Subgroups — C₈₉×#34247 #36412095

Order 22784.

2230 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#126×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2373	C₂	1	4	yes	maximal, complemented
3	C₈₉×#2373	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2345	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2344	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2342	C₂	1	2	yes	maximal
7	C₈₉×#2342	C₂	1	2	yes	maximal
8	C₈₉×#2334	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2313	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#2312	C₂	1	1	yes	maximal, characteristic
11	C₄×#126×C₈₉	C₂	1	4	yes	maximal, complemented
12	C₈₉×#2283	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#2274	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#2273	C₂	1	1	yes	maximal, characteristic
15	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#126×C₈₉	C₄	1	16	yes	complemented
17	C₂×#126×C₈₉	C₂²	1	2	yes	—
18	C₂×#126×C₈₉	C₂²	1	2	yes	—
19	C₂×#126×C₈₉	C₂²	1	2	yes	—
20	C₂×#126×C₈₉	C₂²	1	2	yes	—
21	C₂×#117×C₈₉	C₂²	1	2	yes	—
22	C₂×#117×C₈₉	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
31	#389×C₈₉	C₂²	1	4	yes	complemented
32	#389×C₈₉	C₂²	1	4	yes	complemented
33	#388×C₈₉	C₂²	1	4	yes	—
34	#388×C₈₉	C₂²	1	4	yes	—
35	#388×C₈₉	C₂²	1	4	yes	—
36	#388×C₈₉	C₂²	1	4	yes	—
37	#388×C₈₉	C₂²	1	4	yes	—
38	#388×C₈₉	C₂²	1	4	yes	—
39	#387×C₈₉	C₂²	1	4	yes	complemented
40	#387×C₈₉	C₂²	1	4	yes	complemented
41	#387×C₈₉	C₂²	1	4	yes	complemented
42	#387×C₈₉	C₂²	1	4	yes	complemented
43	#387×C₈₉	C₂²	1	4	yes	complemented
44	#387×C₈₉	C₂²	1	4	yes	complemented
45	#385×C₈₉	C₂²	1	4	yes	—
46	#385×C₈₉	C₂²	1	4	yes	—
47	#383×C₈₉	C₂²	1	2	yes	complemented
48	#382×C₈₉	C₂²	1	2	yes	complemented
49	#381×C₈₉	C₂²	1	2	yes	—
50	#381×C₈₉	C₂²	1	2	yes	—
51	#380×C₈₉	C₂²	1	2	yes	—
52	#380×C₈₉	C₂²	1	2	yes	—
53	#380×C₈₉	C₂²	1	2	yes	—
54	#379×C₈₉	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
56	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
57	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
58	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
59	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
60	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
61	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
62	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
63	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
64	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
65	C₄×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
66	C₄×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
67	C₄×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
68	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
69	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
70	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
71	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
72	C₄×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
73	#376×C₈₉	C₂²	1	2	yes	complemented
74	#376×C₈₉	C₂²	1	2	yes	complemented
75	C₄³×C₈₉	C₂²	1	2	yes	complemented
76	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
80	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
84	#126×C₈₉	C₂×C₄	1	16	yes	complemented
85	#126×C₈₉	C₂×C₄	1	16	yes	complemented
86	#126×C₈₉	C₂×C₄	1	16	yes	complemented
87	#126×C₈₉	C₂×C₄	1	16	yes	complemented
88	#117×C₈₉	—	2	8	no	—
89	#117×C₈₉	—	2	8	no	—
90	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	complemented
91	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	complemented
92	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	complemented
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	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	4	yes	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
