Subgroups — C₈₉×#35377 #36413225

Order 22784.

2914 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#534×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2591	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2591	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2504	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2488	C₂	1	8	yes	maximal, complemented
7	C₈₉×#2472	C₂	1	8	yes	maximal, complemented
8	C₈₉×#2436	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2308	C₂	1	2	yes	maximal, complemented
10	C₂×#386×C₈₉	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2273	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₄×C₈₉	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
15	#534×C₈₉	—	2	8	no	—
16	C₂×#124×C₈₉	C₂²	1	4	yes	complemented
17	C₂×#123×C₈₉	C₂²	1	4	yes	—
18	C₂×C₂²⋊Q₈×C₈₉	—	2	8	no	—
19	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	8	yes	complemented
20	C₂×#121×C₈₉	C₂²	1	2	yes	—
21	C₂×#121×C₈₉	C₂²	1	2	yes	—
22	C₂×#120×C₈₉	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
26	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	complemented
30	#400×C₈₉	C₂²	1	4	yes	complemented
31	#400×C₈₉	C₂²	1	4	yes	complemented
32	#397×C₈₉	C₂²	1	8	yes	complemented
33	#394×C₈₉	C₂²	1	4	yes	complemented
34	#394×C₈₉	C₂²	1	4	yes	complemented
35	#393×C₈₉	C₂²	1	8	yes	complemented
36	#391×C₈₉	C₂²	1	4	yes	complemented
37	#390×C₈₉	C₂²	1	4	yes	complemented
38	#388×C₈₉	C₂²	1	8	yes	complemented
39	#388×C₈₉	C₂²	1	8	yes	complemented
40	#386×C₈₉	C₂²	1	8	yes	—
41	#386×C₈₉	C₂²	1	8	yes	—
42	#386×C₈₉	C₂²	1	4	yes	complemented
43	#386×C₈₉	C₂²	1	4	yes	complemented
44	#386×C₈₉	C₂²	1	4	yes	complemented
45	#386×C₈₉	C₂²	1	4	yes	complemented
46	#382×C₈₉	C₂²	1	4	yes	complemented
47	#381×C₈₉	C₂²	1	4	yes	complemented
48	#379×C₈₉	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	complemented
50	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	complemented
51	C₄×C₂²⋊C₄×C₈₉	C₂²	1	8	yes	complemented
52	C₂×#95×C₈₉	C₂²	1	4	yes	complemented
53	C₂⁵×C₈₉	C₂³	1	2	yes	—
54	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₈₉	—	2	4	no	—
56	C₂²×Q₈×C₈₉	—	4	4	no	—
57	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
58	C₂²×D₄×C₈₉	—	2	8	no	—
59	C₂²×D₄×C₈₉	—	2	8	no	—
60	C₂²×D₄×C₈₉	—	2	8	no	—
61	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
62	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
63	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
64	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
65	C₂³×C₄×C₈₉	—	2	4	no	—
66	C₂³×C₄×C₈₉	—	4	4	no	—
67	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
68	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
70	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
71	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
76	#124×C₈₉	—	4	8	no	—
77	#123×C₈₉	—	2	16	no	—
78	C₂²⋊Q₈×C₈₉	—	2	32	no	—
79	#121×C₈₉	—	2	8	no	—
80	#121×C₈₉	—	2	8	no	—
81	#120×C₈₉	—	2	16	no	—
82	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
85	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
