Subgroups — C₂×C₈₉×#3305 #36438971

Order 22784.

2262 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#555×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#550×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3305	C₂	1	16	yes	maximal, direct product factor
4	C₂×#486×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#482×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#477×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#466×C₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#459×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#449×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#416×C₈₉	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₈₉	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
16	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
17	#555×C₈₉	—	2	4	no	—
18	#555×C₈₉	C₂²	1	8	yes	complemented
19	#550×C₈₉	—	2	4	no	—
20	#550×C₈₉	C₂²	1	8	yes	complemented
21	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#127×C₈₉	—	2	1	no	—
23	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#124×C₈₉	—	2	2	no	—
25	C₂×#121×C₈₉	—	2	1	no	—
26	C₂×#121×C₈₉	—	2	2	no	—
27	C₂×#121×C₈₉	C₂²	1	2	yes	—
28	C₂×#121×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₈₉	—	2	1	no	—
31	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#117×C₈₉	—	2	1	no	—
34	C₂×#117×C₈₉	C₂²	1	2	yes	—
35	#486×C₈₉	C₂²	1	8	yes	complemented
36	#482×C₈₉	C₂²	1	16	yes	complemented
37	#477×C₈₉	C₂²	1	8	yes	complemented
38	#466×C₈₉	C₂²	1	16	yes	complemented
39	#459×C₈₉	C₂²	1	8	yes	complemented
40	#449×C₈₉	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₈₉	C₂²	1	8	yes	complemented
42	C₈×D₄×C₈₉	C₂²	1	8	yes	complemented
43	C₂×#107×C₈₉	C₂²	1	2	yes	—
44	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
46	#416×C₈₉	C₂²	1	16	yes	complemented
47	C₂×#103×C₈₉	C₂²	1	2	yes	—
48	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
49	C₂×#102×C₈₉	C₂²	1	2	yes	—
50	C₂×#102×C₈₉	C₂²	1	2	yes	—
51	C₂×#102×C₈₉	C₂²	1	2	yes	—
52	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×#41×C₈₉	—	2	4	no	—
56	C₂×#41×C₈₉	—	2	8	no	—
57	C₂×#41×C₈₉	—	4	4	no	—
58	C₂×#41×C₈₉	C₂³	1	8	yes	—
59	C₂²×Q₈×C₈₉	—	2	1	no	—
60	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
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	1	no	—
65	C₂²×D₄×C₈₉	—	2	1	no	—
66	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
67	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
68	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₈₉	—	2	1	no	—
70	C₂³×C₄×C₈₉	—	2	2	no	—
71	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₈₉	—	2	4	no	—
73	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
74	C₂×D₈×C₈₉	—	4	4	no	—
75	C₂×D₈×C₈₉	C₂³	1	8	yes	—
76	C₂²×C₈×C₈₉	—	4	4	no	—
77	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
80	C₂²×C₈×C₈₉	C₂³	1	8	yes	—
81	C₄⋊Q₈×C₈₉	—	2	2	no	—
82	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
83	#127×C₈₉	—	2	8	no	—
84	#127×C₈₉	—	2	2	no	—
85	#127×C₈₉	C₂³	1	4	yes	—
86	#124×C₈₉	—	2	16	no	—
87	#121×C₈₉	—	2	8	no	—
88	#121×C₈₉	—	2	4	no	—
89	#121×C₈₉	—	2	4	no	—
90	#121×C₈₉	—	2	16	no	—
91	#121×C₈₉	C₂³	1	8	yes	—
