Subgroups — C₈₉×#62637 #36440485

Order 22784.

2026 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#578×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3417	C₂	1	8	yes	maximal, complemented
4	C₈₉×#3416	C₂	1	8	yes	maximal, complemented
5	C₈₉×#3414	C₂	1	6	yes	maximal, complemented
6	C₂×#510×C₈₉	C₂	1	6	yes	maximal, complemented
7	C₈₉×#3407	C₂	1	1	yes	maximal, characteristic, complemented
8	#585×C₈₉	—	2	1	no	—
9	C₂×2₋¹⁺⁴×C₈₉	—	2	1	no	—
10	#578×C₈₉	—	2	4	no	—
11	#578×C₈₉	C₂²	1	8	yes	complemented
12	#578×C₈₉	C₂²	1	4	yes	complemented
13	#577×C₈₉	—	2	4	no	—
14	#576×C₈₉	C₂²	1	4	yes	complemented
15	#575×C₈₉	—	2	3	no	—
16	C₂×#137×C₈₉	—	2	3	no	—
17	C₂×#135×C₈₉	C₂²	1	6	yes	—
18	C₂×#135×C₈₉	C₂²	1	3	yes	complemented
19	C₂²×Q₁₆×C₈₉	C₂²	1	3	yes	complemented
20	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
21	#510×C₈₉	C₂²	1	12	yes	complemented
22	#510×C₈₉	C₂²	1	24	yes	complemented
23	#510×C₈₉	C₂²	1	12	yes	complemented
24	#509×C₈₉	C₂²	1	4	yes	complemented
25	#509×C₈₉	C₂²	1	12	yes	complemented
26	#508×C₈₉	C₂²	1	24	yes	complemented
27	C₂×Q₃₂×C₈₉	C₂²	1	12	yes	complemented
28	C₂×SD₃₂×C₈₉	C₂²	1	12	yes	complemented
29	#504×C₈₉	C₂²	1	8	yes	complemented
30	C₂×M₃₂×C₈₉	C₂²	1	6	yes	complemented
31	2₋¹⁺⁴×C₈₉	—	2	4	no	—
32	2₋¹⁺⁴×C₈₉	—	2	8	no	—
33	2₋¹⁺⁴×C₈₉	—	4	4	no	—
34	2₋¹⁺⁴×C₈₉	—	4	1	no	—
35	H(5,𝔽₂)×C₈₉	—	2	4	no	—
36	H(5,𝔽₂)×C₈₉	—	4	3	no	—
37	C₂×#41×C₈₉	—	2	6	no	—
38	C₂×#41×C₈₉	—	2	3	no	—
39	C₂×#41×C₈₉	—	4	4	no	—
40	C₂×#41×C₈₉	—	4	3	no	—
41	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₈₉	—	2	3	no	—
43	C₂²×Q₈×C₈₉	—	4	1	no	—
44	#137×C₈₉	—	2	24	no	—
45	#137×C₈₉	—	2	12	no	—
46	AGL(1,ℤ/8)×C₈₉	—	2	12	no	—
47	#135×C₈₉	—	2	6	no	—
48	#135×C₈₉	—	2	6	no	—
49	#135×C₈₉	—	2	6	no	—
50	#135×C₈₉	C₂³	1	12	yes	—
51	#135×C₈₉	C₂³	1	12	yes	—
52	#135×C₈₉	C₂³	1	4	yes	—
53	C₂×Q₁₆×C₈₉	—	2	3	no	—
54	C₂×Q₁₆×C₈₉	—	2	6	no	—
55	C₂×Q₁₆×C₈₉	C₂³	1	12	yes	—
56	C₂×Q₁₆×C₈₉	C₂³	1	6	yes	—
57	C₂×Q₁₆×C₈₉	C₂³	1	3	yes	—
58	C₂×SD₁₆×C₈₉	—	2	6	no	—
59	C₂×SD₁₆×C₈₉	—	2	3	no	—
60	C₂×D₈×C₈₉	C₂³	1	6	yes	—
61	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
62	#131×C₈₉	D₄	1	4	yes	—
63	#131×C₈₉	C₂³	1	4	yes	—
64	C₂×M₁₆×C₈₉	D₄	1	3	yes	—
65	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
66	Q₃₂×C₈₉	C₂³	1	24	yes	—
