Subgroups — C₈₉×#29359 #36407207

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2642	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2331	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2074	C₂	1	2	yes	maximal
4	C₈₉×#2024	C₂	1	2	yes	maximal
5	C₈₉×#1989	C₂	1	2	yes	maximal
6	C₈₉×#1981	C₂	1	2	yes	maximal
7	C₈₉×#1935	C₂	1	2	yes	maximal
8	C₈₉×#1933	C₂	1	2	yes	maximal
9	C₈₉×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×#103×C₈₉	C₂²	1	2	yes	—
15	C₂×#103×C₈₉	C₂²	1	2	yes	—
16	C₂×#103×C₈₉	C₂²	1	2	yes	—
17	C₂×#103×C₈₉	C₂²	1	2	yes	—
18	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#400×C₈₉	—	2	2	no	—
21	#400×C₈₉	—	2	2	no	—
22	#391×C₈₉	—	2	2	no	—
23	#391×C₈₉	C₂²	1	2	yes	—
24	#391×C₈₉	C₂²	1	2	yes	—
25	#389×C₈₉	C₂²	1	2	yes	—
26	#387×C₈₉	—	2	2	no	—
27	#382×C₈₉	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
29	#376×C₈₉	C₂²	1	1	yes	characteristic
30	#340×C₈₉	C₂²	1	2	yes	—
31	#340×C₈₉	C₂²	1	2	yes	—
32	#340×C₈₉	C₂²	1	2	yes	—
33	#340×C₈₉	C₂²	1	2	yes	—
34	#336×C₈₉	C₂²	1	2	yes	—
35	C₂²×Q₈×C₈₉	—	2	1	no	—
36	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₈₉	—	2	4	no	—
40	C₄⋊Q₈×C₈₉	—	2	4	no	—
41	#125×C₈₉	—	2	4	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	2	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×C₄²×C₈₉	—	2	2	no	—
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	—	2	4	no	—
54	#103×C₈₉	—	2	4	no	—
55	#103×C₈₉	—	2	4	no	—
56	#103×C₈₉	—	4	2	no	—
57	#103×C₈₉	—	4	2	no	—
58	#97×C₈₉	—	4	1	no	—
59	C₄×C₈×C₈₉	—	4	1	no	—
60	#95×C₈₉	—	2	2	no	—
61	#95×C₈₉	—	2	1	no	—
62	#95×C₈₉	—	2	1	no	—
63	#95×C₈₉	—	2	2	no	—
64	#95×C₈₉	—	2	2	no	—
65	#95×C₈₉	—	2	4	no	—
66	#95×C₈₉	—	2	2	no	—
67	#95×C₈₉	D₄	1	2	yes	—
68	#95×C₈₉	D₄	1	2	yes	—
69	C₂×Q₈×C₈₉	—	2	4	no	—
70	C₂×Q₈×C₈₉	—	2	2	no	—
71	C₂×Q₈×C₈₉	—	2	2	no	—
72	C₂×Q₈×C₈₉	—	2	2	no	—
73	C₂×Q₈×C₈₉	—	2	2	no	—
74	C₂×Q₈×C₈₉	—	2	2	no	—
75	C₂×Q₈×C₈₉	—	2	2	no	—
76	C₂×Q₈×C₈₉	—	8	2	no	—
77	C₂²×C₄×C₈₉	—	2	2	no	—
78	C₂²×C₄×C₈₉	—	2	2	no	—
79	C₂²×C₄×C₈₉	—	2	2	no	—
80	C₂²×C₄×C₈₉	—	2	2	no	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₄⋊C₄×C₈₉	—	2	4	no	—
93	C₄⋊C₄×C₈₉	—	2	2	no	—
94	C₄⋊C₄×C₈₉	—	2	4	no	—
95	C₄⋊C₄×C₈₉	—	2	2	no	—
96	C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₄⋊C₄×C₈₉	—	4	4	no	—
98	C₄⋊C₄×C₈₉	—	4	4	no	—
99	C₄⋊C₄×C₈₉	—	8	2	no	—
100	C₄⋊C₄×C₈₉	#41	1	4	yes	—
101	C₄²×C₈₉	—	2	2	no	—
102	C₄²×C₈₉	—	2	1	no	—
103	C₄²×C₈₉	—	2	1	no	—
104	C₄²×C₈₉	—	8	2	no	—
105	C₄²×C₈₉	#41	1	4	yes	—
106	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₈₉	—	4	2	no	—
108	Q₈×C₈₉	—	4	2	no	—
109	Q₈×C₈₉	—	4	2	no	—
110	Q₈×C₈₉	—	4	2	no	—
111	Q₈×C₈₉	—	8	4	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	4	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	#124	1	4	yes	—
134	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
136	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
138	C₈×C₈₉	—	4	2	no	—
139	C₈×C₈₉	—	4	2	no	—
140	C₂²×C₈₉	#525	1	1	yes	characteristic
141	C₂²×C₈₉	C₂×#124	1	1	yes	characteristic
142	C₂²×C₈₉	#537	1	1	yes	characteristic
143	C₂²×C₈₉	#577	1	2	yes	—
144	C₂²×C₈₉	#578	1	2	yes	—
145	C₄×C₈₉	—	2	4	no	—
146	C₄×C₈₉	—	4	1	no	—
147	C₄×C₈₉	—	4	4	no	—
148	C₄×C₈₉	—	4	1	no	—
149	C₄×C₈₉	—	4	2	no	—
150	C₄×C₈₉	—	8	2	no	—
151	C₄×C₈₉	—	8	2	no	—
152	C₄×C₈₉	—	8	2	no	—
153	C₄×C₈₉	#556	1	2	yes	—
154	C₄×C₈₉	#559	1	2	yes	—
