Subgroups — C₂×C₈₉×#3057 #36438293

Order 22784.

2870 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#131×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#536×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#535×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#3057	C₂	1	16	yes	maximal, direct product factor
6	C₂×#468×C₈₉	C₂	1	4	yes	maximal, complemented
7	C₂×#465×C₈₉	C₂	1	4	yes	maximal, complemented
8	C₂×#427×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	C₂²×SD₁₆×C₈₉	—	2	8	no	—
15	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂²×SD₁₆×C₈₉	C₂²	1	8	yes	complemented
18	C₂×#131×C₈₉	—	2	8	no	—
19	C₂×#131×C₈₉	C₂²	1	8	yes	complemented
20	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	complemented
22	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₈₉	C₂²	1	2	yes	complemented
24	#536×C₈₉	—	2	4	no	—
25	#536×C₈₉	C₂²	1	8	yes	complemented
26	#535×C₈₉	—	2	4	no	—
27	#535×C₈₉	C₂²	1	8	yes	—
28	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
29	C₂×#127×C₈₉	—	2	1	no	—
30	C₂×#124×C₈₉	—	2	1	no	—
31	C₂×#124×C₈₉	—	2	1	no	—
32	C₂×#123×C₈₉	—	2	2	no	—
33	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	complemented
34	C₂×#121×C₈₉	C₂²	1	4	yes	—
35	C₂×#120×C₈₉	—	2	2	no	—
36	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
38	#468×C₈₉	C₂²	1	32	yes	complemented
39	#465×C₈₉	C₂²	1	32	yes	complemented
40	#427×C₈₉	C₂²	1	8	yes	complemented
41	C₂×#107×C₈₉	C₂²	1	2	yes	complemented
42	C₂×#106×C₈₉	C₂²	1	2	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	4	no	—
49	C₂×#41×C₈₉	—	2	4	no	—
50	C₂×#41×C₈₉	D₄	1	4	yes	—
51	C₂×#41×C₈₉	C₂³	1	4	yes	—
52	C₂²×Q₈×C₈₉	—	2	1	no	—
53	C₂²×Q₈×C₈₉	—	2	2	no	—
54	C₂²×Q₈×C₈₉	—	2	4	no	—
55	C₂²×Q₈×C₈₉	—	2	4	no	—
56	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
58	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
59	C₂²×D₄×C₈₉	—	2	4	no	—
60	C₂²×D₄×C₈₉	—	2	4	no	—
61	C₂²×D₄×C₈₉	—	2	1	no	—
62	C₂²×D₄×C₈₉	—	2	2	no	—
63	C₂²×D₄×C₈₉	—	2	4	no	—
64	C₂²×D₄×C₈₉	D₄	1	2	yes	—
65	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
66	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
67	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
68	C₂³×C₄×C₈₉	—	2	1	no	—
69	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
70	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₈₉	—	2	8	no	—
72	C₂×SD₁₆×C₈₉	—	2	8	no	—
73	C₂×SD₁₆×C₈₉	—	2	32	no	—
74	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
76	#131×C₈₉	—	2	32	no	—
77	C₂×M₁₆×C₈₉	—	2	4	no	—
78	C₂×M₁₆×C₈₉	—	2	8	no	—
79	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
80	C₂×M₁₆×C₈₉	C₂³	1	8	yes	complemented
81	C₂²×C₈×C₈₉	—	2	8	no	—
82	C₂²×C₈×C₈₉	D₄	1	8	yes	complemented
83	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
84	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
87	C₂²×C₈×C₈₉	C₂³	1	8	yes	complemented
88	C₄⋊Q₈×C₈₉	—	2	8	no	—
89	#127×C₈₉	—	2	8	no	—
90	#124×C₈₉	—	2	8	no	—
91	#124×C₈₉	—	2	8	no	—
92	#123×C₈₉	—	2	16	no	—
93	C₂²⋊Q₈×C₈₉	—	2	8	no	—
94	C₂²⋊Q₈×C₈₉	C₂³	1	16	yes	complemented
95	#121×C₈₉	—	2	8	no	—
96	#121×C₈₉	C₂³	1	16	yes	—
97	#120×C₈₉	—	2	16	no	—
