Subgroups — C₂×C₈₉×#3220 #36438639

Order 22784.

2342 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#540×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#535×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3220	C₂	1	16	yes	maximal, direct product factor
4	C₂×#496×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#493×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#490×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#488×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#463×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×#459×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂×#449×C₈₉	C₂	1	2	yes	maximal, complemented
11	C₂×#447×C₈₉	C₂	1	2	yes	maximal, complemented
12	C₂×#433×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	complemented
16	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
17	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
18	C₂²×D₈×C₈₉	C₂²	1	2	yes	complemented
19	#540×C₈₉	—	2	4	no	—
20	#540×C₈₉	C₂²	1	8	yes	complemented
21	#535×C₈₉	—	2	4	no	—
22	#535×C₈₉	C₂²	1	8	yes	complemented
23	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic, complemented
26	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#123×C₈₉	—	2	2	no	—
30	C₂×#121×C₈₉	—	2	2	no	—
31	C₂×#121×C₈₉	—	2	2	no	—
32	C₂×#121×C₈₉	C₂²	1	2	yes	—
33	C₂×#120×C₈₉	—	2	2	no	—
34	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	complemented
35	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
36	#496×C₈₉	C₂²	1	8	yes	complemented
37	#493×C₈₉	C₂²	1	8	yes	complemented
38	#490×C₈₉	C₂²	1	8	yes	complemented
39	#488×C₈₉	C₂²	1	8	yes	complemented
40	#463×C₈₉	C₂²	1	16	yes	complemented
41	#459×C₈₉	C₂²	1	16	yes	complemented
42	#449×C₈₉	C₂²	1	16	yes	complemented
43	#447×C₈₉	C₂²	1	16	yes	complemented
44	#433×C₈₉	C₂²	1	8	yes	complemented
45	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	complemented
46	C₂×#103×C₈₉	C₂²	1	2	yes	complemented
47	C₂×#102×C₈₉	C₂²	1	2	yes	—
48	C₂×#102×C₈₉	C₂²	1	2	yes	—
49	C₂×#102×C₈₉	C₂²	1	2	yes	complemented
50	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
51	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
53	C₂⁵×C₈₉	—	2	1	no	—
54	C₂×#41×C₈₉	—	2	2	no	—
55	C₂×#41×C₈₉	—	4	4	no	—
56	C₂×#41×C₈₉	C₂³	1	4	yes	—
57	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₈₉	—	2	1	no	—
59	C₂²×D₄×C₈₉	—	2	2	no	—
60	C₂²×D₄×C₈₉	—	2	1	no	—
61	C₂²×D₄×C₈₉	—	2	2	no	—
62	C₂²×D₄×C₈₉	—	2	2	no	—
63	C₂²×D₄×C₈₉	—	2	2	no	—
64	C₂²×D₄×C₈₉	—	2	2	no	—
65	C₂²×D₄×C₈₉	—	4	4	no	—
66	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
70	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
71	C₂³×C₄×C₈₉	—	2	1	no	—
72	C₂³×C₄×C₈₉	—	2	2	no	—
73	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×SD₁₆×C₈₉	—	4	4	no	—
75	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	complemented
76	C₂×D₈×C₈₉	—	4	4	no	—
77	C₂×D₈×C₈₉	—	4	4	no	—
78	C₂×D₈×C₈₉	—	4	4	no	—
79	C₂×D₈×C₈₉	C₂³	1	8	yes	—
