	6	yes	complemented
194	#120	—	2	96	no	—
195	#120	—	4	16	no	—
196	C₂×C₂²⋊C₄	—	2	12	no	—
197	C₂×C₂²⋊C₄	—	2	24	no	—
198	C₂×C₂²⋊C₄	C₂³×C₈₉	1	12	yes	—
199	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
200	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	complemented
201	#95	C₂³×C₈₉	1	24	yes	complemented
202	C₂⁴	—	2	12	no	—
203	C₂⁴	—	2	24	no	—
204	C₂⁴	—	2	12	no	—
205	C₂⁴	—	2	12	no	—
206	C₂⁴	—	2	24	no	—
207	C₂⁴	—	4	4	no	—
208	C₂⁴	—	4	12	no	—
209	C₂⁴	—	4	12	no	—
210	C₂⁴	—	4	8	no	—
211	C₂⁴	—	4	96	no	—
212	C₂⁴	—	4	16	no	—
213	C₂⁴	C₂×D₄×C₈₉	1	12	yes	—
214	C₂⁴	C₂×D₄×C₈₉	1	12	yes	complemented
215	C₂⁴	C₂⁴×C₈₉	1	6	yes	—
216	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
217	C₂⁴	C₂⁴×C₈₉	1	3	yes	—
218	#41	—	2	192	no	—
219	C₂×Q₈	—	2	12	no	—
220	C₂×D₄	—	2	192	no	—
221	C₂×D₄	—	2	96	no	—
222	C₂×D₄	—	2	24	no	—
223	C₂×D₄	—	2	24	no	—
224	C₂×D₄	—	2	24	no	—
225	C₂×D₄	—	2	48	no	—
226	C₂×D₄	—	2	48	no	—
227	C₂×D₄	—	2	12	no	—
228	C₂×D₄	—	4	24	no	—
229	C₂×D₄	—	4	96	no	—
230	C₂×D₄	C₂×D₄×C₈₉	1	48	yes	complemented
231	C₂²×C₄	—	2	12	no	—
232	C₂²×C₄	—	2	12	no	—
233	C₂²×C₄	—	2	24	no	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	—	4	12	no	—
236	C₂²×C₄	C₂×D₄×C₈₉	1	12	yes	complemented
237	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
238	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
239	C₂²⋊C₄	—	2	24	no	—
240	C₂²⋊C₄	—	2	48	no	—
241	C₂²⋊C₄	—	4	24	no	—
242	C₂³	—	2	12	no	—
243	C₂³	—	2	48	no	—
244	C₂³	—	2	16	no	—
245	C₂³	—	2	12	no	—
246	C₂³	—	2	12	no	—
247	C₂³	—	2	24	no	—
248	C₂³	—	2	12	no	—
249	C₂³	—	2	24	no	—
250	C₂³	—	4	24	no	—
251	C₂³	—	4	24	no	—
252	C₂³	—	4	96	no	—
253	C₂³	—	4	24	no	—
254	C₂³	—	4	128	no	—
255	C₂³	—	4	24	no	—
256	C₂³	—	8	24	no	—
257	C₂³	—	8	24	no	—
258	C₂³	#120×C₈₉	1	16	yes	complemented
259	C₂³	C₂²×D₄×C₈₉	1	12	yes	—
260	C₂³	C₂²×D₄×C₈₉	1	12	yes	—
261	C₂³	C₂²×D₄×C₈₉	1	6	yes	—
262	C₂³	H(5,𝔽₂)×C₈₉	1	8	yes	complemented
263	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
264	Q₈	—	2	16	no	—
265	D₄	—	2	48	no	—
266	D₄	—	2	96	no	—
267	D₄	—	4	96	no	—
268	C₂×C₄	—	2	48	no	—
269	C₂×C₄	—	2	24	no	—
270	C₂×C₄	—	2	12	no	—
271	C₂×C₄	—	2	12	no	—
272	C₂×C₄	—	2	24	no	—
273	C₂×C₄	—	4	24	no	—
274	C₂×C₄	C₂²×D₄×C₈₉	1	12	yes	—
275	C₂²	—	2	12	no	—
276	C₂²	—	2	24	no	—
277	C₂²	—	2	12	no	—
278	C₂²	—	2	24	no	—
279	C₂²	—	2	24	no	—
280	C₂²	—	4	16	no	—
281	C₂²	—	4	96	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	12	no	—
284	C₂²	—	4	24	no	—
285	C₂²	—	8	24	no	—
286	C₂²	—	8	16	no	—
287	C₂²	C₂×#120×C₈₉	1	12	yes	—
288	C₂²	C₂³×D₄×C₈₉	1	3	yes	—
