Subgroups — C₂×C₈₉×#3217 #36438633

Order 22784.

2834 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#529×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3217	C₂	1	16	yes	maximal, direct product factor
4	C₂×#496×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#490×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#460×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#459×C₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#450×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×#447×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂×#432×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₈₉	—	2	1	no	—
12	C₂³×D₄×C₈₉	—	2	1	no	—
13	C₂³×D₄×C₈₉	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₈₉	C₂²	1	4	yes	complemented
15	C₂²×D₈×C₈₉	C₂²	1	4	yes	complemented
16	C₂×#127×C₈₉	—	2	1	no	—
17	C₂×#127×C₈₉	—	2	1	no	—
18	C₂×#127×C₈₉	—	2	4	no	—
19	C₂×#127×C₈₉	C₂²	1	2	yes	—
20	C₂×#127×C₈₉	C₂²	1	2	yes	—
21	C₂×#127×C₈₉	C₂²	1	8	yes	complemented
22	#529×C₈₉	—	2	4	no	—
23	#529×C₈₉	C₂²	1	8	yes	complemented
24	C₂×#126×C₈₉	—	2	1	no	—
25	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic, complemented
26	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	C₂×#123×C₈₉	—	2	1	no	—
28	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄×D₄×C₈₉	—	2	1	no	—
32	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
33	C₂×#117×C₈₉	—	2	1	no	—
34	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
35	#496×C₈₉	C₂²	1	16	yes	complemented
36	#490×C₈₉	C₂²	1	16	yes	complemented
37	#460×C₈₉	C₂²	1	16	yes	complemented
38	#459×C₈₉	C₂²	1	16	yes	complemented
39	#450×C₈₉	C₂²	1	16	yes	complemented
40	#447×C₈₉	C₂²	1	16	yes	complemented
41	#432×C₈₉	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	complemented
43	C₂×#102×C₈₉	C₂²	1	4	yes	complemented
44	C₂×#102×C₈₉	C₂²	1	4	yes	complemented
45	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
46	C₂×#97×C₈₉	C₂²	1	2	yes	complemented
47	C₂⁵×C₈₉	—	2	2	no	—
48	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₈₉	—	2	1	no	—
50	C₂²×D₄×C₈₉	—	2	4	no	—
51	C₂²×D₄×C₈₉	—	2	2	no	—
52	C₂²×D₄×C₈₉	—	2	2	no	—
53	C₂²×D₄×C₈₉	—	2	2	no	—
54	C₂²×D₄×C₈₉	—	2	2	no	—
55	C₂²×D₄×C₈₉	—	2	2	no	—
56	C₂²×D₄×C₈₉	—	2	8	no	—
57	C₂²×D₄×C₈₉	—	2	8	no	—
58	C₂²×D₄×C₈₉	—	4	8	no	—
59	C₂²×D₄×C₈₉	—	4	8	no	—
60	C₂²×D₄×C₈₉	—	4	8	no	—
61	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
62	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
64	C₂³×C₄×C₈₉	—	2	1	no	—
65	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
67	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×SD₁₆×C₈₉	—	4	8	no	—
69	C₂×SD₁₆×C₈₉	C₂³	1	16	yes	complemented
70	C₂×D₈×C₈₉	—	4	8	no	—
71	C₂×D₈×C₈₉	C₂³	1	16	yes	complemented
72	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
73	#127×C₈₉	—	2	4	no	—
74	#127×C₈₉	—	2	4	no	—
75	#127×C₈₉	—	2	8	no	—
76	#127×C₈₉	—	2	8	no	—
77	#127×C₈₉	C₂³	1	8	yes	—
78	#127×C₈₉	C₂³	1	8	yes	—
79	#126×C₈₉	—	2	8	no	—
80	#125×C₈₉	—	2	2	no	—
81	#125×C₈₉	C₂³	1	4	yes	complemented
82	#124×C₈₉	—	2	2	no	—
83	#124×C₈₉	C₂³	1	4	yes	complemented
