Subgroups — C₈₉×#35301 #36413149

Order 22784.

2826 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2854	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2847	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2628	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2619	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2610	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2598	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2591	C₂	1	2	yes	maximal, complemented
10	C₈₉×#2539	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2406	C₂	1	2	yes	maximal, complemented
12	C₈₉×#2404	C₂	1	2	yes	maximal, complemented
13	C₈₉×#2324	C₂	1	4	yes	maximal, complemented
14	C₈₉×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₈₉	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
17	#532×C₈₉	—	2	8	no	—
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
19	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic, complemented
20	C₂×#124×C₈₉	C₂²	1	2	yes	complemented
21	C₂×#123×C₈₉	C₂²	1	4	yes	—
22	C₂×#121×C₈₉	C₂²	1	4	yes	—
23	C₂×#121×C₈₉	C₂²	1	2	yes	—
24	C₂×#121×C₈₉	C₂²	1	2	yes	—
25	C₂×#120×C₈₉	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	complemented
27	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	complemented
28	C₂×#117×C₈₉	—	2	8	no	—
29	C₂×#117×C₈₉	C₂²	1	2	yes	—
30	C₂×#117×C₈₉	C₂²	1	2	yes	—
31	C₂×#117×C₈₉	C₂²	1	2	yes	—
32	C₂×#117×C₈₉	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
39	#400×C₈₉	C₂²	1	4	yes	complemented
40	#399×C₈₉	C₂²	1	4	yes	—
41	#395×C₈₉	C₂²	1	4	yes	complemented
42	#394×C₈₉	C₂²	1	4	yes	complemented
43	#394×C₈₉	C₂²	1	4	yes	complemented
44	#393×C₈₉	C₂²	1	4	yes	—
45	#392×C₈₉	C₂²	1	4	yes	—
46	#392×C₈₉	C₂²	1	4	yes	—
47	#390×C₈₉	C₂²	1	8	yes	—
48	#390×C₈₉	C₂²	1	4	yes	complemented
49	#389×C₈₉	C₂²	1	4	yes	complemented
50	#388×C₈₉	C₂²	1	8	yes	complemented
51	#388×C₈₉	C₂²	1	8	yes	complemented
52	#386×C₈₉	C₂²	1	8	yes	—
53	#386×C₈₉	C₂²	1	4	yes	—
54	#386×C₈₉	C₂²	1	4	yes	—
55	#386×C₈₉	C₂²	1	8	yes	—
56	#385×C₈₉	C₂²	1	8	yes	complemented
57	#383×C₈₉	C₂²	1	2	yes	complemented
58	#382×C₈₉	C₂²	1	2	yes	complemented
59	#381×C₈₉	C₂²	1	2	yes	—
60	#380×C₈₉	C₂²	1	4	yes	—
61	#380×C₈₉	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
63	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
64	#376×C₈₉	C₂²	1	2	yes	complemented
65	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×#41×C₈₉	—	2	8	no	—
67	C₂×#41×C₈₉	—	4	8	no	—
68	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
69	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
70	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
71	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
72	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
73	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
74	C₂³×C₄×C₈₉	—	2	4	no	—
75	C₂³×C₄×C₈₉	—	2	8	no	—
