Subgroups — C₈₉×#60705 #36438553

Order 22784.

1846 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3455	C₂	1	1	yes	maximal, characteristic
2	C₂×#541×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3178	C₂	1	8	yes	maximal, complemented
4	C₈₉×#3177	C₂	1	4	yes	maximal, complemented
5	C₈₉×#3174	C₂	1	4	yes	maximal, complemented
6	C₈₉×#3166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#3152	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3142	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#3056	C₂	1	4	yes	maximal, complemented
11	C₄×#137×C₈₉	C₂	1	4	yes	maximal, complemented
12	C₈₉×#2873	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#137×C₈₉	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
15	#542×C₈₉	C₂²	1	4	yes	—
16	#541×C₈₉	—	2	4	no	—
17	#541×C₈₉	C₂²	1	4	yes	—
18	#541×C₈₉	C₂²	1	2	yes	—
19	#541×C₈₉	C₂²	1	4	yes	—
20	#540×C₈₉	C₂²	1	2	yes	—
21	#539×C₈₉	—	2	2	no	—
22	#538×C₈₉	—	2	2	no	—
23	#533×C₈₉	C₂²	1	1	yes	characteristic
24	#532×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
26	#529×C₈₉	—	2	4	no	—
27	C₂×#126×C₈₉	—	2	2	no	—
28	C₂×#125×C₈₉	—	2	2	no	—
29	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
30	#525×C₈₉	C₂²	1	2	yes	—
31	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
32	C₄×#41×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
34	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
35	#501×C₈₉	C₂²	1	4	yes	complemented
36	#499×C₈₉	C₂²	1	4	yes	complemented
37	#497×C₈₉	C₂²	1	4	yes	complemented
38	#494×C₈₉	C₂²	1	2	yes	complemented
39	#492×C₈₉	C₂²	1	2	yes	complemented
40	#491×C₈₉	C₂²	1	4	yes	complemented
41	#491×C₈₉	C₂²	1	2	yes	complemented
42	#490×C₈₉	C₂²	1	2	yes	complemented
43	#488×C₈₉	C₂²	1	4	yes	complemented
44	#487×C₈₉	C₂²	1	4	yes	complemented
45	#470×C₈₉	C₂²	1	8	yes	—
46	#468×C₈₉	C₂²	1	8	yes	—
47	#467×C₈₉	C₂²	1	8	yes	—
48	#465×C₈₉	C₂²	1	8	yes	—
49	#441×C₈₉	C₂²	1	8	yes	complemented
50	#440×C₈₉	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₈₉	C₂²	1	8	yes	complemented
52	C₄×SD₁₆×C₈₉	C₂²	1	8	yes	complemented
53	#428×C₈₉	C₂²	1	4	yes	—
54	#419×C₈₉	C₂²	1	4	yes	—
55	#417×C₈₉	C₂²	1	4	yes	—
56	#405×C₈₉	C₂²	1	4	yes	—
57	#405×C₈₉	C₂²	1	4	yes	—
58	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
59	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
60	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
61	C₂×#41×C₈₉	C₂³	1	2	yes	—
62	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
63	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
64	#137×C₈₉	—	2	16	no	—
65	C₂×Q₁₆×C₈₉	C₂³	1	8	yes	—
66	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
67	C₂×M₁₆×C₈₉	—	2	8	no	—
68	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
69	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
70	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
