Subgroups — C₈₉×#61999 #36439847

Order 22784.

2162 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×#3498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3482	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3392	C₂	1	2	yes	maximal, complemented
4	C₈₉×#3358	C₂	1	2	yes	maximal, complemented
5	C₈₉×#3324	C₂	1	6	yes	maximal, complemented
6	C₈₉×#3272	C₂	1	6	yes	maximal, complemented
7	C₈₉×#3254	C₂	1	3	yes	maximal, complemented
8	C₈₉×#3208	C₂	1	3	yes	maximal, complemented
9	C₈₉×#3106	C₂	1	3	yes	maximal, complemented
10	C₈₉×#3038	C₂	1	3	yes	maximal, complemented
11	C₈₉×#2991	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₈₉	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	6	yes	complemented
14	#557×C₈₉	C₂²	1	1	yes	characteristic, complemented
15	#556×C₈₉	C₂²	1	3	yes	complemented
16	#552×C₈₉	—	2	3	no	—
17	#550×C₈₉	—	2	1	no	—
18	#550×C₈₉	C₂²	1	2	yes	—
19	D₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
20	#548×C₈₉	—	2	1	no	—
21	#547×C₈₉	C₂²	1	3	yes	complemented
22	D₄²×C₈₉	—	2	3	no	—
23	D₄²×C₈₉	C₂²	1	6	yes	—
24	#535×C₈₉	—	2	3	no	—
25	#533×C₈₉	C₂²	1	3	yes	complemented
26	#532×C₈₉	C₂²	1	3	yes	—
27	C₂×#127×C₈₉	C₂²	1	3	yes	—
28	#529×C₈₉	—	2	3	no	—
29	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
30	C₄×#41×C₈₉	C₂²	1	3	yes	complemented
31	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
32	#492×C₈₉	C₂²	1	6	yes	complemented
33	#486×C₈₉	C₂²	1	6	yes	complemented
34	#482×C₈₉	C₂²	1	6	yes	complemented
35	#477×C₈₉	C₂²	1	3	yes	complemented
36	#476×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	#474×C₈₉	C₂²	1	6	yes	complemented
38	#468×C₈₉	C₂²	1	12	yes	complemented
39	#461×C₈₉	C₂²	1	6	yes	complemented
40	#460×C₈₉	C₂²	1	6	yes	complemented
41	#460×C₈₉	C₂²	1	3	yes	complemented
42	#460×C₈₉	C₂²	1	6	yes	complemented
43	#459×C₈₉	C₂²	1	6	yes	complemented
44	#459×C₈₉	C₂²	1	3	yes	complemented
45	#450×C₈₉	C₂²	1	12	yes	complemented
46	#449×C₈₉	C₂²	1	6	yes	complemented
47	C₈×Q₈×C₈₉	C₂²	1	2	yes	complemented
48	C₄×SD₁₆×C₈₉	C₂²	1	6	yes	complemented
49	C₄×D₈×C₈₉	C₂²	1	6	yes	complemented
50	#436×C₈₉	C₂²	1	6	yes	complemented
51	#431×C₈₉	C₂²	1	3	yes	complemented
52	#423×C₈₉	C₂²	1	3	yes	complemented
53	#418×C₈₉	C₂²	1	6	yes	complemented
54	H(5,𝔽₂)×C₈₉	—	4	8	no	—
55	C₂×#41×C₈₉	—	2	2	no	—
56	C₂×#41×C₈₉	—	2	3	no	—
57	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
58	C₂×#41×C₈₉	C₂³	1	3	yes	—
59	C₂²×Q₈×C₈₉	—	2	1	no	—
60	C₂²×D₄×C₈₉	—	2	6	no	—
