Subgroups — C₄×#575×C₈₉ #36438006

Order 22784.

2842 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#575×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₈₉	C₂	1	2	yes	maximal, complemented
3	C₄×#137×C₈₉	C₂	1	8	yes	maximal, complemented
4	C₄×AGL(1,ℤ/8)×C₈₉	C₂	1	8	yes	maximal, complemented
5	C₈₉×#2943	C₂	1	4	yes	maximal, complemented
6	C₄×#135×C₈₉	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2911	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2896	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₈₉	C₂²	1	2	yes	—
11	#575×C₈₉	C₄	1	16	yes	direct product factor
12	C₂×#137×C₈₉	C₂²	1	4	yes	—
13	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	4	yes	—
14	C₂×#135×C₈₉	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×#41×C₈₉	—	2	4	no	—
18	C₄×#41×C₈₉	—	2	2	no	—
19	C₄×#41×C₈₉	—	2	2	no	—
20	C₄×#41×C₈₉	C₂²	1	8	yes	—
21	C₄×#41×C₈₉	C₂²	1	4	yes	complemented
22	C₄×#41×C₈₉	C₂²	1	4	yes	complemented
23	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	complemented
24	C₂×C₄×D₄×C₈₉	—	2	4	no	—
25	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	complemented
26	C₂×#117×C₈₉	—	2	2	no	—
27	C₂×#117×C₈₉	C₂²	1	2	yes	—
28	C₂²×C₄²×C₈₉	—	2	2	no	—
29	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
30	#442×C₈₉	C₂²	1	8	yes	complemented
31	#441×C₈₉	C₂²	1	8	yes	complemented
32	#440×C₈₉	C₂²	1	16	yes	complemented
33	C₄×Q₁₆×C₈₉	C₂²	1	8	yes	complemented
34	C₄×SD₁₆×C₈₉	C₂²	1	16	yes	complemented
35	C₄×D₈×C₈₉	C₂²	1	8	yes	complemented
36	#428×C₈₉	C₂²	1	8	yes	complemented
37	#427×C₈₉	C₂²	1	4	yes	complemented
38	#419×C₈₉	C₂²	1	4	yes	complemented
39	#418×C₈₉	C₂²	1	4	yes	complemented
40	#417×C₈₉	C₂²	1	8	yes	—
41	#416×C₈₉	C₂²	1	8	yes	complemented
42	C₄×M₁₆×C₈₉	C₂²	1	8	yes	complemented
43	C₂×#97×C₈₉	C₂²	1	2	yes	complemented
44	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	complemented
45	C₂×#41×C₈₉	—	2	2	no	—
46	C₂×#41×C₈₉	—	2	2	no	—
47	C₂×#41×C₈₉	—	2	8	no	—
48	C₂×#41×C₈₉	C₂×C₄	1	16	yes	complemented
49	C₂×#41×C₈₉	C₂³	1	4	yes	—
50	C₂×#41×C₈₉	C₂³	1	4	yes	—
51	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
52	C₂²×D₄×C₈₉	—	2	2	no	—
53	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
54	C₂³×C₄×C₈₉	—	2	2	no	—
55	C₂³×C₄×C₈₉	—	2	4	no	—
56	C₂³×C₄×C₈₉	D₄	1	2	yes	—
57	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
58	#137×C₈₉	C₂×C₄	1	32	yes	complemented
59	AGL(1,ℤ/8)×C₈₉	C₂×C₄	1	32	yes	complemented
60	#135×C₈₉	C₂×C₄	1	32	yes	complemented
61	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
62	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
63	C₂×D₈×C₈₉	C₂³	1	4	yes	—
64	C₂×M₁₆×C₈₉	—	2	4	no	—
65	C₂×M₁₆×C₈₉	C₂×C₄	1	8	yes	complemented
66	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