76	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
78	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
82	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
83	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈×C₈₉	—	2	4	no	—
86	#127×C₈₉	—	2	4	no	—
87	#124×C₈₉	—	2	8	no	—
88	#123×C₈₉	—	4	8	no	—
89	#121×C₈₉	—	2	8	no	—
90	#121×C₈₉	—	2	8	no	—
91	#121×C₈₉	—	2	16	no	—
92	#120×C₈₉	—	4	8	no	—
93	C₄×D₄×C₈₉	—	2	8	no	—
94	C₄×D₄×C₈₉	—	2	8	no	—
95	#117×C₈₉	—	2	8	no	—
96	#117×C₈₉	—	2	8	no	—
97	#117×C₈₉	—	2	8	no	—
98	#117×C₈₉	—	2	8	no	—
99	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
100	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
110	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
111	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
113	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
120	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
122	C₂×C₄²×C₈₉	—	2	4	no	—
123	C₂×C₄²×C₈₉	D₄	1	8	yes	complemented
124	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
125	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
126	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
127	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
131	#95×C₈₉	C₂³	1	4	yes	—
132	#95×C₈₉	C₂³	1	4	yes	—
133	#95×C₈₉	C₂³	1	4	yes	—
134	#95×C₈₉	C₂³	1	4	yes	—
135	#95×C₈₉	C₂³	1	8	yes	—
136	#95×C₈₉	C₂³	1	8	yes	—
137	C₂⁴×C₈₉	—	2	2	no	—
138	C₂⁴×C₈₉	—	2	4	no	—
139	C₂⁴×C₈₉	—	2	4	no	—
140	C₂⁴×C₈₉	—	4	2	no	—
141	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
142	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
143	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
144	#41×C₈₉	—	4	32	no	—
145	C₂×Q₈×C₈₉	—	2	4	no	—
146	C₂×Q₈×C₈₉	—	4	4	no	—
147	C₂×D₄×C₈₉	—	2	8	no	—
148	C₂×D₄×C₈₉	—	2	8	no	—
149	C₂×D₄×C₈₉	—	2	8	no	—
150	C₂×D₄×C₈₉	—	2	16	no	—
151	C₂×D₄×C₈₉	—	2	4	no	—
152	C₂×D₄×C₈₉	—	2	4	no	—
153	C₂×D₄×C₈₉	—	2	4	no	—
154	C₂×D₄×C₈₉	—	2	8	no	—
155	C₂×D₄×C₈₉	—	4	4	no	—
156	C₂×D₄×C₈₉	—	4	4	no	—
157	C₂×D₄×C₈₉	—	4	8	no	—
158	C₂×D₄×C₈₉	—	4	8	no	—
159	C₂²×C₄×C₈₉	—	2	32	no	—
160	C₂²×C₄×C₈₉	—	2	2	no	—
161	C₂²×C₄×C₈₉	—	2	4	no	—
162	C₂²×C₄×C₈₉	—	2	4	no	—
163	C₂²×C₄×C₈₉	—	2	4	no	—
164	C₂²×C₄×C₈₉	—	2	4	no	—
165	C₂²×C₄×C₈₉	—	2	4	no	—
166	C₂²×C₄×C₈₉	—	2	4	no	—
167	C₂²×C₄×C₈₉	—	2	4	no	—
168	C₂²×C₄×C₈₉	—	2	4	no	—
169	C₂²×C₄×C₈₉	—	2	2	no	—
170	C₂²×C₄×C₈₉	—	2	2	no	—
171	C₂²×C₄×C₈₉	—	2	4	no	—
172	C₂²×C₄×C₈₉	—	2	8	no	—
173	C₂²×C₄×C₈₉	—	4	2	no	—
174	C₂²×C₄×C₈₉	—	4	2	no	—
175	C₂²×C₄×C₈₉	—	4	4	no	—
176	C₂²×C₄×C₈₉	—	4	4	no	—
177	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
178	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₈₉	#41	1	8	yes	complemented
181	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
182	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
184	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
192	C₄⋊C₄×C₈₉	—	2	4	no	—