61	C₂²×D₄×C₈₉	—	2	3	no	—
62	C₂²×D₄×C₈₉	—	2	3	no	—
63	C₂²×D₄×C₈₉	C₂³	1	6	yes	—
64	AGL(1,ℤ/8)×C₈₉	—	4	12	no	—
65	C₂×SD₁₆×C₈₉	C₂³	1	6	yes	—
66	C₂×SD₁₆×C₈₉	C₂³	1	12	yes	—
67	C₂×D₈×C₈₉	C₂³	1	6	yes	—
68	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
69	C₄⋊Q₈×C₈₉	C₂³	1	3	yes	—
70	C₄⋊Q₈×C₈₉	C₂³	1	3	yes	—
71	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
72	#127×C₈₉	—	2	3	no	—
73	#127×C₈₉	—	2	3	no	—
74	#127×C₈₉	C₂³	1	3	yes	—
75	#127×C₈₉	C₂³	1	6	yes	—
76	#126×C₈₉	—	2	6	no	—
77	#125×C₈₉	—	2	3	no	—
78	#125×C₈₉	—	2	3	no	—
79	#124×C₈₉	—	2	3	no	—
80	#124×C₈₉	—	2	3	no	—
81	#123×C₈₉	—	2	6	no	—
82	C₂²⋊Q₈×C₈₉	—	2	3	no	—
83	C₂²⋊Q₈×C₈₉	—	2	3	no	—
84	C₂²⋊Q₈×C₈₉	C₂³	1	6	yes	—
85	#121×C₈₉	—	2	3	no	—
86	#121×C₈₉	—	2	3	no	—
87	#121×C₈₉	—	2	6	no	—
88	#121×C₈₉	—	2	6	no	—
89	#121×C₈₉	C₂³	1	6	yes	—
90	#120×C₈₉	—	2	12	no	—
91	C₄×Q₈×C₈₉	—	2	1	no	—
92	C₄×Q₈×C₈₉	—	2	3	no	—
93	C₄×Q₈×C₈₉	—	2	3	no	—
94	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₈₉	C₂³	1	3	yes	—
96	C₄×Q₈×C₈₉	C₂³	1	3	yes	—
97	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₈₉	—	2	3	no	—
99	C₄×D₄×C₈₉	—	2	3	no	—
100	C₄×D₄×C₈₉	—	2	3	no	—
101	C₄×D₄×C₈₉	D₄	1	3	yes	—
102	C₄×D₄×C₈₉	C₂³	1	3	yes	—
103	C₄×D₄×C₈₉	C₂³	1	6	yes	—
104	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄×D₄×C₈₉	C₂³	1	3	yes	—
106	#117×C₈₉	—	2	3	no	—
107	#117×C₈₉	—	2	3	no	—
108	#117×C₈₉	D₄	1	3	yes	—
109	#117×C₈₉	C₂³	1	3	yes	—
110	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
111	C₂×C₄²×C₈₉	—	2	3	no	—
112	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
113	#107×C₈₉	C₂³	1	6	yes	—
114	#106×C₈₉	C₂³	1	2	yes	—
115	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
116	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
117	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
118	#103×C₈₉	C₂³	1	6	yes	—
119	#102×C₈₉	C₂³	1	6	yes	—
120	#102×C₈₉	C₂³	1	12	yes	—
121	#102×C₈₉	C₂³	1	2	yes	—
122	#102×C₈₉	C₂³	1	6	yes	—
123	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
124	C₄×C₈×C₈₉	C₂³	1	6	yes	—
125	C₂⁴×C₈₉	—	2	6	no	—
126	#41×C₈₉	—	2	4	no	—
127	#41×C₈₉	—	2	6	no	—
128	#41×C₈₉	—	4	3	no	—
129	#41×C₈₉	—	4	12	no	—
130	#41×C₈₉	—	4	8	no	—
131	C₂×Q₈×C₈₉	—	2	3	no	—
132	C₂×Q₈×C₈₉	—	2	1	no	—
133	C₂×Q₈×C₈₉	—	4	4	no	—
134	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₈₉	C₂⁴	1	3	yes	—
136	C₂×D₄×C₈₉	—	2	3	no	—
137	C₂×D₄×C₈₉	—	2	6	no	—
138	C₂×D₄×C₈₉	—	2	3	no	—
139	C₂×D₄×C₈₉	—	2	6	no	—
140	C₂×D₄×C₈₉	—	2	3	no	—
141	C₂×D₄×C₈₉	—	2	6	no	—