67	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
68	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
69	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
70	C₄×Q₈×C₈₉	—	2	4	no	—
71	C₄×Q₈×C₈₉	—	2	4	no	—
72	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
73	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
74	C₄×D₄×C₈₉	—	2	8	no	—
75	C₄×D₄×C₈₉	—	2	8	no	—
76	C₄×D₄×C₈₉	—	2	8	no	—
77	C₄×D₄×C₈₉	—	2	8	no	—
78	C₄×D₄×C₈₉	—	2	4	no	—
79	C₄×D₄×C₈₉	—	2	4	no	—
80	C₄×D₄×C₈₉	C₂³	1	8	yes	—
81	C₄×D₄×C₈₉	C₂³	1	8	yes	—
82	#117×C₈₉	—	2	2	no	—
83	#117×C₈₉	—	2	2	no	—
84	#117×C₈₉	—	2	8	no	—
85	#117×C₈₉	—	2	4	no	—
86	#117×C₈₉	—	2	4	no	—
87	#117×C₈₉	C₂³	1	4	yes	—
88	#117×C₈₉	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
93	C₂×C₄²×C₈₉	—	2	2	no	—
94	C₂×C₄²×C₈₉	—	2	4	no	—
95	C₂×C₄²×C₈₉	—	2	8	no	—
96	C₂×C₄²×C₈₉	—	2	4	no	—
97	C₂×C₄²×C₈₉	—	2	4	no	—
98	C₂×C₄²×C₈₉	D₄	1	2	yes	—
99	C₂×C₄²×C₈₉	D₄	1	2	yes	—
100	C₂×C₄²×C₈₉	D₄	1	2	yes	—
101	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
103	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
104	#107×C₈₉	C₂³	1	8	yes	—
105	#106×C₈₉	C₂³	1	8	yes	—
106	#103×C₈₉	C₂³	1	16	yes	—
107	#102×C₈₉	C₂³	1	16	yes	—
108	#97×C₈₉	C₂³	1	8	yes	—
109	C₄×C₈×C₈₉	C₂³	1	8	yes	—
110	C₂⁴×C₈₉	—	2	2	no	—
111	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
112	#41×C₈₉	—	2	16	no	—
113	#41×C₈₉	—	2	16	no	—
114	#41×C₈₉	—	2	8	no	—
115	#41×C₈₉	—	2	8	no	—
116	#41×C₈₉	C₂²×C₄	1	16	yes	—
117	#41×C₈₉	C₂²×C₄	1	16	yes	complemented
118	C₂×Q₈×C₈₉	—	2	2	no	—
119	C₂×Q₈×C₈₉	—	2	4	no	—
120	C₂×Q₈×C₈₉	C₂²×C₄	1	8	yes	complemented
121	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
122	C₂×D₄×C₈₉	—	2	16	no	—
123	C₂×D₄×C₈₉	—	2	4	no	—
124	C₂×D₄×C₈₉	—	2	4	no	—
125	C₂×D₄×C₈₉	—	2	2	no	—
126	C₂×D₄×C₈₉	—	2	4	no	—
127	C₂×D₄×C₈₉	C₂²×C₄	1	8	yes	complemented
128	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₈₉	—	2	2	no	—
130	C₂²×C₄×C₈₉	—	2	16	no	—
131	C₂²×C₄×C₈₉	—	2	4	no	—
132	C₂²×C₄×C₈₉	—	2	4	no	—
133	C₂²×C₄×C₈₉	—	2	4	no	—
134	C₂²×C₄×C₈₉	—	2	8	no	—
135	C₂²×C₄×C₈₉	—	2	8	no	—
136	C₂²×C₄×C₈₉	—	4	4	no	—
137	C₂²×C₄×C₈₉	—	4	16	no	—
138	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
139	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₈₉	#41	1	4	yes	—
146	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
148	Q₁₆×C₈₉	C₂²×C₄	1	16	yes	complemented
149	SD₁₆×C₈₉	C₂²×C₄	1	32	yes	complemented
150	D₈×C₈₉	C₂²×C₄	1	16	yes	complemented
151	M₁₆×C₈₉	—	2	8	no	—
152	M₁₆×C₈₉	C₂²×C₄	1	16	yes	complemented
153	C₂×C₈×C₈₉	—	2	4	no	—
154	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	complemented
155	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
156	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
