Subgroups — C₂×C₈₉×#3040 #36438262

Order 22784.

2918 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×AGL(1,ℤ/8)×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#124×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#3040	C₂	1	16	yes	maximal, direct product factor
6	C₂×#464×C₈₉	C₂	1	4	yes	maximal, complemented
7	C₂×#463×C₈₉	C₂	1	4	yes	maximal, complemented
8	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#419×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#418×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#137×C₈₉	—	2	8	no	—
15	C₂×#137×C₈₉	C₂²	1	8	yes	complemented
16	C₂×AGL(1,ℤ/8)×C₈₉	—	2	8	no	—
17	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	8	yes	—
18	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	complemented
19	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	complemented
21	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
22	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#124×C₈₉	—	2	1	no	—
25	C₂×#124×C₈₉	—	2	1	no	—
26	C₂×#124×C₈₉	—	2	4	no	—
27	C₂×#124×C₈₉	C₂²	1	4	yes	—
28	C₂×#124×C₈₉	C₂²	1	8	yes	complemented
29	C₄×#41×C₈₉	—	2	4	no	—
30	C₄×#41×C₈₉	C₂²	1	8	yes	complemented
31	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	complemented
32	C₂×C₄×D₄×C₈₉	—	2	2	no	—
33	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	complemented
34	C₂×#117×C₈₉	—	2	1	no	—
35	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₈₉	—	2	2	no	—
37	C₂²×C₄²×C₈₉	—	2	1	no	—
38	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
39	#464×C₈₉	C₂²	1	32	yes	complemented
40	#463×C₈₉	C₂²	1	32	yes	complemented
41	#423×C₈₉	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	complemented
43	#419×C₈₉	C₂²	1	8	yes	complemented
44	#418×C₈₉	C₂²	1	8	yes	complemented
45	C₂×#103×C₈₉	C₂²	1	4	yes	complemented
46	C₂×#102×C₈₉	C₂²	1	4	yes	—
47	C₂⁵×C₈₉	—	2	1	no	—
48	C₂×#41×C₈₉	—	2	2	no	—
49	C₂×#41×C₈₉	—	2	4	no	—
50	C₂×#41×C₈₉	D₄	1	8	yes	—
51	C₂×#41×C₈₉	C₂³	1	4	yes	—
52	C₂²×Q₈×C₈₉	—	2	2	no	—
53	C₂²×Q₈×C₈₉	—	2	1	no	—
54	C₂²×Q₈×C₈₉	—	2	4	no	—
55	C₂²×Q₈×C₈₉	—	2	4	no	—
56	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
57	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
58	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₈₉	—	2	1	no	—
60	C₂²×D₄×C₈₉	—	2	4	no	—
61	C₂²×D₄×C₈₉	—	2	2	no	—
62	C₂²×D₄×C₈₉	—	2	1	no	—
63	C₂²×D₄×C₈₉	—	2	4	no	—
64	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
65	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
66	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₈₉	—	2	1	no	—
68	C₂³×C₄×C₈₉	—	2	1	no	—
69	C₂³×C₄×C₈₉	—	2	2	no	—
70	C₂³×C₄×C₈₉	D₄	1	2	yes	—
71	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
72	#137×C₈₉	—	2	32	no	—
73	AGL(1,ℤ/8)×C₈₉	—	2	32	no	—
74	C₂×Q₁₆×C₈₉	—	2	8	no	—
75	C₂×Q₁₆×C₈₉	C₂³	1	8	yes	complemented