91	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
92	C₂×C₂²⋊C₄×C₈₉	—	4	8	no	—
93	C₂×C₂²⋊C₄×C₈₉	D₄	1	8	yes	—
94	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	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×C₄²×C₈₉	C₂³	1	4	yes	—
105	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
106	#95×C₈₉	C₂³	1	8	yes	—
107	#95×C₈₉	C₂³	1	8	yes	—
108	#95×C₈₉	C₂³	1	4	yes	—
109	#95×C₈₉	C₂³	1	4	yes	—
110	#95×C₈₉	C₂³	1	8	yes	—
111	#95×C₈₉	C₂³	1	8	yes	—
112	C₂⁴×C₈₉	—	2	4	no	—
113	C₂⁴×C₈₉	—	2	4	no	—
114	C₂⁴×C₈₉	—	2	4	no	—
115	C₂⁴×C₈₉	—	4	2	no	—
116	C₂⁴×C₈₉	—	4	2	no	—
117	C₂⁴×C₈₉	—	4	4	no	—
118	C₂⁴×C₈₉	—	4	4	no	—
119	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
120	C₂⁴×C₈₉	#41	1	8	yes	—
121	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
122	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
123	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₂×Q₈×C₈₉	—	2	8	no	—
125	C₂×Q₈×C₈₉	—	2	8	no	—
126	C₂×Q₈×C₈₉	—	4	4	no	—
127	C₂×Q₈×C₈₉	—	4	16	no	—
128	C₂×D₄×C₈₉	—	2	8	no	—
129	C₂×D₄×C₈₉	—	2	8	no	—
130	C₂×D₄×C₈₉	—	2	8	no	—
131	C₂×D₄×C₈₉	—	2	8	no	—
132	C₂×D₄×C₈₉	—	2	8	no	—
133	C₂×D₄×C₈₉	—	2	8	no	—
134	C₂×D₄×C₈₉	—	2	4	no	—
135	C₂×D₄×C₈₉	—	2	4	no	—
136	C₂×D₄×C₈₉	—	4	4	no	—
137	C₂×D₄×C₈₉	—	4	4	no	—
138	C₂×D₄×C₈₉	—	4	32	no	—
139	C₂²×C₄×C₈₉	—	2	2	no	—
140	C₂²×C₄×C₈₉	—	2	4	no	—
141	C₂²×C₄×C₈₉	—	2	2	no	—
142	C₂²×C₄×C₈₉	—	2	2	no	—
143	C₂²×C₄×C₈₉	—	2	2	no	—
144	C₂²×C₄×C₈₉	—	2	8	no	—
145	C₂²×C₄×C₈₉	—	2	4	no	—
146	C₂²×C₄×C₈₉	—	2	4	no	—
147	C₂²×C₄×C₈₉	—	2	4	no	—
148	C₂²×C₄×C₈₉	—	2	8	no	—
149	C₂²×C₄×C₈₉	—	2	8	no	—
150	C₂²×C₄×C₈₉	—	4	4	no	—
151	C₂²×C₄×C₈₉	—	4	2	no	—
152	C₂²×C₄×C₈₉	—	4	2	no	—
153	C₂²×C₄×C₈₉	—	4	2	no	—
154	C₂²×C₄×C₈₉	—	4	2	no	—
155	C₂²×C₄×C₈₉	—	4	2	no	—
156	C₂²×C₄×C₈₉	—	4	4	no	—
157	C₂²×C₄×C₈₉	—	4	4	no	—
158	C₂²×C₄×C₈₉	—	4	16	no	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
163	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
169	C₄⋊C₄×C₈₉	—	2	8	no	—
170	C₄⋊C₄×C₈₉	—	2	8	no	—
171	C₄⋊C₄×C₈₉	—	2	8	no	—
172	C₄⋊C₄×C₈₉	—	2	4	no	—
173	C₄⋊C₄×C₈₉	—	2	4	no	—
174	C₂²⋊C₄×C₈₉	—	2	8	no	—
175	C₂²⋊C₄×C₈₉	—	2	8	no	—
176	C₂²⋊C₄×C₈₉	—	2	8	no	—
177	C₂²⋊C₄×C₈₉	—	2	8	no	—
178	C₂²⋊C₄×C₈₉	—	2	8	no	—
179	C₂²⋊C₄×C₈₉	—	2	16	no	—
180	C₂²⋊C₄×C₈₉	—	4	8	no	—
181	C₂²⋊C₄×C₈₉	—	4	8	no	—
182	C₂²⋊C₄×C₈₉	—	4	8	no	—
183	C₂²⋊C₄×C₈₉	C₂×D₄	1	16	yes	—
184	C₄²×C₈₉	—	4	4	no	—
185	C₄²×C₈₉	—	4	4	no	—
186	C₂³×C₈₉	—	2	2	no	—
187	C₂³×C₈₉	—	2	4	no	—
188	C₂³×C₈₉	—	2	2	no	—
189	C₂³×C₈₉	—	2	4	no	—
190	C₂³×C₈₉	—	2	4	no	—
191	C₂³×C₈₉	—	2	4	no	—
192	C₂³×C₈₉	—	2	4	no	—
193	C₂³×C₈₉	—	2	8	no	—
194	C₂³×C₈₉	—	2	8	no	—
195	C₂³×C₈₉	—	4	4	no	—
196	C₂³×C₈₉	—	4	4	no	—
197	C₂³×C₈₉	—	4	16	no	—
198	C₂³×C₈₉	—	4	16	no	—
199	C₂³×C₈₉	—	4	8	no	—
200	C₂³×C₈₉	—	8	4	no	—
201	C₂³×C₈₉	—	8	4	no	—
202	C₂³×C₈₉	—	8	8	no	—
203	C₂³×C₈₉	#124	1	16	yes	complemented
204	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
205	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
206	C₂³×C₈₉	C₂×#41	1	4	yes	—
207	C₂³×C₈₉	C₂×#41	1	4	yes	—
208	C₂³×C₈₉	C₂×#41	1	4	yes	—
