Subgroups — C₈₉×#21860 #36399708

Order 22784.

1222 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#384×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1990	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1956	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1872	C₂	1	4	yes	maximal
7	C₈₉×#1803	C₂	1	4	yes	maximal, complemented
8	#533×C₈₉	—	2	4	no	—
9	#533×C₈₉	C₄	1	8	yes	complemented
10	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×#125×C₈₉	—	2	1	no	—
13	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
14	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
16	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
17	C₂×#117×C₈₉	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
20	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	#431×C₈₉	—	2	4	no	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
24	C₂×#103×C₈₉	C₂²	1	8	yes	—
25	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
27	#384×C₈₉	—	2	1	no	—
28	#384×C₈₉	—	2	1	no	—
29	#384×C₈₉	C₂²	1	2	yes	—
30	#384×C₈₉	C₂²	1	2	yes	—
31	#340×C₈₉	C₂²	1	8	yes	—
32	C₂²×Q₈×C₈₉	—	2	1	no	—
33	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
34	C₂³×C₄×C₈₉	—	2	2	no	—
35	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₈₉	—	2	2	no	—
40	C₄⋊Q₈×C₈₉	—	2	2	no	—
41	C₄⋊Q₈×C₈₉	C₂×C₄	1	4	yes	complemented
42	C₄⋊Q₈×C₈₉	C₂×C₄	1	4	yes	complemented
43	#125×C₈₉	—	2	8	no	—
44	C₂²⋊Q₈×C₈₉	—	2	8	no	—
45	C₂²⋊Q₈×C₈₉	—	2	4	no	—
46	C₂²⋊Q₈×C₈₉	C₂×C₄	1	8	yes	—
47	C₄×Q₈×C₈₉	—	2	8	no	—
48	C₄×Q₈×C₈₉	—	2	4	no	—
49	C₄×Q₈×C₈₉	C₂×C₄	1	8	yes	complemented
50	#117×C₈₉	—	2	8	no	—
51	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
60	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
61	C₂×C₄⋊C₄×C₈₉	—	4	4	no	—
62	C₂×C₄⋊C₄×C₈₉	—	4	4	no	—
63	C₂×C₄⋊C₄×C₈₉	—	4	4	no	—
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
67	C₂×C₄²×C₈₉	—	2	2	no	—
68	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
69	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₈₉	D₄	1	4	yes	—
72	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	—	2	4	no	—
75	#103×C₈₉	—	4	8	no	—
76	C₂²⋊C₈×C₈₉	—	2	4	no	—
77	C₄×C₈×C₈₉	—	2	4	no	—
78	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×Q₈×C₈₉	—	2	4	no	—
80	C₂×Q₈×C₈₉	—	2	2	no	—
81	C₂×Q₈×C₈₉	—	2	4	no	—
82	C₂×Q₈×C₈₉	—	2	4	no	—
83	C₂×Q₈×C₈₉	—	4	2	no	—
84	C₂×Q₈×C₈₉	—	4	2	no	—
85	C₂×Q₈×C₈₉	C₂²×C₄	1	4	yes	—
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	—	2	2	no	—
92	C₂²×C₄×C₈₉	—	2	4	no	—
93	C₂²×C₄×C₈₉	—	4	4	no	—
94	C₂²×C₄×C₈₉	—	4	4	no	—
95	C₂²×C₄×C₈₉	—	4	4	no	—
96	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
97	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
98	C₂²×C₄×C₈₉	SD₁₆	1	4	yes	—
99	C₂²×C₄×C₈₉	Q₁₆	1	4	yes	—
100	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₈₉	—	2	4	no	—
109	C₂×C₈×C₈₉	—	2	4	no	—
110	C₂×C₈×C₈₉	—	2	4	no	—
111	C₄⋊C₄×C₈₉	—	2	4	no	—
112	C₄⋊C₄×C₈₉	—	2	4	no	—
113	C₄⋊C₄×C₈₉	—	2	4	no	—
114	C₄⋊C₄×C₈₉	—	2	8	no	—
115	C₄⋊C₄×C₈₉	—	2	8	no	—
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	2	8	no	—
118	C₄⋊C₄×C₈₉	—	4	4	no	—
119	C₄⋊C₄×C₈₉	—	4	4	no	—
120	C₄⋊C₄×C₈₉	—	4	4	no	—
121	C₄⋊C₄×C₈₉	—	4	4	no	—
122	C₄⋊C₄×C₈₉	—	4	4	no	—
