Subgroups — C₂×C₈₉×#3262 #36438761

Order 22784.

1882 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#542×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₈₉×#3262	C₂	1	16	yes	maximal, direct product factor
3	C₂×#477×C₈₉	C₂	1	2	yes	maximal, complemented
4	C₂×#474×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#468×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#467×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#466×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#464×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×#463×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂×#424×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#542×C₈₉	—	2	8	no	—
17	#542×C₈₉	C₂²	1	16	yes	complemented
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
19	C₂×#126×C₈₉	—	2	2	no	—
20	C₂×#124×C₈₉	—	2	2	no	—
21	C₂×#124×C₈₉	C₂²	1	2	yes	—
22	C₂×#123×C₈₉	—	2	2	no	—
23	C₂×C₂²⋊Q₈×C₈₉	—	2	2	no	—
24	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
25	C₂×#121×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
28	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
29	#477×C₈₉	C₂²	1	16	yes	complemented
30	#474×C₈₉	C₂²	1	16	yes	complemented
31	#468×C₈₉	C₂²	1	16	yes	complemented
32	#467×C₈₉	C₂²	1	8	yes	complemented
33	#466×C₈₉	C₂²	1	8	yes	—
34	#464×C₈₉	C₂²	1	16	yes	complemented
35	#463×C₈₉	C₂²	1	16	yes	complemented
36	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#107×C₈₉	C₂²	1	2	yes	—
38	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
39	#424×C₈₉	C₂²	1	8	yes	complemented
40	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
41	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
42	C₂×#103×C₈₉	C₂²	1	2	yes	—
43	C₂×#103×C₈₉	C₂²	1	2	yes	—
44	C₂×#102×C₈₉	C₂²	1	2	yes	—
45	C₂×#102×C₈₉	C₂²	1	2	yes	—
46	C₂²×Q₈×C₈₉	—	2	2	no	—
47	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
48	C₂²×D₄×C₈₉	—	2	2	no	—
49	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
50	C₂³×C₄×C₈₉	—	2	2	no	—
51	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	—	4	2	no	—
53	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
54	C₂×SD₁₆×C₈₉	—	4	4	no	—
55	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
56	C₂×D₈×C₈₉	—	4	2	no	—
57	C₂×D₈×C₈₉	C₂³	1	4	yes	—
58	C₂×M₁₆×C₈₉	—	4	2	no	—
59	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
60	C₂²×C₈×C₈₉	—	4	2	no	—
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
65	C₄⋊Q₈×C₈₉	—	2	4	no	—
66	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
67	#126×C₈₉	—	2	16	no	—
68	#124×C₈₉	—	2	16	no	—
69	#124×C₈₉	—	2	4	no	—
70	#124×C₈₉	C₂³	1	8	yes	—
71	#123×C₈₉	—	2	16	no	—
72	C₂²⋊Q₈×C₈₉	—	2	4	no	—
73	C₂²⋊Q₈×C₈₉	—	2	16	no	—
74	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
75	#121×C₈₉	—	2	4	no	—
76	#121×C₈₉	C₂³	1	8	yes	—
77	C₄×Q₈×C₈₉	—	2	4	no	—
78	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
79	C₄×D₄×C₈₉	—	2	4	no	—
80	C₄×D₄×C₈₉	C₂³	1	8	yes	—
81	#117×C₈₉	D₄	1	4	yes	—
82	#117×C₈₉	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
85	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
96	C₂×C₄²×C₈₉	—	2	2	no	—
97	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
