Subgroups — C₈₉×#60325 #36438173

Order 22784.

2130 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2991	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2989	C₂	1	3	yes	maximal, complemented
4	C₈₉×#2988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2975	C₂	1	3	yes	maximal, complemented
6	C₈₉×#2974	C₂	1	6	yes	maximal, complemented
7	C₈₉×#2971	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2965	C₂	1	3	yes	maximal, complemented
9	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×M₁₆×C₈₉	C₂	1	6	yes	maximal, complemented
11	C₂×#436×C₈₉	C₂	1	3	yes	maximal, complemented
12	C₂×H(5,𝔽₂)×C₈₉	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₈₉	C₂²	1	6	yes	complemented
14	#520×C₈₉	C₄	1	2	yes	—
15	#520×C₈₉	C₄	1	3	yes	—
16	#520×C₈₉	C₄	1	1	yes	characteristic
17	#518×C₈₉	C₄	1	6	yes	—
18	#518×C₈₉	C₄	1	3	yes	—
19	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×#41×C₈₉	C₂²	1	3	yes	—
21	C₄×#41×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₈₉	C₂²	1	3	yes	—
23	C₂×C₄×D₄×C₈₉	C₂²	1	6	yes	—
24	#446×C₈₉	C₂²	1	1	yes	characteristic, complemented
25	#446×C₈₉	C₂²	1	3	yes	complemented
26	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
28	C₈×Q₈×C₈₉	C₂²	1	2	yes	complemented
29	#436×C₈₉	C₂²	1	3	yes	complemented
30	#436×C₈₉	C₂²	1	6	yes	complemented
31	#436×C₈₉	C₂²	1	6	yes	complemented
32	#436×C₈₉	C₂²	1	6	yes	complemented
33	#436×C₈₉	C₂²	1	6	yes	complemented
34	#436×C₈₉	C₂²	1	3	yes	complemented
35	#436×C₈₉	C₂²	1	6	yes	complemented
36	#435×C₈₉	C₂²	1	6	yes	complemented
37	#435×C₈₉	C₂²	1	12	yes	complemented
38	#435×C₈₉	C₂²	1	6	yes	complemented
39	C₈×D₄×C₈₉	C₂²	1	3	yes	complemented
40	C₈×D₄×C₈₉	C₂²	1	6	yes	complemented
41	C₈×D₄×C₈₉	C₂²	1	3	yes	complemented
42	#432×C₈₉	C₂²	1	6	yes	complemented
43	#431×C₈₉	C₂²	1	3	yes	complemented
44	#431×C₈₉	C₂²	1	6	yes	complemented
45	#431×C₈₉	C₂²	1	3	yes	complemented
46	#423×C₈₉	C₂²	1	6	yes	complemented
47	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	complemented
48	#407×C₈₉	C₂²	1	12	yes	complemented
49	C₂×C₂²⋊C₈×C₈₉	C₂²	1	6	yes	complemented
50	C₄×M₁₆×C₈₉	C₂²	1	6	yes	complemented
51	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	complemented
52	H(5,𝔽₂)×C₈₉	—	2	8	no	—
53	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
54	C₂×#41×C₈₉	C₂×C₄	1	3	yes	—
55	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₄×C₈₉	C₂×C₄	1	6	yes	—
57	C₂²×D₄×C₈₉	C₂×C₄	1	3	yes	—
58	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
59	C₂×M₁₆×C₈₉	—	4	12	no	—
60	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
61	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
62	C₂×M₁₆×C₈₉	C₂³	1	12	yes	—
63	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
64	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
65	C₂²×C₈×C₈₉	C₂³	1	6	yes	—
66	C₄×Q₈×C₈₉	C₂×C₄	1	3	yes	—
67	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
69	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
72	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