71	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
72	C₄⋊Q₈×C₈₉	—	2	2	no	—
73	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
77	#127×C₈₉	C₂³	1	1	yes	characteristic
78	#126×C₈₉	—	2	4	no	—
79	#126×C₈₉	—	2	8	no	—
80	#126×C₈₉	—	2	8	no	—
81	#126×C₈₉	—	2	4	no	—
82	#125×C₈₉	—	2	8	no	—
83	#125×C₈₉	—	2	8	no	—
84	#125×C₈₉	—	2	4	no	—
85	#125×C₈₉	—	2	2	no	—
86	#125×C₈₉	C₂³	1	2	yes	—
87	#125×C₈₉	C₂³	1	2	yes	—
88	#125×C₈₉	C₂³	1	2	yes	—
89	#124×C₈₉	—	2	4	no	—
90	#124×C₈₉	C₂³	1	2	yes	—
91	#123×C₈₉	—	2	2	no	—
92	#123×C₈₉	—	2	4	no	—
93	#123×C₈₉	—	2	4	no	—
94	C₂²⋊Q₈×C₈₉	—	2	4	no	—
95	C₂²⋊Q₈×C₈₉	—	2	2	no	—
96	C₂²⋊Q₈×C₈₉	—	2	4	no	—
97	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
98	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
99	#121×C₈₉	—	2	2	no	—
100	#121×C₈₉	—	2	4	no	—
101	#121×C₈₉	C₂³	1	4	yes	—
102	#120×C₈₉	—	2	2	no	—
103	C₄×Q₈×C₈₉	—	2	4	no	—
104	C₄×Q₈×C₈₉	—	2	2	no	—
105	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
106	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
107	C₄×D₄×C₈₉	—	2	4	no	—
108	C₄×D₄×C₈₉	—	2	4	no	—
109	C₄×D₄×C₈₉	—	2	2	no	—
110	C₄×D₄×C₈₉	C₂³	1	4	yes	—
111	#117×C₈₉	—	2	2	no	—
112	#117×C₈₉	D₄	1	4	yes	—
113	#117×C₈₉	C₂³	1	2	yes	—
114	#117×C₈₉	C₂³	1	2	yes	—
115	#117×C₈₉	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
121	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
124	C₂×C₄²×C₈₉	—	2	2	no	—
125	C₂×C₄²×C₈₉	—	2	2	no	—
126	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
128	#107×C₈₉	C₂³	1	8	yes	—
129	#106×C₈₉	C₂³	1	8	yes	—
130	#103×C₈₉	C₂³	1	8	yes	—
131	#103×C₈₉	C₂³	1	8	yes	—
132	#103×C₈₉	C₂³	1	8	yes	—
133	#102×C₈₉	C₂³	1	8	yes	—
134	#97×C₈₉	C₂³	1	4	yes	—
135	#97×C₈₉	C₂³	1	2	yes	—
136	#97×C₈₉	C₂³	1	2	yes	—
137	C₄×C₈×C₈₉	C₂³	1	4	yes	—
138	C₄×C₈×C₈₉	C₂³	1	2	yes	—
139	C₄×C₈×C₈₉	C₂³	1	2	yes	—
140	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
141	#41×C₈₉	—	2	8	no	—
142	C₂×Q₈×C₈₉	—	2	2	no	—
143	C₂×Q₈×C₈₉	—	2	2	no	—
144	C₂×Q₈×C₈₉	—	2	8	no	—
145	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
146	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
147	C₂×D₄×C₈₉	—	2	2	no	—
148	C₂×D₄×C₈₉	—	2	4	no	—
149	C₂×D₄×C₈₉	—	2	2	no	—
150	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₈₉	—	2	4	no	—
152	C₂²×C₄×C₈₉	—	2	2	no	—
153	C₂²×C₄×C₈₉	—	2	2	no	—
154	C₂²×C₄×C₈₉	—	2	2	no	—
155	C₂²×C₄×C₈₉	—	2	4	no	—
156	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₈₉	—	2	16	no	—
163	SD₁₆×C₈₉	—	2	16	no	—
164	M₁₆×C₈₉	—	2	4	no	—
165	M₁₆×C₈₉	—	2	8	no	—
166	M₁₆×C₈₉	#41	1	8	yes	—
167	C₂×C₈×C₈₉	—	2	8	no	—
168	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
169	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
170	C₄⋊C₄×C₈₉	—	2	8	no	—
171	C₄⋊C₄×C₈₉	—	2	8	no	—
172	C₄⋊C₄×C₈₉	—	2	4	no	—
173	C₄⋊C₄×C₈₉	—	2	4	no	—
174	C₄⋊C₄×C₈₉	—	2	4	no	—
175	C₄⋊C₄×C₈₉	—	2	4	no	—