98	#117×C₈₉	—	2	2	no	—
99	#117×C₈₉	—	2	2	no	—
100	#117×C₈₉	C₂³	1	4	yes	—
101	#117×C₈₉	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
108	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
109	C₂×C₄²×C₈₉	—	2	1	no	—
110	C₂×C₄²×C₈₉	—	2	1	no	—
111	#107×C₈₉	C₂³	1	8	yes	complemented
112	#106×C₈₉	C₂³	1	8	yes	complemented
113	#103×C₈₉	C₂³	1	16	yes	complemented
114	#102×C₈₉	C₂³	1	16	yes	—
115	C₂⁴×C₈₉	—	2	2	no	—
116	C₂⁴×C₈₉	—	2	4	no	—
117	C₂⁴×C₈₉	—	4	4	no	—
118	C₂⁴×C₈₉	—	8	4	no	—
119	C₂⁴×C₈₉	C₂×D₄	1	2	yes	—
120	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
121	#41×C₈₉	—	2	16	no	—
122	#41×C₈₉	—	2	8	no	—
123	#41×C₈₉	—	2	8	no	—
124	C₂×Q₈×C₈₉	—	2	4	no	—
125	C₂×Q₈×C₈₉	—	2	2	no	—
126	C₂×Q₈×C₈₉	—	2	4	no	—
127	C₂×Q₈×C₈₉	—	2	4	no	—
128	C₂×Q₈×C₈₉	—	2	8	no	—
129	C₂×Q₈×C₈₉	—	2	4	no	—
130	C₂×Q₈×C₈₉	—	2	8	no	—
131	C₂×Q₈×C₈₉	—	4	4	no	—
132	C₂×Q₈×C₈₉	—	4	8	no	—
133	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
134	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
135	C₂×D₄×C₈₉	—	2	8	no	—
136	C₂×D₄×C₈₉	—	2	16	no	—
137	C₂×D₄×C₈₉	—	2	4	no	—
138	C₂×D₄×C₈₉	—	2	4	no	—
139	C₂×D₄×C₈₉	—	2	8	no	—
140	C₂×D₄×C₈₉	—	2	2	no	—
141	C₂×D₄×C₈₉	—	2	4	no	—
142	C₂×D₄×C₈₉	—	2	4	no	—
143	C₂×D₄×C₈₉	—	2	8	no	—
144	C₂×D₄×C₈₉	—	4	4	no	—
145	C₂×D₄×C₈₉	—	4	8	no	—
146	C₂×D₄×C₈₉	—	8	8	no	—
147	C₂×D₄×C₈₉	C₂×D₄	1	8	yes	—
148	C₂×D₄×C₈₉	C₂×D₄	1	4	yes	—
149	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₈₉	—	2	4	no	—
151	C₂²×C₄×C₈₉	—	2	4	no	—
152	C₂²×C₄×C₈₉	—	2	4	no	—
153	C₂²×C₄×C₈₉	—	2	2	no	—
154	C₂²×C₄×C₈₉	—	2	2	no	—
155	C₂²×C₄×C₈₉	—	2	2	no	—
156	C₂²×C₄×C₈₉	—	4	4	no	—
157	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₈₉	#41	1	4	yes	—
163	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
166	SD₁₆×C₈₉	—	2	16	no	—
167	SD₁₆×C₈₉	—	2	16	no	—
168	M₁₆×C₈₉	—	2	8	no	—
169	M₁₆×C₈₉	—	2	16	no	—
170	C₂×C₈×C₈₉	—	2	4	no	—
171	C₂×C₈×C₈₉	—	2	16	no	—
172	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
173	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
174	C₂×C₈×C₈₉	C₂×D₄	1	16	yes	complemented
175	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
176	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
177	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₈₉	—	2	16	no	—
179	C₄⋊C₄×C₈₉	—	2	2	no	—
180	C₄⋊C₄×C₈₉	—	2	2	no	—
181	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
182	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
183	C₂²⋊C₄×C₈₉	—	2	4	no	—
184	C₂²⋊C₄×C₈₉	—	2	4	no	—
185	C₂²⋊C₄×C₈₉	—	2	16	no	—
186	C₂²⋊C₄×C₈₉	—	2	16	no	—
187	C₄²×C₈₉	—	2	4	no	—
188	C₄²×C₈₉	—	2	4	no	—
189	C₂³×C₈₉	—	2	4	no	—
190	C₂³×C₈₉	—	2	4	no	—
191	C₂³×C₈₉	—	2	8	no	—
192	C₂³×C₈₉	—	2	4	no	—
193	C₂³×C₈₉	—	4	8	no	—
194	C₂³×C₈₉	—	4	4	no	—
195	C₂³×C₈₉	—	4	8	no	—
196	C₂³×C₈₉	—	8	8	no	—
197	C₂³×C₈₉	—	8	16	no	—
198	C₂³×C₈₉	#121	1	4	yes	—
199	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
200	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
201	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
202	Q₈×C₈₉	—	2	8	no	—
203	Q₈×C₈₉	—	2	4	no	—
204	Q₈×C₈₉	—	2	4	no	—
205	Q₈×C₈₉	—	4	4	no	—
206	Q₈×C₈₉	—	4	8	no	—