76	C₂×SD₁₆×C₈₉	—	2	8	no	—
77	C₂×SD₁₆×C₈₉	—	2	8	no	—
78	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
79	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	complemented
80	C₂×D₈×C₈₉	—	2	8	no	—
81	C₂×D₈×C₈₉	C₂³	1	8	yes	—
82	C₂×M₁₆×C₈₉	—	2	4	no	—
83	C₂×M₁₆×C₈₉	—	2	4	no	—
84	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
85	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
86	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
87	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
88	#124×C₈₉	—	2	8	no	—
89	#124×C₈₉	—	2	8	no	—
90	#124×C₈₉	—	2	8	no	—
91	#124×C₈₉	C₂³	1	16	yes	—
92	C₄×Q₈×C₈₉	—	2	4	no	—
93	C₄×Q₈×C₈₉	C₂³	1	8	yes	complemented
94	C₄×D₄×C₈₉	—	2	16	no	—
95	C₄×D₄×C₈₉	—	2	4	no	—
96	C₄×D₄×C₈₉	C₂³	1	8	yes	complemented
97	#117×C₈₉	—	2	2	no	—
98	#117×C₈₉	—	2	8	no	—
99	#117×C₈₉	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
104	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
105	C₂×C₂²⋊C₄×C₈₉	—	2	16	no	—
106	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
107	C₂×C₄²×C₈₉	—	2	1	no	—
108	C₂×C₄²×C₈₉	—	2	2	no	—
109	C₂×C₄²×C₈₉	—	2	8	no	—
110	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₈₉	D₄	1	4	yes	—
113	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
114	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
115	C₄⋊C₈×C₈₉	C₂³	1	16	yes	complemented
116	#103×C₈₉	C₂³	1	16	yes	complemented
117	#102×C₈₉	C₂³	1	16	yes	—
118	C₂⁴×C₈₉	—	2	1	no	—
119	C₂⁴×C₈₉	—	2	4	no	—
120	C₂⁴×C₈₉	—	2	2	no	—
121	C₂⁴×C₈₉	—	4	8	no	—
122	C₂⁴×C₈₉	—	4	4	no	—
123	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
124	#41×C₈₉	—	2	16	no	—
125	#41×C₈₉	—	2	8	no	—
126	#41×C₈₉	C₂×D₄	1	16	yes	—
127	C₂×Q₈×C₈₉	—	2	1	no	—
128	C₂×Q₈×C₈₉	—	2	4	no	—
129	C₂×Q₈×C₈₉	—	2	2	no	—
130	C₂×Q₈×C₈₉	—	2	2	no	—
131	C₂×Q₈×C₈₉	—	2	4	no	—
132	C₂×Q₈×C₈₉	—	2	8	no	—
133	C₂×Q₈×C₈₉	—	2	4	no	—
134	C₂×Q₈×C₈₉	—	2	8	no	—
135	C₂×Q₈×C₈₉	—	4	4	no	—
136	C₂×Q₈×C₈₉	—	4	8	no	—
137	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
138	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
139	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
140	C₂×D₄×C₈₉	—	2	8	no	—
141	C₂×D₄×C₈₉	—	2	4	no	—
142	C₂×D₄×C₈₉	—	2	1	no	—
143	C₂×D₄×C₈₉	—	2	4	no	—
144	C₂×D₄×C₈₉	—	2	8	no	—
145	C₂×D₄×C₈₉	—	2	4	no	—
146	C₂×D₄×C₈₉	—	2	2	no	—
147	C₂×D₄×C₈₉	—	2	2	no	—
148	C₂×D₄×C₈₉	—	2	4	no	—
149	C₂×D₄×C₈₉	—	2	8	no	—
150	C₂×D₄×C₈₉	—	4	4	no	—
151	C₂×D₄×C₈₉	—	4	8	no	—
152	C₂×D₄×C₈₉	C₂×D₄	1	4	yes	—
153	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
154	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₈₉	—	2	1	no	—
156	C₂²×C₄×C₈₉	—	2	8	no	—
157	C₂²×C₄×C₈₉	—	2	4	no	—
158	C₂²×C₄×C₈₉	—	2	4	no	—
159	C₂²×C₄×C₈₉	—	2	2	no	—
160	C₂²×C₄×C₈₉	—	2	2	no	—
161	C₂²×C₄×C₈₉	—	2	8	no	—
162	C₂²×C₄×C₈₉	—	4	4	no	—
163	C₂²×C₄×C₈₉	—	4	8	no	—
164	C₂²×C₄×C₈₉	—	4	4	no	—
