Subgroups — C₈₉×#60495 #36438343

Order 22784.

2090 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3084	C₂	1	6	yes	maximal, complemented
4	C₈₉×#3081	C₂	1	6	yes	maximal, complemented
5	C₈₉×#3079	C₂	1	3	yes	maximal, complemented
6	C₈₉×#3077	C₂	1	3	yes	maximal, complemented
7	C₈₉×#3076	C₂	1	3	yes	maximal, complemented
8	C₂×#476×C₈₉	C₂	1	3	yes	maximal, complemented
9	C₈₉×#2912	C₂	1	2	yes	maximal, complemented
10	C₈₉×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2897	C₂	1	2	yes	maximal
12	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#131×C₈₉	C₂²	1	2	yes	—
14	Q₈²×C₈₉	—	2	1	no	—
15	#557×C₈₉	—	2	1	no	—
16	#554×C₈₉	—	2	3	no	—
17	D₄×Q₈×C₈₉	—	2	3	no	—
18	#544×C₈₉	C₂²	1	6	yes	complemented
19	#533×C₈₉	C₂²	1	3	yes	complemented
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	3	yes	complemented
21	#520×C₈₉	—	2	1	no	—
22	#520×C₈₉	C₂²	1	3	yes	—
23	#520×C₈₉	C₂²	1	2	yes	—
24	#520×C₈₉	C₂²	1	1	yes	characteristic
25	#518×C₈₉	—	2	3	no	—
26	#518×C₈₉	C₂²	1	3	yes	—
27	C₄×#41×C₈₉	—	2	1	no	—
28	C₄×#41×C₈₉	C₂²	1	3	yes	—
29	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₈₉	—	2	3	no	—
31	C₂×C₄×D₄×C₈₉	C₂²	1	3	yes	—
32	#479×C₈₉	C₂²	1	6	yes	complemented
33	#478×C₈₉	C₂²	1	6	yes	complemented
34	#478×C₈₉	C₂²	1	12	yes	complemented
35	#477×C₈₉	C₂²	1	3	yes	complemented
36	#476×C₈₉	C₂²	1	3	yes	complemented
37	#476×C₈₉	C₂²	1	6	yes	complemented
38	#476×C₈₉	C₂²	1	12	yes	complemented
39	#475×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	#475×C₈₉	C₂²	1	6	yes	complemented
41	#474×C₈₉	C₂²	1	3	yes	complemented
42	#474×C₈₉	C₂²	1	6	yes	complemented
43	#428×C₈₉	C₂²	1	6	yes	complemented
44	#428×C₈₉	C₂²	1	6	yes	complemented
45	#427×C₈₉	C₂²	1	6	yes	complemented
46	C₂×#106×C₈₉	C₂²	1	6	yes	complemented
47	#424×C₈₉	C₂²	1	6	yes	complemented
48	#423×C₈₉	C₂²	1	3	yes	complemented
49	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	complemented
50	#418×C₈₉	C₂²	1	6	yes	—
51	#417×C₈₉	C₂²	1	6	yes	—
52	#416×C₈₉	C₂²	1	6	yes	—
53	C₂×#102×C₈₉	C₂²	1	6	yes	—
54	H(5,𝔽₂)×C₈₉	—	2	4	no	—
55	H(5,𝔽₂)×C₈₉	Q₈	1	8	yes	complemented
56	C₂×#41×C₈₉	—	2	1	no	—
57	C₂×#41×C₈₉	C₂³	1	3	yes	—
58	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₈₉	—	2	3	no	—
60	C₂²×D₄×C₈₉	—	2	3	no	—
61	C₂²×D₄×C₈₉	C₂³	1	3	yes	—
62	C₂³×C₄×C₈₉	—	2	3	no	—
63	#131×C₈₉	—	2	8	no	—
64	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
65	C₂²×C₈×C₈₉	C₂³	1	6	yes	—
66	C₄⋊Q₈×C₈₉	—	2	6	no	—
67	C₄⋊Q₈×C₈₉	—	2	3	no	—
68	C₄⋊Q₈×C₈₉	—	2	3	no	—
69	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₈₉	C₂³	1	6	yes	—
71	C₄⋊Q₈×C₈₉	C₂³	1	3	yes	—
72	#125×C₈₉	—	2	6	no	—
73	#125×C₈₉	C₂³	1	6	yes	—
74	C₂²⋊Q₈×C₈₉	—	2	3	no	—
75	C₂²⋊Q₈×C₈₉	—	2	3	no	—