142	C₂×D₄×C₈₉	—	2	6	no	—
143	C₂×D₄×C₈₉	—	4	6	no	—
144	C₂×D₄×C₈₉	—	4	6	no	—
145	C₂×D₄×C₈₉	—	4	12	no	—
146	C₂×D₄×C₈₉	—	4	24	no	—
147	C₂×D₄×C₈₉	—	8	6	no	—
148	C₂×D₄×C₈₉	C₂×D₄	1	3	yes	—
149	C₂×D₄×C₈₉	C₂⁴	1	6	yes	—
150	C₂×D₄×C₈₉	C₂⁴	1	1	yes	characteristic
151	C₂²×C₄×C₈₉	—	2	1	no	—
152	C₂²×C₄×C₈₉	—	2	3	no	—
153	C₂²×C₄×C₈₉	—	2	3	no	—
154	C₂²×C₄×C₈₉	—	2	6	no	—
155	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
156	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
157	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
158	SD₁₆×C₈₉	—	4	6	no	—
159	SD₁₆×C₈₉	—	4	12	no	—
160	D₈×C₈₉	—	2	12	no	—
161	M₁₆×C₈₉	—	4	6	no	—
162	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
163	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
164	C₄⋊C₄×C₈₉	—	2	3	no	—
165	C₄⋊C₄×C₈₉	—	2	6	no	—
166	C₄⋊C₄×C₈₉	—	2	3	no	—
167	C₄⋊C₄×C₈₉	—	2	3	no	—
168	C₄⋊C₄×C₈₉	—	2	3	no	—
169	C₄⋊C₄×C₈₉	—	2	3	no	—
170	C₄⋊C₄×C₈₉	—	2	1	no	—
171	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
172	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
174	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
175	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
176	C₂²⋊C₄×C₈₉	—	2	6	no	—
177	C₂²⋊C₄×C₈₉	—	2	3	no	—
178	C₂²⋊C₄×C₈₉	—	2	1	no	—
179	C₂²⋊C₄×C₈₉	—	2	3	no	—
180	C₂²⋊C₄×C₈₉	—	2	6	no	—
181	C₂²⋊C₄×C₈₉	—	2	6	no	—
182	C₂²⋊C₄×C₈₉	C₂×D₄	1	6	yes	—
183	C₄²×C₈₉	—	2	3	no	—
184	C₄²×C₈₉	—	2	3	no	—
185	C₄²×C₈₉	—	2	3	no	—
186	C₄²×C₈₉	C₂×D₄	1	3	yes	—
187	C₄²×C₈₉	C₂×D₄	1	3	yes	—
188	C₄²×C₈₉	C₂⁴	1	3	yes	—
189	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
190	C₂³×C₈₉	—	2	1	no	—
191	C₂³×C₈₉	—	2	6	no	—
192	C₂³×C₈₉	—	4	12	no	—
193	C₂³×C₈₉	—	8	6	no	—
194	C₂³×C₈₉	—	8	6	no	—
195	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
196	Q₈×C₈₉	—	2	3	no	—
197	Q₈×C₈₉	—	2	3	no	—
198	Q₈×C₈₉	—	4	2	no	—
199	Q₈×C₈₉	—	4	6	no	—
200	Q₈×C₈₉	AGL(1,ℤ/8)	1	4	yes	complemented
201	D₄×C₈₉	—	2	6	no	—
202	D₄×C₈₉	—	2	6	no	—
203	D₄×C₈₉	—	2	1	no	—
204	D₄×C₈₉	—	2	6	no	—
205	D₄×C₈₉	—	4	6	no	—
206	D₄×C₈₉	—	4	6	no	—
207	D₄×C₈₉	—	4	12	no	—
208	D₄×C₈₉	—	4	12	no	—
209	D₄×C₈₉	—	4	6	no	—
210	D₄×C₈₉	—	8	6	no	—
211	D₄×C₈₉	—	8	6	no	—
212	D₄×C₈₉	H(5,𝔽₂)	1	2	yes	complemented
213	C₂×C₄×C₈₉	—	2	6	no	—
214	C₂×C₄×C₈₉	—	2	6	no	—
215	C₂×C₄×C₈₉	—	2	3	no	—
216	C₂×C₄×C₈₉	—	2	3	no	—
217	C₂×C₄×C₈₉	—	2	1	no	—
218	C₂×C₄×C₈₉	—	2	3	no	—
219	C₂×C₄×C₈₉	—	2	3	no	—
220	C₂×C₄×C₈₉	—	2	2	no	—
221	C₂×C₄×C₈₉	—	4	6	no	—
222	C₂×C₄×C₈₉	—	4	12	no	—
223	C₂×C₄×C₈₉	—	4	6	no	—