80	C₂×D₈×C₈₉	C₂³	1	8	yes	—
81	C₂×D₈×C₈₉	C₂³	1	8	yes	complemented
82	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
83	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
84	#127×C₈₉	—	2	2	no	—
85	#127×C₈₉	—	2	2	no	—
86	#127×C₈₉	—	2	2	no	—
87	#127×C₈₉	C₂³	1	4	yes	—
88	#127×C₈₉	C₂³	1	4	yes	—
89	#127×C₈₉	C₂³	1	4	yes	complemented
90	#125×C₈₉	—	2	2	no	—
91	#125×C₈₉	C₂³	1	4	yes	complemented
92	#124×C₈₉	—	2	2	no	—
93	#124×C₈₉	—	2	2	no	—
94	#124×C₈₉	C₂³	1	4	yes	—
95	#124×C₈₉	C₂³	1	4	yes	—
96	#123×C₈₉	—	2	16	no	—
97	#121×C₈₉	—	2	4	no	—
98	#121×C₈₉	—	2	16	no	—
99	#121×C₈₉	—	2	16	no	—
100	#121×C₈₉	C₂³	1	8	yes	—
101	#120×C₈₉	—	2	16	no	—
102	C₄×D₄×C₈₉	—	2	4	no	—
103	C₄×D₄×C₈₉	C₂³	1	8	yes	complemented
104	#117×C₈₉	D₄	1	4	yes	—
105	#117×C₈₉	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
114	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₈₉	C₂³	1	8	yes	complemented
117	#103×C₈₉	C₂³	1	8	yes	complemented
118	#102×C₈₉	C₂³	1	8	yes	—
119	#102×C₈₉	C₂³	1	8	yes	—
120	#102×C₈₉	C₂³	1	8	yes	complemented
121	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
122	#97×C₈₉	C₂³	1	4	yes	complemented
123	C₄×C₈×C₈₉	C₂³	1	4	yes	complemented
124	C₂⁴×C₈₉	—	2	2	no	—
125	C₂⁴×C₈₉	—	2	1	no	—
126	C₂⁴×C₈₉	—	2	1	no	—
127	C₂⁴×C₈₉	—	2	1	no	—
128	C₂⁴×C₈₉	—	2	4	no	—
129	C₂⁴×C₈₉	—	4	4	no	—
130	C₂⁴×C₈₉	—	8	4	no	—
131	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
132	#41×C₈₉	—	4	8	no	—
133	#41×C₈₉	—	4	8	no	—
134	C₂×Q₈×C₈₉	—	2	1	no	—
135	C₂×Q₈×C₈₉	—	2	4	no	—
136	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
137	C₂×D₄×C₈₉	—	2	4	no	—
138	C₂×D₄×C₈₉	—	2	8	no	—
139	C₂×D₄×C₈₉	—	2	2	no	—
140	C₂×D₄×C₈₉	—	2	4	no	—
141	C₂×D₄×C₈₉	—	2	1	no	—
142	C₂×D₄×C₈₉	—	2	1	no	—
143	C₂×D₄×C₈₉	—	2	1	no	—
144	C₂×D₄×C₈₉	—	2	8	no	—
145	C₂×D₄×C₈₉	—	2	8	no	—
146	C₂×D₄×C₈₉	—	2	8	no	—
147	C₂×D₄×C₈₉	—	2	4	no	—
148	C₂×D₄×C₈₉	—	2	4	no	—
149	C₂×D₄×C₈₉	—	2	4	no	—
150	C₂×D₄×C₈₉	—	2	4	no	—
151	C₂×D₄×C₈₉	—	2	4	no	—
152	C₂×D₄×C₈₉	—	4	8	no	—
153	C₂×D₄×C₈₉	—	4	4	no	—
154	C₂×D₄×C₈₉	—	4	8	no	—
155	C₂×D₄×C₈₉	—	4	4	no	—
156	C₂×D₄×C₈₉	—	8	4	no	—
157	C₂×D₄×C₈₉	—	8	4	no	—
158	C₂×D₄×C₈₉	—	8	4	no	—
159	C₂×D₄×C₈₉	—	8	4	no	—
160	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
161	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
162	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
163	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₈₉	—	2	4	no	—
165	C₂²×C₄×C₈₉	—	2	1	no	—
166	C₂²×C₄×C₈₉	—	2	2	no	—
167	C₂²×C₄×C₈₉	—	2	4	no	—
168	C₂²×C₄×C₈₉	—	2	8	no	—
169	C₂²×C₄×C₈₉	—	4	4	no	—
170	C₂²×C₄×C₈₉	—	8	4	no	—
171	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
177	SD₁₆×C₈₉	—	4	8	no	—
178	D₈×C₈₉	—	4	8	no	—
179	D₈×C₈₉	—	4	8	no	—
180	D₈×C₈₉	—	4	8	no	—
181	C₂×C₈×C₈₉	—	2	4	no	—
182	C₂×C₈×C₈₉	—	2	4	no	—
183	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
184	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₈₉	—	2	8	no	—
186	C₄⋊C₄×C₈₉	—	2	2	no	—