92	#121×C₈₉	C₂³	1	8	yes	—
93	C₄×Q₈×C₈₉	—	2	2	no	—
94	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
95	C₄×D₄×C₈₉	—	2	8	no	—
96	C₄×D₄×C₈₉	—	2	2	no	—
97	C₄×D₄×C₈₉	D₄	1	4	yes	—
98	C₄×D₄×C₈₉	C₂³	1	4	yes	—
99	C₄×D₄×C₈₉	C₂³	1	4	yes	—
100	#117×C₈₉	—	2	4	no	—
101	#117×C₈₉	—	2	8	no	—
102	#117×C₈₉	C₂³	1	8	yes	—
103	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
104	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
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	1	no	—
113	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
114	C₂×C₄²×C₈₉	—	2	2	no	—
115	C₂×C₄²×C₈₉	—	2	1	no	—
116	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
117	#107×C₈₉	C₂³	1	8	yes	—
118	#106×C₈₉	C₂³	1	4	yes	—
119	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
120	#103×C₈₉	C₂³	1	4	yes	—
121	#103×C₈₉	C₂³	1	8	yes	—
122	#102×C₈₉	C₂³	1	8	yes	—
123	#102×C₈₉	C₂³	1	8	yes	—
124	#102×C₈₉	C₂³	1	4	yes	—
125	#102×C₈₉	C₂³	1	8	yes	—
126	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
127	C₄×C₈×C₈₉	C₂³	1	4	yes	—
128	C₂⁴×C₈₉	—	2	1	no	—
129	C₂⁴×C₈₉	—	2	2	no	—
130	C₂⁴×C₈₉	C₂×D₄	1	2	yes	—
131	#41×C₈₉	—	2	16	no	—
132	#41×C₈₉	—	4	8	no	—
133	#41×C₈₉	—	4	16	no	—
134	C₂×Q₈×C₈₉	—	2	1	no	—
135	C₂×Q₈×C₈₉	—	2	4	no	—
136	C₂×Q₈×C₈₉	—	2	2	no	—
137	C₂×Q₈×C₈₉	—	8	2	no	—
138	C₂×Q₈×C₈₉	#41	1	4	yes	—
139	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
140	C₂×D₄×C₈₉	—	2	8	no	—
141	C₂×D₄×C₈₉	—	2	8	no	—
142	C₂×D₄×C₈₉	—	2	8	no	—
143	C₂×D₄×C₈₉	—	2	4	no	—
144	C₂×D₄×C₈₉	—	2	1	no	—
145	C₂×D₄×C₈₉	—	2	4	no	—
146	C₂×D₄×C₈₉	—	2	2	no	—
147	C₂×D₄×C₈₉	—	2	4	no	—
148	C₂×D₄×C₈₉	—	2	2	no	—
149	C₂×D₄×C₈₉	—	2	4	no	—
150	C₂×D₄×C₈₉	—	2	2	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	—	8	2	no	—
155	C₂×D₄×C₈₉	—	8	4	no	—
156	C₂×D₄×C₈₉	—	8	4	no	—
157	C₂×D₄×C₈₉	C₂×D₄	1	4	yes	—
158	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
159	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₈₉	—	2	1	no	—
161	C₂²×C₄×C₈₉	—	2	4	no	—
162	C₂²×C₄×C₈₉	—	2	4	no	—
163	C₂²×C₄×C₈₉	—	2	1	no	—
164	C₂²×C₄×C₈₉	—	2	1	no	—
165	C₂²×C₄×C₈₉	—	2	2	no	—
166	C₂²×C₄×C₈₉	—	2	4	no	—
167	C₂²×C₄×C₈₉	—	2	8	no	—
168	C₂²×C₄×C₈₉	—	4	8	no	—
169	C₂²×C₄×C₈₉	—	4	4	no	—
170	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
176	SD₁₆×C₈₉	—	2	8	no	—
177	D₈×C₈₉	—	4	8	no	—
178	C₂×C₈×C₈₉	—	2	2	no	—
179	C₂×C₈×C₈₉	—	2	2	no	—
180	C₂×C₈×C₈₉	—	2	4	no	—
181	C₂×C₈×C₈₉	—	4	8	no	—
182	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
183	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
184	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₈₉	—	2	4	no	—
186	C₄⋊C₄×C₈₉	—	2	4	no	—
187	C₄⋊C₄×C₈₉	—	2	1	no	—
188	C₄⋊C₄×C₈₉	—	2	2	no	—
189	C₄⋊C₄×C₈₉	—	2	1	no	—
190	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
191	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
192	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
193	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
194	C₂²⋊C₄×C₈₉	—	2	8	no	—
195	C₂²⋊C₄×C₈₉	—	2	8	no	—
196	C₂²⋊C₄×C₈₉	—	2	8	no	—
197	C₂²⋊C₄×C₈₉	—	2	4	no	—
198	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
199	C₄²×C₈₉	—	2	8	no	—
200	C₄²×C₈₉	—	2	4	no	—
201	C₄²×C₈₉	C₂×D₄	1	2	yes	—