209	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
210	Q₈×C₈₉	—	4	16	no	—
211	D₄×C₈₉	—	4	16	no	—
212	D₄×C₈₉	—	4	16	no	—
213	D₄×C₈₉	—	8	4	no	—
214	D₄×C₈₉	—	8	4	no	—
215	C₂×C₄×C₈₉	—	2	8	no	—
216	C₂×C₄×C₈₉	—	2	2	no	—
217	C₂×C₄×C₈₉	—	2	4	no	—
218	C₂×C₄×C₈₉	—	2	2	no	—
219	C₂×C₄×C₈₉	—	2	4	no	—
220	C₂×C₄×C₈₉	—	2	2	no	—
221	C₂×C₄×C₈₉	—	2	4	no	—
222	C₂×C₄×C₈₉	—	2	2	no	—
223	C₂×C₄×C₈₉	—	2	2	no	—
224	C₂×C₄×C₈₉	—	2	8	no	—
225	C₂×C₄×C₈₉	—	2	4	no	—
226	C₂×C₄×C₈₉	—	2	4	no	—
227	C₂×C₄×C₈₉	—	2	8	no	—
228	C₂×C₄×C₈₉	—	2	4	no	—
229	C₂×C₄×C₈₉	—	2	4	no	—
230	C₂×C₄×C₈₉	—	2	8	no	—
231	C₂×C₄×C₈₉	—	2	4	no	—
232	C₂×C₄×C₈₉	—	4	4	no	—
233	C₂×C₄×C₈₉	—	4	4	no	—
234	C₂×C₄×C₈₉	—	4	16	no	—
235	C₂×C₄×C₈₉	—	4	4	no	—
236	C₂×C₄×C₈₉	—	4	4	no	—
237	C₂×C₄×C₈₉	—	4	4	no	—
238	C₂×C₄×C₈₉	—	4	8	no	—
239	C₂×C₄×C₈₉	—	8	8	no	—
240	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
243	C₂²×C₈₉	—	2	2	no	—
244	C₂²×C₈₉	—	2	4	no	—
245	C₂²×C₈₉	—	2	2	no	—
246	C₂²×C₈₉	—	2	4	no	—
247	C₂²×C₈₉	—	2	8	no	—
248	C₂²×C₈₉	—	2	8	no	—
249	C₂²×C₈₉	—	2	8	no	—
250	C₂²×C₈₉	—	4	16	no	—
251	C₂²×C₈₉	—	4	4	no	—
252	C₂²×C₈₉	—	4	4	no	—
253	C₂²×C₈₉	—	4	4	no	—
254	C₂²×C₈₉	—	4	8	no	—
255	C₂²×C₈₉	—	4	8	no	—
256	C₂²×C₈₉	—	8	16	no	—
257	C₂²×C₈₉	C₂×#124	1	8	yes	—
258	C₂²×C₈₉	C₂×#124	1	4	yes	—
259	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
260	C₂²×C₈₉	C₂²×#41	1	2	yes	—
261	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
262	C₂²×C₈₉	#585	1	2	yes	—
263	C₄×C₈₉	—	2	4	no	—
264	C₄×C₈₉	—	2	4	no	—
265	C₄×C₈₉	—	2	4	no	—
266	C₄×C₈₉	—	4	4	no	—
267	C₄×C₈₉	—	4	2	no	—
268	C₄×C₈₉	—	4	2	no	—
269	C₄×C₈₉	—	4	2	no	—
270	C₄×C₈₉	—	4	8	no	—
271	#35377	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂×C₈₉	—	2	8	no	—
273	C₂×C₈₉	—	2	4	no	—
274	C₂×C₈₉	—	8	4	no	—
275	C₂×C₈₉	C₂²×#124	1	1	yes	characteristic
276	C₂×C₈₉	#3484	1	1	yes	characteristic
277	C₂×C₈₉	#3485	1	1	yes	characteristic
278	C₂×C₈₉	#3505	1	4	yes	—
279	C₂×#534	C₂×C₈₉	1	1	yes	characteristic
280	C₂²×C₂²⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
281	#2591	C₂×C₈₉	1	2	yes	complemented
282	#2591	C₂×C₈₉	1	2	yes	complemented
283	#2504	C₂×C₈₉	1	2	yes	complemented
284	#2488	C₂×C₈₉	1	8	yes	complemented
285	#2472	C₂×C₈₉	1	8	yes	complemented
286	#2436	C₂×C₈₉	1	2	yes	complemented
287	#2308	C₂×C₈₉	1	2	yes	complemented
288	C₂×#386	C₂×C₈₉	1	2	yes	complemented
289	#2273	C₂×C₈₉	1	1	yes	characteristic, complemented
290	C₈₉	#35377	1	1	yes	characteristic, direct product factor, Sylow
291	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
292	C₂³×D₄	C₂²×C₈₉	1	2	yes	—
293	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
294	#534	—	2	8	no	—
295	C₂×#124	C₂²×C₈₉	1	4	yes	complemented
296	C₂×#123	C₂²×C₈₉	1	4	yes	—
297	C₂×C₂²⋊Q₈	—	2	8	no	—
298	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	8	yes	complemented
299	C₂×#121	C₂²×C₈₉	1	2	yes	—
300	C₂×#121	C₂²×C₈₉	1	2	yes	—
301	C₂×#120	C₂²×C₈₉	1	4	yes	—
302	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
303	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
304	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