187	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
188	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
189	C₂²⋊C₄×C₈₉	—	2	8	no	—
190	C₂²⋊C₄×C₈₉	—	2	8	no	—
191	C₂²⋊C₄×C₈₉	—	2	8	no	—
192	C₂²⋊C₄×C₈₉	—	2	8	no	—
193	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
194	C₄²×C₈₉	C₂×D₄	1	2	yes	—
195	C₄²×C₈₉	C₂×D₄	1	2	yes	—
196	C₄²×C₈₉	C₂⁴	1	2	yes	—
197	C₄²×C₈₉	C₂⁴	1	2	yes	—
198	C₂³×C₈₉	—	2	2	no	—
199	C₂³×C₈₉	—	2	4	no	—
200	C₂³×C₈₉	—	2	4	no	—
201	C₂³×C₈₉	—	2	2	no	—
202	C₂³×C₈₉	—	2	2	no	—
203	C₂³×C₈₉	—	2	2	no	—
204	C₂³×C₈₉	—	4	8	no	—
205	C₂³×C₈₉	—	4	2	no	—
206	C₂³×C₈₉	—	4	2	no	—
207	C₂³×C₈₉	—	4	2	no	—
208	C₂³×C₈₉	—	4	4	no	—
209	C₂³×C₈₉	—	4	4	no	—
210	C₂³×C₈₉	—	4	4	no	—
211	C₂³×C₈₉	—	4	4	no	—
212	C₂³×C₈₉	—	4	4	no	—
213	C₂³×C₈₉	—	4	4	no	—
214	C₂³×C₈₉	—	8	8	no	—
215	C₂³×C₈₉	—	8	16	no	—
216	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
217	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
218	Q₈×C₈₉	—	2	4	no	—
219	Q₈×C₈₉	—	4	2	no	—
220	D₄×C₈₉	—	2	4	no	—
221	D₄×C₈₉	—	2	4	no	—
222	D₄×C₈₉	—	2	4	no	—
223	D₄×C₈₉	—	2	4	no	—
224	D₄×C₈₉	—	2	4	no	—
225	D₄×C₈₉	—	4	8	no	—
226	D₄×C₈₉	—	4	8	no	—
227	D₄×C₈₉	—	4	4	no	—
228	D₄×C₈₉	—	4	2	no	—
229	D₄×C₈₉	—	4	2	no	—
230	D₄×C₈₉	—	4	2	no	—
231	D₄×C₈₉	—	8	8	no	—
232	D₄×C₈₉	—	8	8	no	—
233	D₄×C₈₉	—	8	8	no	—
234	D₄×C₈₉	—	8	8	no	—
235	C₂×C₄×C₈₉	—	2	2	no	—
236	C₂×C₄×C₈₉	—	2	4	no	—
237	C₂×C₄×C₈₉	—	2	4	no	—
238	C₂×C₄×C₈₉	—	2	4	no	—
239	C₂×C₄×C₈₉	—	2	4	no	—
240	C₂×C₄×C₈₉	—	2	2	no	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	4	8	no	—
243	C₂×C₄×C₈₉	—	4	2	no	—
244	C₂×C₄×C₈₉	—	4	4	no	—
245	C₂×C₄×C₈₉	—	8	8	no	—
246	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
247	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
248	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
251	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
252	C₈×C₈₉	—	2	4	no	—
253	C₈×C₈₉	—	2	4	no	—
254	C₂²×C₈₉	—	2	2	no	—
255	C₂²×C₈₉	—	2	4	no	—
256	C₂²×C₈₉	—	4	4	no	—
257	C₂²×C₈₉	—	4	2	no	—
258	C₂²×C₈₉	—	4	2	no	—
259	C₂²×C₈₉	—	4	2	no	—
260	C₂²×C₈₉	—	4	4	no	—
261	C₂²×C₈₉	—	4	4	no	—
262	C₂²×C₈₉	—	4	4	no	—
263	C₂²×C₈₉	—	4	4	no	—
264	C₂²×C₈₉	—	4	4	no	—
265	C₂²×C₈₉	—	4	4	no	—
266	C₂²×C₈₉	—	8	16	no	—
267	C₂²×C₈₉	—	8	4	no	—
268	C₂²×C₈₉	—	8	8	no	—
269	C₂²×C₈₉	—	8	4	no	—
270	C₂²×C₈₉	—	8	4	no	—
271	C₂²×C₈₉	#535	1	2	yes	—
272	C₂²×C₈₉	#576	1	4	yes	—
273	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
274	C₄×C₈₉	—	2	2	no	—
275	C₄×C₈₉	—	2	4	no	—
276	C₄×C₈₉	—	2	4	no	—
277	C₄×C₈₉	—	4	2	no	—
278	C₄×C₈₉	—	4	4	no	—
279	C₄×C₈₉	#535	1	2	yes	—
280	C₄×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
281	C₂×#3220	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
282	C₂×C₈₉	—	4	2	no	—
283	C₂×C₈₉	—	8	2	no	—
284	C₂×C₈₉	—	8	4	no	—
285	C₂×C₈₉	—	8	2	no	—
286	C₂×C₈₉	—	8	2	no	—
287	C₂×C₈₉	#3220	1	4	yes	direct product factor
288	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