176	C₄⋊C₄×C₈₉	—	2	2	no	—
177	C₄⋊C₄×C₈₉	—	2	4	no	—
178	C₄⋊C₄×C₈₉	—	2	2	no	—
179	C₄⋊C₄×C₈₉	—	2	4	no	—
180	C₄⋊C₄×C₈₉	—	2	4	no	—
181	C₄⋊C₄×C₈₉	—	2	2	no	—
182	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
183	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
186	C₂²⋊C₄×C₈₉	—	2	8	no	—
187	C₂²⋊C₄×C₈₉	—	2	2	no	—
188	C₂²⋊C₄×C₈₉	—	2	4	no	—
189	C₂²⋊C₄×C₈₉	—	2	4	no	—
190	C₂²⋊C₄×C₈₉	—	2	4	no	—
191	C₂²⋊C₄×C₈₉	—	2	4	no	—
192	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
193	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
194	C₄²×C₈₉	—	2	2	no	—
195	C₄²×C₈₉	—	2	4	no	—
196	C₄²×C₈₉	—	2	8	no	—
197	C₄²×C₈₉	C₂×D₄	1	2	yes	—
198	C₄²×C₈₉	C₂×D₄	1	2	yes	—
199	C₄²×C₈₉	C₂⁴	1	2	yes	—
200	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
201	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
202	C₂³×C₈₉	—	2	2	no	—
203	C₂³×C₈₉	—	2	2	no	—
204	C₂³×C₈₉	—	4	2	no	—
205	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
206	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
207	Q₈×C₈₉	—	2	8	no	—
208	Q₈×C₈₉	—	2	4	no	—
209	Q₈×C₈₉	—	4	4	no	—
210	Q₈×C₈₉	—	8	2	no	—
211	D₄×C₈₉	—	2	4	no	—
212	D₄×C₈₉	—	4	4	no	—
213	D₄×C₈₉	—	8	2	no	—
214	D₄×C₈₉	—	8	4	no	—
215	C₂×C₄×C₈₉	—	2	2	no	—
216	C₂×C₄×C₈₉	—	2	4	no	—
217	C₂×C₄×C₈₉	—	2	2	no	—
218	C₂×C₄×C₈₉	—	2	2	no	—
219	C₂×C₄×C₈₉	—	2	2	no	—
220	C₂×C₄×C₈₉	—	2	4	no	—
221	C₂×C₄×C₈₉	—	2	4	no	—
222	C₂×C₄×C₈₉	—	2	4	no	—
223	C₂×C₄×C₈₉	—	4	2	no	—
224	C₂×C₄×C₈₉	—	4	2	no	—
225	C₂×C₄×C₈₉	—	4	4	no	—
226	C₂×C₄×C₈₉	—	4	4	no	—
227	C₂×C₄×C₈₉	—	4	4	no	—
228	C₂×C₄×C₈₉	—	8	4	no	—
229	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
230	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
235	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
236	C₈×C₈₉	—	2	4	no	—
237	C₈×C₈₉	C₂×#41	1	8	yes	—
238	C₂²×C₈₉	—	2	2	no	—
239	C₂²×C₈₉	—	2	4	no	—
240	C₂²×C₈₉	—	2	2	no	—
241	C₂²×C₈₉	—	4	4	no	—
242	C₂²×C₈₉	—	4	2	no	—
243	C₂²×C₈₉	—	8	2	no	—
244	C₂²×C₈₉	#532	1	2	yes	—
245	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₈₉	—	2	4	no	—
247	C₄×C₈₉	—	2	2	no	—
248	C₄×C₈₉	—	2	2	no	—
249	C₄×C₈₉	—	2	2	no	—
250	C₄×C₈₉	—	4	4	no	—
251	C₄×C₈₉	—	4	4	no	—
252	C₄×C₈₉	—	4	2	no	—
253	C₄×C₈₉	—	4	4	no	—
254	C₄×C₈₉	#532	1	2	yes	—
255	C₄×C₈₉	C₂²×#41	1	2	yes	—
256	#60705	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂×C₈₉	—	2	2	no	—
258	C₂×C₈₉	—	4	2	no	—
259	C₂×C₈₉	—	8	2	no	—
260	C₂×C₈₉	C₂×#532	1	1	yes	characteristic
261	C₂×C₈₉	#3586	1	2	yes	—
262	#3455	C₂×C₈₉	1	1	yes	characteristic
263	C₂×#541	C₂×C₈₉	1	1	yes	characteristic, complemented
264	#3178	C₂×C₈₉	1	8	yes	complemented
265	#3177	C₂×C₈₉	1	4	yes	complemented
266	#3174	C₂×C₈₉	1	4	yes	complemented
267	#3166	C₂×C₈₉	1	1	yes	characteristic, complemented
268	#3152	C₂×C₈₉	1	1	yes	characteristic, complemented
269	#3142	C₂×C₈₉	1	1	yes	characteristic, complemented