84	#123×C₈₉	—	2	8	no	—
85	C₂²⋊Q₈×C₈₉	—	2	2	no	—
86	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
87	#121×C₈₉	—	2	2	no	—
88	#121×C₈₉	C₂³	1	4	yes	—
89	C₄×Q₈×C₈₉	—	2	2	no	—
90	C₄×Q₈×C₈₉	C₂³	1	4	yes	complemented
91	C₄×D₄×C₈₉	—	2	8	no	—
92	C₄×D₄×C₈₉	—	2	2	no	—
93	C₄×D₄×C₈₉	C₂³	1	4	yes	complemented
94	#117×C₈₉	—	2	8	no	—
95	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂×C₄²×C₈₉	—	2	1	no	—
107	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
108	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
109	C₂×C₄²×C₈₉	D₄	1	4	yes	—
110	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
113	C₄⋊C₈×C₈₉	C₂³	1	8	yes	complemented
114	#102×C₈₉	C₂³	1	16	yes	complemented
115	#102×C₈₉	C₂³	1	16	yes	complemented
116	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
117	#97×C₈₉	C₂³	1	8	yes	complemented
118	C₂⁴×C₈₉	—	2	8	no	—
119	C₂⁴×C₈₉	—	2	1	no	—
120	C₂⁴×C₈₉	—	2	4	no	—
121	C₂⁴×C₈₉	—	4	8	no	—
122	C₂⁴×C₈₉	—	4	8	no	—
123	C₂⁴×C₈₉	—	4	8	no	—
124	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Q₈×C₈₉	—	2	1	no	—
126	C₂×Q₈×C₈₉	—	2	4	no	—
127	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
128	C₂×D₄×C₈₉	—	2	4	no	—
129	C₂×D₄×C₈₉	—	2	1	no	—
130	C₂×D₄×C₈₉	—	2	8	no	—
131	C₂×D₄×C₈₉	—	2	4	no	—
132	C₂×D₄×C₈₉	—	2	8	no	—
133	C₂×D₄×C₈₉	—	2	8	no	—
134	C₂×D₄×C₈₉	—	2	8	no	—
135	C₂×D₄×C₈₉	—	2	8	no	—
136	C₂×D₄×C₈₉	—	2	4	no	—
137	C₂×D₄×C₈₉	—	2	8	no	—
138	C₂×D₄×C₈₉	—	2	8	no	—
139	C₂×D₄×C₈₉	—	4	16	no	—
140	C₂×D₄×C₈₉	—	4	8	no	—
141	C₂×D₄×C₈₉	—	4	8	no	—
142	C₂×D₄×C₈₉	—	4	8	no	—
143	C₂×D₄×C₈₉	—	4	8	no	—
144	C₂×D₄×C₈₉	—	4	32	no	—
145	C₂×D₄×C₈₉	—	4	16	no	—
146	C₂×D₄×C₈₉	—	4	32	no	—
147	C₂×D₄×C₈₉	—	4	4	no	—
148	C₂×D₄×C₈₉	—	4	8	no	—
149	C₂×D₄×C₈₉	C₂⁴	1	8	yes	—
150	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₈₉	—	2	4	no	—
152	C₂²×C₄×C₈₉	—	2	4	no	—
153	C₂²×C₄×C₈₉	—	2	4	no	—
154	C₂²×C₄×C₈₉	—	2	1	no	—
155	C₂²×C₄×C₈₉	—	2	1	no	—
156	C₂²×C₄×C₈₉	—	2	1	no	—
157	C₂²×C₄×C₈₉	—	2	4	no	—
158	C₂²×C₄×C₈₉	—	4	4	no	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
169	SD₁₆×C₈₉	—	4	16	no	—
170	D₈×C₈₉	—	4	16	no	—
171	C₂×C₈×C₈₉	—	2	8	no	—
172	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
173	C₄⋊C₄×C₈₉	—	2	4	no	—
174	C₄⋊C₄×C₈₉	—	2	4	no	—
175	C₄⋊C₄×C₈₉	—	2	4	no	—
176	C₄⋊C₄×C₈₉	—	2	4	no	—
177	C₄⋊C₄×C₈₉	—	2	1	no	—
178	C₄⋊C₄×C₈₉	—	2	1	no	—
179	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
180	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
181	C₂²⋊C₄×C₈₉	—	2	4	no	—
182	C₂²⋊C₄×C₈₉	—	2	4	no	—
183	C₂²⋊C₄×C₈₉	—	2	4	no	—
184	C₂²⋊C₄×C₈₉	—	2	4	no	—
185	C₂²⋊C₄×C₈₉	—	2	4	no	—
186	C₄²×C₈₉	—	2	4	no	—
187	C₄²×C₈₉	C₂×D₄	1	4	yes	—
188	C₄²×C₈₉	C₂×D₄	1	2	yes	—
189	C₄²×C₈₉	C₂×D₄	1	2	yes	—
190	C₄²×C₈₉	C₂×D₄	1	4	yes	—
191	C₄²×C₈₉	C₂⁴	1	2	yes	—
192	C₄²×C₈₉	C₂⁴	1	2	yes	—
193	C₂³×C₈₉	—	2	2	no	—
194	C₂³×C₈₉	—	2	2	no	—
195	C₂³×C₈₉	—	2	2	no	—
196	C₂³×C₈₉	—	2	8	no	—
197	C₂³×C₈₉	—	4	16	no	—
198	C₂³×C₈₉	—	4	16	no	—
199	C₂³×C₈₉	—	4	8	no	—
200	C₂³×C₈₉	—	4	2	no	—
201	C₂³×C₈₉	—	4	2	no	—
202	C₂³×C₈₉	—	4	16	no	—
203	C₂³×C₈₉	—	4	8	no	—
204	C₂³×C₈₉	—	4	8	no	—
205	C₂³×C₈₉	—	4	2	no	—
206	C₂³×C₈₉	—	8	32	no	—
