Subgroups — C₈₉×#35254 #36413102

Order 22784.

2762 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#535×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2854	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2624	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2448	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2445	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2418	C₂	1	8	yes	maximal, complemented
8	C₈₉×#2316	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2311	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#2309	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2296	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#390×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	#535×C₈₉	—	2	8	no	—
16	C₂×#127×C₈₉	C₂²	1	2	yes	—
17	C₂×#124×C₈₉	C₂²	1	2	yes	—
18	C₂×#123×C₈₉	C₂²	1	8	yes	complemented
19	C₂×#121×C₈₉	C₂²	1	4	yes	—
20	C₂×#120×C₈₉	C₂²	1	4	yes	—
21	C₄×#41×C₈₉	—	2	8	no	—
22	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	complemented
23	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	complemented
25	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#117×C₈₉	C₂²	1	2	yes	complemented
29	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
31	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₈₉	C₂²	1	2	yes	complemented
33	#399×C₈₉	C₂²	1	8	yes	complemented
34	#397×C₈₉	C₂²	1	8	yes	complemented
35	#394×C₈₉	C₂²	1	8	yes	complemented
36	#394×C₈₉	C₂²	1	8	yes	complemented
37	#391×C₈₉	C₂²	1	2	yes	complemented
38	#390×C₈₉	C₂²	1	4	yes	—
39	#390×C₈₉	C₂²	1	4	yes	—
40	#390×C₈₉	C₂²	1	2	yes	—
41	#390×C₈₉	C₂²	1	2	yes	—
42	#390×C₈₉	C₂²	1	4	yes	—
43	#390×C₈₉	C₂²	1	2	yes	complemented
44	#389×C₈₉	C₂²	1	2	yes	complemented
45	#389×C₈₉	C₂²	1	2	yes	complemented
46	#389×C₈₉	C₂²	1	4	yes	complemented
47	#389×C₈₉	C₂²	1	4	yes	complemented
48	#388×C₈₉	C₂²	1	8	yes	complemented
49	#388×C₈₉	C₂²	1	8	yes	complemented
50	#386×C₈₉	C₂²	1	4	yes	—
51	#386×C₈₉	C₂²	1	8	yes	complemented
52	#385×C₈₉	C₂²	1	4	yes	—
53	#380×C₈₉	C₂²	1	4	yes	—
54	#380×C₈₉	C₂²	1	4	yes	complemented
55	#379×C₈₉	C₂²	1	4	yes	—
56	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
57	C₂⁵×C₈₉	C₂³	1	2	yes	—
58	C₂×#41×C₈₉	—	2	4	no	—
59	C₂×#41×C₈₉	—	2	4	no	—
60	C₂×#41×C₈₉	D₄	1	8	yes	—
61	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₈₉	—	2	4	no	—
63	C₂²×D₄×C₈₉	—	2	4	no	—
64	C₂²×D₄×C₈₉	—	2	4	no	—
65	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
66	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
67	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
68	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
70	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
71	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
72	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
73	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
74	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
75	#127×C₈₉	—	2	8	no	—
76	#124×C₈₉	—	2	8	no	—
77	#123×C₈₉	—	4	16	no	—
78	#121×C₈₉	—	2	16	no	—
79	#120×C₈₉	—	2	16	no	—
80	C₄×Q₈×C₈₉	—	2	8	no	—
81	C₄×D₄×C₈₉	—	2	16	no	—
82	C₄×D₄×C₈₉	—	2	8	no	—
83	#117×C₈₉	—	2	4	no	—
84	#117×C₈₉	—	2	4	no	—
85	#117×C₈₉	—	2	8	no	—