76	C₂²⋊Q₈×C₈₉	—	2	12	no	—
77	C₄×Q₈×C₈₉	—	2	1	no	—
78	C₄×Q₈×C₈₉	—	2	2	no	—
79	C₄×Q₈×C₈₉	—	2	3	no	—
80	C₄×Q₈×C₈₉	—	2	3	no	—
81	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
83	C₄×Q₈×C₈₉	C₂³	1	3	yes	—
84	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄×D₄×C₈₉	—	2	3	no	—
86	C₄×D₄×C₈₉	—	2	6	no	—
87	C₄×D₄×C₈₉	—	2	3	no	—
88	C₄×D₄×C₈₉	—	2	3	no	—
89	C₄×D₄×C₈₉	—	2	3	no	—
90	C₄×D₄×C₈₉	—	2	3	no	—
91	C₄×D₄×C₈₉	—	2	6	no	—
92	C₄×D₄×C₈₉	—	2	3	no	—
93	C₄×D₄×C₈₉	—	2	3	no	—
94	C₄×D₄×C₈₉	D₄	1	3	yes	—
95	C₄×D₄×C₈₉	D₄	1	3	yes	—
96	C₄×D₄×C₈₉	C₂³	1	3	yes	—
97	C₄×D₄×C₈₉	C₂³	1	3	yes	—
98	C₄×D₄×C₈₉	C₂³	1	6	yes	—
99	#117×C₈₉	—	2	3	no	—
100	#117×C₈₉	—	2	3	no	—
101	#117×C₈₉	C₂³	1	6	yes	—
102	#117×C₈₉	C₂³	1	3	yes	—
103	#117×C₈₉	C₂³	1	3	yes	—
104	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
105	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
106	C₂×C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
107	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
108	C₂×C₂²⋊C₄×C₈₉	—	2	3	no	—
109	C₂×C₂²⋊C₄×C₈₉	—	2	6	no	—
110	C₂×C₄²×C₈₉	—	2	3	no	—
111	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
112	#107×C₈₉	C₂³	1	6	yes	—
113	#107×C₈₉	C₂³	1	6	yes	—
114	#106×C₈₉	C₂³	1	6	yes	—
115	#106×C₈₉	C₂³	1	12	yes	—
116	#106×C₈₉	C₂³	1	2	yes	—
117	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
119	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
120	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
121	#103×C₈₉	C₂³	1	2	yes	—
122	#103×C₈₉	C₂³	1	6	yes	—
123	#102×C₈₉	C₂³	1	6	yes	—
124	#102×C₈₉	C₂³	1	12	yes	—
125	#102×C₈₉	C₂³	1	6	yes	—
126	C₂⁴×C₈₉	—	2	3	no	—
127	#41×C₈₉	—	2	8	no	—
128	#41×C₈₉	—	2	6	no	—
129	#41×C₈₉	C₂×Q₈	1	4	yes	—
130	#41×C₈₉	C₂×Q₈	1	12	yes	complemented
131	#41×C₈₉	#41	1	4	yes	—
132	C₂×Q₈×C₈₉	—	2	1	no	—
133	C₂×Q₈×C₈₉	—	2	3	no	—
134	C₂×Q₈×C₈₉	—	2	6	no	—
135	C₂×Q₈×C₈₉	—	8	6	no	—
136	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Q₈×C₈₉	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₈₉	—	2	3	no	—
139	C₂×D₄×C₈₉	—	2	24	no	—
140	C₂×D₄×C₈₉	—	2	3	no	—
141	C₂×D₄×C₈₉	—	2	6	no	—
142	C₂×D₄×C₈₉	C₂×D₄	1	3	yes	—
143	C₂×D₄×C₈₉	C₂×Q₈	1	12	yes	complemented
144	C₂×D₄×C₈₉	C₂⁴	1	3	yes	—
145	C₂²×C₄×C₈₉	—	2	3	no	—
146	C₂²×C₄×C₈₉	—	2	3	no	—
147	C₂²×C₄×C₈₉	—	2	6	no	—
148	C₂²×C₄×C₈₉	—	2	3	no	—
149	C₂²×C₄×C₈₉	—	2	3	no	—
150	C₂²×C₄×C₈₉	—	2	6	no	—
151	C₂²×C₄×C₈₉	—	8	6	no	—
152	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
153	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
155	M₁₆×C₈₉	—	2	12	no	—
156	C₂×C₈×C₈₉	—	2	12	no	—
157	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
158	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
159	C₄⋊C₄×C₈₉	—	2	3	no	—