224	C₂×C₄×C₈₉	—	4	2	no	—
225	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
226	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
228	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
229	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
231	C₈×C₈₉	—	2	6	no	—
232	C₈×C₈₉	—	2	2	no	—
233	C₂²×C₈₉	—	2	3	no	—
234	C₂²×C₈₉	—	2	2	no	—
235	C₂²×C₈₉	—	2	3	no	—
236	C₂²×C₈₉	—	2	3	no	—
237	C₂²×C₈₉	—	4	1	no	—
238	C₂²×C₈₉	—	4	1	no	—
239	C₂²×C₈₉	—	4	6	no	—
240	C₂²×C₈₉	—	4	6	no	—
241	C₂²×C₈₉	—	4	6	no	—
242	C₂²×C₈₉	—	16	12	no	—
243	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
244	C₄×C₈₉	—	2	3	no	—
245	C₄×C₈₉	—	2	1	no	—
246	C₄×C₈₉	—	2	3	no	—
247	C₄×C₈₉	—	4	3	no	—
248	C₄×C₈₉	—	4	2	no	—
249	C₄×C₈₉	—	4	3	no	—
250	C₄×C₈₉	—	4	1	no	—
251	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	6	yes	—
252	C₄×C₈₉	#577	1	2	yes	—
253	C₄×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
254	C₄×C₈₉	#585	1	1	yes	characteristic
255	#61999	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂×C₈₉	—	4	3	no	—
257	C₂×C₈₉	—	4	2	no	—
258	C₂×C₈₉	—	8	6	no	—
259	C₂×C₈₉	#3488	1	1	yes	characteristic
260	C₂×C₈₉	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₂×C₈₉	C₂×#577	1	1	yes	characteristic
262	#3498	C₂×C₈₉	1	1	yes	characteristic, complemented
263	#3482	C₂×C₈₉	1	1	yes	characteristic, complemented
264	#3392	C₂×C₈₉	1	2	yes	complemented
265	#3358	C₂×C₈₉	1	2	yes	complemented
266	#3324	C₂×C₈₉	1	6	yes	complemented
267	#3272	C₂×C₈₉	1	6	yes	complemented
268	#3254	C₂×C₈₉	1	3	yes	complemented
269	#3208	C₂×C₈₉	1	3	yes	complemented
270	#3106	C₂×C₈₉	1	3	yes	complemented
271	#3038	C₂×C₈₉	1	3	yes	complemented
272	#2991	C₂×C₈₉	1	1	yes	characteristic, complemented
273	C₈₉	#61999	1	1	yes	characteristic, direct product factor, Sylow
274	C₂×H(5,𝔽₂)	—	2	1	no	—
275	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	6	yes	complemented
276	#557	C₂²×C₈₉	1	1	yes	characteristic, complemented
277	#556	C₂²×C₈₉	1	3	yes	complemented
278	#552	—	2	3	no	—
279	#550	—	2	1	no	—
280	#550	C₂²×C₈₉	1	2	yes	—
281	D₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
282	#548	—	2	1	no	—
283	#547	C₂²×C₈₉	1	3	yes	complemented
284	D₄²	—	2	3	no	—
285	D₄²	C₂²×C₈₉	1	6	yes	—
286	#535	—	2	3	no	—
287	#533	C₂²×C₈₉	1	3	yes	complemented
288	#532	C₂²×C₈₉	1	3	yes	—
289	C₂×#127	C₂²×C₈₉	1	3	yes	—
290	#529	—	2	3	no	—
291	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
292	C₄×#41	C₂²×C₈₉	1	3	yes	complemented