86	#117×C₈₉	—	2	4	no	—
87	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
97	C₂×C₂²⋊C₄×C₈₉	—	4	8	no	—
98	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
99	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
108	C₂×C₄²×C₈₉	—	2	4	no	—
109	C₂×C₄²×C₈₉	—	2	8	no	—
110	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
111	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
112	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
113	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
114	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
117	#95×C₈₉	C₂³	1	4	yes	—
118	#95×C₈₉	C₂³	1	4	yes	—
119	#95×C₈₉	C₂³	1	8	yes	—
120	#95×C₈₉	C₂³	1	8	yes	—
121	C₂⁴×C₈₉	—	2	4	no	—
122	C₂⁴×C₈₉	—	2	4	no	—
123	C₂⁴×C₈₉	—	2	4	no	—
124	C₂⁴×C₈₉	—	4	4	no	—
125	C₂⁴×C₈₉	—	4	4	no	—
126	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
127	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
128	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
129	#41×C₈₉	—	2	32	no	—
130	C₂×Q₈×C₈₉	—	2	2	no	—
131	C₂×Q₈×C₈₉	—	2	2	no	—
132	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
133	C₂×D₄×C₈₉	—	2	4	no	—
134	C₂×D₄×C₈₉	—	2	4	no	—
135	C₂×D₄×C₈₉	—	2	16	no	—
136	C₂×D₄×C₈₉	—	2	4	no	—
137	C₂×D₄×C₈₉	—	2	4	no	—
138	C₂×D₄×C₈₉	—	2	2	no	—
139	C₂×D₄×C₈₉	—	2	2	no	—
140	C₂×D₄×C₈₉	—	2	4	no	—
141	C₂×D₄×C₈₉	—	2	4	no	—
142	C₂×D₄×C₈₉	—	2	4	no	—
143	C₂×D₄×C₈₉	—	2	4	no	—
144	C₂×D₄×C₈₉	—	4	8	no	—
145	C₂×D₄×C₈₉	—	4	8	no	—
146	C₂×D₄×C₈₉	—	4	16	no	—
147	C₂×D₄×C₈₉	C₂×D₄	1	8	yes	—
148	C₂×D₄×C₈₉	C₂×D₄	1	4	yes	—
149	C₂²×C₄×C₈₉	—	2	4	no	—
150	C₂²×C₄×C₈₉	—	2	4	no	—
151	C₂²×C₄×C₈₉	—	2	8	no	—
152	C₂²×C₄×C₈₉	—	2	4	no	—
153	C₂²×C₄×C₈₉	—	2	4	no	—
154	C₂²×C₄×C₈₉	—	2	4	no	—
155	C₂²×C₄×C₈₉	—	2	8	no	—
156	C₂²×C₄×C₈₉	—	4	4	no	—
157	C₂²×C₄×C₈₉	—	4	2	no	—
158	C₂²×C₄×C₈₉	—	4	4	no	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
160	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₈₉	#41	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	4	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₈₉	—	2	8	no	—
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	4	no	—
178	C₄⋊C₄×C₈₉	—	2	8	no	—
179	C₄⋊C₄×C₈₉	—	4	4	no	—
180	C₄⋊C₄×C₈₉	—	4	8	no	—
181	C₂²⋊C₄×C₈₉	—	2	8	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	16	no	—
186	C₂²⋊C₄×C₈₉	—	2	16	no	—
187	C₂²⋊C₄×C₈₉	—	4	8	no	—
188	C₂²⋊C₄×C₈₉	—	4	8	no	—
189	C₂²⋊C₄×C₈₉	—	4	8	no	—
190	C₂²⋊C₄×C₈₉	—	4	8	no	—
191	C₄²×C₈₉	—	2	2	no	—
192	C₄²×C₈₉	—	2	2	no	—
193	C₄²×C₈₉	—	2	4	no	—
194	C₄²×C₈₉	—	2	8	no	—
195	C₄²×C₈₉	—	2	4	no	—
196	C₄²×C₈₉	—	2	4	no	—
197	C₄²×C₈₉	—	2	4	no	—
198	C₂³×C₈₉	—	2	2	no	—
199	C₂³×C₈₉	—	2	2	no	—
200	C₂³×C₈₉	—	2	2	no	—
201	C₂³×C₈₉	—	2	2	no	—
202	C₂³×C₈₉	—	2	4	no	—
203	C₂³×C₈₉	—	2	4	no	—
204	C₂³×C₈₉	—	2	4	no	—
205	C₂³×C₈₉	—	2	4	no	—
206	C₂³×C₈₉	—	2	4	no	—
207	C₂³×C₈₉	—	2	8	no	—
208	C₂³×C₈₉	—	2	8	no	—
209	C₂³×C₈₉	—	4	16	no	—
210	C₂³×C₈₉	—	4	8	no	—
211	C₂³×C₈₉	—	8	8	no	—
212	C₂³×C₈₉	—	8	4	no	—
213	C₂³×C₈₉	—	8	4	no	—
214	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
215	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
216	C₂³×C₈₉	C₂×#41	1	2	yes	—
217	C₂³×C₈₉	C₂×#41	1	2	yes	—
218	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