202	C₄²×C₈₉	C₂⁴	1	2	yes	—
203	C₂³×C₈₉	—	2	4	no	—
204	C₂³×C₈₉	—	2	4	no	—
205	C₂³×C₈₉	—	2	4	no	—
206	C₂³×C₈₉	—	2	2	no	—
207	C₂³×C₈₉	—	2	4	no	—
208	C₂³×C₈₉	—	4	4	no	—
209	C₂³×C₈₉	—	4	2	no	—
210	C₂³×C₈₉	—	4	2	no	—
211	C₂³×C₈₉	—	4	4	no	—
212	C₂³×C₈₉	—	4	4	no	—
213	C₂³×C₈₉	—	4	2	no	—
214	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
215	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
216	Q₈×C₈₉	—	2	4	no	—
217	Q₈×C₈₉	—	2	2	no	—
218	Q₈×C₈₉	—	8	4	no	—
219	Q₈×C₈₉	C₂×#41	1	4	yes	—
220	D₄×C₈₉	—	2	8	no	—
221	D₄×C₈₉	—	2	2	no	—
222	D₄×C₈₉	—	2	4	no	—
223	D₄×C₈₉	—	2	2	no	—
224	D₄×C₈₉	—	2	4	no	—
225	D₄×C₈₉	—	4	16	no	—
226	D₄×C₈₉	—	4	2	no	—
227	D₄×C₈₉	—	4	4	no	—
228	D₄×C₈₉	—	4	8	no	—
229	D₄×C₈₉	—	8	4	no	—
230	D₄×C₈₉	—	8	8	no	—
231	D₄×C₈₉	—	8	8	no	—
232	C₂×C₄×C₈₉	—	2	4	no	—
233	C₂×C₄×C₈₉	—	2	4	no	—
234	C₂×C₄×C₈₉	—	2	2	no	—
235	C₂×C₄×C₈₉	—	2	2	no	—
236	C₂×C₄×C₈₉	—	2	8	no	—
237	C₂×C₄×C₈₉	—	2	4	no	—
238	C₂×C₄×C₈₉	—	2	4	no	—
239	C₂×C₄×C₈₉	—	2	2	no	—
240	C₂×C₄×C₈₉	—	2	2	no	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	4	16	no	—
243	C₂×C₄×C₈₉	—	4	2	no	—
244	C₂×C₄×C₈₉	—	4	2	no	—
245	C₂×C₄×C₈₉	—	4	4	no	—
246	C₂×C₄×C₈₉	—	4	8	no	—
247	C₂×C₄×C₈₉	#135	1	4	yes	—
248	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
249	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
253	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
254	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
255	C₈×C₈₉	—	2	2	no	—
256	C₈×C₈₉	—	2	2	no	—
257	C₈×C₈₉	—	2	4	no	—
258	C₂²×C₈₉	—	2	4	no	—
259	C₂²×C₈₉	—	2	4	no	—
260	C₂²×C₈₉	—	2	4	no	—
261	C₂²×C₈₉	—	4	8	no	—
262	C₂²×C₈₉	—	4	4	no	—
263	C₂²×C₈₉	—	4	2	no	—
264	C₂²×C₈₉	—	4	2	no	—
265	C₂²×C₈₉	—	4	4	no	—
266	C₂²×C₈₉	—	4	4	no	—
267	C₂²×C₈₉	—	4	2	no	—
268	C₂²×C₈₉	—	8	8	no	—
269	C₂²×C₈₉	—	8	4	no	—
270	C₂²×C₈₉	#546	1	2	yes	—
271	C₂²×C₈₉	C₂×#135	1	2	yes	—
272	C₂²×C₈₉	#576	1	2	yes	—
273	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
274	C₄×C₈₉	—	2	4	no	—
275	C₄×C₈₉	—	2	4	no	—
276	C₄×C₈₉	—	2	2	no	—
277	C₄×C₈₉	—	4	2	no	—
278	C₄×C₈₉	—	4	2	no	—
279	C₄×C₈₉	—	4	4	no	—
280	C₄×C₈₉	#546	1	2	yes	—
281	C₄×C₈₉	C₂×#135	1	4	yes	—
282	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
283	C₄×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
284	C₂×#3305	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
285	C₂×C₈₉	—	4	4	no	—
286	C₂×C₈₉	—	8	4	no	—
287	C₂×C₈₉	—	8	2	no	—
288	C₂×C₈₉	#3305	1	4	yes	direct product factor
289	C₂×C₈₉	C₂×#546	1	1	yes	characteristic
290	C₂×C₈₉	C₂²×#135	1	1	yes	characteristic
291	C₂×C₈₉	C₂×#576	1	1	yes	characteristic
292	C₂×#555	C₂×C₈₉	1	1	yes	characteristic, complemented
293	C₂×#550	C₂×C₈₉	1	1	yes	characteristic, complemented
294	#3305	C₂×C₈₉	1	16	yes	direct product factor
295	C₂×#486	C₂×C₈₉	1	1	yes	characteristic, complemented
296	C₂×#482	C₂×C₈₉	1	2	yes	complemented
297	C₂×#477	C₂×C₈₉	1	1	yes	characteristic, complemented
298	C₂×#466	C₂×C₈₉	1	2	yes	complemented
299	C₂×#459	C₂×C₈₉	1	1	yes	characteristic, complemented
300	C₂×#449	C₂×C₈₉	1	2	yes	complemented
301	C₂×C₄×SD₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
302	C₂×C₈×D₄	C₂×C₈₉	1	1	yes	characteristic, complemented
303	C₂×#416	C₂×C₈₉	1	2	yes	complemented