293	C₂×C₄×Q₈	—	2	1	no	—
294	#492	C₂²×C₈₉	1	6	yes	complemented
295	#486	C₂²×C₈₉	1	6	yes	complemented
296	#482	C₂²×C₈₉	1	6	yes	complemented
297	#477	C₂²×C₈₉	1	3	yes	complemented
298	#476	C₂²×C₈₉	1	1	yes	characteristic, complemented
299	#474	C₂²×C₈₉	1	6	yes	complemented
300	#468	C₂²×C₈₉	1	12	yes	complemented
301	#461	C₂²×C₈₉	1	6	yes	complemented
302	#460	C₂²×C₈₉	1	3	yes	complemented
303	#460	C₂²×C₈₉	1	6	yes	complemented
304	#460	C₂²×C₈₉	1	6	yes	complemented
305	#459	C₂²×C₈₉	1	6	yes	complemented
306	#459	C₂²×C₈₉	1	3	yes	complemented
307	#450	C₂²×C₈₉	1	12	yes	complemented
308	#449	C₂²×C₈₉	1	6	yes	complemented
309	C₈×Q₈	C₂²×C₈₉	1	2	yes	complemented
310	C₄×SD₁₆	C₂²×C₈₉	1	6	yes	complemented
311	C₄×D₈	C₂²×C₈₉	1	6	yes	complemented
312	#436	C₂²×C₈₉	1	6	yes	complemented
313	#431	C₂²×C₈₉	1	3	yes	complemented
314	#423	C₂²×C₈₉	1	3	yes	complemented
315	#418	C₂²×C₈₉	1	6	yes	complemented
316	H(5,𝔽₂)	—	4	8	no	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	—	2	3	no	—
319	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
320	C₂×#41	C₂³×C₈₉	1	3	yes	—
321	C₂²×Q₈	—	2	1	no	—
322	C₂²×D₄	—	2	6	no	—
323	C₂²×D₄	—	2	3	no	—
324	C₂²×D₄	—	2	3	no	—
325	C₂²×D₄	C₂³×C₈₉	1	6	yes	—
326	AGL(1,ℤ/8)	—	4	12	no	—
327	C₂×SD₁₆	C₂³×C₈₉	1	12	yes	—
328	C₂×SD₁₆	C₂³×C₈₉	1	6	yes	—
329	C₂×D₈	C₂³×C₈₉	1	6	yes	—
330	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
331	C₄⋊Q₈	C₂³×C₈₉	1	3	yes	—
332	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₈₉	1	3	yes	—
334	#127	—	2	3	no	—
335	#127	—	2	3	no	—
336	#127	C₂³×C₈₉	1	3	yes	—
337	#127	C₂³×C₈₉	1	6	yes	—
338	#126	—	2	6	no	—
339	#125	—	2	3	no	—
340	#125	—	2	3	no	—
341	#124	—	2	3	no	—
342	#124	—	2	3	no	—
343	#123	—	2	6	no	—
344	C₂²⋊Q₈	—	2	3	no	—
345	C₂²⋊Q₈	—	2	3	no	—
346	C₂²⋊Q₈	C₂³×C₈₉	1	6	yes	—
347	#121	—	2	3	no	—
348	#121	—	2	3	no	—
349	#121	—	2	6	no	—
350	#121	—	2	6	no	—
351	#121	C₂³×C₈₉	1	6	yes	—
352	#120	—	2	12	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	3	no	—
355	C₄×Q₈	—	2	3	no	—
356	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
357	C₄×Q₈	C₂³×C₈₉	1	3	yes	—
358	C₄×Q₈	C₂³×C₈₉	1	3	yes	—
359	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
360	C₄×D₄	—	2	3	no	—
361	C₄×D₄	—	2	3	no	—
362	C₄×D₄	—	2	3	no	—
363	C₄×D₄	D₄×C₈₉	1	3	yes	—
364	C₄×D₄	C₂³×C₈₉	1	1	yes	characteristic
365	C₄×D₄	C₂³×C₈₉	1	3	yes	—
366	C₄×D₄	C₂³×C₈₉	1	6	yes	—
367	C₄×D₄	C₂³×C₈₉	1	3	yes	—
368	#117	—	2	3	no	—
369	#117	—	2	3	no	—
370	#117	D₄×C₈₉	1	3	yes	—
371	#117	C₂³×C₈₉	1	3	yes	—