289	C₂×C₈₉	C₂×#576	1	2	yes	—
290	C₂×#540	C₂×C₈₉	1	1	yes	characteristic, complemented
291	C₂×#535	C₂×C₈₉	1	1	yes	characteristic, complemented
292	#3220	C₂×C₈₉	1	16	yes	direct product factor
293	C₂×#496	C₂×C₈₉	1	1	yes	characteristic, complemented
294	C₂×#493	C₂×C₈₉	1	1	yes	characteristic, complemented
295	C₂×#490	C₂×C₈₉	1	1	yes	characteristic, complemented
296	C₂×#488	C₂×C₈₉	1	1	yes	characteristic, complemented
297	C₂×#463	C₂×C₈₉	1	2	yes	complemented
298	C₂×#459	C₂×C₈₉	1	2	yes	complemented
299	C₂×#449	C₂×C₈₉	1	2	yes	complemented
300	C₂×#447	C₂×C₈₉	1	2	yes	complemented
301	C₂×#433	C₂×C₈₉	1	1	yes	characteristic, complemented
302	C₈₉	C₂×#3220	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
304	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
305	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	complemented
306	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
307	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
308	C₂²×D₈	C₂²×C₈₉	1	2	yes	complemented
309	#540	—	2	4	no	—
310	#540	C₂²×C₈₉	1	8	yes	complemented
311	#535	—	2	4	no	—
312	#535	C₂²×C₈₉	1	8	yes	complemented
313	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
314	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
315	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic, complemented
316	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic, complemented
317	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
318	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
319	C₂×#123	—	2	2	no	—
320	C₂×#121	—	2	2	no	—
321	C₂×#121	—	2	2	no	—
322	C₂×#121	C₂²×C₈₉	1	2	yes	—
323	C₂×#120	—	2	2	no	—
324	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	complemented
325	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
326	#496	C₂²×C₈₉	1	8	yes	complemented
327	#493	C₂²×C₈₉	1	8	yes	complemented
328	#490	C₂²×C₈₉	1	8	yes	complemented
329	#488	C₂²×C₈₉	1	8	yes	complemented
330	#463	C₂²×C₈₉	1	16	yes	complemented
331	#459	C₂²×C₈₉	1	16	yes	complemented
332	#449	C₂²×C₈₉	1	16	yes	complemented
333	#447	C₂²×C₈₉	1	16	yes	complemented
334	#433	C₂²×C₈₉	1	8	yes	complemented
335	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	complemented
336	C₂×#103	C₂²×C₈₉	1	2	yes	complemented
337	C₂×#102	C₂²×C₈₉	1	2	yes	—
338	C₂×#102	C₂²×C₈₉	1	2	yes	—
339	C₂×#102	C₂²×C₈₉	1	2	yes	complemented
340	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
341	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic, complemented
342	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
343	C₂⁵	—	2	1	no	—
344	C₂×#41	—	2	2	no	—
345	C₂×#41	—	4	4	no	—
346	C₂×#41	C₂³×C₈₉	1	4	yes	—
347	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
348	C₂²×D₄	—	2	1	no	—
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	2	1	no	—
351	C₂²×D₄	—	2	2	no	—
352	C₂²×D₄	—	2	2	no	—
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	4	4	no	—
356	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