193	C₄⋊C₄×C₈₉	—	2	4	no	—
194	C₄⋊C₄×C₈₉	—	2	4	no	—
195	C₄⋊C₄×C₈₉	—	2	4	no	—
196	C₄⋊C₄×C₈₉	—	2	4	no	—
197	C₄⋊C₄×C₈₉	—	2	4	no	—
198	C₄⋊C₄×C₈₉	—	2	8	no	—
199	C₄⋊C₄×C₈₉	—	4	8	no	—
200	C₂²⋊C₄×C₈₉	—	2	8	no	—
201	C₂²⋊C₄×C₈₉	—	2	4	no	—
202	C₂²⋊C₄×C₈₉	—	2	4	no	—
203	C₂²⋊C₄×C₈₉	—	2	4	no	—
204	C₂²⋊C₄×C₈₉	—	2	4	no	—
205	C₂²⋊C₄×C₈₉	—	2	8	no	—
206	C₂²⋊C₄×C₈₉	—	2	8	no	—
207	C₂²⋊C₄×C₈₉	—	2	16	no	—
208	C₂²⋊C₄×C₈₉	—	4	8	no	—
209	C₂²⋊C₄×C₈₉	—	4	8	no	—
210	C₂²⋊C₄×C₈₉	—	4	8	no	—
211	C₄²×C₈₉	—	2	4	no	—
212	C₄²×C₈₉	—	2	4	no	—
213	C₄²×C₈₉	—	4	4	no	—
214	C₄²×C₈₉	C₂×D₄	1	4	yes	—
215	C₄²×C₈₉	C₂×D₄	1	4	yes	—
216	C₄²×C₈₉	C₂×D₄	1	4	yes	—
217	C₄²×C₈₉	C₂×D₄	1	4	yes	—
218	C₂³×C₈₉	—	2	2	no	—
219	C₂³×C₈₉	—	2	2	no	—
220	C₂³×C₈₉	—	2	8	no	—
221	C₂³×C₈₉	—	2	2	no	—
222	C₂³×C₈₉	—	2	2	no	—
223	C₂³×C₈₉	—	2	4	no	—
224	C₂³×C₈₉	—	2	4	no	—
225	C₂³×C₈₉	—	2	4	no	—
226	C₂³×C₈₉	—	2	8	no	—
227	C₂³×C₈₉	—	2	4	no	—
228	C₂³×C₈₉	—	2	4	no	—
229	C₂³×C₈₉	—	2	8	no	—
230	C₂³×C₈₉	—	4	16	no	—
231	C₂³×C₈₉	—	4	4	no	—
232	C₂³×C₈₉	—	4	4	no	—
233	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
234	C₂³×C₈₉	C₂×#41	1	4	yes	—
235	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
236	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
237	Q₈×C₈₉	—	4	8	no	—
238	D₄×C₈₉	—	4	8	no	—
239	D₄×C₈₉	—	4	8	no	—
240	D₄×C₈₉	—	4	16	no	—
241	D₄×C₈₉	—	4	8	no	—
242	D₄×C₈₉	—	8	16	no	—
243	C₂×C₄×C₈₉	—	2	16	no	—
244	C₂×C₄×C₈₉	—	2	8	no	—
245	C₂×C₄×C₈₉	—	2	8	no	—
246	C₂×C₄×C₈₉	—	2	2	no	—
247	C₂×C₄×C₈₉	—	2	2	no	—
248	C₂×C₄×C₈₉	—	2	2	no	—
249	C₂×C₄×C₈₉	—	2	2	no	—
250	C₂×C₄×C₈₉	—	2	2	no	—
251	C₂×C₄×C₈₉	—	2	2	no	—
252	C₂×C₄×C₈₉	—	2	2	no	—
253	C₂×C₄×C₈₉	—	2	8	no	—
254	C₂×C₄×C₈₉	—	2	16	no	—
255	C₂×C₄×C₈₉	—	2	2	no	—
256	C₂×C₄×C₈₉	—	2	2	no	—
257	C₂×C₄×C₈₉	—	2	4	no	—
258	C₂×C₄×C₈₉	—	2	8	no	—
259	C₂×C₄×C₈₉	—	2	8	no	—
260	C₂×C₄×C₈₉	—	2	4	no	—
261	C₂×C₄×C₈₉	—	2	4	no	—
262	C₂×C₄×C₈₉	—	2	4	no	—
263	C₂×C₄×C₈₉	—	2	8	no	—
264	C₂×C₄×C₈₉	—	2	4	no	—
265	C₂×C₄×C₈₉	—	4	4	no	—
266	C₂×C₄×C₈₉	—	4	16	no	—
267	C₂×C₄×C₈₉	—	4	4	no	—
268	C₂×C₄×C₈₉	—	4	4	no	—
269	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
276	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
277	C₂²×C₈₉	—	2	2	no	—
278	C₂²×C₈₉	—	2	2	no	—
279	C₂²×C₈₉	—	2	8	no	—
280	C₂²×C₈₉	—	2	4	no	—
281	C₂²×C₈₉	—	2	8	no	—
282	C₂²×C₈₉	—	4	16	no	—
283	C₂²×C₈₉	—	4	2	no	—
284	C₂²×C₈₉	—	4	2	no	—
285	C₂²×C₈₉	—	4	4	no	—
286	C₂²×C₈₉	—	4	4	no	—
287	C₂²×C₈₉	—	4	4	no	—
288	C₂²×C₈₉	—	4	4	no	—
289	C₂²×C₈₉	—	4	8	no	—
290	C₂²×C₈₉	—	4	8	no	—
291	C₂²×C₈₉	#535	1	4	yes	—
292	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
293	C₂²×C₈₉	C₂²×#41	1	2	yes	—
294	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
295	C₂²×C₈₉	#585	1	2	yes	—
296	C₄×C₈₉	—	2	2	no	—
297	C₄×C₈₉	—	2	4	no	—
298	C₄×C₈₉	—	2	2	no	—
299	C₄×C₈₉	—	2	2	no	—
300	C₄×C₈₉	—	2	2	no	—
301	C₄×C₈₉	—	2	2	no	—
302	C₄×C₈₉	—	4	2	no	—
303	C₄×C₈₉	—	4	4	no	—
304	C₄×C₈₉	—	4	2	no	—