372	C₂×C₄⋊C₄	—	2	3	no	—
373	C₂×C₄²	—	2	3	no	—
374	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
375	#107	C₂³×C₈₉	1	6	yes	—
376	#106	C₂³×C₈₉	1	2	yes	—
377	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
378	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
379	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
380	#103	C₂³×C₈₉	1	6	yes	—
381	#102	C₂³×C₈₉	1	6	yes	—
382	#102	C₂³×C₈₉	1	12	yes	—
383	#102	C₂³×C₈₉	1	2	yes	—
384	#102	C₂³×C₈₉	1	6	yes	—
385	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
386	C₄×C₈	C₂³×C₈₉	1	6	yes	—
387	C₂⁴	—	2	6	no	—
388	#41	—	2	4	no	—
389	#41	—	2	6	no	—
390	#41	—	4	3	no	—
391	#41	—	4	12	no	—
392	#41	—	4	8	no	—
393	C₂×Q₈	—	2	3	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	4	4	no	—
396	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
397	C₂×Q₈	C₂⁴×C₈₉	1	3	yes	—
398	C₂×D₄	—	2	3	no	—
399	C₂×D₄	—	2	6	no	—
400	C₂×D₄	—	2	3	no	—
401	C₂×D₄	—	2	6	no	—
402	C₂×D₄	—	2	3	no	—
403	C₂×D₄	—	2	6	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	4	6	no	—
406	C₂×D₄	—	4	6	no	—
407	C₂×D₄	—	4	12	no	—
408	C₂×D₄	—	4	24	no	—
409	C₂×D₄	—	8	6	no	—
410	C₂×D₄	C₂×D₄×C₈₉	1	3	yes	—
411	C₂×D₄	C₂⁴×C₈₉	1	6	yes	—
412	C₂×D₄	C₂⁴×C₈₉	1	1	yes	characteristic
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	3	no	—
415	C₂²×C₄	—	2	3	no	—
416	C₂²×C₄	—	2	6	no	—
417	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
418	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
419	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
420	SD₁₆	—	4	12	no	—
421	SD₁₆	—	4	6	no	—
422	D₈	—	2	12	no	—
423	M₁₆	—	4	6	no	—
424	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
425	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
426	C₄⋊C₄	—	2	3	no	—
427	C₄⋊C₄	—	2	6	no	—
428	C₄⋊C₄	—	2	3	no	—
429	C₄⋊C₄	—	2	3	no	—
430	C₄⋊C₄	—	2	3	no	—
431	C₄⋊C₄	—	2	3	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
434	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
435	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
436	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
437	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
438	C₂²⋊C₄	—	2	6	no	—
439	C₂²⋊C₄	—	2	3	no	—
440	C₂²⋊C₄	—	2	1	no	—
441	C₂²⋊C₄	—	2	3	no	—
442	C₂²⋊C₄	—	2	6	no	—
443	C₂²⋊C₄	—	2	6	no	—
444	C₂²⋊C₄	C₂×D₄×C₈₉	1	6	yes	—
445	C₄²	—	2	3	no	—
446	C₄²	—	2	3	no	—
447	C₄²	—	2	3	no	—
448	C₄²	C₂×D₄×C₈₉	1	3	yes	—
449	C₄²	C₂×D₄×C₈₉	1	3	yes	—
450	C₄²	C₂⁴×C₈₉	1	3	yes	—