165	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
168	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₈₉	#41	1	4	yes	—
172	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
174	Q₁₆×C₈₉	—	2	16	no	—
175	SD₁₆×C₈₉	—	2	16	no	—
176	SD₁₆×C₈₉	—	2	16	no	—
177	D₈×C₈₉	—	2	16	no	—
178	M₁₆×C₈₉	—	2	8	no	—
179	M₁₆×C₈₉	—	2	8	no	—
180	C₂×C₈×C₈₉	—	2	4	no	—
181	C₂×C₈×C₈₉	—	2	4	no	—
182	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
183	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
184	C₄⋊C₄×C₈₉	—	2	8	no	—
185	C₄⋊C₄×C₈₉	—	2	2	no	—
186	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
187	C₂²⋊C₄×C₈₉	—	2	4	no	—
188	C₂²⋊C₄×C₈₉	—	2	16	no	—
189	C₂²⋊C₄×C₈₉	—	2	16	no	—
190	C₂²⋊C₄×C₈₉	—	2	8	no	—
191	C₄²×C₈₉	—	2	4	no	—
192	C₄²×C₈₉	—	2	8	no	—
193	C₄²×C₈₉	C₂×D₄	1	4	yes	—
194	C₄²×C₈₉	C₂×D₄	1	4	yes	—
195	C₄²×C₈₉	C₂×D₄	1	4	yes	—
196	C₄²×C₈₉	C₂⁴	1	4	yes	—
197	C₂³×C₈₉	—	2	4	no	—
198	C₂³×C₈₉	—	2	4	no	—
199	C₂³×C₈₉	—	2	2	no	—
200	C₂³×C₈₉	—	2	4	no	—
201	C₂³×C₈₉	—	4	16	no	—
202	C₂³×C₈₉	—	4	4	no	—
203	C₂³×C₈₉	—	4	8	no	—
204	C₂³×C₈₉	—	4	4	no	—
205	C₂³×C₈₉	—	4	8	no	—
206	C₂³×C₈₉	—	8	16	no	—
207	C₂³×C₈₉	#121	1	4	yes	—
208	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
209	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₈₉	—	2	4	no	—
211	Q₈×C₈₉	—	2	2	no	—
212	Q₈×C₈₉	—	2	4	no	—
213	Q₈×C₈₉	—	2	4	no	—
214	Q₈×C₈₉	—	4	8	no	—
215	Q₈×C₈₉	—	4	4	no	—
216	Q₈×C₈₉	C₂²×D₄	1	4	yes	—
217	D₄×C₈₉	—	2	16	no	—
218	D₄×C₈₉	—	2	4	no	—
219	D₄×C₈₉	—	2	4	no	—
220	D₄×C₈₉	—	2	2	no	—
221	D₄×C₈₉	—	2	4	no	—
222	D₄×C₈₉	—	4	8	no	—
223	D₄×C₈₉	—	4	4	no	—
224	D₄×C₈₉	C₂²×D₄	1	4	yes	—
225	C₂×C₄×C₈₉	—	2	4	no	—
226	C₂×C₄×C₈₉	—	2	4	no	—
227	C₂×C₄×C₈₉	—	2	4	no	—
228	C₂×C₄×C₈₉	—	2	2	no	—
229	C₂×C₄×C₈₉	—	2	16	no	—
230	C₂×C₄×C₈₉	—	2	4	no	—
231	C₂×C₄×C₈₉	—	2	4	no	—
232	C₂×C₄×C₈₉	—	4	8	no	—
233	C₂×C₄×C₈₉	—	4	16	no	—
234	C₂×C₄×C₈₉	—	4	8	no	—
235	C₂×C₄×C₈₉	—	4	4	no	—
236	C₂×C₄×C₈₉	—	4	4	no	—
237	C₂×C₄×C₈₉	#121	1	4	yes	—
238	C₂×C₄×C₈₉	#121	1	8	yes	—
239	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
240	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
246	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
247	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
248	C₈×C₈₉	—	2	4	no	—
249	C₈×C₈₉	—	2	4	no	—
250	C₂²×C₈₉	—	2	8	no	—
251	C₂²×C₈₉	—	2	4	no	—
252	C₂²×C₈₉	—	2	4	no	—
253	C₂²×C₈₉	—	4	8	no	—
254	C₂²×C₈₉	—	4	4	no	—
255	C₂²×C₈₉	—	4	4	no	—
256	C₂²×C₈₉	—	4	8	no	—
257	C₂²×C₈₉	—	8	16	no	—
258	C₂²×C₈₉	—	8	8	no	—
259	C₂²×C₈₉	C₂×#121	1	4	yes	—
260	C₂²×C₈₉	C₂×#121	1	2	yes	—
261	C₂²×C₈₉	#575	1	4	yes	—
262	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
263	C₄×C₈₉	—	2	4	no	—
264	C₄×C₈₉	—	2	4	no	—
265	C₄×C₈₉	—	2	4	no	—
266	C₄×C₈₉	—	4	4	no	—
267	C₄×C₈₉	—	4	4	no	—
268	C₄×C₈₉	C₂×#121	1	4	yes	—
269	C₄×C₈₉	C₂×#121	1	2	yes	—
270	C₄×C₈₉	C₂³×D₄	1	1	yes	characteristic