305	C₄×C₈₉	—	4	8	no	—
306	C₄×C₈₉	#525	1	4	yes	—
307	#35301	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂×C₈₉	—	2	4	no	—
309	C₂×C₈₉	—	4	4	no	—
310	C₂×C₈₉	—	8	4	no	—
311	C₂×C₈₉	C₂×#525	1	1	yes	characteristic
312	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
313	C₂×C₈₉	#3449	1	1	yes	characteristic
314	C₂×C₈₉	#3494	1	4	yes	—
315	C₂×#532	C₂×C₈₉	1	1	yes	characteristic, complemented
316	C₂²×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
317	#2854	C₂×C₈₉	1	2	yes	complemented
318	#2847	C₂×C₈₉	1	2	yes	complemented
319	#2628	C₂×C₈₉	1	2	yes	complemented
320	#2619	C₂×C₈₉	1	2	yes	complemented
321	#2610	C₂×C₈₉	1	4	yes	complemented
322	#2598	C₂×C₈₉	1	4	yes	complemented
323	#2591	C₂×C₈₉	1	2	yes	complemented
324	#2539	C₂×C₈₉	1	2	yes	complemented
325	#2406	C₂×C₈₉	1	2	yes	complemented
326	#2404	C₂×C₈₉	1	2	yes	complemented
327	#2324	C₂×C₈₉	1	4	yes	complemented
328	#2283	C₂×C₈₉	1	1	yes	characteristic, complemented
329	C₈₉	#35301	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²×#41	C₂²×C₈₉	1	2	yes	—
331	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
332	#532	—	2	8	no	—
333	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
334	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic, complemented
335	C₂×#124	C₂²×C₈₉	1	2	yes	complemented
336	C₂×#123	C₂²×C₈₉	1	4	yes	—
337	C₂×#121	C₂²×C₈₉	1	4	yes	—
338	C₂×#121	C₂²×C₈₉	1	2	yes	—
339	C₂×#121	C₂²×C₈₉	1	2	yes	—
340	C₂×#120	C₂²×C₈₉	1	4	yes	—
341	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	complemented
342	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	complemented
343	C₂×#117	—	2	8	no	—
344	C₂×#117	C₂²×C₈₉	1	2	yes	—
345	C₂×#117	C₂²×C₈₉	1	2	yes	—
346	C₂×#117	C₂²×C₈₉	1	2	yes	—
347	C₂×#117	C₂²×C₈₉	1	2	yes	—
348	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
349	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
350	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
351	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
352	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
353	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
354	#400	C₂²×C₈₉	1	4	yes	complemented
355	#399	C₂²×C₈₉	1	4	yes	—
356	#395	C₂²×C₈₉	1	4	yes	complemented
357	#394	C₂²×C₈₉	1	4	yes	complemented
358	#394	C₂²×C₈₉	1	4	yes	complemented
359	#393	C₂²×C₈₉	1	4	yes	—
360	#392	C₂²×C₈₉	1	4	yes	—
361	#392	C₂²×C₈₉	1	4	yes	—
362	#390	C₂²×C₈₉	1	8	yes	—
363	#390	C₂²×C₈₉	1	4	yes	complemented
364	#389	C₂²×C₈₉	1	4	yes	complemented
365	#388	C₂²×C₈₉	1	8	yes	complemented
366	#388	C₂²×C₈₉	1	8	yes	complemented
367	#386	C₂²×C₈₉	1	8	yes	—
368	#386	C₂²×C₈₉	1	4	yes	—
369	#386	C₂²×C₈₉	1	4	yes	—
370	#386	C₂²×C₈₉	1	8	yes	—
371	#385	C₂²×C₈₉	1	8	yes	complemented
372	#383	C₂²×C₈₉	1	2	yes	complemented
373	#382	C₂²×C₈₉	1	2	yes	complemented
374	#381	C₂²×C₈₉	1	2	yes	—
375	#380	C₂²×C₈₉	1	4	yes	—
376	#380	C₂²×C₈₉	1	2	yes	—