219	Q₈×C₈₉	—	2	8	no	—
220	D₄×C₈₉	—	2	16	no	—
221	D₄×C₈₉	—	2	8	no	—
222	D₄×C₈₉	—	4	8	no	—
223	D₄×C₈₉	—	4	8	no	—
224	D₄×C₈₉	—	8	16	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	2	no	—
229	C₂×C₄×C₈₉	—	2	16	no	—
230	C₂×C₄×C₈₉	—	2	2	no	—
231	C₂×C₄×C₈₉	—	2	4	no	—
232	C₂×C₄×C₈₉	—	2	4	no	—
233	C₂×C₄×C₈₉	—	2	4	no	—
234	C₂×C₄×C₈₉	—	2	4	no	—
235	C₂×C₄×C₈₉	—	2	8	no	—
236	C₂×C₄×C₈₉	—	2	4	no	—
237	C₂×C₄×C₈₉	—	2	2	no	—
238	C₂×C₄×C₈₉	—	2	4	no	—
239	C₂×C₄×C₈₉	—	2	4	no	—
240	C₂×C₄×C₈₉	—	2	4	no	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	2	2	no	—
243	C₂×C₄×C₈₉	—	2	4	no	—
244	C₂×C₄×C₈₉	—	4	8	no	—
245	C₂×C₄×C₈₉	—	4	16	no	—
246	C₂×C₄×C₈₉	—	8	4	no	—
247	C₂×C₄×C₈₉	#123	1	16	yes	complemented
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	2	yes	—
251	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
252	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
253	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
254	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
255	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
256	C₂²×C₈₉	—	2	4	no	—
257	C₂²×C₈₉	—	2	4	no	—
258	C₂²×C₈₉	—	2	2	no	—
259	C₂²×C₈₉	—	2	2	no	—
260	C₂²×C₈₉	—	2	4	no	—
261	C₂²×C₈₉	—	2	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	8	no	—
267	C₂²×C₈₉	—	4	8	no	—
268	C₂²×C₈₉	—	4	4	no	—
269	C₂²×C₈₉	—	4	4	no	—
270	C₂²×C₈₉	—	8	16	no	—
271	C₂²×C₈₉	C₂×#123	1	4	yes	—
272	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
273	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
274	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
275	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
276	C₂²×C₈₉	#585	1	2	yes	—
277	C₄×C₈₉	—	2	4	no	—
278	C₄×C₈₉	—	2	4	no	—
279	C₄×C₈₉	—	4	4	no	—
280	C₄×C₈₉	—	4	2	no	—
281	C₄×C₈₉	—	4	4	no	—
282	C₄×C₈₉	—	4	4	no	—
283	C₄×C₈₉	—	4	4	no	—
284	C₄×C₈₉	C₂×#123	1	4	yes	—
285	C₄×C₈₉	C₂×#123	1	4	yes	—
286	#35254	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂×C₈₉	—	2	4	no	—
288	C₂×C₈₉	—	4	4	no	—
289	C₂×C₈₉	—	8	4	no	—
290	C₂×C₈₉	C₂²×#123	1	1	yes	characteristic
291	C₂×C₈₉	#3460	1	2	yes	—
292	C₂×C₈₉	#3488	1	2	yes	—
293	C₂×C₈₉	#3508	1	1	yes	characteristic
294	C₂×C₈₉	#3510	1	1	yes	characteristic
295	C₂×#535	C₂×C₈₉	1	1	yes	characteristic
296	C₂×C₄×#41	C₂×C₈₉	1	1	yes	characteristic, complemented
297	#2854	C₂×C₈₉	1	4	yes	complemented
298	#2624	C₂×C₈₉	1	4	yes	complemented
299	#2448	C₂×C₈₉	1	4	yes	complemented
300	#2445	C₂×C₈₉	1	4	yes	complemented
301	#2418	C₂×C₈₉	1	8	yes	complemented
302	#2316	C₂×C₈₉	1	1	yes	characteristic, complemented
303	#2311	C₂×C₈₉	1	1	yes	characteristic, complemented
304	#2309	C₂×C₈₉	1	1	yes	characteristic, complemented
305	#2296	C₂×C₈₉	1	1	yes	characteristic, complemented
306	C₂×#390	C₂×C₈₉	1	1	yes	characteristic, complemented
307	C₈₉	#35254	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
309	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
310	#535	—	2	8	no	—
311	C₂×#127	C₂²×C₈₉	1	2	yes	—
312	C₂×#124	C₂²×C₈₉	1	2	yes	—
313	C₂×#123	C₂²×C₈₉	1	8	yes	complemented
314	C₂×#121	C₂²×C₈₉	1	4	yes	—
315	C₂×#120	C₂²×C₈₉	1	4	yes	—
316	C₄×#41	—	2	8	no	—
317	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	complemented