73	C₄×D₄×C₈₉	C₂×C₄	1	12	yes	—
74	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
75	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
76	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
77	C₄×D₄×C₈₉	C₂×C₄	1	3	yes	—
78	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
79	C₄×D₄×C₈₉	C₂×C₄	1	3	yes	—
80	C₄×D₄×C₈₉	C₂³	1	3	yes	—
81	C₄×D₄×C₈₉	C₂³	1	6	yes	—
82	C₄×D₄×C₈₉	C₂³	1	3	yes	—
83	C₄×D₄×C₈₉	C₂³	1	6	yes	—
84	#117×C₈₉	C₂×C₄	1	6	yes	—
85	#117×C₈₉	C₂×C₄	1	6	yes	—
86	#117×C₈₉	C₂×C₄	1	3	yes	—
87	#117×C₈₉	C₂×C₄	1	3	yes	—
88	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	6	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
90	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	6	yes	—
91	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	12	yes	—
92	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
93	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
94	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
95	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
96	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
97	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
98	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
99	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
101	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
102	C₂²⋊C₈×C₈₉	C₂³	1	12	yes	—
103	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
104	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
105	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
106	#97×C₈₉	C₂³	1	2	yes	—
107	#97×C₈₉	C₂³	1	6	yes	—
108	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₈₉	C₂³	1	3	yes	—
110	C₄×C₈×C₈₉	C₂³	1	3	yes	—
111	C₄×C₈×C₈₉	C₂³	1	3	yes	—
112	C₄×C₈×C₈₉	C₂³	1	6	yes	—
113	C₂⁴×C₈₉	C₂²×C₄	1	6	yes	—
114	#41×C₈₉	—	2	8	no	—
115	#41×C₈₉	—	2	12	no	—
116	#41×C₈₉	M₁₆	1	8	yes	complemented
117	C₂×Q₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×Q₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×D₄×C₈₉	—	2	24	no	—
120	C₂×D₄×C₈₉	—	2	12	no	—
121	C₂×D₄×C₈₉	C₂²×C₄	1	6	yes	—
122	C₂×D₄×C₈₉	C₂²×C₄	1	6	yes	—
123	C₂×D₄×C₈₉	C₂²×C₄	1	3	yes	—
124	C₂×D₄×C₈₉	C₂²×C₄	1	3	yes	—
125	C₂²×C₄×C₈₉	—	4	6	no	—
126	C₂²×C₄×C₈₉	—	4	6	no	—
127	C₂²×C₄×C₈₉	C₂²×C₄	1	6	yes	—
128	C₂²×C₄×C₈₉	C₂²×C₄	1	6	yes	—
129	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
130	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
131	C₂²×C₄×C₈₉	C₂²×C₄	1	6	yes	—
132	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
133	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
134	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
135	M₁₆×C₈₉	—	2	12	no	—
136	M₁₆×C₈₉	—	4	12	no	—
137	M₁₆×C₈₉	—	4	6	no	—
138	C₂×C₈×C₈₉	—	2	6	no	—
139	C₂×C₈×C₈₉	—	4	3	no	—
140	C₂×C₈×C₈₉	—	4	6	no	—
141	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
142	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
143	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