123	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
124	C₂²⋊C₄×C₈₉	—	2	8	no	—
125	C₄²×C₈₉	—	2	8	no	—
126	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
127	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
128	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
129	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
130	C₄²×C₈₉	C₂×D₄	1	2	yes	—
131	C₄²×C₈₉	C₂×D₄	1	2	yes	—
132	C₂³×C₈₉	—	2	1	no	—
133	C₂³×C₈₉	—	2	1	no	—
134	C₂³×C₈₉	—	2	1	no	—
135	C₂³×C₈₉	—	2	1	no	—
136	C₂³×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
137	C₂³×C₈₉	C₂×SD₁₆	1	2	yes	—
138	C₂³×C₈₉	C₂×Q₁₆	1	2	yes	—
139	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	Q₈×C₈₉	—	2	8	no	—
141	Q₈×C₈₉	—	4	4	no	—
142	Q₈×C₈₉	—	4	8	no	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	1	no	—
145	C₂×C₄×C₈₉	—	2	1	no	—
146	C₂×C₄×C₈₉	—	2	1	no	—
147	C₂×C₄×C₈₉	—	2	4	no	—
148	C₂×C₄×C₈₉	—	2	4	no	—
149	C₂×C₄×C₈₉	—	2	8	no	—
150	C₂×C₄×C₈₉	—	2	4	no	—
151	C₂×C₄×C₈₉	—	2	4	no	—
152	C₂×C₄×C₈₉	—	4	2	no	—
153	C₂×C₄×C₈₉	—	4	2	no	—
154	C₂×C₄×C₈₉	—	4	2	no	—
155	C₂×C₄×C₈₉	—	4	2	no	—
156	C₂×C₄×C₈₉	—	4	4	no	—
157	C₂×C₄×C₈₉	—	4	4	no	—
158	C₂×C₄×C₈₉	—	4	4	no	—
159	C₂×C₄×C₈₉	—	8	8	no	—
160	C₂×C₄×C₈₉	#103	1	16	yes	complemented
161	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
167	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
168	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
169	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
170	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
171	C₂×C₄×C₈₉	#137	1	2	yes	—
172	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
175	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
176	C₈×C₈₉	—	4	4	no	—
177	C₂²×C₈₉	—	2	2	no	—
178	C₂²×C₈₉	—	2	2	no	—
179	C₂²×C₈₉	—	2	2	no	—
180	C₂²×C₈₉	—	2	2	no	—
181	C₂²×C₈₉	—	2	2	no	—
182	C₂²×C₈₉	—	2	2	no	—
183	C₂²×C₈₉	C₂×#103	1	4	yes	—
184	C₂²×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₈₉	#535	1	1	yes	characteristic
186	C₂²×C₈₉	#536	1	1	yes	characteristic
187	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
188	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
189	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
191	C₄×C₈₉	—	2	2	no	—
192	C₄×C₈₉	—	2	2	no	—
193	C₄×C₈₉	—	4	4	no	—
194	C₄×C₈₉	—	4	4	no	—
195	C₄×C₈₉	—	8	4	no	—
196	C₄×C₈₉	C₂×#103	1	4	yes	—
197	C₄×C₈₉	C₂×#103	1	4	yes	—
198	C₄×C₈₉	#417	1	4	yes	—
199	C₄×C₈₉	#520	1	2	yes	—
200	C₄×C₈₉	#537	1	2	yes	—
201	#21860	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₈₉	—	2	4	no	—
203	C₂×C₈₉	#2311	1	1	yes	characteristic
204	C₂×C₈₉	C₂²×#103	1	1	yes	characteristic
205	C₂×C₈₉	C₂×#417	1	1	yes	characteristic
206	C₂×C₈₉	#3208	1	1	yes	characteristic
207	C₂×C₈₉	#3209	1	1	yes	characteristic
208	C₂×C₈₉	#3229	1	1	yes	characteristic
209	C₂×C₈₉	#3230	1	1	yes	characteristic
210	C₂×#533	C₂×C₈₉	1	1	yes	characteristic
211	C₂×#431	C₂×C₈₉	1	1	yes	characteristic
212	C₂×#384	C₂×C₈₉	1	1	yes	characteristic
213	#1990	C₂×C₈₉	1	2	yes	complemented
214	#1956	C₂×C₈₉	1	2	yes	complemented
215	#1872	C₂×C₈₉	1	4	yes	—
216	#1803	C₂×C₈₉	1	4	yes	complemented
217	C₈₉	#21860	1	1	yes	characteristic, direct product factor, Sylow
218	#533	—	2	4	no	—
219	#533	C₄×C₈₉	1	8	yes	complemented