305	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
306	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
307	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
308	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	complemented
309	#400	C₂²×C₈₉	1	4	yes	complemented
310	#400	C₂²×C₈₉	1	4	yes	complemented
311	#397	C₂²×C₈₉	1	8	yes	complemented
312	#394	C₂²×C₈₉	1	4	yes	complemented
313	#394	C₂²×C₈₉	1	4	yes	complemented
314	#393	C₂²×C₈₉	1	8	yes	complemented
315	#391	C₂²×C₈₉	1	4	yes	complemented
316	#390	C₂²×C₈₉	1	4	yes	complemented
317	#388	C₂²×C₈₉	1	8	yes	complemented
318	#388	C₂²×C₈₉	1	8	yes	complemented
319	#386	C₂²×C₈₉	1	8	yes	—
320	#386	C₂²×C₈₉	1	8	yes	—
321	#386	C₂²×C₈₉	1	4	yes	complemented
322	#386	C₂²×C₈₉	1	4	yes	complemented
323	#386	C₂²×C₈₉	1	4	yes	complemented
324	#386	C₂²×C₈₉	1	4	yes	complemented
325	#382	C₂²×C₈₉	1	4	yes	complemented
326	#381	C₂²×C₈₉	1	4	yes	complemented
327	#379	C₂²×C₈₉	1	4	yes	complemented
328	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	complemented
329	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	complemented
330	C₄×C₂²⋊C₄	C₂²×C₈₉	1	8	yes	complemented
331	C₂×#95	C₂²×C₈₉	1	4	yes	complemented
332	C₂⁵	C₂³×C₈₉	1	2	yes	—
333	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
334	C₂²×Q₈	—	2	4	no	—
335	C₂²×Q₈	—	4	4	no	—
336	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
337	C₂²×D₄	—	2	8	no	—
338	C₂²×D₄	—	2	8	no	—
339	C₂²×D₄	—	2	8	no	—
340	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
341	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
342	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
343	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
344	C₂³×C₄	—	2	4	no	—
345	C₂³×C₄	—	4	4	no	—
346	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
348	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
349	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
350	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
355	#124	—	4	8	no	—
356	#123	—	2	16	no	—
357	C₂²⋊Q₈	—	2	32	no	—
358	#121	—	2	8	no	—
359	#121	—	2	8	no	—
360	#120	—	2	16	no	—
361	C₂×C₄⋊C₄	—	2	4	no	—
362	C₂×C₄⋊C₄	—	2	4	no	—
363	C₂×C₄⋊C₄	—	2	4	no	—
364	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
365	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
366	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
367	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
369	C₂×C₂²⋊C₄	—	2	4	no	—
370	C₂×C₂²⋊C₄	—	2	8	no	—
371	C₂×C₂²⋊C₄	—	4	8	no	—
372	C₂×C₂²⋊C₄	D₄×C₈₉	1	8	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
383	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
384	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
385	#95	C₂³×C₈₉	1	8	yes	—
386	#95	C₂³×C₈₉	1	8	yes	—
387	#95	C₂³×C₈₉	1	4	yes	—
388	#95	C₂³×C₈₉	1	4	yes	—
389	#95	C₂³×C₈₉	1	8	yes	—
390	#95	C₂³×C₈₉	1	8	yes	—
391	C₂⁴	—	2	4	no	—
392	C₂⁴	—	2	4	no	—
393	C₂⁴	—	2	4	no	—
394	C₂⁴	—	4	2	no	—
395	C₂⁴	—	4	2	no	—
396	C₂⁴	—	4	4	no	—
397	C₂⁴	—	4	4	no	—
398	C₂⁴	C₂×D₄×C₈₉	1	4	yes	—
399	C₂⁴	#41×C₈₉	1	8	yes	—