144	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
145	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
146	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
147	C₄⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
148	C₄⋊C₄×C₈₉	C₂²×C₄	1	3	yes	—
149	C₄⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
150	C₄⋊C₄×C₈₉	C₂²×C₄	1	3	yes	—
151	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
152	C₄⋊C₄×C₈₉	C₂²×C₄	1	3	yes	—
153	C₄⋊C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
154	C₂²⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
155	C₂²⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
156	C₂²⋊C₄×C₈₉	C₂²×C₄	1	12	yes	—
157	C₂²⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
158	C₂²⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
159	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
160	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
161	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
162	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
163	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
164	C₄²×C₈₉	C₂⁴	1	3	yes	—
165	C₄²×C₈₉	C₂⁴	1	2	yes	—
166	C₂³×C₈₉	—	2	12	no	—
167	C₂³×C₈₉	—	4	6	no	—
168	C₂³×C₈₉	—	4	6	no	—
169	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
170	C₂³×C₈₉	C₂³×C₄	1	3	yes	—
171	Q₈×C₈₉	—	2	2	no	—
172	Q₈×C₈₉	C₂×M₁₆	1	4	yes	complemented
173	D₄×C₈₉	—	2	12	no	—
174	D₄×C₈₉	—	2	12	no	—
175	D₄×C₈₉	—	2	6	no	—
176	D₄×C₈₉	C₂×M₁₆	1	12	yes	complemented
177	C₂×C₄×C₈₉	—	2	12	no	—
178	C₂×C₄×C₈₉	—	2	3	no	—
179	C₂×C₄×C₈₉	—	2	6	no	—
180	C₂×C₄×C₈₉	—	2	3	no	—
181	C₂×C₄×C₈₉	—	2	6	no	—
182	C₂×C₄×C₈₉	—	4	6	no	—
183	C₂×C₄×C₈₉	—	4	12	no	—
184	C₂×C₄×C₈₉	—	4	3	no	—
185	C₂×C₄×C₈₉	—	4	6	no	—
186	C₂×C₄×C₈₉	C₂×M₁₆	1	12	yes	complemented
187	C₂×C₄×C₈₉	C₂³×C₄	1	3	yes	—
188	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
190	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
191	C₂×C₄×C₈₉	C₂³×C₄	1	3	yes	—
192	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
193	C₂×C₄×C₈₉	C₂³×C₄	1	3	yes	—
194	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
196	C₈×C₈₉	—	2	3	no	—
197	C₈×C₈₉	—	2	6	no	—
198	C₈×C₈₉	—	2	1	no	—
199	C₈×C₈₉	—	2	3	no	—
200	C₈×C₈₉	—	2	2	no	—
201	C₈×C₈₉	H(5,𝔽₂)	1	2	yes	complemented
202	C₂²×C₈₉	—	2	3	no	—
203	C₂²×C₈₉	—	2	3	no	—
204	C₂²×C₈₉	—	2	6	no	—
205	C₂²×C₈₉	—	2	6	no	—
206	C₂²×C₈₉	—	2	6	no	—
207	C₂²×C₈₉	—	8	12	no	—
208	C₂²×C₈₉	C₂²×M₁₆	1	6	yes	—
209	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
210	C₄×C₈₉	—	2	3	no	—
211	C₄×C₈₉	—	2	6	no	—
212	C₄×C₈₉	—	2	2	no	—
213	C₄×C₈₉	—	2	1	no	—
214	C₄×C₈₉	—	2	3	no	—
215	C₄×C₈₉	—	2	2	no	—
216	C₄×C₈₉	C₂²×M₁₆	1	6	yes	—
217	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
218	C₄×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
219	C₄×C₈₉	#585	1	1	yes	characteristic
220	#60325	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₈₉	—	2	6	no	—
222	C₂×C₈₉	—	4	6	no	—
223	C₂×C₈₉	C₄×H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₈₉	C₂³×M₁₆	1	1	yes	characteristic