377	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
378	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
379	#376	C₂²×C₈₉	1	2	yes	complemented
380	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
381	C₂×#41	—	2	8	no	—
382	C₂×#41	—	4	8	no	—
383	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
384	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
385	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
386	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
387	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
388	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
389	C₂³×C₄	—	2	4	no	—
390	C₂³×C₄	—	2	8	no	—
391	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
392	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
393	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
394	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
397	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
399	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
400	C₄⋊Q₈	—	2	4	no	—
401	#127	—	2	4	no	—
402	#124	—	2	8	no	—
403	#123	—	4	8	no	—
404	#121	—	2	8	no	—
405	#121	—	2	8	no	—
406	#121	—	2	16	no	—
407	#120	—	4	8	no	—
408	C₄×D₄	—	2	8	no	—
409	C₄×D₄	—	2	8	no	—
410	#117	—	2	8	no	—
411	#117	—	2	8	no	—
412	#117	—	2	8	no	—
413	#117	—	2	8	no	—
414	C₂×C₄⋊C₄	—	2	4	no	—
415	C₂×C₄⋊C₄	—	2	4	no	—
416	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
423	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
424	C₂×C₂²⋊C₄	—	2	4	no	—
425	C₂×C₂²⋊C₄	—	2	4	no	—
426	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
437	C₂×C₄²	—	2	4	no	—
438	C₂×C₄²	D₄×C₈₉	1	8	yes	complemented
439	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
440	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
441	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
442	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
446	#95	C₂³×C₈₉	1	4	yes	—
447	#95	C₂³×C₈₉	1	4	yes	—
448	#95	C₂³×C₈₉	1	4	yes	—
449	#95	C₂³×C₈₉	1	4	yes	—
450	#95	C₂³×C₈₉	1	8	yes	—
451	#95	C₂³×C₈₉	1	8	yes	—
452	C₂⁴	—	2	2	no	—
453	C₂⁴	—	2	4	no	—
454	C₂⁴	—	2	4	no	—
455	C₂⁴	—	4	2	no	—
456	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
457	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
458	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
459	#41	—	4	32	no	—
460	C₂×Q₈	—	2	4	no	—
461	C₂×Q₈	—	4	4	no	—
462	C₂×D₄	—	2	8	no	—
463	C₂×D₄	—	2	8	no	—
464	C₂×D₄	—	2	8	no	—
465	C₂×D₄	—	2	16	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	8	no	—
470	C₂×D₄	—	4	4	no	—
471	C₂×D₄	—	4	4	no	—
472	C₂×D₄	—	4	8	no	—
473	C₂×D₄	—	4	8	no	—
474	C₂²×C₄	—	2	32	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	4	no	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	4	4	no	—
489	C₂²×C₄	—	4	4	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	—	4	2	no	—
492	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