357	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
359	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
360	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
361	C₂³×C₄	—	2	1	no	—
362	C₂³×C₄	—	2	2	no	—
363	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
364	C₂×SD₁₆	—	4	4	no	—
365	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	complemented
366	C₂×D₈	—	4	4	no	—
367	C₂×D₈	—	4	4	no	—
368	C₂×D₈	—	4	4	no	—
369	C₂×D₈	C₂³×C₈₉	1	8	yes	—
370	C₂×D₈	C₂³×C₈₉	1	8	yes	—
371	C₂×D₈	C₂³×C₈₉	1	8	yes	complemented
372	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
373	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
374	#127	—	2	2	no	—
375	#127	—	2	2	no	—
376	#127	—	2	2	no	—
377	#127	C₂³×C₈₉	1	4	yes	—
378	#127	C₂³×C₈₉	1	4	yes	—
379	#127	C₂³×C₈₉	1	4	yes	complemented
380	#125	—	2	2	no	—
381	#125	C₂³×C₈₉	1	4	yes	complemented
382	#124	—	2	2	no	—
383	#124	—	2	2	no	—
384	#124	C₂³×C₈₉	1	4	yes	—
385	#124	C₂³×C₈₉	1	4	yes	—
386	#123	—	2	16	no	—
387	#121	—	2	4	no	—
388	#121	—	2	16	no	—
389	#121	—	2	16	no	—
390	#121	C₂³×C₈₉	1	8	yes	—
391	#120	—	2	16	no	—
392	C₄×D₄	—	2	4	no	—
393	C₄×D₄	C₂³×C₈₉	1	8	yes	complemented
394	#117	D₄×C₈₉	1	4	yes	—
395	#117	C₂³×C₈₉	1	4	yes	—
396	C₂×C₄⋊C₄	—	2	2	no	—
397	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
398	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
399	C₂×C₂²⋊C₄	—	2	2	no	—
400	C₂×C₂²⋊C₄	—	2	2	no	—
401	C₂×C₂²⋊C₄	—	2	2	no	—
402	C₂×C₂²⋊C₄	—	2	2	no	—
403	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
404	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₈₉	1	8	yes	complemented
407	#103	C₂³×C₈₉	1	8	yes	complemented
408	#102	C₂³×C₈₉	1	8	yes	—
409	#102	C₂³×C₈₉	1	8	yes	—
410	#102	C₂³×C₈₉	1	8	yes	complemented
411	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
412	#97	C₂³×C₈₉	1	4	yes	complemented
413	C₄×C₈	C₂³×C₈₉	1	4	yes	complemented
414	C₂⁴	—	2	2	no	—
415	C₂⁴	—	2	1	no	—
416	C₂⁴	—	2	1	no	—
417	C₂⁴	—	2	1	no	—
418	C₂⁴	—	2	4	no	—
419	C₂⁴	—	4	4	no	—
420	C₂⁴	—	8	4	no	—
421	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
422	#41	—	4	8	no	—
423	#41	—	4	8	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	8	no	—
435	C₂×D₄	—	2	8	no	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	4	8	no	—
443	C₂×D₄	—	4	4	no	—
444	C₂×D₄	—	4	8	no	—
445	C₂×D₄	—	4	4	no	—
446	C₂×D₄	—	8	4	no	—
447	C₂×D₄	—	8	4	no	—
448	C₂×D₄	—	8	4	no	—
449	C₂×D₄	—	8	4	no	—
450	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
451	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
452	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
453	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	8	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	—	8	4	no	—
461	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
462	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
463	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