105	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
107	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
108	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	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	2	yes	—
120	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
121	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	complemented
122	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
123	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
124	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
125	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
126	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
127	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
128	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
129	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
130	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
131	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
135	#95×C₈₉	C₂³	1	2	yes	—
136	#95×C₈₉	C₂³	1	4	yes	—
137	#95×C₈₉	C₂³	1	4	yes	—
138	#95×C₈₉	C₂³	1	2	yes	—
139	#95×C₈₉	C₂³	1	4	yes	—
140	#95×C₈₉	C₂³	1	2	yes	—
141	#95×C₈₉	C₂³	1	4	yes	—
142	#95×C₈₉	C₂³	1	4	yes	—
143	#95×C₈₉	C₂³	1	2	yes	—
144	#95×C₈₉	C₂³	1	4	yes	—
145	#95×C₈₉	C₂³	1	4	yes	—
146	#95×C₈₉	C₂³	1	4	yes	—
147	#95×C₈₉	C₂³	1	4	yes	—
148	#95×C₈₉	C₂³	1	4	yes	—
149	C₂⁴×C₈₉	—	2	4	no	—
150	C₂⁴×C₈₉	—	4	2	no	—
151	C₂⁴×C₈₉	—	4	2	no	—
152	C₂⁴×C₈₉	C₂²×C₄	1	4	yes	—
153	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
154	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₈₉	—	2	4	no	—
156	C₂²×C₄×C₈₉	—	2	4	no	—
157	C₂²×C₄×C₈₉	—	2	8	no	—
158	C₂²×C₄×C₈₉	—	4	2	no	—
159	C₂²×C₄×C₈₉	—	4	2	no	—
160	C₂²×C₄×C₈₉	—	4	2	no	—
161	C₂²×C₄×C₈₉	—	4	2	no	—
162	C₂²×C₄×C₈₉	—	4	4	no	—
163	C₂²×C₄×C₈₉	—	4	4	no	—
164	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
165	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
166	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₈₉	—	2	8	no	—
184	C₄⋊C₄×C₈₉	—	2	8	no	—
185	C₄⋊C₄×C₈₉	—	4	4	no	—
186	C₄⋊C₄×C₈₉	—	4	4	no	—
187	C₄⋊C₄×C₈₉	—	4	4	no	—
188	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
189	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
190	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
191	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
192	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
193	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
194	C₂²⋊C₄×C₈₉	—	2	4	no	—
195	C₂²⋊C₄×C₈₉	—	2	4	no	—
196	C₂²⋊C₄×C₈₉	—	2	4	no	—
197	C₂²⋊C₄×C₈₉	—	2	8	no	—
198	C₂²⋊C₄×C₈₉	—	4	4	no	—
199	C₂²⋊C₄×C₈₉	—	4	4	no	—
200	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
201	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
202	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
203	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
204	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
205	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
206	C₂²⋊C₄×C₈₉	#41	1	8	yes	—
207	C₄²×C₈₉	—	2	8	no	—
208	C₄²×C₈₉	—	2	8	no	—
209	C₄²×C₈₉	—	4	2	no	—
210	C₄²×C₈₉	—	4	4	no	—
211	C₄²×C₈₉	—	4	4	no	—
212	C₄²×C₈₉	—	4	4	no	—
213	C₄²×C₈₉	—	4	2	no	—
214	C₄²×C₈₉	—	4	2	no	—
215	C₄²×C₈₉	—	4	2	no	—
216	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
217	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
218	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
219	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
220	C₂³×C₈₉	—	2	2	no	—
221	C₂³×C₈₉	—	2	2	no	—
222	C₂³×C₈₉	—	2	2	no	—
223	C₂³×C₈₉	—	2	2	no	—
224	C₂³×C₈₉	—	2	4	no	—
225	C₂³×C₈₉	—	2	4	no	—
226	C₂³×C₈₉	—	2	8	no	—
227	C₂³×C₈₉	—	4	4	no	—
228	C₂³×C₈₉	—	4	4	no	—