493	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
494	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
495	C₂²×C₄	#41×C₈₉	1	8	yes	complemented
496	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
497	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
503	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
504	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	2	8	no	—
514	C₄⋊C₄	—	4	8	no	—
515	C₂²⋊C₄	—	2	8	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	8	no	—
521	C₂²⋊C₄	—	2	8	no	—
522	C₂²⋊C₄	—	2	16	no	—
523	C₂²⋊C₄	—	4	8	no	—
524	C₂²⋊C₄	—	4	8	no	—
525	C₂²⋊C₄	—	4	8	no	—
526	C₄²	—	2	4	no	—
527	C₄²	—	2	4	no	—
528	C₄²	—	4	4	no	—
529	C₄²	C₂×D₄×C₈₉	1	4	yes	—
530	C₄²	C₂×D₄×C₈₉	1	4	yes	—
531	C₄²	C₂×D₄×C₈₉	1	4	yes	—
532	C₄²	C₂×D₄×C₈₉	1	4	yes	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	8	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	2	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	2	4	no	—
541	C₂³	—	2	8	no	—
542	C₂³	—	2	4	no	—
543	C₂³	—	2	4	no	—
544	C₂³	—	2	8	no	—
545	C₂³	—	4	16	no	—
546	C₂³	—	4	4	no	—
547	C₂³	—	4	4	no	—
548	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
549	C₂³	C₂×#41×C₈₉	1	4	yes	—
550	C₂³	H(5,𝔽₂)×C₈₉	1	4	yes	—
551	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
552	Q₈	—	4	8	no	—
553	D₄	—	4	8	no	—
554	D₄	—	4	8	no	—
555	D₄	—	4	16	no	—
556	D₄	—	4	8	no	—
557	D₄	—	8	16	no	—
558	C₂×C₄	—	2	16	no	—
559	C₂×C₄	—	2	8	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	8	no	—
569	C₂×C₄	—	2	16	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	8	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	4	4	no	—
581	C₂×C₄	—	4	16	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
585	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
586	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
587	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
588	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
589	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
590	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
591	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
592	C₂²	—	2	2	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	2	8	no	—
595	C₂²	—	2	4	no	—
596	C₂²	—	2	8	no	—
597	C₂²	—	4	16	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	4	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	4	4	no	—
603	C₂²	—	4	4	no	—
604	C₂²	—	4	8	no	—
605	C₂²	—	4	8	no	—
606	C₂²	#535×C₈₉	1	4	yes	—
607	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
608	C₂²	C₂²×#41×C₈₉	1	2	yes	—
609	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
610	C₂²	#585×C₈₉	1	2	yes	—
611	C₄	—	2	2	no	—
612	C₄	—	2	4	no	—
613	C₄	—	2	2	no	—
614	C₄	—	2	2	no	—
615	C₄	—	2	2	no	—
616	C₄	—	2	2	no	—
617	C₄	—	4	4	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	8	no	—