466	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
467	SD₁₆	—	4	8	no	—
468	D₈	—	4	8	no	—
469	D₈	—	4	8	no	—
470	D₈	—	4	8	no	—
471	C₂×C₈	—	2	4	no	—
472	C₂×C₈	—	2	4	no	—
473	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
474	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
478	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
479	C₂²⋊C₄	—	2	8	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
484	C₄²	C₂×D₄×C₈₉	1	2	yes	—
485	C₄²	C₂×D₄×C₈₉	1	2	yes	—
486	C₄²	C₂⁴×C₈₉	1	2	yes	—
487	C₄²	C₂⁴×C₈₉	1	2	yes	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	4	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	4	8	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	4	no	—
499	C₂³	—	4	4	no	—
500	C₂³	—	4	4	no	—
501	C₂³	—	4	4	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	8	8	no	—
505	C₂³	—	8	16	no	—
506	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
507	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
508	Q₈	—	2	4	no	—
509	Q₈	—	4	2	no	—
510	D₄	—	2	4	no	—
511	D₄	—	2	4	no	—
512	D₄	—	2	4	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	4	no	—
515	D₄	—	4	8	no	—
516	D₄	—	4	8	no	—
517	D₄	—	4	4	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	2	no	—
521	D₄	—	8	8	no	—
522	D₄	—	8	8	no	—
523	D₄	—	8	8	no	—
524	D₄	—	8	8	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	4	8	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	4	no	—
535	C₂×C₄	—	8	8	no	—
536	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
537	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
538	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
540	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
541	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
542	C₈	—	2	4	no	—
543	C₈	—	2	4	no	—
544	C₂²	—	2	4	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	4	4	no	—
547	C₂²	—	4	4	no	—
548	C₂²	—	4	4	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	4	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	8	16	no	—
557	C₂²	—	8	4	no	—
558	C₂²	—	8	4	no	—
559	C₂²	—	8	4	no	—
560	C₂²	—	8	8	no	—
561	C₂²	#535×C₈₉	1	2	yes	—
562	C₂²	#576×C₈₉	1	4	yes	—
563	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
564	C₄	—	2	4	no	—
565	C₄	—	2	4	no	—
566	C₄	—	2	2	no	—
567	C₄	—	4	4	no	—
568	C₄	—	4	2	no	—
569	C₄	#535×C₈₉	1	2	yes	—
570	C₄	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
571	C₂	—	4	2	no	—
572	C₂	—	8	4	no	—
573	C₂	—	8	2	no	—
574	C₂	—	8	2	no	—
575	C₂	—	8	2	no	—
576	C₂	C₈₉×#3220	1	4	yes	direct product factor
577	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
578	C₂	C₂×#576×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
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×#540×C₈₉	C₂	semidirect
2. C₂×#535×C₈₉	C₂	semidirect
3. C₈₉×#3220	C₂	direct (N × H)
4. C₂×#496×C₈₉	C₂	semidirect
5. C₂×#493×C₈₉	C₂	semidirect
6. C₂×#490×C₈₉	C₂	semidirect
7. C₂×#488×C₈₉	C₂	semidirect
8. C₂×#463×C₈₉	C₂	semidirect
