Subgroups — C₈₉×#23314 #36401162

Order 22784.

792 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#123×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1954	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1939	C₂	1	2	yes	maximal, complemented
4	C₈₉×#1923	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1923	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1916	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1869	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1850	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1797	C₂	1	2	yes	maximal, complemented
10	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#123×C₈₉	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
24	#388×C₈₉	C₄	1	2	yes	—
25	#387×C₈₉	C₄	1	2	yes	—
26	#386×C₈₉	C₄	1	1	yes	characteristic
27	#385×C₈₉	C₄	1	1	yes	characteristic
28	#381×C₈₉	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
31	C₄×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
32	#336×C₈₉	C₂²	1	2	yes	—
33	#336×C₈₉	C₂²	1	2	yes	—
34	#336×C₈₉	C₂²	1	2	yes	—
35	#336×C₈₉	C₂²	1	2	yes	—
36	#336×C₈₉	C₂²	1	2	yes	—
37	C₂²×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	—	4	2	no	—
43	C₂²×C₈×C₈₉	—	4	1	no	—
44	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
48	#123×C₈₉	C₈	1	8	yes	complemented
49	C₄×D₄×C₈₉	—	4	1	no	—
50	C₄×D₄×C₈₉	—	4	1	no	—
51	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	—	4	1	no	—
61	C₂×C₄²×C₈₉	—	4	1	no	—
62	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
65	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₈₉	—	4	2	no	—
67	C₄⋊C₈×C₈₉	—	4	2	no	—
68	C₄⋊C₈×C₈₉	—	4	2	no	—
69	C₄⋊C₈×C₈₉	—	4	1	no	—
70	C₂²⋊C₈×C₈₉	—	4	2	no	—
71	C₂²⋊C₈×C₈₉	—	4	2	no	—
72	C₂²⋊C₈×C₈₉	—	4	2	no	—
73	C₂²⋊C₈×C₈₉	—	4	1	no	—
74	C₄×C₈×C₈₉	—	4	2	no	—
75	C₄×C₈×C₈₉	—	4	1	no	—
76	#95×C₈₉	C₂×C₄	1	2	yes	—
77	#95×C₈₉	C₂×C₄	1	2	yes	—
78	#95×C₈₉	C₂×C₄	1	2	yes	—
79	#95×C₈₉	C₂×C₄	1	2	yes	—
80	#95×C₈₉	C₂×C₄	1	2	yes	—
81	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×D₄×C₈₉	—	4	1	no	—
84	C₂×D₄×C₈₉	—	4	1	no	—
85	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
86	C₂²×C₄×C₈₉	—	2	2	no	—
87	C₂²×C₄×C₈₉	—	2	1	no	—
88	C₂²×C₄×C₈₉	—	2	2	no	—
89	C₂²×C₄×C₈₉	—	2	1	no	—
90	C₂²×C₄×C₈₉	—	2	1	no	—
91	C₂²×C₄×C₈₉	—	2	1	no	—
92	C₂²×C₄×C₈₉	—	2	2	no	—
93	C₂²×C₄×C₈₉	—	2	2	no	—
94	C₂²×C₄×C₈₉	—	4	1	no	—
95	C₂²×C₄×C₈₉	—	4	1	no	—
96	C₂²×C₄×C₈₉	—	4	1	no	—
97	C₂²×C₄×C₈₉	—	4	2	no	—
98	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	complemented
99	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	—	2	2	no	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	—	2	2	no	—
120	C₂×C₈×C₈₉	—	4	4	no	—
121	C₄⋊C₄×C₈₉	—	4	2	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	C₂×C₈	1	8	yes	complemented
124	C₂²⋊C₄×C₈₉	—	4	2	no	—
125	C₂²⋊C₄×C₈₉	—	4	1	no	—
126	C₂²⋊C₄×C₈₉	—	4	1	no	—
127	C₂²⋊C₄×C₈₉	—	4	1	no	—
128	C₂²⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
129	C₂²⋊C₄×C₈₉	C₂×C₈	1	4	yes	complemented
130	C₄²×C₈₉	—	2	2	no	—
131	C₄²×C₈₉	—	2	1	no	—
132	C₄²×C₈₉	—	2	2	no	—
133	C₄²×C₈₉	—	2	2	no	—
134	C₄²×C₈₉	—	2	1	no	—
135	C₄²×C₈₉	—	2	2	no	—
136	C₄²×C₈₉	—	4	2	no	—
137	C₂³×C₈₉	—	2	1	no	—
138	C₂³×C₈₉	—	2	1	no	—