229	C₂³×C₈₉	—	8	8	no	—
230	C₂³×C₈₉	C₂³×C₄	1	4	yes	—
231	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
232	C₂³×C₈₉	C₂×#41	1	4	yes	—
233	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
234	C₂×C₄×C₈₉	—	2	8	no	—
235	C₂×C₄×C₈₉	—	2	8	no	—
236	C₂×C₄×C₈₉	—	2	2	no	—
237	C₂×C₄×C₈₉	—	2	2	no	—
238	C₂×C₄×C₈₉	—	2	2	no	—
239	C₂×C₄×C₈₉	—	2	2	no	—
240	C₂×C₄×C₈₉	—	2	2	no	—
241	C₂×C₄×C₈₉	—	2	2	no	—
242	C₂×C₄×C₈₉	—	2	2	no	—
243	C₂×C₄×C₈₉	—	2	4	no	—
244	C₂×C₄×C₈₉	—	2	4	no	—
245	C₂×C₄×C₈₉	—	2	4	no	—
246	C₂×C₄×C₈₉	—	2	4	no	—
247	C₂×C₄×C₈₉	—	2	2	no	—
248	C₂×C₄×C₈₉	—	2	4	no	—
249	C₂×C₄×C₈₉	—	2	4	no	—
250	C₂×C₄×C₈₉	—	2	4	no	—
251	C₂×C₄×C₈₉	—	2	4	no	—
252	C₂×C₄×C₈₉	—	2	4	no	—
253	C₂×C₄×C₈₉	—	2	4	no	—
254	C₂×C₄×C₈₉	—	2	4	no	—
255	C₂×C₄×C₈₉	—	2	4	no	—
256	C₂×C₄×C₈₉	—	2	4	no	—
257	C₂×C₄×C₈₉	—	4	4	no	—
258	C₂×C₄×C₈₉	—	4	4	no	—
259	C₂×C₄×C₈₉	—	4	4	no	—
260	C₂×C₄×C₈₉	—	4	4	no	—
261	C₂×C₄×C₈₉	—	8	8	no	—
262	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
263	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
264	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
265	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
269	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
270	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
271	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
272	C₂²×C₈₉	—	2	2	no	—
273	C₂²×C₈₉	—	2	2	no	—
274	C₂²×C₈₉	—	2	8	no	—
275	C₂²×C₈₉	—	2	4	no	—
276	C₂²×C₈₉	—	2	4	no	—
277	C₂²×C₈₉	—	2	4	no	—
278	C₂²×C₈₉	—	4	4	no	—
279	C₂²×C₈₉	—	4	4	no	—
280	C₂²×C₈₉	—	8	8	no	—
281	C₂²×C₈₉	C₄×#41	1	4	yes	—
282	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
283	C₂²×C₈₉	C₂²×#41	1	2	yes	—
284	C₂²×C₈₉	#585	1	2	yes	—
285	C₂²×C₈₉	#585	1	2	yes	—
286	C₄×C₈₉	—	2	2	no	—
287	C₄×C₈₉	—	2	4	no	—
288	C₄×C₈₉	—	2	2	no	—
289	C₄×C₈₉	—	2	2	no	—
290	C₄×C₈₉	—	2	2	no	—
291	C₄×C₈₉	—	2	2	no	—
292	C₄×C₈₉	—	2	4	no	—
293	C₄×C₈₉	—	2	4	no	—
294	C₄×C₈₉	—	2	2	no	—
295	C₄×C₈₉	—	2	8	no	—
296	C₄×C₈₉	—	4	4	no	—
297	C₄×C₈₉	—	4	4	no	—
298	C₄×C₈₉	—	4	4	no	—
299	#34247	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
300	C₂×C₈₉	—	2	4	no	—
301	C₂×C₈₉	—	4	4	no	—
302	C₂×C₈₉	C₂×C₄×#41	1	1	yes	characteristic
303	C₂×C₈₉	#3428	1	1	yes	characteristic
304	C₂×C₈₉	#3428	1	1	yes	characteristic
305	C₂×C₈₉	#3521	1	4	yes	—
306	C₂²×#126	C₂×C₈₉	1	1	yes	characteristic
307	#2373	C₂×C₈₉	1	4	yes	complemented
308	#2373	C₂×C₈₉	1	4	yes	complemented
309	#2345	C₂×C₈₉	1	2	yes	complemented
310	#2344	C₂×C₈₉	1	2	yes	complemented
311	#2342	C₂×C₈₉	1	2	yes	—
312	#2342	C₂×C₈₉	1	2	yes	—
313	#2334	C₂×C₈₉	1	4	yes	complemented
314	#2313	C₂×C₈₉	1	1	yes	characteristic, complemented
315	#2312	C₂×C₈₉	1	1	yes	characteristic
316	C₄×#126	C₂×C₈₉	1	4	yes	complemented
317	#2283	C₂×C₈₉	1	1	yes	characteristic, complemented
318	#2274	C₂×C₈₉	1	1	yes	characteristic, complemented
319	#2273	C₂×C₈₉	1	1	yes	characteristic
320	C₄×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
321	C₈₉	#34247	1	1	yes	characteristic, direct product factor, Sylow
322	C₂×#126	C₄×C₈₉	1	16	yes	complemented
323	C₂×#126	C₂²×C₈₉	1	2	yes	—
324	C₂×#126	C₂²×C₈₉	1	2	yes	—
325	C₂×#126	C₂²×C₈₉	1	2	yes	—
326	C₂×#126	C₂²×C₈₉	1	2	yes	—
327	C₂×#117	C₂²×C₈₉	1	2	yes	—
328	C₂×#117	C₂²×C₈₉	1	2	yes	—
329	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
330	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
331	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
332	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
333	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
334	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
335	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
336	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
