Subgroups — C₈₉×#35319 #36413167

Order 22784.

2026 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₈₉	C₂	1	2	yes	maximal
2	C₈₉×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2857	C₂	1	2	yes	maximal
5	C₈₉×#2642	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2621	C₂	1	2	yes	maximal
7	C₈₉×#2506	C₂	1	8	yes	maximal, complemented
8	C₈₉×#2496	C₂	1	8	yes	maximal
9	C₈₉×#2327	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊Q₈×C₈₉	C₂	1	2	yes	maximal, complemented
11	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#541×C₈₉	—	2	16	no	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
16	C₂×#126×C₈₉	C₂²	1	8	yes	—
17	C₂×#125×C₈₉	C₂²	1	2	yes	—
18	C₂×#125×C₈₉	C₂²	1	8	yes	—
19	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	—
21	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
24	#400×C₈₉	C₂²	1	16	yes	—
25	#398×C₈₉	C₂²	1	8	yes	—
26	#397×C₈₉	C₂²	1	8	yes	—
27	#391×C₈₉	C₂²	1	4	yes	—
28	#391×C₈₉	C₂²	1	4	yes	—
29	#389×C₈₉	C₂²	1	8	yes	—
30	#389×C₈₉	C₂²	1	8	yes	—
31	#389×C₈₉	C₂²	1	8	yes	—
32	#387×C₈₉	C₂²	1	16	yes	—
33	#384×C₈₉	C₂²	1	2	yes	—
34	#384×C₈₉	C₂²	1	8	yes	—
35	#382×C₈₉	C₂²	1	2	yes	—
36	#380×C₈₉	C₂²	1	8	yes	—
37	C₄×C₄⋊C₄×C₈₉	C₂²	1	8	yes	—
38	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
39	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
40	#376×C₈₉	C₂²	1	1	yes	characteristic
41	#376×C₈₉	C₂²	1	1	yes	characteristic
42	C₄³×C₈₉	C₂²	1	1	yes	characteristic
43	C₄³×C₈₉	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
45	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
46	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₈₉	—	2	8	no	—
48	C₄⋊Q₈×C₈₉	—	2	8	no	—
49	C₄⋊Q₈×C₈₉	—	2	8	no	—
50	#126×C₈₉	—	2	32	no	—
51	#125×C₈₉	—	2	32	no	—
52	#125×C₈₉	—	2	8	no	—
53	C₂²⋊Q₈×C₈₉	—	2	16	no	—
54	C₄×Q₈×C₈₉	—	2	16	no	—
55	#117×C₈₉	—	2	4	no	—
56	#117×C₈₉	D₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
59	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
66	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
69	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
70	C₂×C₄²×C₈₉	—	2	4	no	—
71	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
73	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
74	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
76	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	#95×C₈₉	C₂³	1	4	yes	—
82	#95×C₈₉	C₂³	1	4	yes	—
83	#95×C₈₉	C₂³	1	16	yes	—
84	#95×C₈₉	C₂³	1	16	yes	—
85	#95×C₈₉	C₂³	1	8	yes	—
86	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
87	C₂×Q₈×C₈₉	—	2	8	no	—
88	C₂×Q₈×C₈₉	—	2	8	no	—
89	C₂×Q₈×C₈₉	—	2	8	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	—	2	8	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
95	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
96	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
97	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
98	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
99	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
100	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
101	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
105	C₄⋊C₄×C₈₉	—	2	8	no	—
106	C₄⋊C₄×C₈₉	—	2	16	no	—
107	C₄⋊C₄×C₈₉	—	2	16	no	—
108	C₄⋊C₄×C₈₉	—	2	16	no	—
109	C₄⋊C₄×C₈₉	—	2	16	no	—
110	C₄⋊C₄×C₈₉	—	2	4	no	—
111	C₄⋊C₄×C₈₉	—	2	16	no	—
112	C₄⋊C₄×C₈₉	—	2	16	no	—
113	C₄⋊C₄×C₈₉	—	2	16	no	—
114	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
115	C₂²⋊C₄×C₈₉	—	2	16	no	—
116	C₂²⋊C₄×C₈₉	—	2	4	no	—
117	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
118	C₄²×C₈₉	—	2	16	no	—
119	C₄²×C₈₉	—	2	4	no	—
120	C₄²×C₈₉	—	2	2	no	—
121	C₄²×C₈₉	—	2	2	no	—
122	C₄²×C₈₉	—	2	8	no	—
123	C₄²×C₈₉	—	2	8	no	—
124	C₄²×C₈₉	C₂×D₄	1	4	yes	—
125	C₄²×C₈₉	C₂×D₄	1	4	yes	—
126	C₄²×C₈₉	#41	1	4	yes	—
127	C₄²×C₈₉	#41	1	4	yes	—
128	C₂³×C₈₉	—	2	1	no	—
129	C₂³×C₈₉	—	2	1	no	—
130	C₂³×C₈₉	—	2	4	no	—
131	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
132	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
133	Q₈×C₈₉	—	4	16	no	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	4	no	—
137	C₂×C₄×C₈₉	—	2	4	no	—
138	C₂×C₄×C₈₉	—	2	4	no	—
139	C₂×C₄×C₈₉	—	2	8	no	—
140	C₂×C₄×C₈₉	—	2	8	no	—
141	C₂×C₄×C₈₉	—	2	8	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	8	no	—
145	C₂×C₄×C₈₉	—	2	8	no	—
146	C₂×C₄×C₈₉	—	2	4	no	—
147	C₂×C₄×C₈₉	—	2	16	no	—
148	C₂×C₄×C₈₉	—	2	4	no	—
149	C₂×C₄×C₈₉	—	4	8	no	—