271	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
272	C₂×#3040	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂×C₈₉	—	2	4	no	—
274	C₂×C₈₉	—	4	4	no	—
275	C₂×C₈₉	—	8	4	no	—
276	C₂×C₈₉	#3040	1	4	yes	direct product factor
277	C₂×C₈₉	C₂²×#121	1	1	yes	characteristic
278	C₂×C₈₉	C₂×#575	1	2	yes	—
279	C₂²×#137	C₂×C₈₉	1	1	yes	characteristic, complemented
280	C₂²×AGL(1,ℤ/8)	C₂×C₈₉	1	1	yes	characteristic
281	C₂²×#124	C₂×C₈₉	1	1	yes	characteristic, complemented
282	C₂×C₄×#41	C₂×C₈₉	1	1	yes	characteristic, complemented
283	#3040	C₂×C₈₉	1	16	yes	direct product factor
284	C₂×#464	C₂×C₈₉	1	4	yes	complemented
285	C₂×#463	C₂×C₈₉	1	4	yes	complemented
286	C₂×#423	C₂×C₈₉	1	1	yes	characteristic, complemented
287	C₂×#419	C₂×C₈₉	1	1	yes	characteristic, complemented
288	C₂×#418	C₂×C₈₉	1	1	yes	characteristic, complemented
289	C₈₉	C₂×#3040	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
291	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
292	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
293	C₂×#137	—	2	8	no	—
294	C₂×#137	C₂²×C₈₉	1	8	yes	complemented
295	C₂×AGL(1,ℤ/8)	—	2	8	no	—
296	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	8	yes	—
297	C₂²×Q₁₆	C₂²×C₈₉	1	2	yes	complemented
298	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
299	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	complemented
300	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
301	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
302	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
303	C₂×#124	—	2	1	no	—
304	C₂×#124	—	2	4	no	—
305	C₂×#124	—	2	1	no	—
306	C₂×#124	C₂²×C₈₉	1	4	yes	—
307	C₂×#124	C₂²×C₈₉	1	8	yes	complemented
308	C₄×#41	—	2	4	no	—
309	C₄×#41	C₂²×C₈₉	1	8	yes	complemented
310	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	complemented
311	C₂×C₄×D₄	—	2	2	no	—
312	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	complemented
313	C₂×#117	—	2	1	no	—
314	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
315	C₂²×C₂²⋊C₄	—	2	2	no	—
316	C₂²×C₄²	—	2	1	no	—
317	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
318	#464	C₂²×C₈₉	1	32	yes	complemented
319	#463	C₂²×C₈₉	1	32	yes	complemented
320	#423	C₂²×C₈₉	1	8	yes	complemented
321	C₂×C₄⋊C₈	C₂²×C₈₉	1	4	yes	complemented
322	#419	C₂²×C₈₉	1	8	yes	complemented
323	#418	C₂²×C₈₉	1	8	yes	complemented
324	C₂×#103	C₂²×C₈₉	1	4	yes	complemented
325	C₂×#102	C₂²×C₈₉	1	4	yes	—
326	C₂⁵	—	2	1	no	—
327	C₂×#41	—	2	2	no	—
328	C₂×#41	—	2	4	no	—
329	C₂×#41	D₄×C₈₉	1	8	yes	—
330	C₂×#41	C₂³×C₈₉	1	4	yes	—
331	C₂²×Q₈	—	2	1	no	—
332	C₂²×Q₈	—	2	2	no	—
333	C₂²×Q₈	—	2	4	no	—
334	C₂²×Q₈	—	2	4	no	—
335	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
336	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
337	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
338	C₂²×D₄	—	2	1	no	—