337	#389	C₂²×C₈₉	1	4	yes	complemented
338	#389	C₂²×C₈₉	1	4	yes	complemented
339	#388	C₂²×C₈₉	1	4	yes	—
340	#388	C₂²×C₈₉	1	4	yes	—
341	#388	C₂²×C₈₉	1	4	yes	—
342	#388	C₂²×C₈₉	1	4	yes	—
343	#388	C₂²×C₈₉	1	4	yes	—
344	#388	C₂²×C₈₉	1	4	yes	—
345	#387	C₂²×C₈₉	1	4	yes	complemented
346	#387	C₂²×C₈₉	1	4	yes	complemented
347	#387	C₂²×C₈₉	1	4	yes	complemented
348	#387	C₂²×C₈₉	1	4	yes	complemented
349	#387	C₂²×C₈₉	1	4	yes	complemented
350	#387	C₂²×C₈₉	1	4	yes	complemented
351	#385	C₂²×C₈₉	1	4	yes	—
352	#385	C₂²×C₈₉	1	4	yes	—
353	#383	C₂²×C₈₉	1	2	yes	complemented
354	#382	C₂²×C₈₉	1	2	yes	complemented
355	#381	C₂²×C₈₉	1	2	yes	—
356	#381	C₂²×C₈₉	1	2	yes	—
357	#380	C₂²×C₈₉	1	2	yes	—
358	#380	C₂²×C₈₉	1	2	yes	—
359	#380	C₂²×C₈₉	1	2	yes	—
360	#379	C₂²×C₈₉	1	2	yes	—
361	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
362	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
363	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
364	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
365	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
366	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
367	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
368	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
369	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
370	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
371	C₄×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
372	C₄×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
373	C₄×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
374	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
375	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
376	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
377	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
378	C₄×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
379	#376	C₂²×C₈₉	1	2	yes	complemented
380	#376	C₂²×C₈₉	1	2	yes	complemented
381	C₄³	C₂²×C₈₉	1	2	yes	complemented
382	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
384	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
385	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
386	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
390	#126	C₂×C₄×C₈₉	1	16	yes	complemented
391	#126	C₂×C₄×C₈₉	1	16	yes	complemented
392	#126	C₂×C₄×C₈₉	1	16	yes	complemented
393	#126	C₂×C₄×C₈₉	1	16	yes	complemented
394	#117	—	2	8	no	—
395	#117	—	2	8	no	—
396	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	8	yes	complemented
397	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	8	yes	complemented
398	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	8	yes	complemented
399	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
400	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
401	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
403	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
404	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
410	C₂×C₂²⋊C₄	—	2	8	no	—
411	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	8	yes	—
412	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	8	yes	—
413	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	8	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
427	C₂×C₄²	C₂×C₄×C₈₉	1	8	yes	complemented
428	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
429	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
430	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
431	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
432	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
433	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
434	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