67	SD₃₂×C₈₉	C₂³	1	24	yes	—
68	SD₃₂×C₈₉	C₂³	1	8	yes	—
69	D₁₆×C₈₉	C₂³	1	8	yes	—
70	M₃₂×C₈₉	C₂³	1	4	yes	—
71	M₃₂×C₈₉	C₂³	1	12	yes	—
72	C₂×C₁₆×C₈₉	C₂³	1	12	yes	—
73	#41×C₈₉	—	2	4	no	—
74	#41×C₈₉	—	2	12	no	—
75	#41×C₈₉	—	2	12	no	—
76	#41×C₈₉	—	4	4	no	—
77	#41×C₈₉	—	4	12	no	—
78	#41×C₈₉	—	4	24	no	—
79	#41×C₈₉	—	4	6	no	—
80	#41×C₈₉	—	4	6	no	—
81	#41×C₈₉	—	4	6	no	—
82	#41×C₈₉	D₈	1	4	yes	—
83	#41×C₈₉	C₂×D₄	1	4	yes	—
84	C₂×Q₈×C₈₉	—	2	3	no	—
85	C₂×Q₈×C₈₉	—	2	6	no	—
86	C₂×Q₈×C₈₉	—	2	12	no	—
87	C₂×Q₈×C₈₉	—	4	8	no	—
88	C₂×Q₈×C₈₉	—	4	4	no	—
89	C₂×Q₈×C₈₉	—	4	3	no	—
90	C₂×Q₈×C₈₉	—	4	6	no	—
91	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
92	C₂×D₄×C₈₉	—	2	1	no	—
93	C₂×D₄×C₈₉	—	2	6	no	—
94	C₂×D₄×C₈₉	—	4	3	no	—
95	C₂×D₄×C₈₉	—	4	6	no	—
96	C₂×D₄×C₈₉	—	4	12	no	—
97	C₂×D₄×C₈₉	D₈	1	3	yes	—
98	C₂²×C₄×C₈₉	—	4	3	no	—
99	C₂²×C₄×C₈₉	—	4	3	no	—
100	C₂²×C₄×C₈₉	—	4	6	no	—
101	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
102	Q₁₆×C₈₉	—	2	3	no	—
103	Q₁₆×C₈₉	—	2	12	no	—
104	Q₁₆×C₈₉	—	2	6	no	—
105	Q₁₆×C₈₉	C₂⁴	1	6	yes	—
106	Q₁₆×C₈₉	C₂⁴	1	4	yes	—
107	SD₁₆×C₈₉	—	2	4	no	—
108	SD₁₆×C₈₉	—	2	12	no	—
109	SD₁₆×C₈₉	—	2	12	no	—
110	D₈×C₈₉	—	2	1	no	—
111	D₈×C₈₉	—	2	6	no	—
112	D₈×C₈₉	C₂⁴	1	2	yes	—
113	D₈×C₈₉	C₂⁴	1	4	yes	—
114	M₁₆×C₈₉	C₂×D₄	1	12	yes	—
115	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
116	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
117	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
118	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₁₆×C₈₉	C₂⁴	1	8	yes	—
120	C₂³×C₈₉	—	4	1	no	—
121	C₂³×C₈₉	—	4	6	no	—
122	C₂³×C₈₉	C₂×D₈	1	3	yes	—
123	Q₈×C₈₉	—	2	4	no	—
124	Q₈×C₈₉	—	2	6	no	—
125	Q₈×C₈₉	—	4	3	no	—
126	Q₈×C₈₉	—	4	12	no	—
127	Q₈×C₈₉	—	4	6	no	—
128	Q₈×C₈₉	C₂×D₈	1	4	yes	—
129	D₄×C₈₉	—	2	4	no	—
130	D₄×C₈₉	—	2	2	no	—
131	D₄×C₈₉	—	4	4	no	—
132	D₄×C₈₉	—	4	12	no	—
133	D₄×C₈₉	—	4	12	no	—
134	D₄×C₈₉	—	4	6	no	—
135	D₄×C₈₉	—	4	1	no	—
136	D₄×C₈₉	C₂×D₈	1	12	yes	—
137	C₂×C₄×C₈₉	—	4	3	no	—
138	C₂×C₄×C₈₉	—	4	6	no	—
139	C₂×C₄×C₈₉	—	4	12	no	—
140	C₂×C₄×C₈₉	—	4	4	no	—
141	C₂×C₄×C₈₉	—	4	12	no	—
142	C₂×C₄×C₈₉	—	4	12	no	—
143	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
144	C₂×C₄×C₈₉	C₂×D₈	1	6	yes	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