207	C₂³×C₈₉	AGL(1,ℤ/8)	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	—	4	2	no	—
212	D₄×C₈₉	—	2	4	no	—
213	D₄×C₈₉	—	2	8	no	—
214	D₄×C₈₉	—	2	8	no	—
215	D₄×C₈₉	—	4	16	no	—
216	D₄×C₈₉	—	4	16	no	—
217	D₄×C₈₉	—	4	16	no	—
218	D₄×C₈₉	—	4	16	no	—
219	D₄×C₈₉	—	4	8	no	—
220	D₄×C₈₉	—	4	2	no	—
221	D₄×C₈₉	—	4	8	no	—
222	D₄×C₈₉	—	4	16	no	—
223	C₂×C₄×C₈₉	—	2	2	no	—
224	C₂×C₄×C₈₉	—	2	8	no	—
225	C₂×C₄×C₈₉	—	2	2	no	—
226	C₂×C₄×C₈₉	—	2	2	no	—
227	C₂×C₄×C₈₉	—	2	2	no	—
228	C₂×C₄×C₈₉	—	2	4	no	—
229	C₂×C₄×C₈₉	—	2	8	no	—
230	C₂×C₄×C₈₉	—	2	4	no	—
231	C₂×C₄×C₈₉	—	2	2	no	—
232	C₂×C₄×C₈₉	—	2	2	no	—
233	C₂×C₄×C₈₉	—	2	2	no	—
234	C₂×C₄×C₈₉	—	4	2	no	—
235	C₂×C₄×C₈₉	—	4	2	no	—
236	C₂×C₄×C₈₉	—	4	2	no	—
237	C₂×C₄×C₈₉	—	4	8	no	—
238	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
246	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
247	C₈×C₈₉	—	2	8	no	—
248	C₂²×C₈₉	—	2	2	no	—
249	C₂²×C₈₉	—	2	8	no	—
250	C₂²×C₈₉	—	2	2	no	—
251	C₂²×C₈₉	—	4	8	no	—
252	C₂²×C₈₉	—	4	2	no	—
253	C₂²×C₈₉	—	4	2	no	—
254	C₂²×C₈₉	—	4	8	no	—
255	C₂²×C₈₉	—	4	8	no	—
256	C₂²×C₈₉	—	4	2	no	—
257	C₂²×C₈₉	—	8	32	no	—
258	C₂²×C₈₉	—	8	4	no	—
259	C₂²×C₈₉	—	8	16	no	—
260	C₂²×C₈₉	#535	1	2	yes	—
261	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	4	yes	—
262	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
263	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
264	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
265	C₄×C₈₉	—	2	2	no	—
266	C₄×C₈₉	—	2	2	no	—
267	C₄×C₈₉	—	2	2	no	—
268	C₄×C₈₉	—	2	2	no	—
269	C₄×C₈₉	—	4	2	no	—
270	C₄×C₈₉	—	4	2	no	—
271	C₄×C₈₉	—	4	2	no	—
272	C₄×C₈₉	#535	1	2	yes	—
273	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	4	yes	—
274	C₄×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
275	C₂×#3217	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂×C₈₉	—	2	4	no	—
277	C₂×C₈₉	—	8	2	no	—
278	C₂×C₈₉	—	8	8	no	—
279	C₂×C₈₉	#3217	1	4	yes	direct product factor
280	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
281	C₂×C₈₉	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₂×C₈₉	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂²×#127	C₂×C₈₉	1	1	yes	characteristic, complemented
284	C₂×#529	C₂×C₈₉	1	1	yes	characteristic, complemented
285	#3217	C₂×C₈₉	1	16	yes	direct product factor
286	C₂×#496	C₂×C₈₉	1	2	yes	complemented
287	C₂×#490	C₂×C₈₉	1	2	yes	complemented
288	C₂×#460	C₂×C₈₉	1	2	yes	complemented
289	C₂×#459	C₂×C₈₉	1	2	yes	complemented
290	C₂×#450	C₂×C₈₉	1	2	yes	complemented
291	C₂×#447	C₂×C₈₉	1	2	yes	complemented
292	C₂×#432	C₂×C₈₉	1	1	yes	characteristic, complemented
293	C₈₉	C₂×#3217	1	1	yes	characteristic, direct product factor, Sylow
294	C₂³×D₄	—	2	1	no	—
295	C₂³×D₄	—	2	1	no	—
296	C₂³×D₄	C₂²×C₈₉	1	2	yes	—
297	C₂²×SD₁₆	C₂²×C₈₉	1	4	yes	complemented
298	C₂²×D₈	C₂²×C₈₉	1	4	yes	complemented
299	C₂×#127	—	2	1	no	—
300	C₂×#127	—	2	1	no	—
301	C₂×#127	—	2	4	no	—
302	C₂×#127	C₂²×C₈₉	1	2	yes	—
303	C₂×#127	C₂²×C₈₉	1	2	yes	—
304	C₂×#127	C₂²×C₈₉	1	8	yes	complemented
305	#529	—	2	4	no	—
306	#529	C₂²×C₈₉	1	8	yes	complemented
307	C₂×#126	—	2	1	no	—
308	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic, complemented