139	C₂³×C₈₉	—	2	2	no	—
140	C₂³×C₈₉	—	2	2	no	—
141	C₂³×C₈₉	—	2	1	no	—
142	C₂³×C₈₉	—	2	2	no	—
143	C₂³×C₈₉	—	2	2	no	—
144	C₂³×C₈₉	—	2	1	no	—
145	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
146	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
147	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
148	D₄×C₈₉	—	4	4	no	—
149	C₂×C₄×C₈₉	—	2	2	no	—
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	2	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	2	1	no	—
155	C₂×C₄×C₈₉	—	2	2	no	—
156	C₂×C₄×C₈₉	—	2	1	no	—
157	C₂×C₄×C₈₉	—	2	1	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	2	2	no	—
160	C₂×C₄×C₈₉	—	2	2	no	—
161	C₂×C₄×C₈₉	—	2	2	no	—
162	C₂×C₄×C₈₉	—	2	2	no	—
163	C₂×C₄×C₈₉	—	2	2	no	—
164	C₂×C₄×C₈₉	—	2	2	no	—
165	C₂×C₄×C₈₉	—	2	2	no	—
166	C₂×C₄×C₈₉	—	2	2	no	—
167	C₂×C₄×C₈₉	—	4	4	no	—
168	C₂×C₄×C₈₉	—	4	4	no	—
169	C₂×C₄×C₈₉	—	4	1	no	—
170	C₂×C₄×C₈₉	—	4	1	no	—
171	C₂×C₄×C₈₉	—	4	2	no	—
172	C₂×C₄×C₈₉	—	4	2	no	—
173	C₂×C₄×C₈₉	—	4	1	no	—
174	C₂×C₄×C₈₉	—	4	1	no	—
175	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
176	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
177	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
178	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
179	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
182	C₈×C₈₉	—	4	2	no	—
183	C₈×C₈₉	—	4	2	no	—
184	C₈×C₈₉	—	4	2	no	—
185	C₈×C₈₉	—	4	2	no	—
186	C₂²×C₈₉	—	2	2	no	—
187	C₂²×C₈₉	—	2	2	no	—
188	C₂²×C₈₉	—	2	4	no	—
189	C₂²×C₈₉	—	2	4	no	—
190	C₂²×C₈₉	—	4	1	no	—
191	C₂²×C₈₉	—	4	1	no	—
192	C₂²×C₈₉	—	4	1	no	—
193	C₂²×C₈₉	—	4	1	no	—
194	C₂²×C₈₉	—	4	2	no	—
195	C₂²×C₈₉	—	4	2	no	—
196	C₂²×C₈₉	#518	1	1	yes	characteristic
197	C₂²×C₈₉	#520	1	2	yes	—
198	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
199	C₂²×C₈₉	#568	1	1	yes	characteristic
200	C₂²×C₈₉	#568	1	2	yes	—
201	C₄×C₈₉	—	2	4	no	—
202	C₄×C₈₉	—	2	2	no	—
203	C₄×C₈₉	—	2	4	no	—
204	C₄×C₈₉	—	4	1	no	—
205	C₄×C₈₉	—	4	1	no	—
206	C₄×C₈₉	—	4	1	no	—
207	C₄×C₈₉	—	4	2	no	—
208	C₄×C₈₉	—	4	2	no	—
209	C₄×C₈₉	#560	1	1	yes	characteristic
210	C₄×C₈₉	#562	1	2	yes	—
211	C₄×C₈₉	#563	1	1	yes	characteristic
212	#23314	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₈₉	—	4	2	no	—
214	C₂×C₈₉	—	4	2	no	—
215	C₂×C₈₉	#2376	1	1	yes	characteristic
216	C₂×C₈₉	#2972	1	1	yes	characteristic
217	C₂×C₈₉	#2978	1	1	yes	characteristic
218	C₂×C₈₉	#2988	1	2	yes	—
219	C₂×C₈₉	#2995	1	2	yes	—
220	C₄×#123	C₂×C₈₉	1	1	yes	characteristic
221	#1954	C₂×C₈₉	1	1	yes	characteristic
222	#1939	C₂×C₈₉	1	2	yes	complemented
223	#1923	C₂×C₈₉	1	2	yes	complemented
224	#1923	C₂×C₈₉	1	2	yes	complemented
225	#1916	C₂×C₈₉	1	2	yes	complemented
226	#1869	C₂×C₈₉	1	1	yes	characteristic
227	#1850	C₂×C₈₉	1	2	yes	complemented
228	#1797	C₂×C₈₉	1	2	yes	complemented
229	C₈₉	#23314	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
231	C₂×#123	C₄×C₈₉	1	1	yes	characteristic
232	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
233	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
234	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
235	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
236	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
238	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