289	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	4	yes	—
290	C₄	—	2	12	no	—
291	C₄	—	4	12	no	—
292	C₂	—	2	12	no	—
293	C₂	—	4	8	no	—
294	C₂	—	4	12	no	—
295	C₂	C₂²×#120×C₈₉	1	1	yes	characteristic
296	C₂	C₈₉×#3484	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
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
1. C₂²×H(5,𝔽₂)×C₈₉	C₂	semidirect
2. C₂²×#120×C₈₉	C₂	semidirect
3. C₈₉×#2404	C₂	semidirect
4. C₈₉×#2403	C₂	semidirect
5. C₈₉×#2308	C₂	semidirect
6. C₂⁶×C₈₉	C₂²	semidirect
8. C₂²×#41×C₈₉	C₂²	semidirect
10. C₂³×D₄×C₈₉	C₂²	semidirect
11. C₂³×D₄×C₈₉	C₂²	semidirect
13. C₂×#120×C₈₉	C₂²	semidirect
14. C₂×#120×C₈₉	C₂²	semidirect
15. C₂²×C₂²⋊C₄×C₈₉	C₂²	semidirect
16. C₂²×C₂²⋊C₄×C₈₉	C₂²	semidirect
17. #393×C₈₉	C₂²	semidirect
18. #392×C₈₉	C₂²	semidirect
19. #386×C₈₉	C₂²	semidirect
20. #386×C₈₉	C₂²	semidirect
21. #381×C₈₉	C₂²	semidirect
22. #379×C₈₉	C₂²	semidirect
28. C₂⁵×C₈₉	C₂³	semidirect
32. C₂²×Q₈×C₈₉	C₂³	semidirect
38. C₂²×D₄×C₈₉	D₄	semidirect
41. C₂²×D₄×C₈₉	C₂³	semidirect
44. C₂³×C₄×C₈₉	C₂³	semidirect
51. C₂×C₂²⋊C₄×C₈₉	C₂³	semidirect
52. #95×C₈₉	C₂³	semidirect
65. C₂⁴×C₈₉	C₂×D₄	semidirect
81. C₂×D₄×C₈₉	C₂×D₄	semidirect
87. C₂²×C₄×C₈₉	C₂×D₄	semidirect
109. C₂³×C₈₉	#120	semidirect
113. C₂³×C₈₉	H(5,𝔽₂)	semidirect
143. #35346	C₈₉	direct (N × H)
149. C₂²×H(5,𝔽₂)	C₂×C₈₉	semidirect
150. C₂²×#120	C₂×C₈₉	semidirect
151. #2404	C₂×C₈₉	semidirect
152. #2403	C₂×C₈₉	semidirect
153. #2308	C₂×C₈₉	semidirect
154. C₈₉	#35346	direct (N × H)
155. C₂⁶	C₂²×C₈₉	semidirect
157. C₂²×#41	C₂²×C₈₉	semidirect
159. C₂³×D₄	C₂²×C₈₉	semidirect
160. C₂³×D₄	C₂²×C₈₉	semidirect
162. C₂×#120	C₂²×C₈₉	semidirect
163. C₂×#120	C₂²×C₈₉	semidirect
164. C₂²×C₂²⋊C₄	C₂²×C₈₉	semidirect
165. C₂²×C₂²⋊C₄	C₂²×C₈₉	semidirect
166. #393	C₂²×C₈₉	semidirect
167. #392	C₂²×C₈₉	semidirect
168. #386	C₂²×C₈₉	semidirect
169. #386	C₂²×C₈₉	semidirect
170. #381	C₂²×C₈₉	semidirect
171. #379	C₂²×C₈₉	semidirect
177. C₂⁵	C₂³×C₈₉	semidirect
181. C₂²×Q₈	C₂³×C₈₉	semidirect
187. C₂²×D₄	D₄×C₈₉	semidirect
190. C₂²×D₄	C₂³×C₈₉	semidirect
193. C₂³×C₄	C₂³×C₈₉	semidirect
200. C₂×C₂²⋊C₄	C₂³×C₈₉	semidirect
201. #95	C₂³×C₈₉	semidirect
214. C₂⁴	C₂×D₄×C₈₉	semidirect
230. C₂×D₄	C₂×D₄×C₈₉	semidirect
236. C₂²×C₄	C₂×D₄×C₈₉	semidirect
258. C₂³	#120×C₈₉	semidirect
262. C₂³	H(5,𝔽₂)×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₈₉	2
C₂²×#120×C₈₉	2
C₈₉×#2404	2
C₈₉×#2403	2
C₈₉×#2308	2
#35346	89

Subgroups — C89×#35346 #36413194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35346 #36413194