Subgroups — C₈₉×#29356 #36407204

Order 22784.

798 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2858	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2073	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2025	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1988	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1982	C₂	1	2	yes	maximal
7	C₈₉×#1936	C₂	1	4	yes	maximal, complemented
8	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×#124×C₈₉	—	2	2	no	—
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×#102×C₈₉	C₂²	1	4	yes	—
15	C₂×#102×C₈₉	C₂²	1	4	yes	—
16	C₂×#97×C₈₉	C₂²	1	2	yes	—
17	#390×C₈₉	—	2	2	no	—
18	#390×C₈₉	C₂²	1	2	yes	—
19	#390×C₈₉	C₂²	1	2	yes	—
20	#389×C₈₉	C₂²	1	2	yes	—
21	#387×C₈₉	—	2	2	no	—
22	#382×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
24	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
25	C₄³×C₈₉	C₂²	1	1	yes	characteristic
26	#340×C₈₉	C₂²	1	4	yes	—
27	#340×C₈₉	C₂²	1	4	yes	—
28	#336×C₈₉	C₂²	1	2	yes	—
29	C₂²×Q₈×C₈₉	—	2	1	no	—
30	C₂²×D₄×C₈₉	—	2	1	no	—
31	C₂²×D₄×C₈₉	—	2	1	no	—
32	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₈₉	—	2	4	no	—
35	#127×C₈₉	—	2	4	no	—
36	#125×C₈₉	—	2	4	no	—
37	#124×C₈₉	—	4	8	no	—
38	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
46	C₂×C₄²×C₈₉	—	2	2	no	—
47	C₂×C₄²×C₈₉	D₄	1	2	yes	—
48	C₂×C₄²×C₈₉	D₄	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₈₉	—	2	4	no	—
53	#102×C₈₉	—	2	8	no	—
54	#102×C₈₉	—	4	4	no	—
55	#97×C₈₉	—	4	2	no	—
56	#95×C₈₉	—	2	2	no	—
57	#95×C₈₉	—	2	2	no	—
58	#95×C₈₉	—	2	2	no	—
59	C₂⁴×C₈₉	—	2	2	no	—
60	C₂×Q₈×C₈₉	—	4	2	no	—
61	C₂×Q₈×C₈₉	—	4	4	no	—
62	C₂×D₄×C₈₉	—	2	4	no	—
63	C₂×D₄×C₈₉	—	2	4	no	—
64	C₂×D₄×C₈₉	—	2	2	no	—
65	C₂×D₄×C₈₉	—	2	2	no	—
66	C₂×D₄×C₈₉	—	4	4	no	—
67	C₂×D₄×C₈₉	—	4	2	no	—
68	C₂×D₄×C₈₉	—	4	2	no	—
69	C₂×D₄×C₈₉	—	4	4	no	—
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	4	no	—
79	C₂×C₈×C₈₉	—	2	4	no	—
80	C₂×C₈×C₈₉	—	2	4	no	—
81	C₄⋊C₄×C₈₉	—	2	8	no	—
82	C₄⋊C₄×C₈₉	—	2	2	no	—
83	C₄⋊C₄×C₈₉	—	2	4	no	—
84	C₄⋊C₄×C₈₉	—	2	2	no	—
85	C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₄⋊C₄×C₈₉	—	4	4	no	—
87	C₄⋊C₄×C₈₉	—	4	2	no	—
88	C₄⋊C₄×C₈₉	#41	1	4	yes	—
89	C₂²⋊C₄×C₈₉	—	4	8	no	—
90	C₂²⋊C₄×C₈₉	—	4	8	no	—
91	C₄²×C₈₉	—	2	4	no	—
92	C₄²×C₈₉	—	2	2	no	—
93	C₄²×C₈₉	—	2	1	no	—
94	C₄²×C₈₉	—	2	1	no	—
95	C₄²×C₈₉	—	4	2	no	—
96	C₄²×C₈₉	—	4	4	no	—
97	C₄²×C₈₉	#41	1	4	yes	—
98	C₂³×C₈₉	—	4	2	no	—
99	C₂³×C₈₉	—	4	2	no	—
100	C₂³×C₈₉	—	4	2	no	—
101	C₂³×C₈₉	—	4	4	no	—
102	C₂³×C₈₉	—	4	4	no	—
103	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₈₉	—	8	4	no	—
105	D₄×C₈₉	—	4	4	no	—
106	D₄×C₈₉	—	4	4	no	—
107	D₄×C₈₉	—	8	4	no	—
108	D₄×C₈₉	—	8	4	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	4	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	#124	1	2	yes	—
125	C₂×C₄×C₈₉	#124	1	2	yes	—
126	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
127	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
128	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
129	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
131	C₈×C₈₉	—	4	4	no	—
132	C₂²×C₈₉	—	8	2	no	—
133	C₂²×C₈₉	—	8	4	no	—
134	C₂²×C₈₉	—	8	2	no	—
135	C₂²×C₈₉	—	8	2	no	—
136	C₂²×C₈₉	—	8	4	no	—
137	C₂²×C₈₉	#525	1	1	yes	characteristic
138	C₂²×C₈₉	C₂×#124	1	1	yes	characteristic
139	C₂²×C₈₉	#537	1	1	yes	characteristic
140	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
141	C₂²×C₈₉	#575	1	2	yes	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	4	2	no	—
144	C₄×C₈₉	—	4	4	no	—
145	C₄×C₈₉	—	4	2	no	—
146	C₄×C₈₉	—	8	2	no	—
147	C₄×C₈₉	—	8	2	no	—
148	C₄×C₈₉	#490	1	4	yes	—
149	C₄×C₈₉	#556	1	2	yes	—
150	C₄×C₈₉	#559	1	2	yes	—
151	#29356	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₈₉	—	16	2	no	—
153	C₂×C₈₉	#2592	1	1	yes	characteristic
154	C₂×C₈₉	#3115	1	2	yes	—
155	C₂×C₈₉	C₂×#490	1	1	yes	characteristic