155	#29359	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₈₉	#2592	1	1	yes	characteristic
157	C₂×C₈₉	#3123	1	1	yes	characteristic
158	C₂×C₈₉	#3127	1	1	yes	characteristic
159	C₂×C₈₉	#3137	1	2	yes	—
160	C₂×C₈₉	#3258	1	2	yes	—
161	#2642	C₂×C₈₉	1	1	yes	characteristic
162	#2331	C₂×C₈₉	1	1	yes	characteristic
163	#2074	C₂×C₈₉	1	2	yes	—
164	#2024	C₂×C₈₉	1	2	yes	—
165	#1989	C₂×C₈₉	1	2	yes	—
166	#1981	C₂×C₈₉	1	2	yes	—
167	#1935	C₂×C₈₉	1	2	yes	—
168	#1933	C₂×C₈₉	1	2	yes	—
169	#1768	C₂×C₈₉	1	1	yes	characteristic
170	C₈₉	#29359	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
173	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
174	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
175	C₂×#103	C₂²×C₈₉	1	2	yes	—
176	C₂×#103	C₂²×C₈₉	1	2	yes	—
177	C₂×#103	C₂²×C₈₉	1	2	yes	—
178	C₂×#103	C₂²×C₈₉	1	2	yes	—
179	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
181	#400	—	2	2	no	—
182	#400	—	2	2	no	—
183	#391	—	2	2	no	—
184	#391	C₂²×C₈₉	1	2	yes	—
185	#391	C₂²×C₈₉	1	2	yes	—
186	#389	C₂²×C₈₉	1	2	yes	—
187	#387	—	2	2	no	—
188	#382	C₂²×C₈₉	1	1	yes	characteristic
189	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
190	#376	C₂²×C₈₉	1	1	yes	characteristic
191	#340	C₂²×C₈₉	1	2	yes	—
192	#340	C₂²×C₈₉	1	2	yes	—
193	#340	C₂²×C₈₉	1	2	yes	—
194	#340	C₂²×C₈₉	1	2	yes	—
195	#336	C₂²×C₈₉	1	2	yes	—
196	C₂²×Q₈	—	2	1	no	—
197	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
198	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
199	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	#125	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	#103	—	2	4	no	—
216	#103	—	2	4	no	—
217	#103	—	4	2	no	—
218	#103	—	4	2	no	—
219	#97	—	4	1	no	—
220	C₄×C₈	—	4	1	no	—
221	#95	—	2	2	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	4	no	—
227	#95	—	2	2	no	—
228	#95	D₄×C₈₉	1	2	yes	—
229	#95	D₄×C₈₉	1	2	yes	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	8	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	8	2	no	—
261	C₄⋊C₄	#41×C₈₉	1	4	yes	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	8	2	no	—
266	C₄²	#41×C₈₉	1	4	yes	—
267	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	8	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	#124×C₈₉	1	4	yes	—
295	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
296	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
297	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
298	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₂²	#525×C₈₉	1	1	yes	characteristic
302	C₂²	C₂×#124×C₈₉	1	1	yes	characteristic
303	C₂²	#537×C₈₉	1	1	yes	characteristic
304	C₂²	#577×C₈₉	1	2	yes	—
305	C₂²	#578×C₈₉	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	4	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	#556×C₈₉	1	2	yes	—
315	C₄	#559×C₈₉	1	2	yes	—
316	C₂	C₈₉×#2592	1	1	yes	characteristic
317	C₂	C₈₉×#3123	1	1	yes	characteristic
318	C₂	C₈₉×#3127	1	1	yes	characteristic
319	C₂	C₈₉×#3137	1	2	yes	—
320	C₂	C₈₉×#3258	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
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
155. #29359	C₈₉	direct (N × H)
170. C₈₉	#29359	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×#2642	2
C₈₉×#2331	2
C₈₉×#2074	2
C₈₉×#2024	2
C₈₉×#1989	2
C₈₉×#1981	2
C₈₉×#1935	2
C₈₉×#1933	2
C₈₉×#1768	2
#29359	89

Subgroups — C89×#29359 #36407207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29359 #36407207