239	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
240	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
241	C₂×C₂²⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
243	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
244	#388	C₄×C₈₉	1	2	yes	—
245	#387	C₄×C₈₉	1	2	yes	—
246	#386	C₄×C₈₉	1	1	yes	characteristic
247	#385	C₄×C₈₉	1	1	yes	characteristic
248	#381	C₄×C₈₉	1	1	yes	characteristic
249	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
250	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
251	C₄×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
252	#336	C₂²×C₈₉	1	2	yes	—
253	#336	C₂²×C₈₉	1	2	yes	—
254	#336	C₂²×C₈₉	1	2	yes	—
255	#336	C₂²×C₈₉	1	2	yes	—
256	#336	C₂²×C₈₉	1	2	yes	—
257	C₂²×D₄	C₂×C₄×C₈₉	1	1	yes	characteristic
258	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
259	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
260	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
261	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
262	C₂²×C₈	—	4	2	no	—
263	C₂²×C₈	—	4	1	no	—
264	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
265	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
266	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
267	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
268	#123	C₈×C₈₉	1	8	yes	complemented
269	C₄×D₄	—	4	1	no	—
270	C₄×D₄	—	4	1	no	—
271	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
273	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
274	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
275	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
276	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
278	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
279	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
280	C₂×C₄²	—	4	1	no	—
281	C₂×C₄²	—	4	1	no	—
282	C₂×C₄²	C₂×C₄×C₈₉	1	2	yes	—
283	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
285	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
286	C₄⋊C₈	—	4	2	no	—
287	C₄⋊C₈	—	4	2	no	—
288	C₄⋊C₈	—	4	2	no	—
289	C₄⋊C₈	—	4	1	no	—
290	C₂²⋊C₈	—	4	2	no	—
291	C₂²⋊C₈	—	4	2	no	—
292	C₂²⋊C₈	—	4	2	no	—
293	C₂²⋊C₈	—	4	1	no	—
294	C₄×C₈	—	4	2	no	—
295	C₄×C₈	—	4	1	no	—
296	#95	C₂×C₄×C₈₉	1	2	yes	—
297	#95	C₂×C₄×C₈₉	1	2	yes	—
298	#95	C₂×C₄×C₈₉	1	2	yes	—
299	#95	C₂×C₄×C₈₉	1	2	yes	—
300	#95	C₂×C₄×C₈₉	1	2	yes	—
301	C₂⁴	C₂²×C₄×C₈₉	1	1	yes	characteristic
302	C₂⁴	C₂²×C₄×C₈₉	1	1	yes	characteristic
303	C₂×D₄	—	4	1	no	—
304	C₂×D₄	—	4	1	no	—
305	C₂×D₄	C₂×C₈×C₈₉	1	4	yes	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	4	1	no	—
315	C₂²×C₄	—	4	1	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	C₂×C₈×C₈₉	1	4	yes	complemented
319	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
320	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
321	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
322	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
323	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
324	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
325	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
326	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