318	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
319	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	complemented
320	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
321	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
322	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
323	C₂×#117	C₂²×C₈₉	1	2	yes	complemented
324	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
325	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
326	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
327	C₂²×C₄²	C₂²×C₈₉	1	2	yes	complemented
328	#399	C₂²×C₈₉	1	8	yes	complemented
329	#397	C₂²×C₈₉	1	8	yes	complemented
330	#394	C₂²×C₈₉	1	8	yes	complemented
331	#394	C₂²×C₈₉	1	8	yes	complemented
332	#391	C₂²×C₈₉	1	2	yes	complemented
333	#390	C₂²×C₈₉	1	4	yes	—
334	#390	C₂²×C₈₉	1	4	yes	—
335	#390	C₂²×C₈₉	1	2	yes	—
336	#390	C₂²×C₈₉	1	2	yes	—
337	#390	C₂²×C₈₉	1	4	yes	—
338	#390	C₂²×C₈₉	1	2	yes	complemented
339	#389	C₂²×C₈₉	1	4	yes	complemented
340	#389	C₂²×C₈₉	1	4	yes	complemented
341	#389	C₂²×C₈₉	1	2	yes	complemented
342	#389	C₂²×C₈₉	1	2	yes	complemented
343	#388	C₂²×C₈₉	1	8	yes	complemented
344	#388	C₂²×C₈₉	1	8	yes	complemented
345	#386	C₂²×C₈₉	1	4	yes	—
346	#386	C₂²×C₈₉	1	8	yes	complemented
347	#385	C₂²×C₈₉	1	4	yes	—
348	#380	C₂²×C₈₉	1	4	yes	—
349	#380	C₂²×C₈₉	1	4	yes	complemented
350	#379	C₂²×C₈₉	1	4	yes	—
351	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
352	C₂⁵	C₂³×C₈₉	1	2	yes	—
353	C₂×#41	—	2	4	no	—
354	C₂×#41	—	2	4	no	—
355	C₂×#41	D₄×C₈₉	1	8	yes	—
356	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
357	C₂²×D₄	—	2	4	no	—
358	C₂²×D₄	—	2	4	no	—
359	C₂²×D₄	—	2	4	no	—
360	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
361	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
362	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
363	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
364	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
365	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
366	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
367	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
368	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
369	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
370	#127	—	2	8	no	—
371	#124	—	2	8	no	—
372	#123	—	4	16	no	—
373	#121	—	2	16	no	—
374	#120	—	2	16	no	—
375	C₄×Q₈	—	2	8	no	—
376	C₄×D₄	—	2	16	no	—
377	C₄×D₄	—	2	8	no	—
378	#117	—	2	4	no	—
379	#117	—	2	4	no	—
380	#117	—	2	8	no	—
381	#117	—	2	4	no	—
382	C₂×C₄⋊C₄	—	2	4	no	—
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
387	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
391	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
392	C₂×C₂²⋊C₄	—	4	8	no	—
393	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
403	C₂×C₄²	—	2	4	no	—
404	C₂×C₄²	—	2	8	no	—
405	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
406	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
407	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
408	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
410	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
411	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