150	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
151	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
152	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
153	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
154	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
155	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
156	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
157	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
158	C₂×C₄×C₈₉	2₋¹⁺⁴	1	8	yes	—
159	C₂²×C₈₉	—	2	2	no	—
160	C₂²×C₈₉	—	2	4	no	—
161	C₂²×C₈₉	—	4	2	no	—
162	C₂²×C₈₉	—	4	2	no	—
163	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
164	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
165	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
166	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	4	yes	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	4	no	—
169	C₄×C₈₉	—	2	4	no	—
170	C₄×C₈₉	—	2	4	no	—
171	C₄×C₈₉	—	4	8	no	—
172	C₄×C₈₉	—	4	2	no	—
173	C₄×C₈₉	—	4	8	no	—
174	C₄×C₈₉	—	4	2	no	—
175	C₄×C₈₉	#532	1	4	yes	—
176	C₄×C₈₉	#541	1	8	yes	complemented
177	#35319	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₈₉	—	4	2	no	—
179	C₂×C₈₉	C₂×#532	1	1	yes	characteristic
180	C₂×C₈₉	C₂×#541	1	2	yes	—
181	C₂×C₈₉	#3499	1	2	yes	—
182	C₂×C₈₉	#3509	1	2	yes	—
183	C₂×#541	C₂×C₈₉	1	2	yes	—
184	#2868	C₂×C₈₉	1	1	yes	characteristic, complemented
185	#2859	C₂×C₈₉	1	1	yes	characteristic, complemented
186	#2857	C₂×C₈₉	1	2	yes	—
187	#2642	C₂×C₈₉	1	2	yes	complemented
188	#2621	C₂×C₈₉	1	2	yes	—
189	#2506	C₂×C₈₉	1	8	yes	complemented
190	#2496	C₂×C₈₉	1	8	yes	—
191	#2327	C₂×C₈₉	1	2	yes	complemented
192	C₄×C₄⋊Q₈	C₂×C₈₉	1	2	yes	complemented
193	C₄×#117	C₂×C₈₉	1	1	yes	characteristic
194	C₈₉	#35319	1	1	yes	characteristic, direct product factor, Sylow
195	#541	—	2	16	no	—
196	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
197	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
198	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
199	C₂×#126	C₂²×C₈₉	1	8	yes	—
200	C₂×#125	C₂²×C₈₉	1	2	yes	—
201	C₂×#125	C₂²×C₈₉	1	8	yes	—
202	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
203	C₂×C₄×Q₈	C₂²×C₈₉	1	4	yes	—
204	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
205	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
206	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
207	#400	C₂²×C₈₉	1	16	yes	—
208	#398	C₂²×C₈₉	1	8	yes	—
209	#397	C₂²×C₈₉	1	8	yes	—
210	#391	C₂²×C₈₉	1	4	yes	—
211	#391	C₂²×C₈₉	1	4	yes	—
212	#389	C₂²×C₈₉	1	8	yes	—
213	#389	C₂²×C₈₉	1	8	yes	—
214	#389	C₂²×C₈₉	1	8	yes	—
215	#387	C₂²×C₈₉	1	16	yes	—
216	#384	C₂²×C₈₉	1	2	yes	—
217	#384	C₂²×C₈₉	1	8	yes	—
218	#382	C₂²×C₈₉	1	2	yes	—
219	#380	C₂²×C₈₉	1	8	yes	—
220	C₄×C₄⋊C₄	C₂²×C₈₉	1	8	yes	—
221	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
222	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
223	#376	C₂²×C₈₉	1	1	yes	characteristic
224	#376	C₂²×C₈₉	1	1	yes	characteristic
225	C₄³	C₂²×C₈₉	1	1	yes	characteristic
226	C₄³	C₂²×C₈₉	1	1	yes	characteristic
227	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
228	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
229	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
230	C₄⋊Q₈	—	2	8	no	—
231	C₄⋊Q₈	—	2	8	no	—
232	C₄⋊Q₈	—	2	8	no	—
233	#126	—	2	32	no	—
234	#125	—	2	32	no	—
235	#125	—	2	8	no	—
236	C₂²⋊Q₈	—	2	16	no	—
237	C₄×Q₈	—	2	16	no	—
238	#117	—	2	4	no	—
239	#117	D₄×C₈₉	1	8	yes	—
240	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
241	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
242	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
243	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
244	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
245	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
246	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
247	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
248	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
249	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
250	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
251	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
252	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
253	C₂×C₄²	—	2	4	no	—
254	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
256	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
257	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