157	C₄⋊C₄×C₈₉	—	2	8	no	—
158	C₄⋊C₄×C₈₉	—	2	8	no	—
159	C₄⋊C₄×C₈₉	—	2	4	no	—
160	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
161	C₂²⋊C₄×C₈₉	—	2	4	no	—
162	C₂²⋊C₄×C₈₉	—	2	4	no	—
163	C₂²⋊C₄×C₈₉	—	2	8	no	—
164	C₂²⋊C₄×C₈₉	—	2	8	no	—
165	C₄²×C₈₉	—	2	4	no	—
166	C₄²×C₈₉	—	2	4	no	—
167	C₄²×C₈₉	—	2	8	no	—
168	C₄²×C₈₉	—	2	8	no	—
169	C₄²×C₈₉	C₂×D₄	1	2	yes	—
170	C₄²×C₈₉	C₂×D₄	1	4	yes	—
171	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₈₉	C₂×D₄	1	4	yes	—
173	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₈₉	C₂⁴	1	2	yes	—
175	C₄²×C₈₉	C₂⁴	1	2	yes	—
176	C₂³×C₈₉	—	2	8	no	—
177	C₂³×C₈₉	—	2	4	no	—
178	C₂³×C₈₉	—	4	2	no	—
179	C₂³×C₈₉	—	4	8	no	—
180	C₂³×C₈₉	C₄×D₄	1	4	yes	—
181	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
182	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
183	Q₈×C₈₉	—	2	8	no	—
184	Q₈×C₈₉	—	2	4	no	—
185	Q₈×C₈₉	C₂³×C₄	1	8	yes	—
186	D₄×C₈₉	—	2	16	no	—
187	D₄×C₈₉	—	2	16	no	—
188	D₄×C₈₉	—	2	8	no	—
189	D₄×C₈₉	—	2	4	no	—
190	D₄×C₈₉	C₂³×C₄	1	8	yes	—
191	C₂×C₄×C₈₉	—	2	8	no	—
192	C₂×C₄×C₈₉	—	2	4	no	—
193	C₂×C₄×C₈₉	—	2	16	no	—
194	C₂×C₄×C₈₉	—	2	16	no	—
195	C₂×C₄×C₈₉	—	2	8	no	—
196	C₂×C₄×C₈₉	—	2	4	no	—
197	C₂×C₄×C₈₉	—	2	8	no	—
198	C₂×C₄×C₈₉	—	4	2	no	—
199	C₂×C₄×C₈₉	—	4	4	no	—
200	C₂×C₄×C₈₉	—	4	16	no	—
201	C₂×C₄×C₈₉	—	4	8	no	—
202	C₂×C₄×C₈₉	—	4	16	no	—
203	C₂×C₄×C₈₉	C₄×D₄	1	8	yes	—
204	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
205	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
206	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
207	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
208	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
209	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
213	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
214	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
215	C₈×C₈₉	—	2	4	no	—
216	C₈×C₈₉	C₂³×C₄	1	8	yes	—
217	C₂²×C₈₉	—	2	4	no	—
218	C₂²×C₈₉	—	2	2	no	—
219	C₂²×C₈₉	—	2	8	no	—
220	C₂²×C₈₉	—	4	4	no	—
221	C₂²×C₈₉	—	4	16	no	—
222	C₂²×C₈₉	—	4	8	no	—
223	C₂²×C₈₉	C₂×C₄×D₄	1	4	yes	—
224	C₂²×C₈₉	C₂×C₄×D₄	1	2	yes	—
225	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
226	C₄×C₈₉	—	2	8	no	—
227	C₄×C₈₉	—	2	4	no	—
228	C₄×C₈₉	—	2	2	no	—
229	C₄×C₈₉	—	4	8	no	—
230	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
231	C₄×C₈₉	C₂×C₄×D₄	1	4	yes	—
232	C₄×C₈₉	#575	1	4	yes	direct product factor
233	C₄×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
234	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
235	C₄×#575	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂×C₈₉	—	2	4	no	—
237	C₂×C₈₉	—	2	2	no	—
238	C₂×C₈₉	—	4	8	no	—
239	C₂×C₈₉	C₂²×C₄×D₄	1	1	yes	characteristic
240	C₂×C₈₉	C₂×#575	1	2	yes	—
241	C₂×#575	C₂×C₈₉	1	1	yes	characteristic