225	C₂×C₈₉	#3573	1	1	yes	characteristic
226	C₄×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic
227	#2991	C₂×C₈₉	1	2	yes	complemented
228	#2989	C₂×C₈₉	1	3	yes	complemented
229	#2988	C₂×C₈₉	1	1	yes	characteristic, complemented
230	#2975	C₂×C₈₉	1	3	yes	complemented
231	#2974	C₂×C₈₉	1	6	yes	complemented
232	#2971	C₂×C₈₉	1	2	yes	complemented
233	#2965	C₂×C₈₉	1	3	yes	complemented
234	C₈×#41	C₂×C₈₉	1	1	yes	characteristic, complemented
235	D₄×M₁₆	C₂×C₈₉	1	6	yes	complemented
236	C₂×#436	C₂×C₈₉	1	3	yes	complemented
237	C₈₉	#60325	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×H(5,𝔽₂)	C₄×C₈₉	1	1	yes	characteristic
239	C₂²×M₁₆	C₂²×C₈₉	1	6	yes	complemented
240	#520	C₄×C₈₉	1	2	yes	—
241	#520	C₄×C₈₉	1	3	yes	—
242	#520	C₄×C₈₉	1	1	yes	characteristic
243	#518	C₄×C₈₉	1	6	yes	—
244	#518	C₄×C₈₉	1	3	yes	—
245	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
246	C₄×#41	C₂²×C₈₉	1	3	yes	—
247	C₄×#41	C₂²×C₈₉	1	2	yes	—
248	C₂×C₄×D₄	C₂²×C₈₉	1	3	yes	—
249	C₂×C₄×D₄	C₂²×C₈₉	1	6	yes	—
250	#446	C₂²×C₈₉	1	1	yes	characteristic, complemented
251	#446	C₂²×C₈₉	1	3	yes	complemented
252	C₈×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
253	C₈×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
254	C₈×Q₈	C₂²×C₈₉	1	2	yes	complemented
255	#436	C₂²×C₈₉	1	3	yes	complemented
256	#436	C₂²×C₈₉	1	6	yes	complemented
257	#436	C₂²×C₈₉	1	6	yes	complemented
258	#436	C₂²×C₈₉	1	6	yes	complemented
259	#436	C₂²×C₈₉	1	6	yes	complemented
260	#436	C₂²×C₈₉	1	3	yes	complemented
261	#436	C₂²×C₈₉	1	6	yes	complemented
262	#435	C₂²×C₈₉	1	6	yes	complemented
263	#435	C₂²×C₈₉	1	12	yes	complemented
264	#435	C₂²×C₈₉	1	6	yes	complemented
265	C₈×D₄	C₂²×C₈₉	1	3	yes	complemented
266	C₈×D₄	C₂²×C₈₉	1	6	yes	complemented
267	C₈×D₄	C₂²×C₈₉	1	3	yes	complemented
268	#432	C₂²×C₈₉	1	6	yes	complemented
269	#431	C₂²×C₈₉	1	3	yes	complemented
270	#431	C₂²×C₈₉	1	6	yes	complemented
271	#431	C₂²×C₈₉	1	3	yes	complemented
272	#423	C₂²×C₈₉	1	6	yes	complemented
273	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	complemented
274	#407	C₂²×C₈₉	1	12	yes	complemented
275	C₂×C₂²⋊C₈	C₂²×C₈₉	1	6	yes	complemented
276	C₄×M₁₆	C₂²×C₈₉	1	6	yes	complemented
277	C₂×C₄×C₈	C₂²×C₈₉	1	3	yes	complemented
278	H(5,𝔽₂)	—	2	8	no	—
279	C₂×#41	C₂×C₄×C₈₉	1	2	yes	—
280	C₂×#41	C₂×C₄×C₈₉	1	3	yes	—
281	C₂×#41	C₂×C₄×C₈₉	1	1	yes	characteristic
282	C₂²×D₄	C₂×C₄×C₈₉	1	6	yes	—
283	C₂²×D₄	C₂×C₄×C₈₉	1	3	yes	—
284	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
285	C₂×M₁₆	—	4	12	no	—
286	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
287	C₂×M₁₆	C₂³×C₈₉	1	12	yes	—
288	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
289	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
290	C₂²×C₈	C₂³×C₈₉	1	6	yes	—
291	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
292	C₄×Q₈	C₂×C₄×C₈₉	1	3	yes	—
293	C₄×Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
294	C₄×Q₈	C₂×C₄×C₈₉	1	2	yes	—