98	#107×C₈₉	C₂³	1	4	yes	—
99	#107×C₈₉	C₂³	1	4	yes	—
100	#107×C₈₉	C₂³	1	8	yes	—
101	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
102	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
103	#103×C₈₉	C₂³	1	8	yes	—
104	#103×C₈₉	C₂³	1	8	yes	—
105	#102×C₈₉	C₂³	1	8	yes	—
106	#102×C₈₉	C₂³	1	8	yes	—
107	C₂⁴×C₈₉	—	2	2	no	—
108	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×Q₈×C₈₉	—	2	2	no	—
110	C₂×Q₈×C₈₉	—	2	8	no	—
111	C₂×Q₈×C₈₉	—	2	4	no	—
112	C₂×Q₈×C₈₉	—	2	4	no	—
113	C₂×Q₈×C₈₉	—	8	4	no	—
114	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
115	C₂×D₄×C₈₉	—	2	8	no	—
116	C₂×D₄×C₈₉	—	2	2	no	—
117	C₂×D₄×C₈₉	—	2	4	no	—
118	C₂×D₄×C₈₉	—	2	4	no	—
119	C₂×D₄×C₈₉	—	8	4	no	—
120	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
121	C₂²×C₄×C₈₉	—	2	2	no	—
122	C₂²×C₄×C₈₉	—	2	2	no	—
123	C₂²×C₄×C₈₉	—	2	2	no	—
124	C₂²×C₄×C₈₉	—	2	2	no	—
125	C₂²×C₄×C₈₉	—	2	2	no	—
126	C₂²×C₄×C₈₉	—	2	8	no	—
127	C₂²×C₄×C₈₉	—	8	4	no	—
128	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	Q₁₆×C₈₉	—	4	4	no	—
135	SD₁₆×C₈₉	—	4	8	no	—
136	D₈×C₈₉	—	4	4	no	—
137	M₁₆×C₈₉	—	4	4	no	—
138	C₂×C₈×C₈₉	—	2	2	no	—
139	C₂×C₈×C₈₉	—	2	2	no	—
140	C₂×C₈×C₈₉	—	2	4	no	—
141	C₂×C₈×C₈₉	—	4	4	no	—
142	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
143	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
144	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
145	C₄⋊C₄×C₈₉	—	2	8	no	—
146	C₄⋊C₄×C₈₉	—	2	8	no	—
147	C₄⋊C₄×C₈₉	—	2	8	no	—
148	C₄⋊C₄×C₈₉	—	2	2	no	—
149	C₄⋊C₄×C₈₉	—	2	2	no	—
150	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
151	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
152	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
153	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
154	C₂²⋊C₄×C₈₉	—	2	8	no	—
155	C₂²⋊C₄×C₈₉	—	2	8	no	—
156	C₂²⋊C₄×C₈₉	—	2	8	no	—
157	C₂²⋊C₄×C₈₉	—	2	8	no	—
158	C₂²⋊C₄×C₈₉	—	2	8	no	—
159	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
160	C₄²×C₈₉	—	2	8	no	—
161	C₄²×C₈₉	C₂×D₄	1	4	yes	—
162	C₄²×C₈₉	C₂⁴	1	4	yes	—
163	C₂³×C₈₉	—	2	2	no	—
164	C₂³×C₈₉	—	2	2	no	—
165	C₂³×C₈₉	—	2	2	no	—
166	C₂³×C₈₉	—	2	4	no	—
167	C₂³×C₈₉	—	4	4	no	—
168	C₂³×C₈₉	—	4	4	no	—
169	C₂³×C₈₉	—	4	4	no	—
170	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
171	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
172	Q₈×C₈₉	—	2	4	no	—
173	Q₈×C₈₉	—	2	4	no	—
174	Q₈×C₈₉	—	4	4	no	—
175	Q₈×C₈₉	—	8	8	no	—
176	D₄×C₈₉	—	2	4	no	—
177	D₄×C₈₉	—	2	4	no	—
178	D₄×C₈₉	—	4	4	no	—
179	D₄×C₈₉	—	8	8	no	—
180	C₂×C₄×C₈₉	—	2	2	no	—
181	C₂×C₄×C₈₉	—	2	4	no	—
182	C₂×C₄×C₈₉	—	2	4	no	—
183	C₂×C₄×C₈₉	—	2	4	no	—
184	C₂×C₄×C₈₉	—	2	2	no	—
185	C₂×C₄×C₈₉	—	2	2	no	—
186	C₂×C₄×C₈₉	—	2	4	no	—
187	C₂×C₄×C₈₉	—	2	4	no	—
188	C₂×C₄×C₈₉	—	2	4	no	—
189	C₂×C₄×C₈₉	—	4	4	no	—
190	C₂×C₄×C₈₉	—	4	4	no	—
191	C₂×C₄×C₈₉	—	4	4	no	—
192	C₂×C₄×C₈₉	—	8	8	no	—
193	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
194	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
195	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
196	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
198	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