258	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
259	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
260	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
261	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
264	#95	C₂³×C₈₉	1	4	yes	—
265	#95	C₂³×C₈₉	1	4	yes	—
266	#95	C₂³×C₈₉	1	16	yes	—
267	#95	C₂³×C₈₉	1	16	yes	—
268	#95	C₂³×C₈₉	1	8	yes	—
269	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
270	C₂×Q₈	—	2	8	no	—
271	C₂×Q₈	—	2	8	no	—
272	C₂×Q₈	—	2	8	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	2	8	no	—
277	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
278	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
279	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
280	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
281	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
282	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
283	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
284	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
285	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
286	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
287	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
288	C₄⋊C₄	—	2	8	no	—
289	C₄⋊C₄	—	2	16	no	—
290	C₄⋊C₄	—	2	16	no	—
291	C₄⋊C₄	—	2	16	no	—
292	C₄⋊C₄	—	2	16	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	16	no	—
295	C₄⋊C₄	—	2	16	no	—
296	C₄⋊C₄	—	2	16	no	—
297	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
298	C₂²⋊C₄	—	2	16	no	—
299	C₂²⋊C₄	—	2	4	no	—
300	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
301	C₄²	—	2	16	no	—
302	C₄²	—	2	4	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	2	8	no	—
306	C₄²	—	2	8	no	—
307	C₄²	C₂×D₄×C₈₉	1	4	yes	—
308	C₄²	C₂×D₄×C₈₉	1	4	yes	—
309	C₄²	#41×C₈₉	1	4	yes	—
310	C₄²	#41×C₈₉	1	4	yes	—
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	2	4	no	—
314	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
315	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
316	Q₈	—	4	16	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	8	no	—
323	C₂×C₄	—	2	8	no	—
324	C₂×C₄	—	2	8	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	8	no	—
328	C₂×C₄	—	2	8	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	16	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	4	8	no	—
333	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
334	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
335	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
336	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
337	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
338	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
339	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
340	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
341	C₂×C₄	2₋¹⁺⁴×C₈₉	1	8	yes	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	4	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
347	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
348	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
349	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	4	yes	—
350	C₄	—	2	2	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	4	no	—
353	C₄	—	2	4	no	—
354	C₄	—	4	8	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	8	no	—
357	C₄	—	4	2	no	—
358	C₄	#532×C₈₉	1	4	yes	—
359	C₄	#541×C₈₉	1	8	yes	complemented
360	C₂	—	4	2	no	—
361	C₂	C₂×#532×C₈₉	1	1	yes	characteristic
362	C₂	C₂×#541×C₈₉	1	2	yes	—
363	C₂	C₈₉×#3499	1	2	yes	—
364	C₂	C₈₉×#3509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁴×C₈₉	C₂⁴
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. C₈₉×#2868	C₂	semidirect
3. C₈₉×#2859	C₂	semidirect
5. C₈₉×#2642	C₂	semidirect
7. C₈₉×#2506	C₂	semidirect
9. C₈₉×#2327	C₂	semidirect
10. C₄×C₄⋊Q₈×C₈₉	C₂	semidirect
176. C₄×C₈₉	#541	semidirect
177. #35319	C₈₉	direct (N × H)
184. #2868	C₂×C₈₉	semidirect
185. #2859	C₂×C₈₉	semidirect
187. #2642	C₂×C₈₉	semidirect
189. #2506	C₂×C₈₉	semidirect
191. #2327	C₂×C₈₉	semidirect
192. C₄×C₄⋊Q₈	C₂×C₈₉	semidirect
194. C₈₉	#35319	direct (N × H)
359. C₄	#541×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#541×C₈₉	2
C₈₉×#2868	2
C₈₉×#2859	2
C₈₉×#2857	2
C₈₉×#2642	2
C₈₉×#2621	2
C₈₉×#2506	2
C₈₉×#2496	2
C₈₉×#2327	2
C₄×C₄⋊Q₈×C₈₉	2
C₄×#117×C₈₉	2
#35319	89

Subgroups — C89×#35319 #36413167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35319 #36413167