412	#95	C₂³×C₈₉	1	4	yes	—
413	#95	C₂³×C₈₉	1	4	yes	—
414	#95	C₂³×C₈₉	1	8	yes	—
415	#95	C₂³×C₈₉	1	8	yes	—
416	C₂⁴	—	2	4	no	—
417	C₂⁴	—	2	4	no	—
418	C₂⁴	—	2	4	no	—
419	C₂⁴	—	4	4	no	—
420	C₂⁴	—	4	4	no	—
421	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
422	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
423	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
424	#41	—	2	32	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	16	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	4	8	no	—
440	C₂×D₄	—	4	8	no	—
441	C₂×D₄	—	4	16	no	—
442	C₂×D₄	C₂×D₄×C₈₉	1	8	yes	—
443	C₂×D₄	C₂×D₄×C₈₉	1	4	yes	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	8	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	8	no	—
451	C₂²×C₄	—	4	4	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	4	no	—
454	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
455	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
456	C₂²×C₄	#41×C₈₉	1	4	yes	—
457	C₂²×C₄	#41×C₈₉	1	4	yes	—
458	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
461	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
463	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
464	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	4	8	no	—
475	C₄⋊C₄	—	4	4	no	—
476	C₂²⋊C₄	—	2	8	no	—
477	C₂²⋊C₄	—	2	4	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	16	no	—
481	C₂²⋊C₄	—	2	16	no	—
482	C₂²⋊C₄	—	4	8	no	—
483	C₂²⋊C₄	—	4	8	no	—
484	C₂²⋊C₄	—	4	8	no	—
485	C₂²⋊C₄	—	4	8	no	—
486	C₄²	—	2	2	no	—
487	C₄²	—	2	2	no	—
488	C₄²	—	2	4	no	—
489	C₄²	—	2	8	no	—
490	C₄²	—	2	4	no	—
491	C₄²	—	2	4	no	—
492	C₄²	—	2	4	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	2	8	no	—
503	C₂³	—	2	8	no	—
504	C₂³	—	4	16	no	—
505	C₂³	—	4	8	no	—
506	C₂³	—	8	8	no	—
507	C₂³	—	8	4	no	—
508	C₂³	—	8	4	no	—
509	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
510	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
511	C₂³	C₂×#41×C₈₉	1	2	yes	—
512	C₂³	C₂×#41×C₈₉	1	2	yes	—
513	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
514	Q₈	—	2	8	no	—
515	D₄	—	2	16	no	—
516	D₄	—	2	8	no	—
517	D₄	—	4	8	no	—
518	D₄	—	4	8	no	—
519	D₄	—	8	16	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	16	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	8	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	—	4	16	no	—
541	C₂×C₄	—	8	4	no	—
542	C₂×C₄	#123×C₈₉	1	16	yes	complemented
543	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
544	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
545	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
546	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
547	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
548	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
549	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
550	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
551	C₂²	—	2	4	no	—
552	C₂²	—	2	4	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	4	no	—
556	C₂²	—	2	8	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	8	no	—
562	C₂²	—	4	8	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	8	16	no	—
566	C₂²	C₂×#123×C₈₉	1	4	yes	—