160	C₄⋊C₄×C₈₉	—	2	3	no	—
161	C₄⋊C₄×C₈₉	—	2	3	no	—
162	C₄⋊C₄×C₈₉	—	2	6	no	—
163	C₄⋊C₄×C₈₉	—	2	6	no	—
164	C₄⋊C₄×C₈₉	—	2	1	no	—
165	C₄⋊C₄×C₈₉	—	2	1	no	—
166	C₄⋊C₄×C₈₉	—	2	6	no	—
167	C₄⋊C₄×C₈₉	—	2	6	no	—
168	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
170	C₄⋊C₄×C₈₉	C₂⁴	1	6	yes	—
171	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
172	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
173	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
174	C₂²⋊C₄×C₈₉	—	2	3	no	—
175	C₂²⋊C₄×C₈₉	—	2	6	no	—
176	C₂²⋊C₄×C₈₉	—	2	3	no	—
177	C₂²⋊C₄×C₈₉	—	2	6	no	—
178	C₂²⋊C₄×C₈₉	—	2	3	no	—
179	C₂²⋊C₄×C₈₉	C₂×D₄	1	6	yes	—
180	C₄²×C₈₉	—	2	1	no	—
181	C₄²×C₈₉	—	2	3	no	—
182	C₄²×C₈₉	—	2	3	no	—
183	C₄²×C₈₉	—	2	6	no	—
184	C₄²×C₈₉	C₂×D₄	1	3	yes	—
185	C₄²×C₈₉	C₂×D₄	1	3	yes	—
186	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
187	C₄²×C₈₉	C₂⁴	1	3	yes	—
188	C₂³×C₈₉	—	2	3	no	—
189	C₂³×C₈₉	—	2	12	no	—
190	C₂³×C₈₉	—	8	3	no	—
191	C₂³×C₈₉	—	8	3	no	—
192	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
193	Q₈×C₈₉	—	2	1	no	—
194	Q₈×C₈₉	—	4	6	no	—
195	Q₈×C₈₉	—	4	2	no	—
196	Q₈×C₈₉	—	8	6	no	—
197	Q₈×C₈₉	C₂²⋊Q₈	1	4	yes	—
198	Q₈×C₈₉	C₂²×Q₈	1	2	yes	—
199	D₄×C₈₉	—	2	12	no	—
200	D₄×C₈₉	—	2	12	no	—
201	D₄×C₈₉	—	2	3	no	—
202	D₄×C₈₉	C₂²⋊Q₈	1	12	yes	—
203	D₄×C₈₉	C₂²×Q₈	1	6	yes	—
204	C₂×C₄×C₈₉	—	2	1	no	—
205	C₂×C₄×C₈₉	—	2	12	no	—
206	C₂×C₄×C₈₉	—	2	6	no	—
207	C₂×C₄×C₈₉	—	2	1	no	—
208	C₂×C₄×C₈₉	—	2	2	no	—
209	C₂×C₄×C₈₉	—	2	3	no	—
210	C₂×C₄×C₈₉	—	2	6	no	—
211	C₂×C₄×C₈₉	—	4	3	no	—
212	C₂×C₄×C₈₉	—	4	6	no	—
213	C₂×C₄×C₈₉	—	8	3	no	—
214	C₂×C₄×C₈₉	—	8	6	no	—
215	C₂×C₄×C₈₉	—	8	3	no	—
216	C₂×C₄×C₈₉	—	8	6	no	—
217	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
218	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
220	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
221	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₈₉	C₂²×Q₈	1	6	yes	—
223	C₂×C₄×C₈₉	C₂×#41	1	6	yes	—
224	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
225	C₈×C₈₉	—	2	2	no	—
226	C₈×C₈₉	—	2	6	no	—
227	C₂²×C₈₉	—	2	3	no	—
228	C₂²×C₈₉	—	2	3	no	—
229	C₂²×C₈₉	—	2	6	no	—
230	C₂²×C₈₉	—	4	3	no	—
231	C₂²×C₈₉	—	4	3	no	—
232	C₂²×C₈₉	—	8	12	no	—
233	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	6	yes	—
234	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
235	C₄×C₈₉	—	2	1	no	—
236	C₄×C₈₉	—	2	3	no	—
237	C₄×C₈₉	—	2	2	no	—
238	C₄×C₈₉	—	2	3	no	—
239	C₄×C₈₉	—	4	1	no	—
240	C₄×C₈₉	—	4	6	no	—
241	C₄×C₈₉	—	4	3	no	—
242	C₄×C₈₉	—	4	2	no	—
243	C₄×C₈₉	C₂×C₂²⋊Q₈	1	6	yes	—
244	C₄×C₈₉	#577	1	2	yes	—
245	C₄×C₈₉	C₂³×Q₈	1	1	yes	characteristic
246	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
247	#60495	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂×C₈₉	—	2	6	no	—
