Subgroups — C₈₉×#22385 #36400233

Order 22784.

822 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#124×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1984	C₂	1	2	yes	maximal, complemented
3	C₈₉×#1923	C₂	1	2	yes	maximal, complemented
4	C₈₉×#1923	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1869	C₂	1	2	yes	maximal
6	C₈₉×#1850	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₂²⋊C₄×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
11	C₂×#124×C₈₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×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×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#391×C₈₉	C₄	1	1	yes	characteristic
23	#390×C₈₉	C₄	1	1	yes	characteristic
24	#388×C₈₉	C₄	1	2	yes	—
25	#388×C₈₉	C₄	1	2	yes	—
26	#382×C₈₉	C₄	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
29	C₄³×C₈₉	C₂²	1	1	yes	characteristic
30	#336×C₈₉	C₂²	1	1	yes	characteristic
31	#336×C₈₉	C₂²	1	2	yes	—
32	#336×C₈₉	C₂²	1	1	yes	characteristic
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×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	—	2	4	no	—
41	C₂²×C₈×C₈₉	—	4	2	no	—
42	C₂²×C₈×C₈₉	—	4	2	no	—
43	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
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	#124×C₈₉	C₈	1	8	yes	complemented
48	C₄×Q₈×C₈₉	—	4	2	no	—
49	C₄×D₄×C₈₉	—	4	2	no	—
50	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
66	C₄⋊C₈×C₈₉	—	2	4	no	—
67	C₄⋊C₈×C₈₉	—	4	2	no	—
68	C₂²⋊C₈×C₈₉	—	2	4	no	—
69	C₂²⋊C₈×C₈₉	—	4	2	no	—
70	C₂²⋊C₈×C₈₉	—	4	2	no	—
71	C₄×C₈×C₈₉	—	4	1	no	—
72	C₄×C₈×C₈₉	—	4	1	no	—
73	C₄×C₈×C₈₉	—	4	2	no	—
74	#95×C₈₉	C₂×C₄	1	2	yes	—
75	#95×C₈₉	C₂×C₄	1	2	yes	—
76	#95×C₈₉	C₂×C₄	1	2	yes	—
77	#95×C₈₉	C₂×C₄	1	2	yes	—
78	C₂⁴×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂×Q₈×C₈₉	—	4	1	no	—
80	C₂×Q₈×C₈₉	—	4	1	no	—
81	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	complemented
82	C₂×D₄×C₈₉	—	4	1	no	—
83	C₂×D₄×C₈₉	—	4	1	no	—
84	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
85	C₂²×C₄×C₈₉	—	2	2	no	—
86	C₂²×C₄×C₈₉	—	2	2	no	—
87	C₂²×C₄×C₈₉	—	2	2	no	—
88	C₂²×C₄×C₈₉	—	2	2	no	—
89	C₂²×C₄×C₈₉	—	2	2	no	—
90	C₂²×C₄×C₈₉	—	2	2	no	—
91	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
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	—	4	8	no	—
113	C₂×C₈×C₈₉	#41	1	4	yes	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	2	no	—
117	C₂²⋊C₄×C₈₉	—	2	4	no	—
118	C₂²⋊C₄×C₈₉	—	4	2	no	—
119	C₂²⋊C₄×C₈₉	—	4	2	no	—
120	C₂²⋊C₄×C₈₉	C₂×C₈	1	8	yes	complemented
121	C₂²⋊C₄×C₈₉	C₂×C₈	1	8	yes	complemented
122	C₄²×C₈₉	—	2	2	no	—
123	C₄²×C₈₉	—	2	2	no	—
124	C₄²×C₈₉	—	2	1	no	—
125	C₄²×C₈₉	—	2	2	no	—
126	C₄²×C₈₉	—	2	2	no	—
127	C₄²×C₈₉	—	2	2	no	—
128	C₄²×C₈₉	—	2	1	no	—
129	C₄²×C₈₉	—	4	1	no	—
130	C₄²×C₈₉	—	4	1	no	—
131	C₄²×C₈₉	C₂×C₈	1	4	yes	complemented
132	C₂³×C₈₉	—	2	2	no	—
133	C₂³×C₈₉	—	2	2	no	—
134	C₂³×C₈₉	—	2	2	no	—
135	C₂³×C₈₉	—	2	2	no	—
136	C₂³×C₈₉	—	2	2	no	—
137	C₂³×C₈₉	—	2	2	no	—
138	C₂³×C₈₉	C₂²×C₈	1	4	yes	—