242	C₂×C₄×#41	C₂×C₈₉	1	2	yes	complemented
243	C₄×#137	C₂×C₈₉	1	8	yes	complemented
244	C₄×AGL(1,ℤ/8)	C₂×C₈₉	1	8	yes	complemented
245	#2943	C₂×C₈₉	1	4	yes	complemented
246	C₄×#135	C₂×C₈₉	1	4	yes	complemented
247	#2911	C₂×C₈₉	1	1	yes	characteristic, complemented
248	#2896	C₂×C₈₉	1	2	yes	complemented
249	C₂×C₄×M₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
250	C₈₉	C₄×#575	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²×#41	C₂²×C₈₉	1	2	yes	—
252	#575	C₄×C₈₉	1	16	yes	direct product factor
253	C₂×#137	C₂²×C₈₉	1	4	yes	—
254	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	4	yes	—
255	C₂×#135	C₂²×C₈₉	1	4	yes	—
256	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
257	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
258	C₄×#41	—	2	4	no	—
259	C₄×#41	—	2	2	no	—
260	C₄×#41	—	2	2	no	—
261	C₄×#41	C₂²×C₈₉	1	8	yes	—
262	C₄×#41	C₂²×C₈₉	1	4	yes	complemented
263	C₄×#41	C₂²×C₈₉	1	4	yes	complemented
264	C₂×C₄×Q₈	C₂²×C₈₉	1	4	yes	complemented
265	C₂×C₄×D₄	—	2	4	no	—
266	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	complemented
267	C₂×#117	—	2	2	no	—
268	C₂×#117	C₂²×C₈₉	1	2	yes	—
269	C₂²×C₄²	—	2	2	no	—
270	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
271	#442	C₂²×C₈₉	1	8	yes	complemented
272	#441	C₂²×C₈₉	1	8	yes	complemented
273	#440	C₂²×C₈₉	1	16	yes	complemented
274	C₄×Q₁₆	C₂²×C₈₉	1	8	yes	complemented
275	C₄×SD₁₆	C₂²×C₈₉	1	16	yes	complemented
276	C₄×D₈	C₂²×C₈₉	1	8	yes	complemented
277	#428	C₂²×C₈₉	1	8	yes	complemented
278	#427	C₂²×C₈₉	1	4	yes	complemented
279	#419	C₂²×C₈₉	1	4	yes	complemented
280	#418	C₂²×C₈₉	1	4	yes	complemented
281	#417	C₂²×C₈₉	1	8	yes	—
282	#416	C₂²×C₈₉	1	8	yes	complemented
283	C₄×M₁₆	C₂²×C₈₉	1	8	yes	complemented
284	C₂×#97	C₂²×C₈₉	1	2	yes	complemented
285	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	complemented
286	C₂×#41	—	2	2	no	—
287	C₂×#41	—	2	2	no	—
288	C₂×#41	—	2	8	no	—
289	C₂×#41	C₂×C₄×C₈₉	1	16	yes	complemented
290	C₂×#41	C₂³×C₈₉	1	4	yes	—
291	C₂×#41	C₂³×C₈₉	1	4	yes	—
292	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
293	C₂²×D₄	—	2	2	no	—
294	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
295	C₂³×C₄	—	2	2	no	—
296	C₂³×C₄	—	2	4	no	—
297	C₂³×C₄	D₄×C₈₉	1	2	yes	—
298	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
299	#137	C₂×C₄×C₈₉	1	32	yes	complemented
300	AGL(1,ℤ/8)	C₂×C₄×C₈₉	1	32	yes	complemented
301	#135	C₂×C₄×C₈₉	1	32	yes	complemented
302	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
303	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
304	C₂×D₈	C₂³×C₈₉	1	4	yes	—
305	C₂×M₁₆	—	2	4	no	—
306	C₂×M₁₆	C₂×C₄×C₈₉	1	8	yes	complemented
307	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
308	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