295	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
296	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
297	C₄×D₄	C₂×C₄×C₈₉	1	6	yes	—
298	C₄×D₄	C₂×C₄×C₈₉	1	6	yes	—
299	C₄×D₄	C₂×C₄×C₈₉	1	12	yes	—
300	C₄×D₄	C₂×C₄×C₈₉	1	6	yes	—
301	C₄×D₄	C₂×C₄×C₈₉	1	6	yes	—
302	C₄×D₄	C₂×C₄×C₈₉	1	6	yes	—
303	C₄×D₄	C₂×C₄×C₈₉	1	3	yes	—
304	C₄×D₄	C₂×C₄×C₈₉	1	6	yes	—
305	C₄×D₄	C₂×C₄×C₈₉	1	3	yes	—
306	C₄×D₄	C₂³×C₈₉	1	3	yes	—
307	C₄×D₄	C₂³×C₈₉	1	6	yes	—
308	C₄×D₄	C₂³×C₈₉	1	3	yes	—
309	C₄×D₄	C₂³×C₈₉	1	6	yes	—
310	#117	C₂×C₄×C₈₉	1	6	yes	—
311	#117	C₂×C₄×C₈₉	1	6	yes	—
312	#117	C₂×C₄×C₈₉	1	3	yes	—
313	#117	C₂×C₄×C₈₉	1	3	yes	—
314	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	6	yes	—
315	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
316	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	6	yes	—
317	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	12	yes	—
318	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
319	C₂×C₄²	C₂³×C₈₉	1	6	yes	—
320	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
321	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
322	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
323	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
324	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
325	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
326	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
327	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
328	C₂²⋊C₈	C₂³×C₈₉	1	12	yes	—
329	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
330	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
331	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
332	#97	C₂³×C₈₉	1	2	yes	—
333	#97	C₂³×C₈₉	1	6	yes	—
334	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
335	C₄×C₈	C₂³×C₈₉	1	3	yes	—
336	C₄×C₈	C₂³×C₈₉	1	3	yes	—
337	C₄×C₈	C₂³×C₈₉	1	3	yes	—
338	C₄×C₈	C₂³×C₈₉	1	6	yes	—
339	C₂⁴	C₂²×C₄×C₈₉	1	6	yes	—
340	#41	—	2	8	no	—
341	#41	—	2	12	no	—
342	#41	M₁₆×C₈₉	1	8	yes	complemented
343	C₂×Q₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
344	C₂×Q₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
345	C₂×D₄	—	2	24	no	—
346	C₂×D₄	—	2	12	no	—
347	C₂×D₄	C₂²×C₄×C₈₉	1	6	yes	—
348	C₂×D₄	C₂²×C₄×C₈₉	1	6	yes	—
349	C₂×D₄	C₂²×C₄×C₈₉	1	3	yes	—
350	C₂×D₄	C₂²×C₄×C₈₉	1	3	yes	—
351	C₂²×C₄	—	4	6	no	—
352	C₂²×C₄	—	4	6	no	—
353	C₂²×C₄	C₂²×C₄×C₈₉	1	6	yes	—
354	C₂²×C₄	C₂²×C₄×C₈₉	1	6	yes	—
355	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
356	C₂²×C₄	C₂²×C₄×C₈₉	1	12	yes	—
357	C₂²×C₄	C₂²×C₄×C₈₉	1	6	yes	—
358	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
359	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
360	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
361	M₁₆	—	2	12	no	—
362	M₁₆	—	4	12	no	—
363	M₁₆	—	4	6	no	—
364	C₂×C₈	—	2	6	no	—
365	C₂×C₈	—	4	6	no	—
366	C₂×C₈	—	4	3	no	—
367	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
368	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