567	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
568	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
569	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
570	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
571	C₂²	#585×C₈₉	1	2	yes	—
572	C₄	—	2	4	no	—
573	C₄	—	2	4	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	4	no	—
577	C₄	—	4	4	no	—
578	C₄	—	4	4	no	—
579	C₄	C₂×#123×C₈₉	1	4	yes	—
580	C₄	C₂×#123×C₈₉	1	4	yes	—
581	C₂	—	2	4	no	—
582	C₂	—	4	4	no	—
583	C₂	—	8	4	no	—
584	C₂	C₂²×#123×C₈₉	1	1	yes	characteristic
585	C₂	C₈₉×#3460	1	2	yes	—
586	C₂	C₈₉×#3488	1	2	yes	—
587	C₂	C₈₉×#3508	1	1	yes	characteristic
588	C₂	C₈₉×#3510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#535×C₈₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×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. C₈₉×#2854	C₂	semidirect
4. C₈₉×#2624	C₂	semidirect
5. C₈₉×#2448	C₂	semidirect
6. C₈₉×#2445	C₂	semidirect
7. C₈₉×#2418	C₂	semidirect
8. C₈₉×#2316	C₂	semidirect
9. C₈₉×#2311	C₂	semidirect
10. C₈₉×#2309	C₂	semidirect
11. C₈₉×#2296	C₂	semidirect
12. C₂×#390×C₈₉	C₂	semidirect
18. C₂×#123×C₈₉	C₂²	semidirect
22. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
24. C₂×C₄×D₄×C₈₉	C₂²	semidirect
28. C₂×#117×C₈₉	C₂²	semidirect
32. C₂²×C₄²×C₈₉	C₂²	semidirect
33. #399×C₈₉	C₂²	semidirect
34. #397×C₈₉	C₂²	semidirect
35. #394×C₈₉	C₂²	semidirect
36. #394×C₈₉	C₂²	semidirect
37. #391×C₈₉	C₂²	semidirect
43. #390×C₈₉	C₂²	semidirect
44. #389×C₈₉	C₂²	semidirect
45. #389×C₈₉	C₂²	semidirect
46. #389×C₈₉	C₂²	semidirect
47. #389×C₈₉	C₂²	semidirect
48. #388×C₈₉	C₂²	semidirect
49. #388×C₈₉	C₂²	semidirect
51. #386×C₈₉	C₂²	semidirect
54. #380×C₈₉	C₂²	semidirect
56. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
247. C₂×C₄×C₈₉	#123	semidirect
286. #35254	C₈₉	direct (N × H)
296. C₂×C₄×#41	C₂×C₈₉	semidirect
297. #2854	C₂×C₈₉	semidirect
298. #2624	C₂×C₈₉	semidirect
299. #2448	C₂×C₈₉	semidirect
300. #2445	C₂×C₈₉	semidirect
301. #2418	C₂×C₈₉	semidirect
302. #2316	C₂×C₈₉	semidirect
303. #2311	C₂×C₈₉	semidirect
304. #2309	C₂×C₈₉	semidirect
305. #2296	C₂×C₈₉	semidirect
306. C₂×#390	C₂×C₈₉	semidirect
307. C₈₉	#35254	direct (N × H)
313. C₂×#123	C₂²×C₈₉	semidirect
317. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
319. C₂×C₄×D₄	C₂²×C₈₉	semidirect
323. C₂×#117	C₂²×C₈₉	semidirect
327. C₂²×C₄²	C₂²×C₈₉	semidirect
328. #399	C₂²×C₈₉	semidirect
329. #397	C₂²×C₈₉	semidirect
330. #394	C₂²×C₈₉	semidirect
331. #394	C₂²×C₈₉	semidirect
332. #391	C₂²×C₈₉	semidirect
338. #390	C₂²×C₈₉	semidirect
339. #389	C₂²×C₈₉	semidirect
340. #389	C₂²×C₈₉	semidirect
341. #389	C₂²×C₈₉	semidirect
342. #389	C₂²×C₈₉	semidirect
343. #388	C₂²×C₈₉	semidirect
344. #388	C₂²×C₈₉	semidirect
346. #386	C₂²×C₈₉	semidirect
349. #380	C₂²×C₈₉	semidirect
351. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
542. C₂×C₄	#123×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#535×C₈₉	2
C₂×C₄×#41×C₈₉	2
C₈₉×#2854	2
C₈₉×#2624	2
C₈₉×#2448	2
C₈₉×#2445	2
C₈₉×#2418	2
C₈₉×#2316	2
C₈₉×#2311	2
C₈₉×#2309	2
C₈₉×#2296	2
C₂×#390×C₈₉	2
#35254	89

Subgroups — C89×#35254 #36413102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35254 #36413102