400	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
401	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
402	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
403	C₂×Q₈	—	2	8	no	—
404	C₂×Q₈	—	2	8	no	—
405	C₂×Q₈	—	4	4	no	—
406	C₂×Q₈	—	4	16	no	—
407	C₂×D₄	—	2	8	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	4	4	no	—
416	C₂×D₄	—	4	4	no	—
417	C₂×D₄	—	4	32	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	8	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	8	no	—
429	C₂²×C₄	—	4	4	no	—
430	C₂²×C₄	—	4	2	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	—	4	2	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	4	no	—
437	C₂²×C₄	—	4	16	no	—
438	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
439	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
442	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
445	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	8	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	8	no	—
456	C₂²⋊C₄	—	2	8	no	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	16	no	—
459	C₂²⋊C₄	—	4	8	no	—
460	C₂²⋊C₄	—	4	8	no	—
461	C₂²⋊C₄	—	4	8	no	—
462	C₂²⋊C₄	C₂×D₄×C₈₉	1	16	yes	—
463	C₄²	—	4	4	no	—
464	C₄²	—	4	4	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	4	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	4	no	—
472	C₂³	—	2	8	no	—
473	C₂³	—	2	8	no	—
474	C₂³	—	4	4	no	—
475	C₂³	—	4	4	no	—
476	C₂³	—	4	16	no	—
477	C₂³	—	4	16	no	—
478	C₂³	—	4	8	no	—
479	C₂³	—	8	4	no	—
480	C₂³	—	8	4	no	—
481	C₂³	—	8	8	no	—
482	C₂³	#124×C₈₉	1	16	yes	complemented
483	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
484	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
485	C₂³	C₂×#41×C₈₉	1	4	yes	—
486	C₂³	C₂×#41×C₈₉	1	4	yes	—
487	C₂³	C₂×#41×C₈₉	1	4	yes	—
488	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
489	Q₈	—	4	16	no	—
490	D₄	—	4	16	no	—
491	D₄	—	4	16	no	—
492	D₄	—	8	4	no	—
493	D₄	—	8	4	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	8	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	4	4	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	16	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	8	8	no	—
519	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
520	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
521	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	2	8	no	—
527	C₂²	—	2	8	no	—
528	C₂²	—	2	8	no	—
529	C₂²	—	4	16	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	4	8	no	—
534	C₂²	—	4	8	no	—
535	C₂²	—	8	16	no	—
536	C₂²	C₂×#124×C₈₉	1	8	yes	—
537	C₂²	C₂×#124×C₈₉	1	4	yes	—
538	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
539	C₂²	C₂²×#41×C₈₉	1	2	yes	—
540	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
541	C₂²	#585×C₈₉	1	2	yes	—
542	C₄	—	2	4	no	—
543	C₄	—	2	4	no	—
544	C₄	—	2	4	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	2	no	—
549	C₄	—	4	8	no	—
550	C₂	—	2	8	no	—
551	C₂	—	2	4	no	—
552	C₂	—	8	4	no	—
553	C₂	C₂²×#124×C₈₉	1	1	yes	characteristic
554	C₂	C₈₉×#3484	1	1	yes	characteristic
555	C₂	C₈₉×#3485	1	1	yes	characteristic