199	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
200	C₈×C₈₉	—	2	2	no	—
201	C₈×C₈₉	—	2	2	no	—
202	C₈×C₈₉	—	2	4	no	—
203	C₂²×C₈₉	—	2	2	no	—
204	C₂²×C₈₉	—	2	2	no	—
205	C₂²×C₈₉	—	2	2	no	—
206	C₂²×C₈₉	—	4	4	no	—
207	C₂²×C₈₉	—	4	4	no	—
208	C₂²×C₈₉	—	4	4	no	—
209	C₂²×C₈₉	—	4	4	no	—
210	C₂²×C₈₉	—	8	8	no	—
211	C₂²×C₈₉	#537	1	2	yes	—
212	C₂²×C₈₉	#576	1	2	yes	—
213	C₂²×C₈₉	#578	1	2	yes	—
214	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
215	C₄×C₈₉	—	2	2	no	—
216	C₄×C₈₉	—	2	4	no	—
217	C₄×C₈₉	—	2	4	no	—
218	C₄×C₈₉	—	4	4	no	—
219	C₄×C₈₉	—	4	4	no	—
220	C₄×C₈₉	—	4	4	no	—
221	C₄×C₈₉	#537	1	2	yes	—
222	C₄×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
223	C₄×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
224	C₂×#3262	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂×C₈₉	—	4	2	no	—
226	C₂×C₈₉	—	8	4	no	—
227	C₂×C₈₉	#3262	1	4	yes	direct product factor
228	C₂×C₈₉	C₂×#537	1	1	yes	characteristic
229	C₂×C₈₉	C₂×#576	1	1	yes	characteristic
230	C₂×C₈₉	C₂×#578	1	1	yes	characteristic
231	C₂×#542	C₂×C₈₉	1	2	yes	complemented
232	#3262	C₂×C₈₉	1	16	yes	direct product factor
233	C₂×#477	C₂×C₈₉	1	2	yes	complemented
234	C₂×#474	C₂×C₈₉	1	2	yes	complemented
235	C₂×#468	C₂×C₈₉	1	2	yes	complemented
236	C₂×#467	C₂×C₈₉	1	1	yes	characteristic, complemented
237	C₂×#466	C₂×C₈₉	1	1	yes	characteristic
238	C₂×#464	C₂×C₈₉	1	2	yes	complemented
239	C₂×#463	C₂×C₈₉	1	2	yes	complemented
240	C₂×#424	C₂×C₈₉	1	1	yes	characteristic, complemented
241	C₈₉	C₂×#3262	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
243	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
244	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
245	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
246	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
247	#542	—	2	8	no	—
248	#542	C₂²×C₈₉	1	16	yes	complemented
249	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
250	C₂×#126	—	2	2	no	—
251	C₂×#124	—	2	2	no	—
252	C₂×#124	C₂²×C₈₉	1	2	yes	—
253	C₂×#123	—	2	2	no	—
254	C₂×C₂²⋊Q₈	—	2	2	no	—
255	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
256	C₂×#121	C₂²×C₈₉	1	2	yes	—
257	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
258	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
259	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
260	#477	C₂²×C₈₉	1	16	yes	complemented
261	#474	C₂²×C₈₉	1	16	yes	complemented
262	#468	C₂²×C₈₉	1	16	yes	complemented
263	#467	C₂²×C₈₉	1	8	yes	complemented
264	#466	C₂²×C₈₉	1	8	yes	—
265	#464	C₂²×C₈₉	1	16	yes	complemented
266	#463	C₂²×C₈₉	1	16	yes	complemented
267	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
268	C₂×#107	C₂²×C₈₉	1	2	yes	—
269	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
270	#424	C₂²×C₈₉	1	8	yes	complemented
271	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
272	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
273	C₂×#103	C₂²×C₈₉	1	2	yes	—
274	C₂×#103	C₂²×C₈₉	1	2	yes	—
275	C₂×#102	C₂²×C₈₉	1	2	yes	—
276	C₂×#102	C₂²×C₈₉	1	2	yes	—
277	C₂²×Q₈	—	2	2	no	—
278	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
279	C₂²×D₄	—	2	2	no	—
280	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