207	D₄×C₈₉	—	2	16	no	—
208	D₄×C₈₉	—	2	8	no	—
209	D₄×C₈₉	—	2	4	no	—
210	D₄×C₈₉	—	2	4	no	—
211	D₄×C₈₉	—	4	4	no	—
212	D₄×C₈₉	—	4	8	no	—
213	D₄×C₈₉	—	8	16	no	—
214	C₂×C₄×C₈₉	—	2	4	no	—
215	C₂×C₄×C₈₉	—	2	8	no	—
216	C₂×C₄×C₈₉	—	2	4	no	—
217	C₂×C₄×C₈₉	—	2	4	no	—
218	C₂×C₄×C₈₉	—	2	4	no	—
219	C₂×C₄×C₈₉	—	2	4	no	—
220	C₂×C₄×C₈₉	—	2	4	no	—
221	C₂×C₄×C₈₉	—	4	8	no	—
222	C₂×C₄×C₈₉	—	4	4	no	—
223	C₂×C₄×C₈₉	—	4	4	no	—
224	C₂×C₄×C₈₉	—	4	4	no	—
225	C₂×C₄×C₈₉	#121	1	4	yes	—
226	C₂×C₄×C₈₉	#121	1	8	yes	—
227	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
228	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
232	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
233	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
234	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
235	C₈×C₈₉	—	2	4	no	—
236	C₈×C₈₉	C₂²×D₄	1	4	yes	—
237	C₈×C₈₉	C₂²×D₄	1	4	yes	—
238	C₂²×C₈₉	—	2	8	no	—
239	C₂²×C₈₉	—	2	4	no	—
240	C₂²×C₈₉	—	2	4	no	—
241	C₂²×C₈₉	—	2	8	no	—
242	C₂²×C₈₉	—	4	8	no	—
243	C₂²×C₈₉	—	4	4	no	—
244	C₂²×C₈₉	—	4	8	no	—
245	C₂²×C₈₉	—	8	16	no	—
246	C₂²×C₈₉	—	8	8	no	—
247	C₂²×C₈₉	C₂×#121	1	4	yes	—
248	C₂²×C₈₉	C₂×#121	1	2	yes	—
249	C₂²×C₈₉	#577	1	4	yes	—
250	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
251	C₄×C₈₉	—	2	4	no	—
252	C₄×C₈₉	—	4	4	no	—
253	C₄×C₈₉	—	4	4	no	—
254	C₄×C₈₉	—	4	4	no	—
255	C₄×C₈₉	C₂×#121	1	2	yes	—
256	C₄×C₈₉	C₂×#121	1	4	yes	—
257	C₄×C₈₉	C₂³×D₄	1	1	yes	characteristic
258	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
259	C₂×#3057	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂×C₈₉	—	2	4	no	—
261	C₂×C₈₉	—	4	4	no	—
262	C₂×C₈₉	—	8	4	no	—
263	C₂×C₈₉	#3057	1	4	yes	direct product factor
264	C₂×C₈₉	C₂²×#121	1	1	yes	characteristic
265	C₂×C₈₉	C₂×#577	1	2	yes	—
266	C₂³×SD₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
267	C₂²×#131	C₂×C₈₉	1	1	yes	characteristic, complemented
268	C₂×#536	C₂×C₈₉	1	1	yes	characteristic, complemented
269	C₂×#535	C₂×C₈₉	1	1	yes	characteristic
270	#3057	C₂×C₈₉	1	16	yes	direct product factor
271	C₂×#468	C₂×C₈₉	1	4	yes	complemented
272	C₂×#465	C₂×C₈₉	1	4	yes	complemented
273	C₂×#427	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₂×#419	C₂×C₈₉	1	1	yes	characteristic, complemented
275	C₂×#418	C₂×C₈₉	1	1	yes	characteristic, complemented
276	C₈₉	C₂×#3057	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
278	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
279	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
280	C₂²×SD₁₆	—	2	8	no	—
281	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
282	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
283	C₂²×SD₁₆	C₂²×C₈₉	1	8	yes	complemented
284	C₂×#131	—	2	8	no	—
285	C₂×#131	C₂²×C₈₉	1	8	yes	complemented
286	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
287	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	complemented
288	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
289	C₂³×C₈	C₂²×C₈₉	1	2	yes	complemented
290	#536	—	2	4	no	—
291	#536	C₂²×C₈₉	1	8	yes	complemented
292	#535	—	2	4	no	—
293	#535	C₂²×C₈₉	1	8	yes	—
294	C₂×C₄⋊Q₈	—	2	1	no	—