309	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic, complemented
310	C₂×#123	—	2	1	no	—
311	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
312	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
313	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
314	C₂×C₄×D₄	—	2	1	no	—
315	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
316	C₂×#117	—	2	1	no	—
317	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
318	#496	C₂²×C₈₉	1	16	yes	complemented
319	#490	C₂²×C₈₉	1	16	yes	complemented
320	#460	C₂²×C₈₉	1	16	yes	complemented
321	#459	C₂²×C₈₉	1	16	yes	complemented
322	#450	C₂²×C₈₉	1	16	yes	complemented
323	#447	C₂²×C₈₉	1	16	yes	complemented
324	#432	C₂²×C₈₉	1	8	yes	complemented
325	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	complemented
326	C₂×#102	C₂²×C₈₉	1	4	yes	complemented
327	C₂×#102	C₂²×C₈₉	1	4	yes	complemented
328	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
329	C₂×#97	C₂²×C₈₉	1	2	yes	complemented
330	C₂⁵	—	2	2	no	—
331	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	4	no	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	—	2	2	no	—
337	C₂²×D₄	—	2	2	no	—
338	C₂²×D₄	—	2	2	no	—
339	C₂²×D₄	—	2	8	no	—
340	C₂²×D₄	—	2	8	no	—
341	C₂²×D₄	—	4	8	no	—
342	C₂²×D₄	—	4	8	no	—
343	C₂²×D₄	—	4	8	no	—
344	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
345	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
346	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
349	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
351	C₂×SD₁₆	—	4	8	no	—
352	C₂×SD₁₆	C₂³×C₈₉	1	16	yes	complemented
353	C₂×D₈	—	4	8	no	—
354	C₂×D₈	C₂³×C₈₉	1	16	yes	complemented
355	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
356	#127	—	2	4	no	—
357	#127	—	2	4	no	—
358	#127	—	2	8	no	—
359	#127	—	2	8	no	—
360	#127	C₂³×C₈₉	1	8	yes	—
361	#127	C₂³×C₈₉	1	8	yes	—
362	#126	—	2	8	no	—
363	#125	—	2	2	no	—
364	#125	C₂³×C₈₉	1	4	yes	complemented
365	#124	—	2	2	no	—
366	#124	C₂³×C₈₉	1	4	yes	complemented
367	#123	—	2	8	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
370	#121	—	2	2	no	—
371	#121	C₂³×C₈₉	1	4	yes	—
372	C₄×Q₈	—	2	2	no	—
373	C₄×Q₈	C₂³×C₈₉	1	4	yes	complemented
374	C₄×D₄	—	2	8	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	C₂³×C₈₉	1	4	yes	complemented
377	#117	—	2	8	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₄²	—	2	1	no	—
390	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
391	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
392	C₂×C₄²	D₄×C₈₉	1	4	yes	—
393	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
396	C₄⋊C₈	C₂³×C₈₉	1	8	yes	complemented
397	#102	C₂³×C₈₉	1	16	yes	complemented
398	#102	C₂³×C₈₉	1	16	yes	complemented
399	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
400	#97	C₂³×C₈₉	1	8	yes	complemented
401	C₂⁴	—	2	1	no	—
402	C₂⁴	—	2	8	no	—
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	4	8	no	—
405	C₂⁴	—	4	8	no	—
406	C₂⁴	—	4	8	no	—
407	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	4	16	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	8	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	4	8	no	—