139	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
140	Q₈×C₈₉	—	4	4	no	—
141	D₄×C₈₉	—	4	4	no	—
142	C₂×C₄×C₈₉	—	2	1	no	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	1	no	—
146	C₂×C₄×C₈₉	—	2	1	no	—
147	C₂×C₄×C₈₉	—	2	1	no	—
148	C₂×C₄×C₈₉	—	2	1	no	—
149	C₂×C₄×C₈₉	—	2	1	no	—
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	2	no	—
152	C₂×C₄×C₈₉	—	2	1	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	2	no	—
157	C₂×C₄×C₈₉	—	2	2	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	2	2	no	—
160	C₂×C₄×C₈₉	—	2	1	no	—
161	C₂×C₄×C₈₉	—	2	1	no	—
162	C₂×C₄×C₈₉	—	2	1	no	—
163	C₂×C₄×C₈₉	—	2	2	no	—
164	C₂×C₄×C₈₉	—	4	2	no	—
165	C₂×C₄×C₈₉	—	4	2	no	—
166	C₂×C₄×C₈₉	—	4	2	no	—
167	C₂×C₄×C₈₉	—	4	2	no	—
168	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
169	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
170	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
171	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
172	C₂×C₄×C₈₉	#131	1	2	yes	—
173	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
174	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
176	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
177	C₈×C₈₉	—	2	4	no	—
178	C₈×C₈₉	—	4	2	no	—
179	C₈×C₈₉	—	4	2	no	—
180	C₈×C₈₉	—	4	2	no	—
181	C₂²×C₈₉	—	2	4	no	—
182	C₂²×C₈₉	—	2	4	no	—
183	C₂²×C₈₉	—	2	4	no	—
184	C₂²×C₈₉	—	4	2	no	—
185	C₂²×C₈₉	—	4	2	no	—
186	C₂²×C₈₉	—	4	2	no	—
187	C₂²×C₈₉	—	4	2	no	—
188	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
189	C₂²×C₈₉	#518	1	1	yes	characteristic
190	C₂²×C₈₉	#520	1	1	yes	characteristic
191	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
192	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
193	C₂²×C₈₉	#568	1	1	yes	characteristic
194	C₂²×C₈₉	#568	1	1	yes	characteristic
195	C₄×C₈₉	—	2	2	no	—
196	C₄×C₈₉	—	2	2	no	—
197	C₄×C₈₉	—	2	4	no	—
198	C₄×C₈₉	—	2	2	no	—
199	C₄×C₈₉	—	2	2	no	—
200	C₄×C₈₉	—	4	2	no	—
201	C₄×C₈₉	—	4	1	no	—
202	C₄×C₈₉	—	4	1	no	—
203	C₄×C₈₉	—	4	2	no	—
204	C₄×C₈₉	#538	1	2	yes	—
205	C₄×C₈₉	#542	1	2	yes	—
206	#22385	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₈₉	—	4	4	no	—
208	C₂×C₈₉	#2337	1	1	yes	characteristic
209	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
210	C₂×C₈₉	#2967	1	1	yes	characteristic
211	C₂×C₈₉	#2972	1	1	yes	characteristic
212	C₂×C₈₉	#2981	1	1	yes	characteristic
213	C₂×C₈₉	#2988	1	1	yes	characteristic
214	C₂×C₈₉	#2992	1	1	yes	characteristic
215	C₄×#124	C₂×C₈₉	1	1	yes	characteristic
216	#1984	C₂×C₈₉	1	2	yes	complemented
217	#1923	C₂×C₈₉	1	2	yes	complemented
218	#1923	C₂×C₈₉	1	2	yes	complemented
219	#1869	C₂×C₈₉	1	2	yes	—
220	#1850	C₂×C₈₉	1	2	yes	complemented
221	#1839	C₂×C₈₉	1	1	yes	characteristic, complemented
222	C₄×C₄⋊C₈	C₂×C₈₉	1	1	yes	characteristic, complemented
223	C₈×C₂²⋊C₄	C₂×C₈₉	1	2	yes	complemented
224	C₈₉	#22385	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³×C₈	C₂²×C₈₉	1	2	yes	—
226	C₂×#124	C₄×C₈₉	1	1	yes	characteristic
227	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