156	C₂×C₈₉	#3140	1	1	yes	characteristic
157	C₂×C₈₉	#3251	1	2	yes	—
158	#2858	C₂×C₈₉	1	1	yes	characteristic
159	C₄×#125	C₂×C₈₉	1	1	yes	characteristic, complemented
160	#2073	C₂×C₈₉	1	2	yes	complemented
161	#2025	C₂×C₈₉	1	2	yes	complemented
162	#1988	C₂×C₈₉	1	2	yes	complemented
163	#1982	C₂×C₈₉	1	2	yes	—
164	#1936	C₂×C₈₉	1	4	yes	complemented
165	#1767	C₂×C₈₉	1	1	yes	characteristic, complemented
166	C₈₉	#29356	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
168	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
169	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
170	C₂×#124	—	2	2	no	—
171	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
172	C₂×#102	C₂²×C₈₉	1	4	yes	—
173	C₂×#102	C₂²×C₈₉	1	4	yes	—
174	C₂×#97	C₂²×C₈₉	1	2	yes	—
175	#390	—	2	2	no	—
176	#390	C₂²×C₈₉	1	2	yes	—
177	#390	C₂²×C₈₉	1	2	yes	—
178	#389	C₂²×C₈₉	1	2	yes	—
179	#387	—	2	2	no	—
180	#382	C₂²×C₈₉	1	1	yes	characteristic
181	C₄×C₄⋊C₄	—	2	2	no	—
182	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
183	C₄³	C₂²×C₈₉	1	1	yes	characteristic
184	#340	C₂²×C₈₉	1	4	yes	—
185	#340	C₂²×C₈₉	1	4	yes	—
186	#336	C₂²×C₈₉	1	2	yes	—
187	C₂²×Q₈	—	2	1	no	—
188	C₂²×D₄	—	2	1	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
191	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
192	C₄⋊Q₈	—	2	4	no	—
193	#127	—	2	4	no	—
194	#125	—	2	4	no	—
195	#124	—	4	8	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	D₄×C₈₉	1	2	yes	—
206	C₂×C₄²	D₄×C₈₉	1	2	yes	—
207	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
210	C₄⋊C₈	—	2	4	no	—
211	#102	—	2	8	no	—
212	#102	—	4	4	no	—
213	#97	—	4	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	C₂⁴	—	2	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	#41×C₈₉	1	4	yes	—
247	C₂²⋊C₄	—	4	8	no	—
248	C₂²⋊C₄	—	4	8	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	4	no	—
255	C₄²	#41×C₈₉	1	4	yes	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
262	Q₈	—	8	4	no	—
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	4	no	—
266	D₄	—	8	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	#124×C₈₉	1	2	yes	—
283	C₂×C₄	#124×C₈₉	1	2	yes	—
284	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
285	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
286	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
287	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
288	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
289	C₈	—	4	4	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	4	no	—
295	C₂²	#525×C₈₉	1	1	yes	characteristic
296	C₂²	C₂×#124×C₈₉	1	1	yes	characteristic
297	C₂²	#537×C₈₉	1	1	yes	characteristic
298	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
299	C₂²	#575×C₈₉	1	2	yes	—
300	C₄	—	2	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	#490×C₈₉	1	4	yes	—
307	C₄	#556×C₈₉	1	2	yes	—
308	C₄	#559×C₈₉	1	2	yes	—
309	C₂	—	16	2	no	—
310	C₂	C₈₉×#2592	1	1	yes	characteristic
311	C₂	C₈₉×#3115	1	2	yes	—
312	C₂	C₂×#490×C₈₉	1	1	yes	characteristic
313	C₂	C₈₉×#3140	1	1	yes	characteristic
314	C₂	C₈₉×#3251	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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. C₄×#125×C₈₉	C₂	semidirect
3. C₈₉×#2073	C₂	semidirect
4. C₈₉×#2025	C₂	semidirect
5. C₈₉×#1988	C₂	semidirect
7. C₈₉×#1936	C₂	semidirect
8. C₈₉×#1767	C₂	semidirect
151. #29356	C₈₉	direct (N × H)
159. C₄×#125	C₂×C₈₉	semidirect
160. #2073	C₂×C₈₉	semidirect
161. #2025	C₂×C₈₉	semidirect
162. #1988	C₂×C₈₉	semidirect
164. #1936	C₂×C₈₉	semidirect
165. #1767	C₂×C₈₉	semidirect
166. C₈₉	#29356	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×#2858	2
C₄×#125×C₈₉	2
C₈₉×#2073	2
C₈₉×#2025	2
C₈₉×#1988	2
C₈₉×#1982	2
C₈₉×#1936	2
C₈₉×#1767	2
#29356	89

Subgroups — C89×#29356 #36407204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29356 #36407204