249	C₂×C₈₉	—	4	6	no	—
250	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₈₉	C₂×#577	1	1	yes	characteristic
252	C₂×C₈₉	#3585	1	1	yes	characteristic
253	#3503	C₂×C₈₉	1	1	yes	characteristic, complemented
254	C₄×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic
255	#3084	C₂×C₈₉	1	6	yes	complemented
256	#3081	C₂×C₈₉	1	6	yes	complemented
257	#3079	C₂×C₈₉	1	3	yes	complemented
258	#3077	C₂×C₈₉	1	3	yes	complemented
259	#3076	C₂×C₈₉	1	3	yes	complemented
260	C₂×#476	C₂×C₈₉	1	3	yes	complemented
261	#2912	C₂×C₈₉	1	2	yes	complemented
262	#2906	C₂×C₈₉	1	1	yes	characteristic, complemented
263	#2897	C₂×C₈₉	1	2	yes	—
264	C₈₉	#60495	1	1	yes	characteristic, direct product factor, Sylow
265	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	1	yes	characteristic
266	C₂×#131	C₂²×C₈₉	1	2	yes	—
267	Q₈²	—	2	1	no	—
268	#557	—	2	1	no	—
269	#554	—	2	3	no	—
270	D₄×Q₈	—	2	3	no	—
271	#544	C₂²×C₈₉	1	6	yes	complemented
272	#533	C₂²×C₈₉	1	3	yes	complemented
273	C₂×C₄⋊Q₈	C₂²×C₈₉	1	3	yes	complemented
274	#520	—	2	1	no	—
275	#520	C₂²×C₈₉	1	3	yes	—
276	#520	C₂²×C₈₉	1	2	yes	—
277	#520	C₂²×C₈₉	1	1	yes	characteristic
278	#518	—	2	3	no	—
279	#518	C₂²×C₈₉	1	3	yes	—
280	C₄×#41	—	2	1	no	—
281	C₄×#41	C₂²×C₈₉	1	3	yes	—
282	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
283	C₂×C₄×D₄	—	2	3	no	—
284	C₂×C₄×D₄	C₂²×C₈₉	1	3	yes	—
285	#479	C₂²×C₈₉	1	6	yes	complemented
286	#478	C₂²×C₈₉	1	6	yes	complemented
287	#478	C₂²×C₈₉	1	12	yes	complemented
288	#477	C₂²×C₈₉	1	3	yes	complemented
289	#476	C₂²×C₈₉	1	3	yes	complemented
290	#476	C₂²×C₈₉	1	6	yes	complemented
291	#476	C₂²×C₈₉	1	12	yes	complemented
292	#475	C₂²×C₈₉	1	1	yes	characteristic, complemented
293	#475	C₂²×C₈₉	1	6	yes	complemented
294	#474	C₂²×C₈₉	1	3	yes	complemented
295	#474	C₂²×C₈₉	1	6	yes	complemented
296	#428	C₂²×C₈₉	1	6	yes	complemented
297	#428	C₂²×C₈₉	1	6	yes	complemented
298	#427	C₂²×C₈₉	1	6	yes	complemented
299	C₂×#106	C₂²×C₈₉	1	6	yes	complemented
300	#424	C₂²×C₈₉	1	6	yes	complemented
301	#423	C₂²×C₈₉	1	3	yes	complemented
302	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	complemented
303	#418	C₂²×C₈₉	1	6	yes	—
304	#417	C₂²×C₈₉	1	6	yes	—
305	#416	C₂²×C₈₉	1	6	yes	—
306	C₂×#102	C₂²×C₈₉	1	6	yes	—
307	H(5,𝔽₂)	—	2	4	no	—
308	H(5,𝔽₂)	Q₈×C₈₉	1	8	yes	complemented
309	C₂×#41	—	2	1	no	—
310	C₂×#41	C₂³×C₈₉	1	3	yes	—
311	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
312	C₂²×Q₈	—	2	3	no	—
313	C₂²×D₄	—	2	3	no	—
314	C₂²×D₄	C₂³×C₈₉	1	3	yes	—
315	C₂³×C₄	—	2	3	no	—
316	#131	—	2	8	no	—
317	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
318	C₂²×C₈	C₂³×C₈₉	1	6	yes	—
319	C₄⋊Q₈	—	2	6	no	—
320	C₄⋊Q₈	—	2	3	no	—