281	C₂³×C₄	—	2	2	no	—
282	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
283	C₂×Q₁₆	—	4	2	no	—
284	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
285	C₂×SD₁₆	—	4	4	no	—
286	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
287	C₂×D₈	—	4	2	no	—
288	C₂×D₈	C₂³×C₈₉	1	4	yes	—
289	C₂×M₁₆	—	4	2	no	—
290	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
291	C₂²×C₈	—	4	2	no	—
292	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
294	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
295	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
296	C₄⋊Q₈	—	2	4	no	—
297	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
298	#126	—	2	16	no	—
299	#124	—	2	16	no	—
300	#124	—	2	4	no	—
301	#124	C₂³×C₈₉	1	8	yes	—
302	#123	—	2	16	no	—
303	C₂²⋊Q₈	—	2	4	no	—
304	C₂²⋊Q₈	—	2	16	no	—
305	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
306	#121	—	2	4	no	—
307	#121	C₂³×C₈₉	1	8	yes	—
308	C₄×Q₈	—	2	4	no	—
309	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
310	C₄×D₄	—	2	4	no	—
311	C₄×D₄	C₂³×C₈₉	1	8	yes	—
312	#117	D₄×C₈₉	1	4	yes	—
313	#117	C₂³×C₈₉	1	4	yes	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
318	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
319	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	2	no	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₂²⋊C₄	—	2	2	no	—
326	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
327	C₂×C₄²	—	2	2	no	—
328	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
329	#107	C₂³×C₈₉	1	4	yes	—
330	#107	C₂³×C₈₉	1	4	yes	—
331	#107	C₂³×C₈₉	1	8	yes	—
332	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
333	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
334	#103	C₂³×C₈₉	1	8	yes	—
335	#103	C₂³×C₈₉	1	8	yes	—
336	#102	C₂³×C₈₉	1	8	yes	—
337	#102	C₂³×C₈₉	1	8	yes	—
338	C₂⁴	—	2	2	no	—
339	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	8	no	—
342	C₂×Q₈	—	2	4	no	—
343	C₂×Q₈	—	2	4	no	—
344	C₂×Q₈	—	8	4	no	—
345	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
346	C₂×D₄	—	2	8	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	8	4	no	—
351	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	8	no	—
358	C₂²×C₄	—	8	4	no	—
359	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
360	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
361	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
363	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
364	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
365	Q₁₆	—	4	4	no	—
366	SD₁₆	—	4	8	no	—
367	D₈	—	4	4	no	—
368	M₁₆	—	4	4	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	4	no	—
372	C₂×C₈	—	4	4	no	—
373	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
374	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
375	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
382	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
383	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
384	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
385	C₂²⋊C₄	—	2	8	no	—
386	C₂²⋊C₄	—	2	8	no	—
387	C₂²⋊C₄	—	2	8	no	—
388	C₂²⋊C₄	—	2	8	no	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
391	C₄²	—	2	8	no	—