295	C₂×#127	—	2	1	no	—
296	C₂×#124	—	2	1	no	—
297	C₂×#124	—	2	1	no	—
298	C₂×#123	—	2	2	no	—
299	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	complemented
300	C₂×#121	C₂²×C₈₉	1	4	yes	—
301	C₂×#120	—	2	2	no	—
302	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
303	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
304	#468	C₂²×C₈₉	1	32	yes	complemented
305	#465	C₂²×C₈₉	1	32	yes	complemented
306	#427	C₂²×C₈₉	1	8	yes	complemented
307	C₂×#107	C₂²×C₈₉	1	2	yes	complemented
308	C₂×#106	C₂²×C₈₉	1	2	yes	complemented
309	#419	C₂²×C₈₉	1	8	yes	complemented
310	#418	C₂²×C₈₉	1	8	yes	complemented
311	C₂×#103	C₂²×C₈₉	1	4	yes	complemented
312	C₂×#102	C₂²×C₈₉	1	4	yes	—
313	C₂⁵	—	2	1	no	—
314	C₂×#41	—	2	4	no	—
315	C₂×#41	—	2	4	no	—
316	C₂×#41	D₄×C₈₉	1	4	yes	—
317	C₂×#41	C₂³×C₈₉	1	4	yes	—
318	C₂²×Q₈	—	2	1	no	—
319	C₂²×Q₈	—	2	2	no	—
320	C₂²×Q₈	—	2	4	no	—
321	C₂²×Q₈	—	2	4	no	—
322	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
323	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
324	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
325	C₂²×D₄	—	2	4	no	—
326	C₂²×D₄	—	2	4	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	—	2	2	no	—
329	C₂²×D₄	—	2	4	no	—
330	C₂²×D₄	D₄×C₈₉	1	2	yes	—
331	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
332	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
333	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
334	C₂³×C₄	—	2	1	no	—
335	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
336	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
337	C₂×SD₁₆	—	2	8	no	—
338	C₂×SD₁₆	—	2	8	no	—
339	C₂×SD₁₆	—	2	32	no	—
340	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
341	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
342	#131	—	2	32	no	—
343	C₂×M₁₆	—	2	4	no	—
344	C₂×M₁₆	—	2	8	no	—
345	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
346	C₂×M₁₆	C₂³×C₈₉	1	8	yes	complemented
347	C₂²×C₈	—	2	8	no	—
348	C₂²×C₈	D₄×C₈₉	1	8	yes	complemented
349	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
350	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
353	C₂²×C₈	C₂³×C₈₉	1	8	yes	complemented
354	C₄⋊Q₈	—	2	8	no	—
355	#127	—	2	8	no	—
356	#124	—	2	8	no	—
357	#124	—	2	8	no	—
358	#123	—	2	16	no	—
359	C₂²⋊Q₈	—	2	8	no	—
360	C₂²⋊Q₈	C₂³×C₈₉	1	16	yes	complemented
361	#121	—	2	8	no	—
362	#121	C₂³×C₈₉	1	16	yes	—
363	#120	—	2	16	no	—
364	#117	—	2	2	no	—
365	#117	—	2	2	no	—
366	#117	C₂³×C₈₉	1	4	yes	—
367	#117	C₂³×C₈₉	1	4	yes	—
368	C₂×C₄⋊C₄	—	2	4	no	—
369	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
371	C₂×C₂²⋊C₄	—	2	1	no	—
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	4	no	—
374	C₂×C₂²⋊C₄	—	2	4	no	—
375	C₂×C₄²	—	2	1	no	—
376	C₂×C₄²	—	2	1	no	—
377	#107	C₂³×C₈₉	1	8	yes	complemented
378	#106	C₂³×C₈₉	1	8	yes	complemented
379	#103	C₂³×C₈₉	1	16	yes	complemented
380	#102	C₂³×C₈₉	1	16	yes	—
381	C₂⁴	—	2	2	no	—
382	C₂⁴	—	2	4	no	—
383	C₂⁴	—	4	4	no	—
384	C₂⁴	—	8	4	no	—
385	C₂⁴	C₂×D₄×C₈₉	1	2	yes	—
386	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
387	#41	—	2	16	no	—
388	#41	—	2	8	no	—
389	#41	—	2	8	no	—
390	C₂×Q₈	—	2	4	no	—