304	C₈₉	C₂×#3305	1	1	yes	characteristic, direct product factor, Sylow
305	C₂²×#41	C₂²×C₈₉	1	2	yes	—
306	C₂²×SD₁₆	C₂²×C₈₉	1	1	yes	characteristic
307	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
308	C₂³×C₈	C₂²×C₈₉	1	2	yes	—
309	#555	—	2	4	no	—
310	#555	C₂²×C₈₉	1	8	yes	complemented
311	#550	—	2	4	no	—
312	#550	C₂²×C₈₉	1	8	yes	complemented
313	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
314	C₂×#127	—	2	1	no	—
315	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
316	C₂×#124	—	2	2	no	—
317	C₂×#121	—	2	1	no	—
318	C₂×#121	—	2	2	no	—
319	C₂×#121	C₂²×C₈₉	1	2	yes	—
320	C₂×#121	C₂²×C₈₉	1	2	yes	—
321	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
322	C₂×C₄×D₄	—	2	1	no	—
323	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
324	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
325	C₂×#117	—	2	1	no	—
326	C₂×#117	C₂²×C₈₉	1	2	yes	—
327	#486	C₂²×C₈₉	1	8	yes	complemented
328	#482	C₂²×C₈₉	1	16	yes	complemented
329	#477	C₂²×C₈₉	1	8	yes	complemented
330	#466	C₂²×C₈₉	1	16	yes	complemented
331	#459	C₂²×C₈₉	1	8	yes	complemented
332	#449	C₂²×C₈₉	1	16	yes	complemented
333	C₄×SD₁₆	C₂²×C₈₉	1	8	yes	complemented
334	C₈×D₄	C₂²×C₈₉	1	8	yes	complemented
335	C₂×#107	C₂²×C₈₉	1	2	yes	—
336	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
337	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
338	#416	C₂²×C₈₉	1	16	yes	complemented
339	C₂×#103	C₂²×C₈₉	1	2	yes	—
340	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
341	C₂×#102	C₂²×C₈₉	1	2	yes	—
342	C₂×#102	C₂²×C₈₉	1	2	yes	—
343	C₂×#102	C₂²×C₈₉	1	2	yes	—
344	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
345	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
346	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
347	C₂×#41	—	2	4	no	—
348	C₂×#41	—	2	8	no	—
349	C₂×#41	—	4	4	no	—
350	C₂×#41	C₂³×C₈₉	1	8	yes	—
351	C₂²×Q₈	—	2	1	no	—
352	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	2	2	no	—
356	C₂²×D₄	—	2	1	no	—
357	C₂²×D₄	—	2	1	no	—
358	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
359	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
360	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
361	C₂³×C₄	—	2	1	no	—
362	C₂³×C₄	—	2	2	no	—
363	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
364	C₂×SD₁₆	—	2	4	no	—
365	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
366	C₂×D₈	—	4	4	no	—
367	C₂×D₈	C₂³×C₈₉	1	8	yes	—
368	C₂²×C₈	—	4	4	no	—
369	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
370	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
371	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₈₉	1	8	yes	—
373	C₄⋊Q₈	—	2	2	no	—
374	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
375	#127	—	2	8	no	—
376	#127	—	2	2	no	—
377	#127	C₂³×C₈₉	1	4	yes	—
378	#124	—	2	16	no	—
379	#121	—	2	8	no	—
380	#121	—	2	4	no	—
381	#121	—	2	4	no	—
382	#121	—	2	16	no	—
383	#121	C₂³×C₈₉	1	8	yes	—
384	#121	C₂³×C₈₉	1	8	yes	—
385	C₄×Q₈	—	2	2	no	—
386	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
387	C₄×D₄	—	2	8	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	D₄×C₈₉	1	4	yes	—
390	C₄×D₄	C₂³×C₈₉	1	4	yes	—