309	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
310	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
311	C₄×Q₈	—	2	4	no	—
312	C₄×Q₈	—	2	4	no	—
313	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
314	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
315	C₄×D₄	—	2	8	no	—
316	C₄×D₄	—	2	8	no	—
317	C₄×D₄	—	2	8	no	—
318	C₄×D₄	—	2	8	no	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	—	2	4	no	—
321	C₄×D₄	C₂³×C₈₉	1	8	yes	—
322	C₄×D₄	C₂³×C₈₉	1	8	yes	—
323	#117	—	2	2	no	—
324	#117	—	2	2	no	—
325	#117	—	2	8	no	—
326	#117	—	2	4	no	—
327	#117	—	2	4	no	—
328	#117	C₂³×C₈₉	1	4	yes	—
329	#117	C₂³×C₈₉	1	4	yes	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	4	no	—
334	C₂×C₄²	—	2	2	no	—
335	C₂×C₄²	—	2	4	no	—
336	C₂×C₄²	—	2	8	no	—
337	C₂×C₄²	—	2	4	no	—
338	C₂×C₄²	—	2	4	no	—
339	C₂×C₄²	D₄×C₈₉	1	2	yes	—
340	C₂×C₄²	D₄×C₈₉	1	2	yes	—
341	C₂×C₄²	D₄×C₈₉	1	2	yes	—
342	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
343	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
345	#107	C₂³×C₈₉	1	8	yes	—
346	#106	C₂³×C₈₉	1	8	yes	—
347	#103	C₂³×C₈₉	1	16	yes	—
348	#102	C₂³×C₈₉	1	16	yes	—
349	#97	C₂³×C₈₉	1	8	yes	—
350	C₄×C₈	C₂³×C₈₉	1	8	yes	—
351	C₂⁴	—	2	2	no	—
352	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
353	#41	—	2	16	no	—
354	#41	—	2	16	no	—
355	#41	—	2	8	no	—
356	#41	—	2	8	no	—
357	#41	C₂²×C₄×C₈₉	1	16	yes	—
358	#41	C₂²×C₄×C₈₉	1	16	yes	complemented
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	C₂²×C₄×C₈₉	1	8	yes	complemented
362	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
363	C₂×D₄	—	2	16	no	—
364	C₂×D₄	—	2	4	no	—
365	C₂×D₄	—	2	4	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	C₂²×C₄×C₈₉	1	8	yes	complemented
369	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	16	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	8	no	—
376	C₂²×C₄	—	2	8	no	—
377	C₂²×C₄	—	4	4	no	—
378	C₂²×C₄	—	4	16	no	—
379	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
380	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
381	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
383	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
384	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
386	C₂²×C₄	#41×C₈₉	1	4	yes	—
387	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
389	Q₁₆	C₂²×C₄×C₈₉	1	16	yes	complemented
390	SD₁₆	C₂²×C₄×C₈₉	1	32	yes	complemented
391	D₈	C₂²×C₄×C₈₉	1	16	yes	complemented
392	M₁₆	—	2	8	no	—
393	M₁₆	C₂²×C₄×C₈₉	1	16	yes	complemented
394	C₂×C₈	—	2	4	no	—
395	C₂×C₈	C₂²×C₄×C₈₉	1	8	yes	complemented
396	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
397	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
398	C₄⋊C₄	—	2	8	no	—