391	C₂×Q₈	—	2	2	no	—
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	—	2	4	no	—
394	C₂×Q₈	—	2	8	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	8	no	—
397	C₂×Q₈	—	4	4	no	—
398	C₂×Q₈	—	4	8	no	—
399	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
400	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	16	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	8	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	8	no	—
412	C₂×D₄	—	8	8	no	—
413	C₂×D₄	C₂×D₄×C₈₉	1	8	yes	—
414	C₂×D₄	C₂×D₄×C₈₉	1	4	yes	—
415	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
424	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
428	C₂²×C₄	#41×C₈₉	1	4	yes	—
429	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
430	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
431	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
432	SD₁₆	—	2	16	no	—
433	SD₁₆	—	2	16	no	—
434	M₁₆	—	2	8	no	—
435	M₁₆	—	2	16	no	—
436	C₂×C₈	—	2	4	no	—
437	C₂×C₈	—	2	16	no	—
438	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
439	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
440	C₂×C₈	C₂×D₄×C₈₉	1	16	yes	complemented
441	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
442	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
443	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
444	C₄⋊C₄	—	2	16	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
448	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	16	no	—
452	C₂²⋊C₄	—	2	16	no	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	4	no	—
455	C₂³	—	2	4	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	8	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	4	8	no	—
460	C₂³	—	4	4	no	—
461	C₂³	—	4	8	no	—
462	C₂³	—	8	8	no	—
463	C₂³	—	8	16	no	—
464	C₂³	#121×C₈₉	1	4	yes	—
465	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
466	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
467	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
468	Q₈	—	2	8	no	—
469	Q₈	—	2	4	no	—
470	Q₈	—	2	4	no	—
471	Q₈	—	4	4	no	—
472	Q₈	—	4	8	no	—
473	D₄	—	2	16	no	—
474	D₄	—	2	8	no	—
475	D₄	—	2	4	no	—
476	D₄	—	2	4	no	—
477	D₄	—	4	4	no	—
478	D₄	—	4	8	no	—
479	D₄	—	8	16	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	8	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	4	8	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	—	4	4	no	—
491	C₂×C₄	#121×C₈₉	1	4	yes	—
492	C₂×C₄	#121×C₈₉	1	8	yes	—
493	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
494	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
495	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
497	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
498	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
499	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
500	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
501	C₈	—	2	4	no	—
502	C₈	C₂²×D₄×C₈₉	1	4	yes	—
503	C₈	C₂²×D₄×C₈₉	1	4	yes	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	8	no	—
506	C₂²	—	2	8	no	—
507	C₂²	—	2	4	no	—
508	C₂²	—	4	8	no	—
509	C₂²	—	4	8	no	—
510	C₂²	—	4	4	no	—
511	C₂²	—	8	16	no	—