327	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	4	4	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	C₂×C₈×C₈₉	1	8	yes	complemented
344	C₂²⋊C₄	—	4	1	no	—
345	C₂²⋊C₄	—	4	1	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	4	1	no	—
348	C₂²⋊C₄	C₂×C₈×C₈₉	1	8	yes	—
349	C₂²⋊C₄	C₂×C₈×C₈₉	1	4	yes	complemented
350	C₄²	—	2	2	no	—
351	C₄²	—	2	1	no	—
352	C₄²	—	2	2	no	—
353	C₄²	—	2	2	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	4	2	no	—
357	C₂³	—	2	1	no	—
358	C₂³	—	2	1	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	2	2	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	1	no	—
365	C₂³	C₂²×C₈×C₈₉	1	2	yes	—
366	C₂³	C₂²×C₈×C₈₉	1	2	yes	—
367	C₂³	C₂³×C₄×C₈₉	1	1	yes	characteristic
368	D₄	—	4	4	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	C₂²×C₈×C₈₉	1	4	yes	—
396	C₂×C₄	C₂²×C₈×C₈₉	1	4	yes	—
397	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
398	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
399	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
400	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
401	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
402	C₈	—	4	2	no	—
403	C₈	—	4	2	no	—
404	C₈	—	4	2	no	—
405	C₈	—	4	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	4	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	4	1	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	2	no	—
416	C₂²	#518×C₈₉	1	1	yes	characteristic
417	C₂²	#520×C₈₉	1	2	yes	—
418	C₂²	C₂³×C₈×C₈₉	1	1	yes	characteristic
419	C₂²	#568×C₈₉	1	1	yes	characteristic
420	C₂²	#568×C₈₉	1	2	yes	—
421	C₄	—	2	4	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	4	no	—
424	C₄	—	4	1	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	1	no	—
428	C₄	—	4	2	no	—
429	C₄	#560×C₈₉	1	1	yes	characteristic
430	C₄	#562×C₈₉	1	2	yes	—
431	C₄	#563×C₈₉	1	1	yes	characteristic
432	C₂	—	4	2	no	—
433	C₂	—	4	2	no	—
434	C₂	C₈₉×#2376	1	1	yes	characteristic
435	C₂	C₈₉×#2972	1	1	yes	characteristic
436	C₂	C₈₉×#2978	1	1	yes	characteristic
437	C₂	C₈₉×#2988	1	2	yes	—
438	C₂	C₈₉×#2995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₈₉	C₂⁴
commutator	C₂²	C₂³×C₈×C₈₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
socle	C₂³×C₈₉	C₂³×C₄
prime radical	C₂²×D₄×C₈₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#1939	C₂	semidirect
4. C₈₉×#1923	C₂	semidirect
5. C₈₉×#1923	C₂	semidirect
6. C₈₉×#1916	C₂	semidirect
8. C₈₉×#1850	C₂	semidirect
9. C₈₉×#1797	C₂	semidirect
48. #123×C₈₉	C₈	semidirect
98. C₂²×C₄×C₈₉	C₂×C₈	semidirect
123. C₄⋊C₄×C₈₉	C₂×C₈	semidirect
129. C₂²⋊C₄×C₈₉	C₂×C₈	semidirect
212. #23314	C₈₉	direct (N × H)
222. #1939	C₂×C₈₉	semidirect
223. #1923	C₂×C₈₉	semidirect
224. #1923	C₂×C₈₉	semidirect
225. #1916	C₂×C₈₉	semidirect
227. #1850	C₂×C₈₉	semidirect
228. #1797	C₂×C₈₉	semidirect
229. C₈₉	#23314	direct (N × H)
268. #123	C₈×C₈₉	semidirect
318. C₂²×C₄	C₂×C₈×C₈₉	semidirect
343. C₄⋊C₄	C₂×C₈×C₈₉	semidirect
349. C₂²⋊C₄	C₂×C₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#123×C₈₉	2
C₈₉×#1954	2
C₈₉×#1939	2
C₈₉×#1923	2
C₈₉×#1916	2
C₈₉×#1869	2
C₈₉×#1850	2
C₈₉×#1797	2
#23314	89

Subgroups — C89×#23314 #36401162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23314 #36401162