369	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
370	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
371	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
372	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
373	C₄⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
374	C₄⋊C₄	C₂²×C₄×C₈₉	1	3	yes	—
375	C₄⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
376	C₄⋊C₄	C₂²×C₄×C₈₉	1	3	yes	—
377	C₄⋊C₄	C₂²×C₄×C₈₉	1	2	yes	—
378	C₄⋊C₄	C₂²×C₄×C₈₉	1	3	yes	—
379	C₄⋊C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
380	C₂²⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
381	C₂²⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
382	C₂²⋊C₄	C₂²×C₄×C₈₉	1	12	yes	—
383	C₂²⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
384	C₂²⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
385	C₄²	C₂²×C₄×C₈₉	1	6	yes	—
386	C₄²	C₂²×C₄×C₈₉	1	3	yes	—
387	C₄²	C₂²×C₄×C₈₉	1	3	yes	—
388	C₄²	C₂²×C₄×C₈₉	1	6	yes	—
389	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
390	C₄²	C₂⁴×C₈₉	1	3	yes	—
391	C₄²	C₂⁴×C₈₉	1	2	yes	—
392	C₂³	—	2	12	no	—
393	C₂³	—	4	6	no	—
394	C₂³	—	4	6	no	—
395	C₂³	C₂³×C₄×C₈₉	1	6	yes	—
396	C₂³	C₂³×C₄×C₈₉	1	3	yes	—
397	Q₈	—	2	2	no	—
398	Q₈	C₂×M₁₆×C₈₉	1	4	yes	complemented
399	D₄	—	2	12	no	—
400	D₄	—	2	12	no	—
401	D₄	—	2	6	no	—
402	D₄	C₂×M₁₆×C₈₉	1	12	yes	complemented
403	C₂×C₄	—	2	12	no	—
404	C₂×C₄	—	2	3	no	—
405	C₂×C₄	—	2	6	no	—
406	C₂×C₄	—	2	3	no	—
407	C₂×C₄	—	2	6	no	—
408	C₂×C₄	—	4	6	no	—
409	C₂×C₄	—	4	12	no	—
410	C₂×C₄	—	4	3	no	—
411	C₂×C₄	—	4	6	no	—
412	C₂×C₄	C₂×M₁₆×C₈₉	1	12	yes	complemented
413	C₂×C₄	C₂³×C₄×C₈₉	1	3	yes	—
414	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
415	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
416	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
417	C₂×C₄	C₂³×C₄×C₈₉	1	3	yes	—
418	C₂×C₄	C₂³×C₄×C₈₉	1	6	yes	—
419	C₂×C₄	C₂³×C₄×C₈₉	1	3	yes	—
420	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
421	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
422	C₈	—	2	3	no	—
423	C₈	—	2	6	no	—
424	C₈	—	2	1	no	—
425	C₈	—	2	3	no	—
426	C₈	—	2	2	no	—
427	C₈	H(5,𝔽₂)×C₈₉	1	2	yes	complemented
428	C₂²	—	2	3	no	—
429	C₂²	—	2	3	no	—
430	C₂²	—	2	6	no	—
431	C₂²	—	2	6	no	—
432	C₂²	—	2	6	no	—
433	C₂²	—	8	12	no	—
434	C₂²	C₂²×M₁₆×C₈₉	1	6	yes	—
435	C₂²	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
436	C₄	—	2	3	no	—
437	C₄	—	2	6	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	3	no	—
441	C₄	—	2	2	no	—
442	C₄	C₂²×M₁₆×C₈₉	1	6	yes	—
443	C₄	C₂²×M₁₆×C₈₉	1	2	yes	—
444	C₄	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
445	C₄	#585×C₈₉	1	1	yes	characteristic
446	C₂	—	2	6	no	—
447	C₂	—	4	6	no	—
448	C₂	C₄×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
449	C₂	C₂³×M₁₆×C₈₉	1	1	yes	characteristic
450	C₂	C₈₉×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂⁴×C₄
prime radical	C₂×H(5,𝔽₂)×C₈₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2991	C₂	semidirect
3. C₈₉×#2989	C₂	semidirect
4. C₈₉×#2988	C₂	semidirect