321	C₄⋊Q₈	—	2	3	no	—
322	C₄⋊Q₈	C₂³×C₈₉	1	3	yes	—
323	C₄⋊Q₈	C₂³×C₈₉	1	6	yes	—
324	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
325	#125	—	2	6	no	—
326	#125	C₂³×C₈₉	1	6	yes	—
327	C₂²⋊Q₈	—	2	3	no	—
328	C₂²⋊Q₈	—	2	3	no	—
329	C₂²⋊Q₈	—	2	12	no	—
330	C₄×Q₈	—	2	1	no	—
331	C₄×Q₈	—	2	2	no	—
332	C₄×Q₈	—	2	3	no	—
333	C₄×Q₈	—	2	3	no	—
334	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
335	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
336	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
337	C₄×Q₈	C₂³×C₈₉	1	3	yes	—
338	C₄×D₄	—	2	3	no	—
339	C₄×D₄	—	2	6	no	—
340	C₄×D₄	—	2	3	no	—
341	C₄×D₄	—	2	3	no	—
342	C₄×D₄	—	2	3	no	—
343	C₄×D₄	—	2	3	no	—
344	C₄×D₄	—	2	6	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	—	2	3	no	—
347	C₄×D₄	D₄×C₈₉	1	3	yes	—
348	C₄×D₄	D₄×C₈₉	1	3	yes	—
349	C₄×D₄	C₂³×C₈₉	1	3	yes	—
350	C₄×D₄	C₂³×C₈₉	1	6	yes	—
351	C₄×D₄	C₂³×C₈₉	1	3	yes	—
352	#117	—	2	3	no	—
353	#117	—	2	3	no	—
354	#117	C₂³×C₈₉	1	3	yes	—
355	#117	C₂³×C₈₉	1	6	yes	—
356	#117	C₂³×C₈₉	1	3	yes	—
357	C₂×C₄⋊C₄	—	2	3	no	—
358	C₂×C₄⋊C₄	—	2	3	no	—
359	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
360	C₂×C₄⋊C₄	C₂³×C₈₉	1	6	yes	—
361	C₂×C₂²⋊C₄	—	2	3	no	—
362	C₂×C₂²⋊C₄	—	2	6	no	—
363	C₂×C₄²	—	2	3	no	—
364	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
365	#107	C₂³×C₈₉	1	6	yes	—
366	#107	C₂³×C₈₉	1	6	yes	—
367	#106	C₂³×C₈₉	1	6	yes	—
368	#106	C₂³×C₈₉	1	12	yes	—
369	#106	C₂³×C₈₉	1	2	yes	—
370	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
371	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
372	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
373	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
374	#103	C₂³×C₈₉	1	2	yes	—
375	#103	C₂³×C₈₉	1	6	yes	—
376	#102	C₂³×C₈₉	1	6	yes	—
377	#102	C₂³×C₈₉	1	12	yes	—
378	#102	C₂³×C₈₉	1	6	yes	—
379	C₂⁴	—	2	3	no	—
380	#41	—	2	8	no	—
381	#41	—	2	6	no	—
382	#41	C₂×Q₈×C₈₉	1	4	yes	—
383	#41	C₂×Q₈×C₈₉	1	12	yes	complemented
384	#41	#41×C₈₉	1	4	yes	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	2	3	no	—
387	C₂×Q₈	—	2	6	no	—
388	C₂×Q₈	—	8	6	no	—
389	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
390	C₂×Q₈	C₂⁴×C₈₉	1	1	yes	characteristic
391	C₂×D₄	—	2	3	no	—
392	C₂×D₄	—	2	24	no	—
393	C₂×D₄	—	2	3	no	—
394	C₂×D₄	—	2	6	no	—
395	C₂×D₄	C₂×D₄×C₈₉	1	3	yes	—
396	C₂×D₄	C₂×Q₈×C₈₉	1	12	yes	complemented
397	C₂×D₄	C₂⁴×C₈₉	1	3	yes	—
398	C₂²×C₄	—	2	3	no	—
399	C₂²×C₄	—	2	3	no	—
400	C₂²×C₄	—	2	6	no	—
401	C₂²×C₄	—	2	3	no	—
402	C₂²×C₄	—	2	3	no	—
403	C₂²×C₄	—	2	6	no	—
404	C₂²×C₄	—	8	6	no	—
405	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
406	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
407	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