229	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
230	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
231	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
232	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
233	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
234	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
237	#391	C₄×C₈₉	1	1	yes	characteristic
238	#390	C₄×C₈₉	1	1	yes	characteristic
239	#388	C₄×C₈₉	1	2	yes	—
240	#388	C₄×C₈₉	1	2	yes	—
241	#382	C₄×C₈₉	1	1	yes	characteristic
242	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
243	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
244	C₄³	C₂²×C₈₉	1	1	yes	characteristic
245	#336	C₂²×C₈₉	1	1	yes	characteristic
246	#336	C₂²×C₈₉	1	2	yes	—
247	#336	C₂²×C₈₉	1	1	yes	characteristic
248	#336	C₂²×C₈₉	1	2	yes	—
249	#336	C₂²×C₈₉	1	2	yes	—
250	#336	C₂²×C₈₉	1	2	yes	—
251	#336	C₂²×C₈₉	1	2	yes	—
252	C₂²×Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
253	C₂²×D₄	C₂×C₄×C₈₉	1	1	yes	characteristic
254	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
255	C₂²×C₈	—	2	4	no	—
256	C₂²×C₈	—	4	2	no	—
257	C₂²×C₈	—	4	2	no	—
258	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
259	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
260	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
261	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
262	#124	C₈×C₈₉	1	8	yes	complemented
263	C₄×Q₈	—	4	2	no	—
264	C₄×D₄	—	4	2	no	—
265	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
268	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
270	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
271	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
272	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
273	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
274	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
275	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
276	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
279	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
281	C₄⋊C₈	—	2	4	no	—
282	C₄⋊C₈	—	4	2	no	—
283	C₂²⋊C₈	—	2	4	no	—
284	C₂²⋊C₈	—	4	2	no	—
285	C₂²⋊C₈	—	4	2	no	—
286	C₄×C₈	—	4	1	no	—
287	C₄×C₈	—	4	1	no	—
288	C₄×C₈	—	4	2	no	—
289	#95	C₂×C₄×C₈₉	1	2	yes	—
290	#95	C₂×C₄×C₈₉	1	2	yes	—
291	#95	C₂×C₄×C₈₉	1	2	yes	—
292	#95	C₂×C₄×C₈₉	1	2	yes	—
293	C₂⁴	C₂²×C₄×C₈₉	1	2	yes	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	C₂×C₈×C₈₉	1	4	yes	complemented
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	C₂×C₈×C₈₉	1	4	yes	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
313	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
315	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	4	8	no	—
328	C₂×C₈	#41×C₈₉	1	4	yes	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	—	2	4	no	—
333	C₂²⋊C₄	—	4	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₂²⋊C₄	C₂×C₈×C₈₉	1	8	yes	complemented
336	C₂²⋊C₄	C₂×C₈×C₈₉	1	8	yes	complemented
337	C₄²	—	2	2	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	4	1	no	—