435	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
436	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
437	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
441	#95	C₂³×C₈₉	1	2	yes	—
442	#95	C₂³×C₈₉	1	4	yes	—
443	#95	C₂³×C₈₉	1	4	yes	—
444	#95	C₂³×C₈₉	1	2	yes	—
445	#95	C₂³×C₈₉	1	4	yes	—
446	#95	C₂³×C₈₉	1	2	yes	—
447	#95	C₂³×C₈₉	1	4	yes	—
448	#95	C₂³×C₈₉	1	4	yes	—
449	#95	C₂³×C₈₉	1	2	yes	—
450	#95	C₂³×C₈₉	1	4	yes	—
451	#95	C₂³×C₈₉	1	4	yes	—
452	#95	C₂³×C₈₉	1	4	yes	—
453	#95	C₂³×C₈₉	1	4	yes	—
454	#95	C₂³×C₈₉	1	4	yes	—
455	C₂⁴	—	2	4	no	—
456	C₂⁴	—	4	2	no	—
457	C₂⁴	—	4	2	no	—
458	C₂⁴	C₂²×C₄×C₈₉	1	4	yes	—
459	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
460	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	4	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
471	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
472	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
473	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
474	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
475	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
487	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	8	no	—
491	C₄⋊C₄	—	4	4	no	—
492	C₄⋊C₄	—	4	4	no	—
493	C₄⋊C₄	—	4	4	no	—
494	C₄⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
495	C₄⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
496	C₄⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
497	C₄⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
498	C₄⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
499	C₄⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	4	4	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
507	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
508	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
509	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
510	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
511	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
512	C₂²⋊C₄	#41×C₈₉	1	8	yes	—
513	C₄²	—	2	8	no	—
514	C₄²	—	2	8	no	—
515	C₄²	—	4	2	no	—
516	C₄²	—	4	4	no	—
517	C₄²	—	4	4	no	—
518	C₄²	—	4	4	no	—
519	C₄²	—	4	2	no	—
520	C₄²	—	4	2	no	—
521	C₄²	—	4	2	no	—
522	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
523	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
524	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
525	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	4	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	8	no	—
533	C₂³	—	4	4	no	—
534	C₂³	—	4	4	no	—
535	C₂³	—	8	8	no	—
536	C₂³	C₂³×C₄×C₈₉	1	4	yes	—
537	C₂³	C₂³×C₄×C₈₉	1	2	yes	—
538	C₂³	C₂×#41×C₈₉	1	4	yes	—
539	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
540	C₂×C₄	—	2	8	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	4	4	no	—
566	C₂×C₄	—	4	4	no	—
567	C₂×C₄	—	8	8	no	—
568	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
569	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
570	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
571	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
572	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
573	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
574	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
575	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
576	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
577	C₂×C₄	C₂×#41×C₈₉	1	8	yes	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	2	8	no	—
581	C₂²	—	2	4	no	—
582	C₂²	—	2	4	no	—
583	C₂²	—	2	4	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	4	no	—
586	C₂²	—	8	8	no	—
587	C₂²	C₄×#41×C₈₉	1	4	yes	—
588	C₂²	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
589	C₂²	C₂²×#41×C₈₉	1	2	yes	—
590	C₂²	#585×C₈₉	1	2	yes	—