408	M₁₆	—	2	12	no	—
409	C₂×C₈	—	2	12	no	—
410	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
411	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
412	C₄⋊C₄	—	2	3	no	—
413	C₄⋊C₄	—	2	3	no	—
414	C₄⋊C₄	—	2	3	no	—
415	C₄⋊C₄	—	2	6	no	—
416	C₄⋊C₄	—	2	6	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	6	no	—
420	C₄⋊C₄	—	2	6	no	—
421	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
422	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
423	C₄⋊C₄	C₂⁴×C₈₉	1	6	yes	—
424	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
425	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
426	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
427	C₂²⋊C₄	—	2	3	no	—
428	C₂²⋊C₄	—	2	6	no	—
429	C₂²⋊C₄	—	2	3	no	—
430	C₂²⋊C₄	—	2	6	no	—
431	C₂²⋊C₄	—	2	3	no	—
432	C₂²⋊C₄	C₂×D₄×C₈₉	1	6	yes	—
433	C₄²	—	2	1	no	—
434	C₄²	—	2	3	no	—
435	C₄²	—	2	3	no	—
436	C₄²	—	2	6	no	—
437	C₄²	C₂×D₄×C₈₉	1	3	yes	—
438	C₄²	C₂×D₄×C₈₉	1	3	yes	—
439	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
440	C₄²	C₂⁴×C₈₉	1	3	yes	—
441	C₂³	—	2	3	no	—
442	C₂³	—	2	12	no	—
443	C₂³	—	8	3	no	—
444	C₂³	—	8	3	no	—
445	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
446	Q₈	—	2	1	no	—
447	Q₈	—	4	6	no	—
448	Q₈	—	4	2	no	—
449	Q₈	—	8	6	no	—
450	Q₈	C₂²⋊Q₈×C₈₉	1	4	yes	—
451	Q₈	C₂²×Q₈×C₈₉	1	2	yes	—
452	D₄	—	2	12	no	—
453	D₄	—	2	12	no	—
454	D₄	—	2	3	no	—
455	D₄	C₂²⋊Q₈×C₈₉	1	12	yes	—
456	D₄	C₂²×Q₈×C₈₉	1	6	yes	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	12	no	—
459	C₂×C₄	—	2	6	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	3	no	—
463	C₂×C₄	—	2	6	no	—
464	C₂×C₄	—	4	3	no	—
465	C₂×C₄	—	4	6	no	—
466	C₂×C₄	—	8	3	no	—
467	C₂×C₄	—	8	6	no	—
468	C₂×C₄	—	8	3	no	—
469	C₂×C₄	—	8	6	no	—
470	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
471	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
473	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
474	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
475	C₂×C₄	C₂²×Q₈×C₈₉	1	6	yes	—
476	C₂×C₄	C₂×#41×C₈₉	1	6	yes	—
477	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
478	C₈	—	2	2	no	—
479	C₈	—	2	6	no	—
480	C₂²	—	2	3	no	—
481	C₂²	—	2	3	no	—
482	C₂²	—	2	6	no	—
483	C₂²	—	4	3	no	—
484	C₂²	—	4	3	no	—
485	C₂²	—	8	12	no	—
486	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	6	yes	—
487	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
488	C₄	—	2	3	no	—
489	C₄	—	2	1	no	—
490	C₄	—	2	3	no	—
491	C₄	—	2	2	no	—
492	C₄	—	4	3	no	—
493	C₄	—	4	2	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	6	no	—
496	C₄	C₂×C₂²⋊Q₈×C₈₉	1	6	yes	—
497	C₄	#577×C₈₉	1	2	yes	—
498	C₄	C₂³×Q₈×C₈₉	1	1	yes	characteristic
499	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