339	C₂²×D₄	—	2	4	no	—
340	C₂²×D₄	—	2	1	no	—
341	C₂²×D₄	—	2	2	no	—
342	C₂²×D₄	—	2	4	no	—
343	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
344	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
345	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
346	C₂³×C₄	—	2	1	no	—
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	—	2	2	no	—
349	C₂³×C₄	D₄×C₈₉	1	2	yes	—
350	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
351	#137	—	2	32	no	—
352	AGL(1,ℤ/8)	—	2	32	no	—
353	C₂×Q₁₆	—	2	8	no	—
354	C₂×Q₁₆	C₂³×C₈₉	1	8	yes	complemented
355	C₂×SD₁₆	—	2	8	no	—
356	C₂×SD₁₆	—	2	8	no	—
357	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
358	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	complemented
359	C₂×D₈	—	2	8	no	—
360	C₂×D₈	C₂³×C₈₉	1	8	yes	—
361	C₂×M₁₆	—	2	4	no	—
362	C₂×M₁₆	—	2	4	no	—
363	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
364	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
365	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
366	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
367	#124	—	2	8	no	—
368	#124	—	2	8	no	—
369	#124	—	2	8	no	—
370	#124	C₂³×C₈₉	1	16	yes	—
371	C₄×Q₈	—	2	4	no	—
372	C₄×Q₈	C₂³×C₈₉	1	8	yes	complemented
373	C₄×D₄	—	2	16	no	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	C₂³×C₈₉	1	8	yes	complemented
376	#117	—	2	2	no	—
377	#117	—	2	8	no	—
378	#117	C₂³×C₈₉	1	4	yes	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	4	no	—
383	C₂×C₂²⋊C₄	—	2	4	no	—
384	C₂×C₂²⋊C₄	—	2	16	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	—	2	2	no	—
388	C₂×C₄²	—	2	8	no	—
389	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
390	C₂×C₄²	D₄×C₈₉	1	4	yes	—
391	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
393	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
394	C₄⋊C₈	C₂³×C₈₉	1	16	yes	complemented
395	#103	C₂³×C₈₉	1	16	yes	complemented
396	#102	C₂³×C₈₉	1	16	yes	—
397	C₂⁴	—	2	1	no	—
398	C₂⁴	—	2	4	no	—
399	C₂⁴	—	2	2	no	—
400	C₂⁴	—	4	8	no	—
401	C₂⁴	—	4	4	no	—
402	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
403	#41	—	2	16	no	—
404	#41	—	2	8	no	—
405	#41	C₂×D₄×C₈₉	1	16	yes	—
406	C₂×Q₈	—	2	1	no	—
407	C₂×Q₈	—	2	4	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	4	no	—
411	C₂×Q₈	—	2	8	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	8	no	—
414	C₂×Q₈	—	4	4	no	—
415	C₂×Q₈	—	4	8	no	—
416	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
417	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
418	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	4	4	no	—
430	C₂×D₄	—	4	8	no	—
431	C₂×D₄	C₂×D₄×C₈₉	1	4	yes	—
432	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
433	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	8	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	8	no	—
441	C₂²×C₄	—	4	4	no	—
442	C₂²×C₄	—	4	8	no	—