147	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₈₉	C₂²×D₄	1	6	yes	—
149	C₈×C₈₉	C₂⁵	1	1	yes	characteristic
150	C₂²×C₈₉	—	4	3	no	—
151	C₂²×C₈₉	—	4	2	no	—
152	C₂²×C₈₉	—	4	4	no	—
153	C₂²×C₈₉	—	8	2	no	—
154	C₂²×C₈₉	—	16	6	no	—
155	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
156	C₂²×C₈₉	C₂²×D₈	1	6	yes	—
157	C₄×C₈₉	—	4	4	no	—
158	C₄×C₈₉	—	4	6	no	—
159	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₈₉	C₂²×D₈	1	6	yes	—
161	C₄×C₈₉	C₂³×D₄	1	1	yes	characteristic
162	#62637	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₈₉	—	2	1	no	—
164	C₂×C₈₉	—	2	6	no	—
165	C₂×C₈₉	—	8	2	no	—
166	C₂×C₈₉	—	8	4	no	—
167	C₂×C₈₉	C₂³×D₈	1	1	yes	characteristic
168	#3584	C₂×C₈₉	1	1	yes	characteristic, complemented
169	C₂×#578	C₂×C₈₉	1	1	yes	characteristic, complemented
170	#3417	C₂×C₈₉	1	8	yes	complemented
171	#3416	C₂×C₈₉	1	8	yes	complemented
172	#3414	C₂×C₈₉	1	6	yes	complemented
173	C₂×#510	C₂×C₈₉	1	6	yes	complemented
174	#3407	C₂×C₈₉	1	1	yes	characteristic, complemented
175	C₈₉	#62637	1	1	yes	characteristic, direct product factor, Sylow
176	#585	—	2	1	no	—
177	C₂×2₋¹⁺⁴	—	2	1	no	—
178	#578	—	2	4	no	—
179	#578	C₂²×C₈₉	1	4	yes	complemented
180	#578	C₂²×C₈₉	1	8	yes	complemented
181	#577	—	2	4	no	—
182	#576	C₂²×C₈₉	1	4	yes	complemented
183	#575	—	2	3	no	—
184	C₂×#137	—	2	3	no	—
185	C₂×#135	C₂²×C₈₉	1	6	yes	—
186	C₂×#135	C₂²×C₈₉	1	3	yes	complemented
187	C₂²×Q₁₆	C₂²×C₈₉	1	3	yes	complemented
188	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
189	#510	C₂²×C₈₉	1	12	yes	complemented
190	#510	C₂²×C₈₉	1	24	yes	complemented
191	#510	C₂²×C₈₉	1	12	yes	complemented
192	#509	C₂²×C₈₉	1	4	yes	complemented
193	#509	C₂²×C₈₉	1	12	yes	complemented
194	#508	C₂²×C₈₉	1	24	yes	complemented
195	C₂×Q₃₂	C₂²×C₈₉	1	12	yes	complemented
196	C₂×SD₃₂	C₂²×C₈₉	1	12	yes	complemented
197	#504	C₂²×C₈₉	1	8	yes	complemented
198	C₂×M₃₂	C₂²×C₈₉	1	6	yes	complemented
199	2₋¹⁺⁴	—	2	4	no	—
200	2₋¹⁺⁴	—	2	8	no	—
201	2₋¹⁺⁴	—	4	4	no	—
202	2₋¹⁺⁴	—	4	1	no	—
203	H(5,𝔽₂)	—	2	4	no	—
204	H(5,𝔽₂)	—	4	3	no	—
205	C₂×#41	—	2	6	no	—
206	C₂×#41	—	2	3	no	—
207	C₂×#41	—	4	4	no	—
208	C₂×#41	—	4	3	no	—
209	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
210	C₂²×Q₈	—	2	3	no	—
211	C₂²×Q₈	—	4	1	no	—
212	#137	—	2	24	no	—
213	#137	—	2	12	no	—