399	C₄⋊C₄	—	2	8	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	2	8	no	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₄²	—	2	4	no	—
407	C₄²	—	2	4	no	—
408	C₄²	—	2	8	no	—
409	C₄²	—	2	8	no	—
410	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
411	C₄²	C₂×D₄×C₈₉	1	2	yes	—
412	C₄²	C₂×D₄×C₈₉	1	4	yes	—
413	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
414	C₄²	C₂×D₄×C₈₉	1	4	yes	—
415	C₄²	C₂⁴×C₈₉	1	2	yes	—
416	C₄²	C₂⁴×C₈₉	1	2	yes	—
417	C₂³	—	2	8	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	8	no	—
421	C₂³	C₄×D₄×C₈₉	1	4	yes	—
422	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
423	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
424	Q₈	—	2	8	no	—
425	Q₈	—	2	4	no	—
426	Q₈	C₂³×C₄×C₈₉	1	8	yes	—
427	D₄	—	2	16	no	—
428	D₄	—	2	16	no	—
429	D₄	—	2	8	no	—
430	D₄	—	2	4	no	—
431	D₄	C₂³×C₄×C₈₉	1	8	yes	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	16	no	—
435	C₂×C₄	—	2	16	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	16	no	—
442	C₂×C₄	—	4	8	no	—
443	C₂×C₄	—	4	16	no	—
444	C₂×C₄	C₄×D₄×C₈₉	1	8	yes	—
445	C₂×C₄	C₄×D₄×C₈₉	1	4	yes	—
446	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
447	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
448	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
449	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
450	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
453	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
454	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
455	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
456	C₈	—	2	4	no	—
457	C₈	C₂³×C₄×C₈₉	1	8	yes	—
458	C₂²	—	2	4	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	8	no	—
461	C₂²	—	4	4	no	—
462	C₂²	—	4	16	no	—
463	C₂²	—	4	8	no	—
464	C₂²	C₂×C₄×D₄×C₈₉	1	4	yes	—
465	C₂²	C₂×C₄×D₄×C₈₉	1	2	yes	—
466	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
467	C₄	—	2	4	no	—
468	C₄	—	2	2	no	—
469	C₄	—	2	8	no	—
470	C₄	—	4	8	no	—
471	C₄	C₂×C₄×D₄×C₈₉	1	2	yes	—
472	C₄	C₂×C₄×D₄×C₈₉	1	4	yes	—
473	C₄	#575×C₈₉	1	4	yes	direct product factor
474	C₄	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
475	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
476	C₂	—	2	4	no	—
477	C₂	—	2	2	no	—
478	C₂	—	4	8	no	—
479	C₂	C₂²×C₄×D₄×C₈₉	1	1	yes	characteristic
480	C₂	C₂×#575×C₈₉	1	2	yes	—

	subgroup	G/H
center	C₄²×C₈₉	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂³×D₄
prime radical	C₂×#575×C₈₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₈₉	C₂	semidirect
3. C₄×#137×C₈₉	C₂	semidirect
4. C₄×AGL(1,ℤ/8)×C₈₉	C₂	semidirect
5. C₈₉×#2943	C₂	semidirect
6. C₄×#135×C₈₉	C₂	semidirect
7. C₈₉×#2911	C₂	semidirect
8. C₈₉×#2896	C₂	semidirect
9. C₂×C₄×M₁₆×C₈₉	C₂	semidirect
11. #575×C₈₉	C₄	direct (N × H)