391	C₄×D₄	C₂³×C₈₉	1	4	yes	—
392	#117	—	2	4	no	—
393	#117	—	2	8	no	—
394	#117	C₂³×C₈₉	1	8	yes	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
401	C₂×C₂²⋊C₄	—	2	2	no	—
402	C₂×C₂²⋊C₄	—	2	2	no	—
403	C₂×C₂²⋊C₄	—	2	2	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
406	C₂×C₄²	—	2	2	no	—
407	C₂×C₄²	—	2	1	no	—
408	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
409	#107	C₂³×C₈₉	1	8	yes	—
410	#106	C₂³×C₈₉	1	4	yes	—
411	C₄⋊C₈	C₂³×C₈₉	1	4	yes	—
412	#103	C₂³×C₈₉	1	4	yes	—
413	#103	C₂³×C₈₉	1	8	yes	—
414	#102	C₂³×C₈₉	1	8	yes	—
415	#102	C₂³×C₈₉	1	8	yes	—
416	#102	C₂³×C₈₉	1	4	yes	—
417	#102	C₂³×C₈₉	1	8	yes	—
418	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
419	C₄×C₈	C₂³×C₈₉	1	4	yes	—
420	C₂⁴	—	2	1	no	—
421	C₂⁴	—	2	2	no	—
422	C₂⁴	C₂×D₄×C₈₉	1	2	yes	—
423	#41	—	2	16	no	—
424	#41	—	4	8	no	—
425	#41	—	4	16	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	8	2	no	—
430	C₂×Q₈	#41×C₈₉	1	4	yes	—
431	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
432	C₂×D₄	—	2	8	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	8	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	4	8	no	—
445	C₂×D₄	—	4	4	no	—
446	C₂×D₄	—	8	2	no	—
447	C₂×D₄	—	8	4	no	—
448	C₂×D₄	—	8	4	no	—
449	C₂×D₄	C₂×D₄×C₈₉	1	4	yes	—
450	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
451	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	8	no	—
460	C₂²×C₄	—	4	8	no	—
461	C₂²×C₄	—	4	4	no	—
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	4	yes	—
465	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
468	SD₁₆	—	2	8	no	—
469	D₈	—	4	8	no	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	4	no	—
473	C₂×C₈	—	4	8	no	—
474	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
475	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
476	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
483	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
484	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
485	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
491	C₄²	—	2	8	no	—
492	C₄²	—	2	4	no	—
493	C₄²	C₂×D₄×C₈₉	1	2	yes	—
494	C₄²	C₂⁴×C₈₉	1	2	yes	—
495	C₂³	—	2	4	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	4	4	no	—
501	C₂³	—	4	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	4	4	no	—
505	C₂³	—	4	2	no	—
506	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
507	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
508	Q₈	—	2	4	no	—
509	Q₈	—	2	2	no	—
510	Q₈	—	8	4	no	—
511	Q₈	C₂×#41×C₈₉	1	4	yes	—
512	D₄	—	2	8	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	4	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	4	no	—
517	D₄	—	4	16	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	4	no	—
520	D₄	—	4	8	no	—
521	D₄	—	8	4	no	—
522	D₄	—	8	8	no	—
523	D₄	—	8	8	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	8	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	4	16	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	4	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	#135×C₈₉	1	4	yes	—
540	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
541	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