556	C₂	C₈₉×#3505	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#534×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. C₂²×C₂²⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2591	C₂	semidirect
4. C₈₉×#2591	C₂	semidirect
5. C₈₉×#2504	C₂	semidirect
6. C₈₉×#2488	C₂	semidirect
7. C₈₉×#2472	C₂	semidirect
8. C₈₉×#2436	C₂	semidirect
9. C₈₉×#2308	C₂	semidirect
10. C₂×#386×C₈₉	C₂	semidirect
11. C₈₉×#2273	C₂	semidirect
12. C₂³×Q₈×C₈₉	C₂²	semidirect
16. C₂×#124×C₈₉	C₂²	semidirect
19. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
24. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₈₉	C₂²	semidirect
30. #400×C₈₉	C₂²	semidirect
31. #400×C₈₉	C₂²	semidirect
32. #397×C₈₉	C₂²	semidirect
33. #394×C₈₉	C₂²	semidirect
34. #394×C₈₉	C₂²	semidirect
35. #393×C₈₉	C₂²	semidirect
36. #391×C₈₉	C₂²	semidirect
37. #390×C₈₉	C₂²	semidirect
38. #388×C₈₉	C₂²	semidirect
39. #388×C₈₉	C₂²	semidirect
42. #386×C₈₉	C₂²	semidirect
43. #386×C₈₉	C₂²	semidirect
44. #386×C₈₉	C₂²	semidirect
45. #386×C₈₉	C₂²	semidirect
46. #382×C₈₉	C₂²	semidirect
47. #381×C₈₉	C₂²	semidirect
48. #379×C₈₉	C₂²	semidirect
49. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
51. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
52. C₂×#95×C₈₉	C₂²	semidirect
203. C₂³×C₈₉	#124	semidirect
271. #35377	C₈₉	direct (N × H)
280. C₂²×C₂²⋊Q₈	C₂×C₈₉	semidirect
281. #2591	C₂×C₈₉	semidirect
282. #2591	C₂×C₈₉	semidirect
283. #2504	C₂×C₈₉	semidirect
284. #2488	C₂×C₈₉	semidirect
285. #2472	C₂×C₈₉	semidirect
286. #2436	C₂×C₈₉	semidirect
287. #2308	C₂×C₈₉	semidirect
288. C₂×#386	C₂×C₈₉	semidirect
289. #2273	C₂×C₈₉	semidirect
290. C₈₉	#35377	direct (N × H)
291. C₂³×Q₈	C₂²×C₈₉	semidirect
295. C₂×#124	C₂²×C₈₉	semidirect
298. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
302. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
303. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
304. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
308. C₂²×C₂²⋊C₄	C₂²×C₈₉	semidirect
309. #400	C₂²×C₈₉	semidirect
310. #400	C₂²×C₈₉	semidirect
311. #397	C₂²×C₈₉	semidirect
312. #394	C₂²×C₈₉	semidirect
313. #394	C₂²×C₈₉	semidirect
314. #393	C₂²×C₈₉	semidirect
315. #391	C₂²×C₈₉	semidirect
316. #390	C₂²×C₈₉	semidirect
317. #388	C₂²×C₈₉	semidirect
318. #388	C₂²×C₈₉	semidirect
321. #386	C₂²×C₈₉	semidirect
322. #386	C₂²×C₈₉	semidirect
323. #386	C₂²×C₈₉	semidirect
324. #386	C₂²×C₈₉	semidirect
325. #382	C₂²×C₈₉	semidirect
326. #381	C₂²×C₈₉	semidirect
327. #379	C₂²×C₈₉	semidirect
328. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
329. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
330. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
331. C₂×#95	C₂²×C₈₉	semidirect
482. C₂³	#124×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#534×C₈₉	2
C₂²×C₂²⋊Q₈×C₈₉	2
C₈₉×#2591	2
C₈₉×#2504	2
C₈₉×#2488	2
C₈₉×#2472	2
C₈₉×#2436	2
C₈₉×#2308	2
C₂×#386×C₈₉	2
C₈₉×#2273	2
#35377	89

Subgroups — C89×#35377 #36413225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35377 #36413225