451	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
452	C₂³	—	2	1	no	—
453	C₂³	—	2	6	no	—
454	C₂³	—	4	12	no	—
455	C₂³	—	8	6	no	—
456	C₂³	—	8	6	no	—
457	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
458	Q₈	—	2	3	no	—
459	Q₈	—	2	3	no	—
460	Q₈	—	4	2	no	—
461	Q₈	—	4	6	no	—
462	Q₈	AGL(1,ℤ/8)×C₈₉	1	4	yes	complemented
463	D₄	—	2	6	no	—
464	D₄	—	2	6	no	—
465	D₄	—	2	1	no	—
466	D₄	—	2	6	no	—
467	D₄	—	4	6	no	—
468	D₄	—	4	6	no	—
469	D₄	—	4	12	no	—
470	D₄	—	4	12	no	—
471	D₄	—	4	6	no	—
472	D₄	—	8	6	no	—
473	D₄	—	8	6	no	—
474	D₄	H(5,𝔽₂)×C₈₉	1	2	yes	complemented
475	C₂×C₄	—	2	6	no	—
476	C₂×C₄	—	2	6	no	—
477	C₂×C₄	—	2	3	no	—
478	C₂×C₄	—	2	3	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	3	no	—
481	C₂×C₄	—	2	3	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	4	6	no	—
484	C₂×C₄	—	4	12	no	—
485	C₂×C₄	—	4	6	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
488	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
489	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
490	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
491	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
492	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
493	C₈	—	2	6	no	—
494	C₈	—	2	2	no	—
495	C₂²	—	2	3	no	—
496	C₂²	—	2	3	no	—
497	C₂²	—	2	3	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	6	no	—
502	C₂²	—	4	6	no	—
503	C₂²	—	4	6	no	—
504	C₂²	—	16	12	no	—
505	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
506	C₄	—	2	1	no	—
507	C₄	—	2	3	no	—
508	C₄	—	2	3	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	3	no	—
511	C₄	—	4	1	no	—
512	C₄	—	4	3	no	—
513	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	6	yes	—
514	C₄	#577×C₈₉	1	2	yes	—
515	C₄	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
516	C₄	#585×C₈₉	1	1	yes	characteristic
517	C₂	—	4	3	no	—
518	C₂	—	4	2	no	—
519	C₂	—	8	6	no	—
520	C₂	C₈₉×#3488	1	1	yes	characteristic
521	C₂	C₂²×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
522	C₂	C₂×#577×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3498	C₂	semidirect
2. C₈₉×#3482	C₂	semidirect
3. C₈₉×#3392	C₂	semidirect
4. C₈₉×#3358	C₂	semidirect
5. C₈₉×#3324	C₂	semidirect
6. C₈₉×#3272	C₂	semidirect
7. C₈₉×#3254	C₂	semidirect
8. C₈₉×#3208	C₂	semidirect
9. C₈₉×#3106	C₂	semidirect
10. C₈₉×#3038	C₂	semidirect
11. C₈₉×#2991	C₂	semidirect
13. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
14. #557×C₈₉	C₂²	semidirect
15. #556×C₈₉	C₂²	semidirect