345	C₄²	—	4	1	no	—
346	C₄²	C₂×C₈×C₈₉	1	4	yes	complemented
347	C₂³	—	2	2	no	—
348	C₂³	—	2	2	no	—
349	C₂³	—	2	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	2	2	no	—
352	C₂³	—	2	2	no	—
353	C₂³	C₂²×C₈×C₈₉	1	4	yes	—
354	C₂³	C₂³×C₄×C₈₉	1	1	yes	characteristic
355	Q₈	—	4	4	no	—
356	D₄	—	4	4	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
384	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
385	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
386	C₂×C₄	C₂²×C₈×C₈₉	1	4	yes	—
387	C₂×C₄	#131×C₈₉	1	2	yes	—
388	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
389	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
390	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
391	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
392	C₈	—	2	4	no	—
393	C₈	—	4	2	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	2	no	—
396	C₂²	—	2	4	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	2	4	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	4	2	no	—
403	C₂²	C₄×#41×C₈₉	1	1	yes	characteristic
404	C₂²	#518×C₈₉	1	1	yes	characteristic
405	C₂²	#520×C₈₉	1	1	yes	characteristic
406	C₂²	C₂³×C₈×C₈₉	1	1	yes	characteristic
407	C₂²	C₂×#131×C₈₉	1	1	yes	characteristic
408	C₂²	#568×C₈₉	1	1	yes	characteristic
409	C₂²	#568×C₈₉	1	1	yes	characteristic
410	C₄	—	2	2	no	—
411	C₄	—	2	2	no	—
412	C₄	—	2	4	no	—
413	C₄	—	2	2	no	—
414	C₄	—	2	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	1	no	—
417	C₄	—	4	1	no	—
418	C₄	—	4	2	no	—
419	C₄	#538×C₈₉	1	2	yes	—
420	C₄	#542×C₈₉	1	2	yes	—
421	C₂	—	4	4	no	—
422	C₂	C₈₉×#2337	1	1	yes	characteristic
423	C₂	C₈×#41×C₈₉	1	1	yes	characteristic
424	C₂	C₈₉×#2967	1	1	yes	characteristic
425	C₂	C₈₉×#2972	1	1	yes	characteristic
426	C₂	C₈₉×#2981	1	1	yes	characteristic
427	C₂	C₈₉×#2988	1	1	yes	characteristic
428	C₂	C₈₉×#2992	1	1	yes	characteristic

	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
2. C₈₉×#1984	C₂	semidirect
3. C₈₉×#1923	C₂	semidirect
4. C₈₉×#1923	C₂	semidirect
6. C₈₉×#1850	C₂	semidirect
7. C₈₉×#1839	C₂	semidirect
8. C₄×C₄⋊C₈×C₈₉	C₂	semidirect
9. C₈×C₂²⋊C₄×C₈₉	C₂	semidirect
47. #124×C₈₉	C₈	semidirect
81. C₂×Q₈×C₈₉	C₂×C₈	semidirect
120. C₂²⋊C₄×C₈₉	C₂×C₈	semidirect
121. C₂²⋊C₄×C₈₉	C₂×C₈	semidirect
131. C₄²×C₈₉	C₂×C₈	semidirect
206. #22385	C₈₉	direct (N × H)
216. #1984	C₂×C₈₉	semidirect
217. #1923	C₂×C₈₉	semidirect
218. #1923	C₂×C₈₉	semidirect
220. #1850	C₂×C₈₉	semidirect
221. #1839	C₂×C₈₉	semidirect
222. C₄×C₄⋊C₈	C₂×C₈₉	semidirect
223. C₈×C₂²⋊C₄	C₂×C₈₉	semidirect
224. C₈₉	#22385	direct (N × H)
262. #124	C₈×C₈₉	semidirect
296. C₂×Q₈	C₂×C₈×C₈₉	semidirect
335. C₂²⋊C₄	C₂×C₈×C₈₉	semidirect
336. C₂²⋊C₄	C₂×C₈×C₈₉	semidirect
346. C₄²	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×#124×C₈₉	2
C₈₉×#1984	2
C₈₉×#1923	2
C₈₉×#1869	2
C₈₉×#1850	2
C₈₉×#1839	2
C₄×C₄⋊C₈×C₈₉	2
C₈×C₂²⋊C₄×C₈₉	2
#22385	89

Subgroups — C89×#22385 #36400233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#22385 #36400233