427	C₂×D₄	—	4	32	no	—
428	C₂×D₄	—	4	16	no	—
429	C₂×D₄	—	4	32	no	—
430	C₂×D₄	—	4	4	no	—
431	C₂×D₄	—	4	8	no	—
432	C₂×D₄	C₂⁴×C₈₉	1	8	yes	—
433	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	4	4	no	—
442	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
444	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
448	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
449	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
452	SD₁₆	—	4	16	no	—
453	D₈	—	4	16	no	—
454	C₂×C₈	—	2	8	no	—
455	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
463	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₄²	—	2	4	no	—
470	C₄²	C₂×D₄×C₈₉	1	4	yes	—
471	C₄²	C₂×D₄×C₈₉	1	2	yes	—
472	C₄²	C₂×D₄×C₈₉	1	2	yes	—
473	C₄²	C₂×D₄×C₈₉	1	4	yes	—
474	C₄²	C₂⁴×C₈₉	1	2	yes	—
475	C₄²	C₂⁴×C₈₉	1	2	yes	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	8	no	—
480	C₂³	—	4	16	no	—
481	C₂³	—	4	16	no	—
482	C₂³	—	4	8	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	16	no	—
486	C₂³	—	4	8	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	8	32	no	—
490	C₂³	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
491	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
492	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
493	Q₈	—	2	4	no	—
494	Q₈	—	4	2	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	8	no	—
497	D₄	—	2	8	no	—
498	D₄	—	4	16	no	—
499	D₄	—	4	16	no	—
500	D₄	—	4	16	no	—
501	D₄	—	4	16	no	—
502	D₄	—	4	8	no	—
503	D₄	—	4	2	no	—
504	D₄	—	4	8	no	—
505	D₄	—	4	16	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
522	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
523	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
524	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
525	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
526	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
528	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
529	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
530	C₈	—	2	8	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	8	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	8	no	—
536	C₂²	—	4	8	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	8	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	8	32	no	—
541	C₂²	—	8	16	no	—
542	C₂²	—	8	4	no	—
543	C₂²	#535×C₈₉	1	2	yes	—
544	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	4	yes	—
545	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
546	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
547	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	2	no	—
551	C₄	—	2	2	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	#535×C₈₉	1	2	yes	—
556	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	4	yes	—
557	C₄	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
558	C₂	—	2	4	no	—
559	C₂	—	8	2	no	—
560	C₂	—	8	8	no	—
561	C₂	C₈₉×#3217	1	4	yes	direct product factor
562	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
563	C₂	C₂²×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