19. D₄×Q₈×C₈₉	C₂²	semidirect
21. #547×C₈₉	C₂²	semidirect
25. #533×C₈₉	C₂²	semidirect
30. C₄×#41×C₈₉	C₂²	semidirect
32. #492×C₈₉	C₂²	semidirect
33. #486×C₈₉	C₂²	semidirect
34. #482×C₈₉	C₂²	semidirect
35. #477×C₈₉	C₂²	semidirect
36. #476×C₈₉	C₂²	semidirect
37. #474×C₈₉	C₂²	semidirect
38. #468×C₈₉	C₂²	semidirect
39. #461×C₈₉	C₂²	semidirect
40. #460×C₈₉	C₂²	semidirect
41. #460×C₈₉	C₂²	semidirect
42. #460×C₈₉	C₂²	semidirect
43. #459×C₈₉	C₂²	semidirect
44. #459×C₈₉	C₂²	semidirect
45. #450×C₈₉	C₂²	semidirect
46. #449×C₈₉	C₂²	semidirect
47. C₈×Q₈×C₈₉	C₂²	semidirect
48. C₄×SD₁₆×C₈₉	C₂²	semidirect
49. C₄×D₈×C₈₉	C₂²	semidirect
50. #436×C₈₉	C₂²	semidirect
51. #431×C₈₉	C₂²	semidirect
52. #423×C₈₉	C₂²	semidirect
53. #418×C₈₉	C₂²	semidirect
200. Q₈×C₈₉	AGL(1,ℤ/8)	semidirect
212. D₄×C₈₉	H(5,𝔽₂)	semidirect
255. #61999	C₈₉	direct (N × H)
262. #3498	C₂×C₈₉	semidirect
263. #3482	C₂×C₈₉	semidirect
264. #3392	C₂×C₈₉	semidirect
265. #3358	C₂×C₈₉	semidirect
266. #3324	C₂×C₈₉	semidirect
267. #3272	C₂×C₈₉	semidirect
268. #3254	C₂×C₈₉	semidirect
269. #3208	C₂×C₈₉	semidirect
270. #3106	C₂×C₈₉	semidirect
271. #3038	C₂×C₈₉	semidirect
272. #2991	C₂×C₈₉	semidirect
273. C₈₉	#61999	direct (N × H)
275. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
276. #557	C₂²×C₈₉	semidirect
277. #556	C₂²×C₈₉	semidirect
281. D₄×Q₈	C₂²×C₈₉	semidirect
283. #547	C₂²×C₈₉	semidirect
287. #533	C₂²×C₈₉	semidirect
292. C₄×#41	C₂²×C₈₉	semidirect
294. #492	C₂²×C₈₉	semidirect
295. #486	C₂²×C₈₉	semidirect
296. #482	C₂²×C₈₉	semidirect
297. #477	C₂²×C₈₉	semidirect
298. #476	C₂²×C₈₉	semidirect
299. #474	C₂²×C₈₉	semidirect
300. #468	C₂²×C₈₉	semidirect
301. #461	C₂²×C₈₉	semidirect
302. #460	C₂²×C₈₉	semidirect
303. #460	C₂²×C₈₉	semidirect
304. #460	C₂²×C₈₉	semidirect
305. #459	C₂²×C₈₉	semidirect
306. #459	C₂²×C₈₉	semidirect
307. #450	C₂²×C₈₉	semidirect
308. #449	C₂²×C₈₉	semidirect
309. C₈×Q₈	C₂²×C₈₉	semidirect
310. C₄×SD₁₆	C₂²×C₈₉	semidirect
311. C₄×D₈	C₂²×C₈₉	semidirect
312. #436	C₂²×C₈₉	semidirect
313. #431	C₂²×C₈₉	semidirect
314. #423	C₂²×C₈₉	semidirect
315. #418	C₂²×C₈₉	semidirect
462. Q₈	AGL(1,ℤ/8)×C₈₉	semidirect
474. D₄	H(5,𝔽₂)×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3498	2
C₈₉×#3482	2
C₈₉×#3392	2
C₈₉×#3358	2
C₈₉×#3324	2
C₈₉×#3272	2
C₈₉×#3254	2
C₈₉×#3208	2
C₈₉×#3106	2
C₈₉×#3038	2
C₈₉×#2991	2
#61999	89

Subgroups — C89×#61999 #36439847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#61999 #36439847