542	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
543	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
544	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
545	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
546	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
547	C₈	—	2	2	no	—
548	C₈	—	2	2	no	—
549	C₈	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	2	4	no	—
553	C₂²	—	4	8	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	4	4	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	8	8	no	—
561	C₂²	—	8	4	no	—
562	C₂²	#546×C₈₉	1	2	yes	—
563	C₂²	C₂×#135×C₈₉	1	2	yes	—
564	C₂²	#576×C₈₉	1	2	yes	—
565	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
566	C₄	—	2	2	no	—
567	C₄	—	2	4	no	—
568	C₄	—	2	4	no	—
569	C₄	—	4	2	no	—
570	C₄	—	4	2	no	—
571	C₄	—	4	4	no	—
572	C₄	#546×C₈₉	1	2	yes	—
573	C₄	C₂×#135×C₈₉	1	4	yes	—
574	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
575	C₄	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
576	C₂	—	4	4	no	—
577	C₂	—	8	2	no	—
578	C₂	—	8	4	no	—
579	C₂	C₈₉×#3305	1	4	yes	direct product factor
580	C₂	C₂×#546×C₈₉	1	1	yes	characteristic
581	C₂	C₂²×#135×C₈₉	1	1	yes	characteristic
582	C₂	C₂×#576×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 ◁ C₂×C₄×D₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#555×C₈₉	C₂	semidirect
2. C₂×#550×C₈₉	C₂	semidirect
3. C₈₉×#3305	C₂	direct (N × H)
4. C₂×#486×C₈₉	C₂	semidirect
5. C₂×#482×C₈₉	C₂	semidirect
6. C₂×#477×C₈₉	C₂	semidirect
7. C₂×#466×C₈₉	C₂	semidirect
8. C₂×#459×C₈₉	C₂	semidirect
9. C₂×#449×C₈₉	C₂	semidirect
10. C₂×C₄×SD₁₆×C₈₉	C₂	semidirect
11. C₂×C₈×D₄×C₈₉	C₂	semidirect
12. C₂×#416×C₈₉	C₂	semidirect
18. #555×C₈₉	C₂²	semidirect
20. #550×C₈₉	C₂²	semidirect
35. #486×C₈₉	C₂²	semidirect
36. #482×C₈₉	C₂²	semidirect
37. #477×C₈₉	C₂²	semidirect
38. #466×C₈₉	C₂²	semidirect
39. #459×C₈₉	C₂²	semidirect
40. #449×C₈₉	C₂²	semidirect
41. C₄×SD₁₆×C₈₉	C₂²	semidirect
42. C₈×D₄×C₈₉	C₂²	semidirect
46. #416×C₈₉	C₂²	semidirect
284. C₂×#3305	C₈₉	direct (N × H)
288. C₂×C₈₉	#3305	direct (N × H)
292. C₂×#555	C₂×C₈₉	semidirect
293. C₂×#550	C₂×C₈₉	semidirect
294. #3305	C₂×C₈₉	direct (N × H)
295. C₂×#486	C₂×C₈₉	semidirect
296. C₂×#482	C₂×C₈₉	semidirect
297. C₂×#477	C₂×C₈₉	semidirect
298. C₂×#466	C₂×C₈₉	semidirect
299. C₂×#459	C₂×C₈₉	semidirect
300. C₂×#449	C₂×C₈₉	semidirect
301. C₂×C₄×SD₁₆	C₂×C₈₉	semidirect
302. C₂×C₈×D₄	C₂×C₈₉	semidirect
303. C₂×#416	C₂×C₈₉	semidirect
304. C₈₉	C₂×#3305	direct (N × H)
310. #555	C₂²×C₈₉	semidirect
312. #550	C₂²×C₈₉	semidirect
327. #486	C₂²×C₈₉	semidirect
328. #482	C₂²×C₈₉	semidirect
329. #477	C₂²×C₈₉	semidirect
330. #466	C₂²×C₈₉	semidirect
331. #459	C₂²×C₈₉	semidirect
332. #449	C₂²×C₈₉	semidirect
333. C₄×SD₁₆	C₂²×C₈₉	semidirect
334. C₈×D₄	C₂²×C₈₉	semidirect
338. #416	C₂²×C₈₉	semidirect
579. C₂	C₈₉×#3305	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×#555×C₈₉	2
C₂×#550×C₈₉	2
C₈₉×#3305	2
C₂×#486×C₈₉	2
C₂×#482×C₈₉	2
C₂×#477×C₈₉	2
C₂×#466×C₈₉	2
C₂×#459×C₈₉	2
C₂×#449×C₈₉	2
C₂×C₄×SD₁₆×C₈₉	2
C₂×C₈×D₄×C₈₉	2
C₂×#416×C₈₉	2
C₂×#3305	89

Subgroups — C2×C89×#3305 #36438971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3305 #36438971