621	C₄	#525×C₈₉	1	4	yes	—
622	C₂	—	2	4	no	—
623	C₂	—	4	4	no	—
624	C₂	—	8	4	no	—
625	C₂	C₂×#525×C₈₉	1	1	yes	characteristic
626	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
627	C₂	C₈₉×#3449	1	1	yes	characteristic
628	C₂	C₈₉×#3494	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
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
1. C₂×#532×C₈₉	C₂	semidirect
2. C₂²×#117×C₈₉	C₂	semidirect
3. C₈₉×#2854	C₂	semidirect
4. C₈₉×#2847	C₂	semidirect
5. C₈₉×#2628	C₂	semidirect
6. C₈₉×#2619	C₂	semidirect
7. C₈₉×#2610	C₂	semidirect
8. C₈₉×#2598	C₂	semidirect
9. C₈₉×#2591	C₂	semidirect
10. C₈₉×#2539	C₂	semidirect
11. C₈₉×#2406	C₂	semidirect
12. C₈₉×#2404	C₂	semidirect
13. C₈₉×#2324	C₂	semidirect
14. C₈₉×#2283	C₂	semidirect
18. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
19. C₂×#127×C₈₉	C₂²	semidirect
20. C₂×#124×C₈₉	C₂²	semidirect
26. C₂×C₄×D₄×C₈₉	C₂²	semidirect
27. C₂×C₄×D₄×C₈₉	C₂²	semidirect
39. #400×C₈₉	C₂²	semidirect
41. #395×C₈₉	C₂²	semidirect
42. #394×C₈₉	C₂²	semidirect
43. #394×C₈₉	C₂²	semidirect
48. #390×C₈₉	C₂²	semidirect
49. #389×C₈₉	C₂²	semidirect
50. #388×C₈₉	C₂²	semidirect
51. #388×C₈₉	C₂²	semidirect
56. #385×C₈₉	C₂²	semidirect
57. #383×C₈₉	C₂²	semidirect
58. #382×C₈₉	C₂²	semidirect
62. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
64. #376×C₈₉	C₂²	semidirect
123. C₂×C₄²×C₈₉	D₄	semidirect
180. C₂²×C₄×C₈₉	#41	semidirect
307. #35301	C₈₉	direct (N × H)
315. C₂×#532	C₂×C₈₉	semidirect
316. C₂²×#117	C₂×C₈₉	semidirect
317. #2854	C₂×C₈₉	semidirect
318. #2847	C₂×C₈₉	semidirect
319. #2628	C₂×C₈₉	semidirect
320. #2619	C₂×C₈₉	semidirect
321. #2610	C₂×C₈₉	semidirect
322. #2598	C₂×C₈₉	semidirect
323. #2591	C₂×C₈₉	semidirect
324. #2539	C₂×C₈₉	semidirect
325. #2406	C₂×C₈₉	semidirect
326. #2404	C₂×C₈₉	semidirect
327. #2324	C₂×C₈₉	semidirect
328. #2283	C₂×C₈₉	semidirect
329. C₈₉	#35301	direct (N × H)
333. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
334. C₂×#127	C₂²×C₈₉	semidirect
335. C₂×#124	C₂²×C₈₉	semidirect
341. C₂×C₄×D₄	C₂²×C₈₉	semidirect
342. C₂×C₄×D₄	C₂²×C₈₉	semidirect
354. #400	C₂²×C₈₉	semidirect
356. #395	C₂²×C₈₉	semidirect
357. #394	C₂²×C₈₉	semidirect
358. #394	C₂²×C₈₉	semidirect
363. #390	C₂²×C₈₉	semidirect
364. #389	C₂²×C₈₉	semidirect
365. #388	C₂²×C₈₉	semidirect
366. #388	C₂²×C₈₉	semidirect
371. #385	C₂²×C₈₉	semidirect
372. #383	C₂²×C₈₉	semidirect
373. #382	C₂²×C₈₉	semidirect
377. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
379. #376	C₂²×C₈₉	semidirect
438. C₂×C₄²	D₄×C₈₉	semidirect
495. C₂²×C₄	#41×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₈₉	2
C₂²×#117×C₈₉	2
C₈₉×#2854	2
C₈₉×#2847	2
C₈₉×#2628	2
C₈₉×#2619	2
C₈₉×#2610	2
C₈₉×#2598	2
C₈₉×#2591	2
C₈₉×#2539	2
C₈₉×#2406	2
C₈₉×#2404	2
C₈₉×#2324	2
C₈₉×#2283	2
#35301	89

Subgroups — C89×#35301 #36413149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35301 #36413149