512	C₂²	—	8	8	no	—
513	C₂²	C₂×#121×C₈₉	1	4	yes	—
514	C₂²	C₂×#121×C₈₉	1	2	yes	—
515	C₂²	#577×C₈₉	1	4	yes	—
516	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
517	C₄	—	2	4	no	—
518	C₄	—	4	4	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	4	no	—
521	C₄	C₂×#121×C₈₉	1	2	yes	—
522	C₄	C₂×#121×C₈₉	1	4	yes	—
523	C₄	C₂³×D₄×C₈₉	1	1	yes	characteristic
524	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
525	C₂	—	2	4	no	—
526	C₂	—	4	4	no	—
527	C₂	—	8	4	no	—
528	C₂	C₈₉×#3057	1	4	yes	direct product factor
529	C₂	C₂²×#121×C₈₉	1	1	yes	characteristic
530	C₂	C₂×#577×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂×#535×C₈₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×SD₁₆×C₈₉	C₂	semidirect
2. C₂²×#131×C₈₉	C₂	semidirect
3. C₂×#536×C₈₉	C₂	semidirect
5. C₈₉×#3057	C₂	direct (N × H)
6. C₂×#468×C₈₉	C₂	semidirect
7. C₂×#465×C₈₉	C₂	semidirect
8. C₂×#427×C₈₉	C₂	semidirect
9. C₂×#419×C₈₉	C₂	semidirect
10. C₂×#418×C₈₉	C₂	semidirect
17. C₂²×SD₁₆×C₈₉	C₂²	semidirect
19. C₂×#131×C₈₉	C₂²	semidirect
21. C₂²×M₁₆×C₈₉	C₂²	semidirect
23. C₂³×C₈×C₈₉	C₂²	semidirect
25. #536×C₈₉	C₂²	semidirect
33. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
38. #468×C₈₉	C₂²	semidirect
39. #465×C₈₉	C₂²	semidirect
40. #427×C₈₉	C₂²	semidirect
41. C₂×#107×C₈₉	C₂²	semidirect
42. C₂×#106×C₈₉	C₂²	semidirect
43. #419×C₈₉	C₂²	semidirect
44. #418×C₈₉	C₂²	semidirect
45. C₂×#103×C₈₉	C₂²	semidirect
80. C₂×M₁₆×C₈₉	C₂³	semidirect
82. C₂²×C₈×C₈₉	D₄	semidirect
87. C₂²×C₈×C₈₉	C₂³	semidirect
94. C₂²⋊Q₈×C₈₉	C₂³	semidirect
111. #107×C₈₉	C₂³	semidirect
112. #106×C₈₉	C₂³	semidirect
113. #103×C₈₉	C₂³	semidirect
174. C₂×C₈×C₈₉	C₂×D₄	semidirect
259. C₂×#3057	C₈₉	direct (N × H)
263. C₂×C₈₉	#3057	direct (N × H)
266. C₂³×SD₁₆	C₂×C₈₉	semidirect
267. C₂²×#131	C₂×C₈₉	semidirect
268. C₂×#536	C₂×C₈₉	semidirect
270. #3057	C₂×C₈₉	direct (N × H)
271. C₂×#468	C₂×C₈₉	semidirect
272. C₂×#465	C₂×C₈₉	semidirect
273. C₂×#427	C₂×C₈₉	semidirect
274. C₂×#419	C₂×C₈₉	semidirect
275. C₂×#418	C₂×C₈₉	semidirect
276. C₈₉	C₂×#3057	direct (N × H)
283. C₂²×SD₁₆	C₂²×C₈₉	semidirect
285. C₂×#131	C₂²×C₈₉	semidirect
287. C₂²×M₁₆	C₂²×C₈₉	semidirect
289. C₂³×C₈	C₂²×C₈₉	semidirect
291. #536	C₂²×C₈₉	semidirect
299. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
304. #468	C₂²×C₈₉	semidirect
305. #465	C₂²×C₈₉	semidirect
306. #427	C₂²×C₈₉	semidirect
307. C₂×#107	C₂²×C₈₉	semidirect
308. C₂×#106	C₂²×C₈₉	semidirect
309. #419	C₂²×C₈₉	semidirect
310. #418	C₂²×C₈₉	semidirect
311. C₂×#103	C₂²×C₈₉	semidirect
346. C₂×M₁₆	C₂³×C₈₉	semidirect
348. C₂²×C₈	D₄×C₈₉	semidirect
353. C₂²×C₈	C₂³×C₈₉	semidirect
360. C₂²⋊Q₈	C₂³×C₈₉	semidirect
377. #107	C₂³×C₈₉	semidirect
378. #106	C₂³×C₈₉	semidirect
379. #103	C₂³×C₈₉	semidirect
440. C₂×C₈	C₂×D₄×C₈₉	semidirect
528. C₂	C₈₉×#3057	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×SD₁₆×C₈₉	2
C₂²×#131×C₈₉	2
C₂×#536×C₈₉	2
C₂×#535×C₈₉	2
C₈₉×#3057	2
C₂×#468×C₈₉	2
C₂×#465×C₈₉	2
C₂×#427×C₈₉	2
C₂×#419×C₈₉	2
C₂×#418×C₈₉	2
C₂×#3057	89

Subgroups — C2×C89×#3057 #36438293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3057 #36438293