591	C₂²	#585×C₈₉	1	2	yes	—
592	C₄	—	2	2	no	—
593	C₄	—	2	4	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	4	no	—
599	C₄	—	2	4	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	8	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	4	no	—
604	C₄	—	4	4	no	—
605	C₂	—	2	4	no	—
606	C₂	—	4	4	no	—
607	C₂	C₂×C₄×#41×C₈₉	1	1	yes	characteristic
608	C₂	C₈₉×#3428	1	1	yes	characteristic
609	C₂	C₈₉×#3428	1	1	yes	characteristic
610	C₂	C₈₉×#3521	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁵×C₈₉	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2373	C₂	semidirect
3. C₈₉×#2373	C₂	semidirect
4. C₈₉×#2345	C₂	semidirect
5. C₈₉×#2344	C₂	semidirect
8. C₈₉×#2334	C₂	semidirect
9. C₈₉×#2313	C₂	semidirect
11. C₄×#126×C₈₉	C₂	semidirect
12. C₈₉×#2283	C₂	semidirect
13. C₈₉×#2274	C₂	semidirect
15. C₄×#117×C₈₉	C₂	semidirect
16. C₂×#126×C₈₉	C₄	semidirect
31. #389×C₈₉	C₂²	semidirect
32. #389×C₈₉	C₂²	semidirect
39. #387×C₈₉	C₂²	semidirect
40. #387×C₈₉	C₂²	semidirect
41. #387×C₈₉	C₂²	semidirect
42. #387×C₈₉	C₂²	semidirect
43. #387×C₈₉	C₂²	semidirect
44. #387×C₈₉	C₂²	semidirect
47. #383×C₈₉	C₂²	semidirect
48. #382×C₈₉	C₂²	semidirect
55. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
56. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
57. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
58. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
59. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
60. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
61. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
73. #376×C₈₉	C₂²	semidirect
74. #376×C₈₉	C₂²	semidirect
75. C₄³×C₈₉	C₂²	semidirect
84. #126×C₈₉	C₂×C₄	semidirect
85. #126×C₈₉	C₂×C₄	semidirect
86. #126×C₈₉	C₂×C₄	semidirect
87. #126×C₈₉	C₂×C₄	semidirect
90. C₂×C₄⋊C₄×C₈₉	C₂×C₄	semidirect
91. C₂×C₄⋊C₄×C₈₉	C₂×C₄	semidirect
92. C₂×C₄⋊C₄×C₈₉	C₂×C₄	semidirect
121. C₂×C₄²×C₈₉	C₂×C₄	semidirect
299. #34247	C₈₉	direct (N × H)
307. #2373	C₂×C₈₉	semidirect
308. #2373	C₂×C₈₉	semidirect
309. #2345	C₂×C₈₉	semidirect
310. #2344	C₂×C₈₉	semidirect
313. #2334	C₂×C₈₉	semidirect
314. #2313	C₂×C₈₉	semidirect
316. C₄×#126	C₂×C₈₉	semidirect
317. #2283	C₂×C₈₉	semidirect
318. #2274	C₂×C₈₉	semidirect
320. C₄×#117	C₂×C₈₉	semidirect
321. C₈₉	#34247	direct (N × H)
322. C₂×#126	C₄×C₈₉	semidirect
337. #389	C₂²×C₈₉	semidirect
338. #389	C₂²×C₈₉	semidirect
345. #387	C₂²×C₈₉	semidirect
346. #387	C₂²×C₈₉	semidirect
347. #387	C₂²×C₈₉	semidirect
348. #387	C₂²×C₈₉	semidirect
349. #387	C₂²×C₈₉	semidirect
350. #387	C₂²×C₈₉	semidirect
353. #383	C₂²×C₈₉	semidirect
354. #382	C₂²×C₈₉	semidirect
361. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
362. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
363. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
364. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
365. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
366. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
367. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
379. #376	C₂²×C₈₉	semidirect
380. #376	C₂²×C₈₉	semidirect
381. C₄³	C₂²×C₈₉	semidirect
390. #126	C₂×C₄×C₈₉	semidirect
391. #126	C₂×C₄×C₈₉	semidirect
392. #126	C₂×C₄×C₈₉	semidirect
393. #126	C₂×C₄×C₈₉	semidirect
396. C₂×C₄⋊C₄	C₂×C₄×C₈₉	semidirect
397. C₂×C₄⋊C₄	C₂×C₄×C₈₉	semidirect
398. C₂×C₄⋊C₄	C₂×C₄×C₈₉	semidirect
427. C₂×C₄²	C₂×C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#126×C₈₉	2
C₈₉×#2373	2
C₈₉×#2345	2
C₈₉×#2344	2
C₈₉×#2342	2
C₈₉×#2334	2
C₈₉×#2313	2
C₈₉×#2312	2
C₄×#126×C₈₉	2
C₈₉×#2283	2
C₈₉×#2274	2
C₈₉×#2273	2
C₄×#117×C₈₉	2
#34247	89

Subgroups — C89×#34247 #36412095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#34247 #36412095