270	#3139	C₂×C₈₉	1	1	yes	characteristic, complemented
271	#3056	C₂×C₈₉	1	4	yes	complemented
272	C₄×#137	C₂×C₈₉	1	4	yes	complemented
273	#2873	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₈₉	#60705	1	1	yes	characteristic, direct product factor, Sylow
275	C₂×#137	C₂²×C₈₉	1	4	yes	—
276	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
277	#542	C₂²×C₈₉	1	4	yes	—
278	#541	—	2	4	no	—
279	#541	C₂²×C₈₉	1	4	yes	—
280	#541	C₂²×C₈₉	1	2	yes	—
281	#541	C₂²×C₈₉	1	4	yes	—
282	#540	C₂²×C₈₉	1	2	yes	—
283	#539	—	2	2	no	—
284	#538	—	2	2	no	—
285	#533	C₂²×C₈₉	1	1	yes	characteristic
286	#532	C₂²×C₈₉	1	1	yes	characteristic
287	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
288	#529	—	2	4	no	—
289	C₂×#126	—	2	2	no	—
290	C₂×#125	—	2	2	no	—
291	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
292	#525	C₂²×C₈₉	1	2	yes	—
293	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
294	C₄×#41	C₂²×C₈₉	1	2	yes	—
295	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
296	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
297	#501	C₂²×C₈₉	1	4	yes	complemented
298	#499	C₂²×C₈₉	1	4	yes	complemented
299	#497	C₂²×C₈₉	1	4	yes	complemented
300	#494	C₂²×C₈₉	1	2	yes	complemented
301	#492	C₂²×C₈₉	1	2	yes	complemented
302	#491	C₂²×C₈₉	1	4	yes	complemented
303	#491	C₂²×C₈₉	1	2	yes	complemented
304	#490	C₂²×C₈₉	1	2	yes	complemented
305	#488	C₂²×C₈₉	1	4	yes	complemented
306	#487	C₂²×C₈₉	1	4	yes	complemented
307	#470	C₂²×C₈₉	1	8	yes	—
308	#468	C₂²×C₈₉	1	8	yes	—
309	#467	C₂²×C₈₉	1	8	yes	—
310	#465	C₂²×C₈₉	1	8	yes	—
311	#441	C₂²×C₈₉	1	8	yes	complemented
312	#440	C₂²×C₈₉	1	8	yes	complemented
313	C₄×Q₁₆	C₂²×C₈₉	1	8	yes	complemented
314	C₄×SD₁₆	C₂²×C₈₉	1	8	yes	complemented
315	#428	C₂²×C₈₉	1	4	yes	—
316	#419	C₂²×C₈₉	1	4	yes	—
317	#417	C₂²×C₈₉	1	4	yes	—
318	#405	C₂²×C₈₉	1	4	yes	—
319	#405	C₂²×C₈₉	1	4	yes	—
320	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
321	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
322	C₄×M₁₆	C₂²×C₈₉	1	2	yes	—
323	C₂×#41	C₂³×C₈₉	1	2	yes	—
324	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
325	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
326	#137	—	2	16	no	—
327	C₂×Q₁₆	C₂³×C₈₉	1	8	yes	—
328	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
329	C₂×M₁₆	—	2	8	no	—
330	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
331	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
332	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
333	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
334	C₄⋊Q₈	—	2	2	no	—
335	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
336	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
337	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
338	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
339	#127	C₂³×C₈₉	1	1	yes	characteristic
340	#126	—	2	4	no	—