220	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
221	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
222	C₂×#125	—	2	1	no	—
223	C₂×C₂²⋊Q₈	—	2	1	no	—
224	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
225	C₂×C₄×Q₈	—	2	1	no	—
226	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
227	C₂×#117	—	2	1	no	—
228	C₂²×C₄⋊C₄	—	2	1	no	—
229	C₂²×C₄⋊C₄	—	2	1	no	—
230	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
231	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
232	#431	—	2	4	no	—
233	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
234	C₂×#103	C₂²×C₈₉	1	8	yes	—
235	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
237	#384	—	2	1	no	—
238	#384	—	2	1	no	—
239	#384	C₂²×C₈₉	1	2	yes	—
240	#384	C₂²×C₈₉	1	2	yes	—
241	#340	C₂²×C₈₉	1	8	yes	—
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
244	C₂³×C₄	—	2	2	no	—
245	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
246	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
247	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
249	C₄⋊Q₈	—	2	2	no	—
250	C₄⋊Q₈	—	2	2	no	—
251	C₄⋊Q₈	C₂×C₄×C₈₉	1	4	yes	complemented
252	C₄⋊Q₈	C₂×C₄×C₈₉	1	4	yes	complemented
253	#125	—	2	8	no	—
254	C₂²⋊Q₈	—	2	8	no	—
255	C₂²⋊Q₈	—	2	4	no	—
256	C₂²⋊Q₈	C₂×C₄×C₈₉	1	8	yes	—
257	C₄×Q₈	—	2	8	no	—
258	C₄×Q₈	—	2	4	no	—
259	C₄×Q₈	C₂×C₄×C₈₉	1	8	yes	complemented
260	#117	—	2	8	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	—	4	4	no	—
272	C₂×C₄⋊C₄	—	4	4	no	—
273	C₂×C₄⋊C₄	—	4	4	no	—
274	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
275	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₄²	—	2	2	no	—
278	C₂×C₄²	C₂×C₄×C₈₉	1	4	yes	—
279	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
280	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
281	C₂×C₄²	D₄×C₈₉	1	4	yes	—
282	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
284	C₄⋊C₈	—	2	4	no	—
285	#103	—	4	8	no	—
286	C₂²⋊C₈	—	2	4	no	—
287	C₄×C₈	—	2	4	no	—
288	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
289	C₂×Q₈	—	2	4	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	4	no	—
292	C₂×Q₈	—	2	4	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂×Q₈	—	4	2	no	—
295	C₂×Q₈	C₂²×C₄×C₈₉	1	4	yes	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	4	no	—
303	C₂²×C₄	—	4	4	no	—
304	C₂²×C₄	—	4	4	no	—
305	C₂²×C₄	—	4	4	no	—
306	C₂²×C₄	C₂²⋊C₄×C₈₉	1	4	yes	—
307	C₂²×C₄	C₂²⋊C₄×C₈₉	1	4	yes	—
308	C₂²×C₄	SD₁₆×C₈₉	1	4	yes	—
309	C₂²×C₄	Q₁₆×C₈₉	1	4	yes	—
310	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
311	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
312	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
317	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	2	4	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	2	8	no	—
325	C₄⋊C₄	—	2	8	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	8	no	—
328	C₄⋊C₄	—	4	4	no	—
329	C₄⋊C₄	—	4	4	no	—
330	C₄⋊C₄	—	4	4	no	—
331	C₄⋊C₄	—	4	4	no	—
332	C₄⋊C₄	—	4	4	no	—
333	C₄⋊C₄	C₂²×C₄×C₈₉	1	4	yes	—
334	C₂²⋊C₄	—	2	8	no	—
335	C₄²	—	2	8	no	—
336	C₄²	C₂²⋊C₄×C₈₉	1	4	yes	—
337	C₄²	C₂²⋊C₄×C₈₉	1	4	yes	—
338	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
339	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