9. C₂×#459×C₈₉	C₂	semidirect
10. C₂×#449×C₈₉	C₂	semidirect
11. C₂×#447×C₈₉	C₂	semidirect
12. C₂×#433×C₈₉	C₂	semidirect
15. C₂²×SD₁₆×C₈₉	C₂²	semidirect
18. C₂²×D₈×C₈₉	C₂²	semidirect
20. #540×C₈₉	C₂²	semidirect
22. #535×C₈₉	C₂²	semidirect
25. C₂×#127×C₈₉	C₂²	semidirect
26. C₂×#125×C₈₉	C₂²	semidirect
34. C₂×C₄×D₄×C₈₉	C₂²	semidirect
36. #496×C₈₉	C₂²	semidirect
37. #493×C₈₉	C₂²	semidirect
38. #490×C₈₉	C₂²	semidirect
39. #488×C₈₉	C₂²	semidirect
40. #463×C₈₉	C₂²	semidirect
41. #459×C₈₉	C₂²	semidirect
42. #449×C₈₉	C₂²	semidirect
43. #447×C₈₉	C₂²	semidirect
44. #433×C₈₉	C₂²	semidirect
45. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
46. C₂×#103×C₈₉	C₂²	semidirect
49. C₂×#102×C₈₉	C₂²	semidirect
51. C₂×#97×C₈₉	C₂²	semidirect
52. C₂×C₄×C₈×C₈₉	C₂²	semidirect
75. C₂×SD₁₆×C₈₉	C₂³	semidirect
81. C₂×D₈×C₈₉	C₂³	semidirect
89. #127×C₈₉	C₂³	semidirect
91. #125×C₈₉	C₂³	semidirect
103. C₄×D₄×C₈₉	C₂³	semidirect
116. C₄⋊C₈×C₈₉	C₂³	semidirect
117. #103×C₈₉	C₂³	semidirect
120. #102×C₈₉	C₂³	semidirect
122. #97×C₈₉	C₂³	semidirect
123. C₄×C₈×C₈₉	C₂³	semidirect
281. C₂×#3220	C₈₉	direct (N × H)
287. C₂×C₈₉	#3220	direct (N × H)
290. C₂×#540	C₂×C₈₉	semidirect
291. C₂×#535	C₂×C₈₉	semidirect
292. #3220	C₂×C₈₉	direct (N × H)
293. C₂×#496	C₂×C₈₉	semidirect
294. C₂×#493	C₂×C₈₉	semidirect
295. C₂×#490	C₂×C₈₉	semidirect
296. C₂×#488	C₂×C₈₉	semidirect
297. C₂×#463	C₂×C₈₉	semidirect
298. C₂×#459	C₂×C₈₉	semidirect
299. C₂×#449	C₂×C₈₉	semidirect
300. C₂×#447	C₂×C₈₉	semidirect
301. C₂×#433	C₂×C₈₉	semidirect
302. C₈₉	C₂×#3220	direct (N × H)
305. C₂²×SD₁₆	C₂²×C₈₉	semidirect
308. C₂²×D₈	C₂²×C₈₉	semidirect
310. #540	C₂²×C₈₉	semidirect
312. #535	C₂²×C₈₉	semidirect
315. C₂×#127	C₂²×C₈₉	semidirect
316. C₂×#125	C₂²×C₈₉	semidirect
324. C₂×C₄×D₄	C₂²×C₈₉	semidirect
326. #496	C₂²×C₈₉	semidirect
327. #493	C₂²×C₈₉	semidirect
328. #490	C₂²×C₈₉	semidirect
329. #488	C₂²×C₈₉	semidirect
330. #463	C₂²×C₈₉	semidirect
331. #459	C₂²×C₈₉	semidirect
332. #449	C₂²×C₈₉	semidirect
333. #447	C₂²×C₈₉	semidirect
334. #433	C₂²×C₈₉	semidirect
335. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
336. C₂×#103	C₂²×C₈₉	semidirect
339. C₂×#102	C₂²×C₈₉	semidirect
341. C₂×#97	C₂²×C₈₉	semidirect
342. C₂×C₄×C₈	C₂²×C₈₉	semidirect
365. C₂×SD₁₆	C₂³×C₈₉	semidirect
371. C₂×D₈	C₂³×C₈₉	semidirect
379. #127	C₂³×C₈₉	semidirect
381. #125	C₂³×C₈₉	semidirect
393. C₄×D₄	C₂³×C₈₉	semidirect
406. C₄⋊C₈	C₂³×C₈₉	semidirect
407. #103	C₂³×C₈₉	semidirect
410. #102	C₂³×C₈₉	semidirect
412. #97	C₂³×C₈₉	semidirect
413. C₄×C₈	C₂³×C₈₉	semidirect
576. C₂	C₈₉×#3220	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×#540×C₈₉	2
C₂×#535×C₈₉	2
C₈₉×#3220	2
C₂×#496×C₈₉	2
C₂×#493×C₈₉	2
C₂×#490×C₈₉	2
C₂×#488×C₈₉	2
C₂×#463×C₈₉	2
C₂×#459×C₈₉	2
C₂×#449×C₈₉	2
C₂×#447×C₈₉	2
C₂×#433×C₈₉	2
C₂×#3220	89

Subgroups — C2×C89×#3220 #36438639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3220 #36438639