500	C₂	—	2	6	no	—
501	C₂	—	4	6	no	—
502	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
503	C₂	C₂×#577×C₈₉	1	1	yes	characteristic
504	C₂	C₈₉×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂×H(5,𝔽₂)×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3503	C₂	semidirect
3. C₈₉×#3084	C₂	semidirect
4. C₈₉×#3081	C₂	semidirect
5. C₈₉×#3079	C₂	semidirect
6. C₈₉×#3077	C₂	semidirect
7. C₈₉×#3076	C₂	semidirect
8. C₂×#476×C₈₉	C₂	semidirect
9. C₈₉×#2912	C₂	semidirect
10. C₈₉×#2906	C₂	semidirect
18. #544×C₈₉	C₂²	semidirect
19. #533×C₈₉	C₂²	semidirect
20. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
32. #479×C₈₉	C₂²	semidirect
33. #478×C₈₉	C₂²	semidirect
34. #478×C₈₉	C₂²	semidirect
35. #477×C₈₉	C₂²	semidirect
36. #476×C₈₉	C₂²	semidirect
37. #476×C₈₉	C₂²	semidirect
38. #476×C₈₉	C₂²	semidirect
39. #475×C₈₉	C₂²	semidirect
40. #475×C₈₉	C₂²	semidirect
41. #474×C₈₉	C₂²	semidirect
42. #474×C₈₉	C₂²	semidirect
43. #428×C₈₉	C₂²	semidirect
44. #428×C₈₉	C₂²	semidirect
45. #427×C₈₉	C₂²	semidirect
46. C₂×#106×C₈₉	C₂²	semidirect
47. #424×C₈₉	C₂²	semidirect
48. #423×C₈₉	C₂²	semidirect
49. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
55. H(5,𝔽₂)×C₈₉	Q₈	semidirect
130. #41×C₈₉	C₂×Q₈	semidirect
143. C₂×D₄×C₈₉	C₂×Q₈	semidirect
247. #60495	C₈₉	direct (N × H)
253. #3503	C₂×C₈₉	semidirect
255. #3084	C₂×C₈₉	semidirect
256. #3081	C₂×C₈₉	semidirect
257. #3079	C₂×C₈₉	semidirect
258. #3077	C₂×C₈₉	semidirect
259. #3076	C₂×C₈₉	semidirect
260. C₂×#476	C₂×C₈₉	semidirect
261. #2912	C₂×C₈₉	semidirect
262. #2906	C₂×C₈₉	semidirect
264. C₈₉	#60495	direct (N × H)
271. #544	C₂²×C₈₉	semidirect
272. #533	C₂²×C₈₉	semidirect
273. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
285. #479	C₂²×C₈₉	semidirect
286. #478	C₂²×C₈₉	semidirect
287. #478	C₂²×C₈₉	semidirect
288. #477	C₂²×C₈₉	semidirect
289. #476	C₂²×C₈₉	semidirect
290. #476	C₂²×C₈₉	semidirect
291. #476	C₂²×C₈₉	semidirect
292. #475	C₂²×C₈₉	semidirect
293. #475	C₂²×C₈₉	semidirect
294. #474	C₂²×C₈₉	semidirect
295. #474	C₂²×C₈₉	semidirect
296. #428	C₂²×C₈₉	semidirect
297. #428	C₂²×C₈₉	semidirect
298. #427	C₂²×C₈₉	semidirect
299. C₂×#106	C₂²×C₈₉	semidirect
300. #424	C₂²×C₈₉	semidirect
301. #423	C₂²×C₈₉	semidirect
302. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
308. H(5,𝔽₂)	Q₈×C₈₉	semidirect
383. #41	C₂×Q₈×C₈₉	semidirect
396. C₂×D₄	C₂×Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3503	2
C₄×H(5,𝔽₂)×C₈₉	2
C₈₉×#3084	2
C₈₉×#3081	2
C₈₉×#3079	2
C₈₉×#3077	2
C₈₉×#3076	2
C₂×#476×C₈₉	2
C₈₉×#2912	2
C₈₉×#2906	2
C₈₉×#2897	2
#60495	89

Subgroups — C89×#60495 #36438343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60495 #36438343