340	C₄²	C₂×D₄×C₈₉	1	2	yes	—
341	C₄²	C₂×D₄×C₈₉	1	2	yes	—
342	C₂³	—	2	1	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	C₂×C₂²⋊C₄×C₈₉	1	2	yes	—
347	C₂³	C₂×SD₁₆×C₈₉	1	2	yes	—
348	C₂³	C₂×Q₁₆×C₈₉	1	2	yes	—
349	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
350	Q₈	—	2	8	no	—
351	Q₈	—	4	8	no	—
352	Q₈	—	4	4	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	8	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	4	no	—
369	C₂×C₄	—	8	8	no	—
370	C₂×C₄	#103×C₈₉	1	16	yes	complemented
371	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	2	yes	—
372	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	2	yes	—
373	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	2	yes	—
374	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	2	yes	—
375	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	2	yes	—
376	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
377	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
378	C₂×C₄	C₂×Q₁₆×C₈₉	1	2	yes	—
379	C₂×C₄	C₂×Q₁₆×C₈₉	1	2	yes	—
380	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
381	C₂×C₄	#137×C₈₉	1	2	yes	—
382	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
383	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
384	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
385	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
386	C₈	—	4	4	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	C₂×#103×C₈₉	1	4	yes	—
394	C₂²	C₂²×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
395	C₂²	#535×C₈₉	1	1	yes	characteristic
396	C₂²	#536×C₈₉	1	1	yes	characteristic
397	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
398	C₂²	C₂²×Q₁₆×C₈₉	1	1	yes	characteristic
399	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
400	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
401	C₄	—	2	2	no	—
402	C₄	—	2	2	no	—
403	C₄	—	4	4	no	—
404	C₄	—	4	4	no	—
405	C₄	—	8	4	no	—
406	C₄	C₂×#103×C₈₉	1	4	yes	—
407	C₄	C₂×#103×C₈₉	1	4	yes	—
408	C₄	#417×C₈₉	1	4	yes	—
409	C₄	#520×C₈₉	1	2	yes	—
410	C₄	#537×C₈₉	1	2	yes	—
411	C₂	—	2	4	no	—
412	C₂	C₈₉×#2311	1	1	yes	characteristic
413	C₂	C₂²×#103×C₈₉	1	1	yes	characteristic
414	C₂	C₂×#417×C₈₉	1	1	yes	characteristic
415	C₂	C₈₉×#3208	1	1	yes	characteristic
416	C₂	C₈₉×#3209	1	1	yes	characteristic
417	C₂	C₈₉×#3229	1	1	yes	characteristic
418	C₂	C₈₉×#3230	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈₉×#1990	C₂	semidirect
5. C₈₉×#1956	C₂	semidirect
7. C₈₉×#1803	C₂	semidirect
9. #533×C₈₉	C₄	semidirect
41. C₄⋊Q₈×C₈₉	C₂×C₄	semidirect
42. C₄⋊Q₈×C₈₉	C₂×C₄	semidirect
49. C₄×Q₈×C₈₉	C₂×C₄	semidirect
160. C₂×C₄×C₈₉	#103	semidirect
201. #21860	C₈₉	direct (N × H)
213. #1990	C₂×C₈₉	semidirect
214. #1956	C₂×C₈₉	semidirect
216. #1803	C₂×C₈₉	semidirect
217. C₈₉	#21860	direct (N × H)
219. #533	C₄×C₈₉	semidirect
251. C₄⋊Q₈	C₂×C₄×C₈₉	semidirect
252. C₄⋊Q₈	C₂×C₄×C₈₉	semidirect
259. C₄×Q₈	C₂×C₄×C₈₉	semidirect
370. C₂×C₄	#103×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₈₉	2
C₂×#431×C₈₉	2
C₂×#384×C₈₉	2
C₈₉×#1990	2
C₈₉×#1956	2
C₈₉×#1872	2
C₈₉×#1803	2
#21860	89

Subgroups — C89×#21860 #36399708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#21860 #36399708