564	C₂	C₂²×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic

	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 ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#127×C₈₉	C₂	semidirect
2. C₂×#529×C₈₉	C₂	semidirect
3. C₈₉×#3217	C₂	direct (N × H)
4. C₂×#496×C₈₉	C₂	semidirect
5. C₂×#490×C₈₉	C₂	semidirect
6. C₂×#460×C₈₉	C₂	semidirect
7. C₂×#459×C₈₉	C₂	semidirect
8. C₂×#450×C₈₉	C₂	semidirect
9. C₂×#447×C₈₉	C₂	semidirect
10. C₂×#432×C₈₉	C₂	semidirect
14. C₂²×SD₁₆×C₈₉	C₂²	semidirect
15. C₂²×D₈×C₈₉	C₂²	semidirect
21. C₂×#127×C₈₉	C₂²	semidirect
23. #529×C₈₉	C₂²	semidirect
25. C₂×#125×C₈₉	C₂²	semidirect
26. C₂×#124×C₈₉	C₂²	semidirect
30. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
32. C₂×C₄×D₄×C₈₉	C₂²	semidirect
35. #496×C₈₉	C₂²	semidirect
36. #490×C₈₉	C₂²	semidirect
37. #460×C₈₉	C₂²	semidirect
38. #459×C₈₉	C₂²	semidirect
39. #450×C₈₉	C₂²	semidirect
40. #447×C₈₉	C₂²	semidirect
41. #432×C₈₉	C₂²	semidirect
42. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
43. C₂×#102×C₈₉	C₂²	semidirect
44. C₂×#102×C₈₉	C₂²	semidirect
46. C₂×#97×C₈₉	C₂²	semidirect
69. C₂×SD₁₆×C₈₉	C₂³	semidirect
71. C₂×D₈×C₈₉	C₂³	semidirect
81. #125×C₈₉	C₂³	semidirect
83. #124×C₈₉	C₂³	semidirect
90. C₄×Q₈×C₈₉	C₂³	semidirect
93. C₄×D₄×C₈₉	C₂³	semidirect
113. C₄⋊C₈×C₈₉	C₂³	semidirect
114. #102×C₈₉	C₂³	semidirect
115. #102×C₈₉	C₂³	semidirect
117. #97×C₈₉	C₂³	semidirect
275. C₂×#3217	C₈₉	direct (N × H)
279. C₂×C₈₉	#3217	direct (N × H)
283. C₂²×#127	C₂×C₈₉	semidirect
284. C₂×#529	C₂×C₈₉	semidirect
285. #3217	C₂×C₈₉	direct (N × H)
286. C₂×#496	C₂×C₈₉	semidirect
287. C₂×#490	C₂×C₈₉	semidirect
288. C₂×#460	C₂×C₈₉	semidirect
289. C₂×#459	C₂×C₈₉	semidirect
290. C₂×#450	C₂×C₈₉	semidirect
291. C₂×#447	C₂×C₈₉	semidirect
292. C₂×#432	C₂×C₈₉	semidirect
293. C₈₉	C₂×#3217	direct (N × H)
297. C₂²×SD₁₆	C₂²×C₈₉	semidirect
298. C₂²×D₈	C₂²×C₈₉	semidirect
304. C₂×#127	C₂²×C₈₉	semidirect
306. #529	C₂²×C₈₉	semidirect
308. C₂×#125	C₂²×C₈₉	semidirect
309. C₂×#124	C₂²×C₈₉	semidirect
313. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
315. C₂×C₄×D₄	C₂²×C₈₉	semidirect
318. #496	C₂²×C₈₉	semidirect
319. #490	C₂²×C₈₉	semidirect
320. #460	C₂²×C₈₉	semidirect
321. #459	C₂²×C₈₉	semidirect
322. #450	C₂²×C₈₉	semidirect
323. #447	C₂²×C₈₉	semidirect
324. #432	C₂²×C₈₉	semidirect
325. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
326. C₂×#102	C₂²×C₈₉	semidirect
327. C₂×#102	C₂²×C₈₉	semidirect
329. C₂×#97	C₂²×C₈₉	semidirect
352. C₂×SD₁₆	C₂³×C₈₉	semidirect
354. C₂×D₈	C₂³×C₈₉	semidirect
364. #125	C₂³×C₈₉	semidirect
366. #124	C₂³×C₈₉	semidirect
373. C₄×Q₈	C₂³×C₈₉	semidirect
376. C₄×D₄	C₂³×C₈₉	semidirect
396. C₄⋊C₈	C₂³×C₈₉	semidirect
397. #102	C₂³×C₈₉	semidirect
398. #102	C₂³×C₈₉	semidirect
400. #97	C₂³×C₈₉	semidirect
561. C₂	C₈₉×#3217	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×#127×C₈₉	2
C₂×#529×C₈₉	2
C₈₉×#3217	2
C₂×#496×C₈₉	2
C₂×#490×C₈₉	2
C₂×#460×C₈₉	2
C₂×#459×C₈₉	2
C₂×#450×C₈₉	2
C₂×#447×C₈₉	2
C₂×#432×C₈₉	2
C₂×#3217	89

Subgroups — C2×C89×#3217 #36438633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3217 #36438633