341	#126	—	2	8	no	—
342	#126	—	2	8	no	—
343	#126	—	2	4	no	—
344	#125	—	2	8	no	—
345	#125	—	2	8	no	—
346	#125	—	2	4	no	—
347	#125	—	2	2	no	—
348	#125	C₂³×C₈₉	1	2	yes	—
349	#125	C₂³×C₈₉	1	2	yes	—
350	#125	C₂³×C₈₉	1	2	yes	—
351	#124	—	2	4	no	—
352	#124	C₂³×C₈₉	1	2	yes	—
353	#123	—	2	2	no	—
354	#123	—	2	4	no	—
355	#123	—	2	4	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	2	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
360	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
361	#121	—	2	2	no	—
362	#121	—	2	4	no	—
363	#121	C₂³×C₈₉	1	4	yes	—
364	#120	—	2	2	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×Q₈	—	2	2	no	—
367	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
368	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	4	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	C₂³×C₈₉	1	4	yes	—
373	#117	—	2	2	no	—
374	#117	D₄×C₈₉	1	4	yes	—
375	#117	C₂³×C₈₉	1	2	yes	—
376	#117	C₂³×C₈₉	1	2	yes	—
377	#117	C₂³×C₈₉	1	2	yes	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
386	C₂×C₄²	—	2	2	no	—
387	C₂×C₄²	—	2	2	no	—
388	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
390	#107	C₂³×C₈₉	1	8	yes	—
391	#106	C₂³×C₈₉	1	8	yes	—
392	#103	C₂³×C₈₉	1	8	yes	—
393	#103	C₂³×C₈₉	1	8	yes	—
394	#103	C₂³×C₈₉	1	8	yes	—
395	#102	C₂³×C₈₉	1	8	yes	—
396	#97	C₂³×C₈₉	1	4	yes	—
397	#97	C₂³×C₈₉	1	2	yes	—
398	#97	C₂³×C₈₉	1	2	yes	—
399	C₄×C₈	C₂³×C₈₉	1	4	yes	—
400	C₄×C₈	C₂³×C₈₉	1	2	yes	—
401	C₄×C₈	C₂³×C₈₉	1	2	yes	—
402	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
403	#41	—	2	8	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	8	no	—
407	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
408	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
422	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
424	Q₁₆	—	2	16	no	—
425	SD₁₆	—	2	16	no	—
426	M₁₆	—	2	4	no	—
427	M₁₆	—	2	8	no	—
428	M₁₆	#41×C₈₉	1	8	yes	—
429	C₂×C₈	—	2	8	no	—
430	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
431	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
432	C₄⋊C₄	—	2	8	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
445	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
446	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
447	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
455	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
456	C₄²	—	2	2	no	—
457	C₄²	—	2	4	no	—
458	C₄²	—	2	8	no	—
459	C₄²	C₂×D₄×C₈₉	1	2	yes	—
460	C₄²	C₂×D₄×C₈₉	1	2	yes	—
461	C₄²	C₂⁴×C₈₉	1	2	yes	—
462	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
463	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	4	2	no	—
467	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
468	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
469	Q₈	—	2	8	no	—
470	Q₈	—	2	4	no	—
471	Q₈	—	4	4	no	—
472	Q₈	—	8	2	no	—
473	D₄	—	2	4	no	—
474	D₄	—	4	4	no	—