443	C₂²×C₄	—	4	4	no	—
444	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
445	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
446	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
447	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
449	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
450	C₂²×C₄	#41×C₈₉	1	4	yes	—
451	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
452	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
453	Q₁₆	—	2	16	no	—
454	SD₁₆	—	2	16	no	—
455	SD₁₆	—	2	16	no	—
456	D₈	—	2	16	no	—
457	M₁₆	—	2	8	no	—
458	M₁₆	—	2	8	no	—
459	C₂×C₈	—	2	4	no	—
460	C₂×C₈	—	2	4	no	—
461	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
462	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
463	C₄⋊C₄	—	2	8	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	16	no	—
468	C₂²⋊C₄	—	2	16	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₄²	—	2	4	no	—
471	C₄²	—	2	8	no	—
472	C₄²	C₂×D₄×C₈₉	1	4	yes	—
473	C₄²	C₂×D₄×C₈₉	1	4	yes	—
474	C₄²	C₂×D₄×C₈₉	1	4	yes	—
475	C₄²	C₂⁴×C₈₉	1	4	yes	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	4	16	no	—
481	C₂³	—	4	4	no	—
482	C₂³	—	4	8	no	—
483	C₂³	—	4	4	no	—
484	C₂³	—	4	8	no	—
485	C₂³	—	8	16	no	—
486	C₂³	#121×C₈₉	1	4	yes	—
487	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
488	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
489	Q₈	—	2	4	no	—
490	Q₈	—	2	2	no	—
491	Q₈	—	2	4	no	—
492	Q₈	—	2	4	no	—
493	Q₈	—	4	8	no	—
494	Q₈	—	4	4	no	—
495	Q₈	C₂²×D₄×C₈₉	1	4	yes	—
496	D₄	—	2	16	no	—
497	D₄	—	2	4	no	—
498	D₄	—	2	4	no	—
499	D₄	—	2	2	no	—
500	D₄	—	2	4	no	—
501	D₄	—	4	8	no	—
502	D₄	—	4	4	no	—
503	D₄	C₂²×D₄×C₈₉	1	4	yes	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	16	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	4	8	no	—
512	C₂×C₄	—	4	16	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	#121×C₈₉	1	4	yes	—
517	C₂×C₄	#121×C₈₉	1	8	yes	—
518	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
519	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
520	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
521	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
523	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
524	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
525	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
526	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
527	C₈	—	2	4	no	—
528	C₈	—	2	4	no	—
529	C₂²	—	2	4	no	—
530	C₂²	—	2	8	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	4	8	no	—
534	C₂²	—	4	8	no	—
535	C₂²	—	4	4	no	—
536	C₂²	—	8	16	no	—
537	C₂²	—	8	8	no	—
538	C₂²	C₂×#121×C₈₉	1	4	yes	—
539	C₂²	C₂×#121×C₈₉	1	2	yes	—
540	C₂²	#575×C₈₉	1	4	yes	—
541	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
542	C₄	—	2	4	no	—
543	C₄	—	2	4	no	—
544	C₄	—	2	4	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	4	no	—