21. C₄×#41×C₈₉	C₂²	semidirect
22. C₄×#41×C₈₉	C₂²	semidirect
23. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
25. C₂×C₄×D₄×C₈₉	C₂²	semidirect
30. #442×C₈₉	C₂²	semidirect
31. #441×C₈₉	C₂²	semidirect
32. #440×C₈₉	C₂²	semidirect
33. C₄×Q₁₆×C₈₉	C₂²	semidirect
34. C₄×SD₁₆×C₈₉	C₂²	semidirect
35. C₄×D₈×C₈₉	C₂²	semidirect
36. #428×C₈₉	C₂²	semidirect
37. #427×C₈₉	C₂²	semidirect
38. #419×C₈₉	C₂²	semidirect
39. #418×C₈₉	C₂²	semidirect
41. #416×C₈₉	C₂²	semidirect
42. C₄×M₁₆×C₈₉	C₂²	semidirect
43. C₂×#97×C₈₉	C₂²	semidirect
44. C₂×C₄×C₈×C₈₉	C₂²	semidirect
48. C₂×#41×C₈₉	C₂×C₄	semidirect
58. #137×C₈₉	C₂×C₄	semidirect
59. AGL(1,ℤ/8)×C₈₉	C₂×C₄	semidirect
60. #135×C₈₉	C₂×C₄	semidirect
65. C₂×M₁₆×C₈₉	C₂×C₄	semidirect
117. #41×C₈₉	C₂²×C₄	semidirect
120. C₂×Q₈×C₈₉	C₂²×C₄	semidirect
127. C₂×D₄×C₈₉	C₂²×C₄	semidirect
148. Q₁₆×C₈₉	C₂²×C₄	semidirect
149. SD₁₆×C₈₉	C₂²×C₄	semidirect
150. D₈×C₈₉	C₂²×C₄	semidirect
152. M₁₆×C₈₉	C₂²×C₄	semidirect
154. C₂×C₈×C₈₉	C₂²×C₄	semidirect
232. C₄×C₈₉	#575	direct (N × H)
235. C₄×#575	C₈₉	direct (N × H)
242. C₂×C₄×#41	C₂×C₈₉	semidirect
243. C₄×#137	C₂×C₈₉	semidirect
244. C₄×AGL(1,ℤ/8)	C₂×C₈₉	semidirect
245. #2943	C₂×C₈₉	semidirect
246. C₄×#135	C₂×C₈₉	semidirect
247. #2911	C₂×C₈₉	semidirect
248. #2896	C₂×C₈₉	semidirect
249. C₂×C₄×M₁₆	C₂×C₈₉	semidirect
250. C₈₉	C₄×#575	direct (N × H)
252. #575	C₄×C₈₉	direct (N × H)
262. C₄×#41	C₂²×C₈₉	semidirect
263. C₄×#41	C₂²×C₈₉	semidirect
264. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
266. C₂×C₄×D₄	C₂²×C₈₉	semidirect
271. #442	C₂²×C₈₉	semidirect
272. #441	C₂²×C₈₉	semidirect
273. #440	C₂²×C₈₉	semidirect
274. C₄×Q₁₆	C₂²×C₈₉	semidirect
275. C₄×SD₁₆	C₂²×C₈₉	semidirect
276. C₄×D₈	C₂²×C₈₉	semidirect
277. #428	C₂²×C₈₉	semidirect
278. #427	C₂²×C₈₉	semidirect
279. #419	C₂²×C₈₉	semidirect
280. #418	C₂²×C₈₉	semidirect
282. #416	C₂²×C₈₉	semidirect
283. C₄×M₁₆	C₂²×C₈₉	semidirect
284. C₂×#97	C₂²×C₈₉	semidirect
285. C₂×C₄×C₈	C₂²×C₈₉	semidirect
289. C₂×#41	C₂×C₄×C₈₉	semidirect
299. #137	C₂×C₄×C₈₉	semidirect
300. AGL(1,ℤ/8)	C₂×C₄×C₈₉	semidirect
301. #135	C₂×C₄×C₈₉	semidirect
306. C₂×M₁₆	C₂×C₄×C₈₉	semidirect
358. #41	C₂²×C₄×C₈₉	semidirect
361. C₂×Q₈	C₂²×C₄×C₈₉	semidirect
368. C₂×D₄	C₂²×C₄×C₈₉	semidirect
389. Q₁₆	C₂²×C₄×C₈₉	semidirect
390. SD₁₆	C₂²×C₄×C₈₉	semidirect
391. D₈	C₂²×C₄×C₈₉	semidirect
393. M₁₆	C₂²×C₄×C₈₉	semidirect
395. C₂×C₈	C₂²×C₄×C₈₉	semidirect
473. C₄	#575×C₈₉	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×#575×C₈₉	2
C₂×C₄×#41×C₈₉	2
C₄×#137×C₈₉	2
C₄×AGL(1,ℤ/8)×C₈₉	2
C₈₉×#2943	2
C₄×#135×C₈₉	2
C₈₉×#2911	2
C₈₉×#2896	2
C₂×C₄×M₁₆×C₈₉	2
C₄×#575	89

Subgroups — C4×#575×C89 #36438006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#575×C₈₉ #36438006