475	D₄	—	8	2	no	—
476	D₄	—	8	4	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	—	8	4	no	—
491	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
493	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
495	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
496	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
497	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
498	C₈	—	2	4	no	—
499	C₈	C₂×#41×C₈₉	1	8	yes	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	4	4	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	#532×C₈₉	1	2	yes	—
507	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
508	C₄	—	2	4	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	2	no	—
515	C₄	—	4	4	no	—
516	C₄	#532×C₈₉	1	2	yes	—
517	C₄	C₂²×#41×C₈₉	1	2	yes	—
518	C₂	—	2	2	no	—
519	C₂	—	4	2	no	—
520	C₂	—	8	2	no	—
521	C₂	C₂×#532×C₈₉	1	1	yes	characteristic
522	C₂	C₈₉×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₈₉×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#541×C₈₉	C₂	semidirect
3. C₈₉×#3178	C₂	semidirect
4. C₈₉×#3177	C₂	semidirect
5. C₈₉×#3174	C₂	semidirect
6. C₈₉×#3166	C₂	semidirect
7. C₈₉×#3152	C₂	semidirect
8. C₈₉×#3142	C₂	semidirect
9. C₈₉×#3139	C₂	semidirect
10. C₈₉×#3056	C₂	semidirect
11. C₄×#137×C₈₉	C₂	semidirect
12. C₈₉×#2873	C₂	semidirect
35. #501×C₈₉	C₂²	semidirect
36. #499×C₈₉	C₂²	semidirect
37. #497×C₈₉	C₂²	semidirect
38. #494×C₈₉	C₂²	semidirect
39. #492×C₈₉	C₂²	semidirect
40. #491×C₈₉	C₂²	semidirect
41. #491×C₈₉	C₂²	semidirect
42. #490×C₈₉	C₂²	semidirect
43. #488×C₈₉	C₂²	semidirect
44. #487×C₈₉	C₂²	semidirect
49. #441×C₈₉	C₂²	semidirect
50. #440×C₈₉	C₂²	semidirect
51. C₄×Q₁₆×C₈₉	C₂²	semidirect
52. C₄×SD₁₆×C₈₉	C₂²	semidirect
256. #60705	C₈₉	direct (N × H)
263. C₂×#541	C₂×C₈₉	semidirect
264. #3178	C₂×C₈₉	semidirect
265. #3177	C₂×C₈₉	semidirect
266. #3174	C₂×C₈₉	semidirect
267. #3166	C₂×C₈₉	semidirect
268. #3152	C₂×C₈₉	semidirect
269. #3142	C₂×C₈₉	semidirect
270. #3139	C₂×C₈₉	semidirect
271. #3056	C₂×C₈₉	semidirect
272. C₄×#137	C₂×C₈₉	semidirect
273. #2873	C₂×C₈₉	semidirect
274. C₈₉	#60705	direct (N × H)
297. #501	C₂²×C₈₉	semidirect
298. #499	C₂²×C₈₉	semidirect
299. #497	C₂²×C₈₉	semidirect
300. #494	C₂²×C₈₉	semidirect
301. #492	C₂²×C₈₉	semidirect
302. #491	C₂²×C₈₉	semidirect
303. #491	C₂²×C₈₉	semidirect
304. #490	C₂²×C₈₉	semidirect
305. #488	C₂²×C₈₉	semidirect
306. #487	C₂²×C₈₉	semidirect
311. #441	C₂²×C₈₉	semidirect
312. #440	C₂²×C₈₉	semidirect
313. C₄×Q₁₆	C₂²×C₈₉	semidirect
314. C₄×SD₁₆	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3455	2
C₂×#541×C₈₉	2
C₈₉×#3178	2
C₈₉×#3177	2
C₈₉×#3174	2
C₈₉×#3166	2
C₈₉×#3152	2
C₈₉×#3142	2
C₈₉×#3139	2
C₈₉×#3056	2
C₄×#137×C₈₉	2
C₈₉×#2873	2
#60705	89

Subgroups — C89×#60705 #36438553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60705 #36438553