5. C₈₉×#2975	C₂	semidirect
6. C₈₉×#2974	C₂	semidirect
7. C₈₉×#2971	C₂	semidirect
8. C₈₉×#2965	C₂	semidirect
9. C₈×#41×C₈₉	C₂	semidirect
10. D₄×M₁₆×C₈₉	C₂	semidirect
11. C₂×#436×C₈₉	C₂	semidirect
13. C₂²×M₁₆×C₈₉	C₂²	semidirect
24. #446×C₈₉	C₂²	semidirect
25. #446×C₈₉	C₂²	semidirect
26. C₈×Q₈×C₈₉	C₂²	semidirect
27. C₈×Q₈×C₈₉	C₂²	semidirect
28. C₈×Q₈×C₈₉	C₂²	semidirect
29. #436×C₈₉	C₂²	semidirect
30. #436×C₈₉	C₂²	semidirect
31. #436×C₈₉	C₂²	semidirect
32. #436×C₈₉	C₂²	semidirect
33. #436×C₈₉	C₂²	semidirect
34. #436×C₈₉	C₂²	semidirect
35. #436×C₈₉	C₂²	semidirect
36. #435×C₈₉	C₂²	semidirect
37. #435×C₈₉	C₂²	semidirect
38. #435×C₈₉	C₂²	semidirect
39. C₈×D₄×C₈₉	C₂²	semidirect
40. C₈×D₄×C₈₉	C₂²	semidirect
41. C₈×D₄×C₈₉	C₂²	semidirect
42. #432×C₈₉	C₂²	semidirect
43. #431×C₈₉	C₂²	semidirect
44. #431×C₈₉	C₂²	semidirect
45. #431×C₈₉	C₂²	semidirect
46. #423×C₈₉	C₂²	semidirect
47. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
48. #407×C₈₉	C₂²	semidirect
49. C₂×C₂²⋊C₈×C₈₉	C₂²	semidirect
50. C₄×M₁₆×C₈₉	C₂²	semidirect
51. C₂×C₄×C₈×C₈₉	C₂²	semidirect
116. #41×C₈₉	M₁₆	semidirect
172. Q₈×C₈₉	C₂×M₁₆	semidirect
176. D₄×C₈₉	C₂×M₁₆	semidirect
186. C₂×C₄×C₈₉	C₂×M₁₆	semidirect
201. C₈×C₈₉	H(5,𝔽₂)	semidirect
220. #60325	C₈₉	direct (N × H)
227. #2991	C₂×C₈₉	semidirect
228. #2989	C₂×C₈₉	semidirect
229. #2988	C₂×C₈₉	semidirect
230. #2975	C₂×C₈₉	semidirect
231. #2974	C₂×C₈₉	semidirect
232. #2971	C₂×C₈₉	semidirect
233. #2965	C₂×C₈₉	semidirect
234. C₈×#41	C₂×C₈₉	semidirect
235. D₄×M₁₆	C₂×C₈₉	semidirect
236. C₂×#436	C₂×C₈₉	semidirect
237. C₈₉	#60325	direct (N × H)
239. C₂²×M₁₆	C₂²×C₈₉	semidirect
250. #446	C₂²×C₈₉	semidirect
251. #446	C₂²×C₈₉	semidirect
252. C₈×Q₈	C₂²×C₈₉	semidirect
253. C₈×Q₈	C₂²×C₈₉	semidirect
254. C₈×Q₈	C₂²×C₈₉	semidirect
255. #436	C₂²×C₈₉	semidirect
256. #436	C₂²×C₈₉	semidirect
257. #436	C₂²×C₈₉	semidirect
258. #436	C₂²×C₈₉	semidirect
259. #436	C₂²×C₈₉	semidirect
260. #436	C₂²×C₈₉	semidirect
261. #436	C₂²×C₈₉	semidirect
262. #435	C₂²×C₈₉	semidirect
263. #435	C₂²×C₈₉	semidirect
264. #435	C₂²×C₈₉	semidirect
265. C₈×D₄	C₂²×C₈₉	semidirect
266. C₈×D₄	C₂²×C₈₉	semidirect
267. C₈×D₄	C₂²×C₈₉	semidirect
268. #432	C₂²×C₈₉	semidirect
269. #431	C₂²×C₈₉	semidirect
270. #431	C₂²×C₈₉	semidirect
271. #431	C₂²×C₈₉	semidirect
272. #423	C₂²×C₈₉	semidirect
273. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
274. #407	C₂²×C₈₉	semidirect
275. C₂×C₂²⋊C₈	C₂²×C₈₉	semidirect
276. C₄×M₁₆	C₂²×C₈₉	semidirect
277. C₂×C₄×C₈	C₂²×C₈₉	semidirect
342. #41	M₁₆×C₈₉	semidirect
398. Q₈	C₂×M₁₆×C₈₉	semidirect
402. D₄	C₂×M₁₆×C₈₉	semidirect
412. C₂×C₄	C₂×M₁₆×C₈₉	semidirect
427. C₈	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
C₄×H(5,𝔽₂)×C₈₉	2
C₈₉×#2991	2
C₈₉×#2989	2
C₈₉×#2988	2
C₈₉×#2975	2
C₈₉×#2974	2
C₈₉×#2971	2
C₈₉×#2965	2
C₈×#41×C₈₉	2
D₄×M₁₆×C₈₉	2
C₂×#436×C₈₉	2
#60325	89

Subgroups — C89×#60325 #36438173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60325 #36438173