547	C₄	C₂×#121×C₈₉	1	4	yes	—
548	C₄	C₂×#121×C₈₉	1	2	yes	—
549	C₄	C₂³×D₄×C₈₉	1	1	yes	characteristic
550	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
551	C₂	—	2	4	no	—
552	C₂	—	4	4	no	—
553	C₂	—	8	4	no	—
554	C₂	C₈₉×#3040	1	4	yes	direct product factor
555	C₂	C₂²×#121×C₈₉	1	1	yes	characteristic
556	C₂	C₂×#575×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂²×AGL(1,ℤ/8)×C₈₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#137×C₈₉	C₂	semidirect
3. C₂²×#124×C₈₉	C₂	semidirect
4. C₂×C₄×#41×C₈₉	C₂	semidirect
5. C₈₉×#3040	C₂	direct (N × H)
6. C₂×#464×C₈₉	C₂	semidirect
7. C₂×#463×C₈₉	C₂	semidirect
8. C₂×#423×C₈₉	C₂	semidirect
9. C₂×#419×C₈₉	C₂	semidirect
10. C₂×#418×C₈₉	C₂	semidirect
15. C₂×#137×C₈₉	C₂²	semidirect
18. C₂²×Q₁₆×C₈₉	C₂²	semidirect
20. C₂²×SD₁₆×C₈₉	C₂²	semidirect
28. C₂×#124×C₈₉	C₂²	semidirect
30. C₄×#41×C₈₉	C₂²	semidirect
31. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
33. C₂×C₄×D₄×C₈₉	C₂²	semidirect
39. #464×C₈₉	C₂²	semidirect
40. #463×C₈₉	C₂²	semidirect
41. #423×C₈₉	C₂²	semidirect
42. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
43. #419×C₈₉	C₂²	semidirect
44. #418×C₈₉	C₂²	semidirect
45. C₂×#103×C₈₉	C₂²	semidirect
75. C₂×Q₁₆×C₈₉	C₂³	semidirect
79. C₂×SD₁₆×C₈₉	C₂³	semidirect
93. C₄×Q₈×C₈₉	C₂³	semidirect
96. C₄×D₄×C₈₉	C₂³	semidirect
115. C₄⋊C₈×C₈₉	C₂³	semidirect
116. #103×C₈₉	C₂³	semidirect
272. C₂×#3040	C₈₉	direct (N × H)
276. C₂×C₈₉	#3040	direct (N × H)
279. C₂²×#137	C₂×C₈₉	semidirect
281. C₂²×#124	C₂×C₈₉	semidirect
282. C₂×C₄×#41	C₂×C₈₉	semidirect
283. #3040	C₂×C₈₉	direct (N × H)
284. C₂×#464	C₂×C₈₉	semidirect
285. C₂×#463	C₂×C₈₉	semidirect
286. C₂×#423	C₂×C₈₉	semidirect
287. C₂×#419	C₂×C₈₉	semidirect
288. C₂×#418	C₂×C₈₉	semidirect
289. C₈₉	C₂×#3040	direct (N × H)
294. C₂×#137	C₂²×C₈₉	semidirect
297. C₂²×Q₁₆	C₂²×C₈₉	semidirect
299. C₂²×SD₁₆	C₂²×C₈₉	semidirect
307. C₂×#124	C₂²×C₈₉	semidirect
309. C₄×#41	C₂²×C₈₉	semidirect
310. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
312. C₂×C₄×D₄	C₂²×C₈₉	semidirect
318. #464	C₂²×C₈₉	semidirect
319. #463	C₂²×C₈₉	semidirect
320. #423	C₂²×C₈₉	semidirect
321. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
322. #419	C₂²×C₈₉	semidirect
323. #418	C₂²×C₈₉	semidirect
324. C₂×#103	C₂²×C₈₉	semidirect
354. C₂×Q₁₆	C₂³×C₈₉	semidirect
358. C₂×SD₁₆	C₂³×C₈₉	semidirect
372. C₄×Q₈	C₂³×C₈₉	semidirect
375. C₄×D₄	C₂³×C₈₉	semidirect
394. C₄⋊C₈	C₂³×C₈₉	semidirect
395. #103	C₂³×C₈₉	semidirect
554. C₂	C₈₉×#3040	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#137×C₈₉	2
C₂²×AGL(1,ℤ/8)×C₈₉	2
C₂²×#124×C₈₉	2
C₂×C₄×#41×C₈₉	2
C₈₉×#3040	2
C₂×#464×C₈₉	2
C₂×#463×C₈₉	2
C₂×#423×C₈₉	2
C₂×#419×C₈₉	2
C₂×#418×C₈₉	2
C₂×#3040	89

Subgroups — C2×C89×#3040 #36438262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3040 #36438262