392	C₄²	C₂×D₄×C₈₉	1	4	yes	—
393	C₄²	C₂⁴×C₈₉	1	4	yes	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	4	4	no	—
399	C₂³	—	4	4	no	—
400	C₂³	—	4	4	no	—
401	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
402	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
403	Q₈	—	2	4	no	—
404	Q₈	—	2	4	no	—
405	Q₈	—	4	4	no	—
406	Q₈	—	8	8	no	—
407	D₄	—	2	4	no	—
408	D₄	—	2	4	no	—
409	D₄	—	4	4	no	—
410	D₄	—	8	8	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	8	8	no	—
424	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
425	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
426	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
427	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
428	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
429	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
430	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	2	4	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	8	8	no	—
442	C₂²	#537×C₈₉	1	2	yes	—
443	C₂²	#576×C₈₉	1	2	yes	—
444	C₂²	#578×C₈₉	1	2	yes	—
445	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
446	C₄	—	2	4	no	—
447	C₄	—	2	4	no	—
448	C₄	—	2	2	no	—
449	C₄	—	4	4	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	4	no	—
452	C₄	#537×C₈₉	1	2	yes	—
453	C₄	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
454	C₄	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
455	C₂	—	4	2	no	—
456	C₂	—	8	4	no	—
457	C₂	C₈₉×#3262	1	4	yes	direct product factor
458	C₂	C₂×#537×C₈₉	1	1	yes	characteristic
459	C₂	C₂×#576×C₈₉	1	1	yes	characteristic
460	C₂	C₂×#578×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂×#466×C₈₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#542×C₈₉	C₂	semidirect
2. C₈₉×#3262	C₂	direct (N × H)
3. C₂×#477×C₈₉	C₂	semidirect
4. C₂×#474×C₈₉	C₂	semidirect
5. C₂×#468×C₈₉	C₂	semidirect
6. C₂×#467×C₈₉	C₂	semidirect
8. C₂×#464×C₈₉	C₂	semidirect
9. C₂×#463×C₈₉	C₂	semidirect
10. C₂×#424×C₈₉	C₂	semidirect
17. #542×C₈₉	C₂²	semidirect
29. #477×C₈₉	C₂²	semidirect
30. #474×C₈₉	C₂²	semidirect
31. #468×C₈₉	C₂²	semidirect
32. #467×C₈₉	C₂²	semidirect
34. #464×C₈₉	C₂²	semidirect
35. #463×C₈₉	C₂²	semidirect
39. #424×C₈₉	C₂²	semidirect
224. C₂×#3262	C₈₉	direct (N × H)
227. C₂×C₈₉	#3262	direct (N × H)
231. C₂×#542	C₂×C₈₉	semidirect
232. #3262	C₂×C₈₉	direct (N × H)
233. C₂×#477	C₂×C₈₉	semidirect
234. C₂×#474	C₂×C₈₉	semidirect
235. C₂×#468	C₂×C₈₉	semidirect
236. C₂×#467	C₂×C₈₉	semidirect
238. C₂×#464	C₂×C₈₉	semidirect
239. C₂×#463	C₂×C₈₉	semidirect
240. C₂×#424	C₂×C₈₉	semidirect
241. C₈₉	C₂×#3262	direct (N × H)
248. #542	C₂²×C₈₉	semidirect
260. #477	C₂²×C₈₉	semidirect
261. #474	C₂²×C₈₉	semidirect
262. #468	C₂²×C₈₉	semidirect
263. #467	C₂²×C₈₉	semidirect
265. #464	C₂²×C₈₉	semidirect
266. #463	C₂²×C₈₉	semidirect
270. #424	C₂²×C₈₉	semidirect
457. C₂	C₈₉×#3262	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#542×C₈₉	2
C₈₉×#3262	2
C₂×#477×C₈₉	2
C₂×#474×C₈₉	2
C₂×#468×C₈₉	2
C₂×#467×C₈₉	2
C₂×#466×C₈₉	2
C₂×#464×C₈₉	2
C₂×#463×C₈₉	2
C₂×#424×C₈₉	2
C₂×#3262	89

Subgroups — C2×C89×#3262 #36438761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3262 #36438761