214	AGL(1,ℤ/8)	—	2	12	no	—
215	#135	—	2	6	no	—
216	#135	—	2	6	no	—
217	#135	—	2	6	no	—
218	#135	C₂³×C₈₉	1	4	yes	—
219	#135	C₂³×C₈₉	1	12	yes	—
220	#135	C₂³×C₈₉	1	12	yes	—
221	C₂×Q₁₆	—	2	3	no	—
222	C₂×Q₁₆	—	2	6	no	—
223	C₂×Q₁₆	C₂³×C₈₉	1	3	yes	—
224	C₂×Q₁₆	C₂³×C₈₉	1	12	yes	—
225	C₂×Q₁₆	C₂³×C₈₉	1	6	yes	—
226	C₂×SD₁₆	—	2	6	no	—
227	C₂×SD₁₆	—	2	3	no	—
228	C₂×D₈	C₂³×C₈₉	1	6	yes	—
229	C₂×D₈	C₂³×C₈₉	1	1	yes	characteristic
230	#131	D₄×C₈₉	1	4	yes	—
231	#131	C₂³×C₈₉	1	4	yes	—
232	C₂×M₁₆	D₄×C₈₉	1	3	yes	—
233	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
234	Q₃₂	C₂³×C₈₉	1	24	yes	—
235	SD₃₂	C₂³×C₈₉	1	24	yes	—
236	SD₃₂	C₂³×C₈₉	1	8	yes	—
237	D₁₆	C₂³×C₈₉	1	8	yes	—
238	M₃₂	C₂³×C₈₉	1	4	yes	—
239	M₃₂	C₂³×C₈₉	1	12	yes	—
240	C₂×C₁₆	C₂³×C₈₉	1	12	yes	—
241	#41	—	2	4	no	—
242	#41	—	2	12	no	—
243	#41	—	2	12	no	—
244	#41	—	4	4	no	—
245	#41	—	4	12	no	—
246	#41	—	4	24	no	—
247	#41	—	4	6	no	—
248	#41	—	4	6	no	—
249	#41	—	4	6	no	—
250	#41	D₈×C₈₉	1	4	yes	—
251	#41	C₂×D₄×C₈₉	1	4	yes	—
252	C₂×Q₈	—	2	3	no	—
253	C₂×Q₈	—	2	6	no	—
254	C₂×Q₈	—	2	12	no	—
255	C₂×Q₈	—	4	8	no	—
256	C₂×Q₈	—	4	4	no	—
257	C₂×Q₈	—	4	3	no	—
258	C₂×Q₈	—	4	6	no	—
259	C₂×Q₈	D₈×C₈₉	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	6	no	—
262	C₂×D₄	—	4	3	no	—
263	C₂×D₄	—	4	6	no	—
264	C₂×D₄	—	4	12	no	—
265	C₂×D₄	D₈×C₈₉	1	3	yes	—
266	C₂²×C₄	—	4	3	no	—
267	C₂²×C₄	—	4	3	no	—
268	C₂²×C₄	—	4	6	no	—
269	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
270	Q₁₆	—	2	3	no	—
271	Q₁₆	—	2	12	no	—
272	Q₁₆	—	2	6	no	—
273	Q₁₆	C₂⁴×C₈₉	1	6	yes	—
274	Q₁₆	C₂⁴×C₈₉	1	4	yes	—
275	SD₁₆	—	2	4	no	—
276	SD₁₆	—	2	12	no	—
277	SD₁₆	—	2	12	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	6	no	—
280	D₈	C₂⁴×C₈₉	1	2	yes	—
281	D₈	C₂⁴×C₈₉	1	4	yes	—
282	M₁₆	C₂×D₄×C₈₉	1	12	yes	—
283	C₂×C₈	C₂×D₄×C₈₉	1	3	yes	—
284	C₂×C₈	C₂×D₄×C₈₉	1	6	yes	—
285	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
286	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
287	C₁₆	C₂⁴×C₈₉	1	8	yes	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	6	no	—
290	C₂³	C₂×D₈×C₈₉	1	3	yes	—
291	Q₈	—	2	4	no	—
292	Q₈	—	2	6	no	—
293	Q₈	—	4	3	no	—
294	Q₈	—	4	12	no	—
295	Q₈	—	4	6	no	—
296	Q₈	C₂×D₈×C₈₉	1	4	yes	—
297	D₄	—	2	4	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	12	no	—
301	D₄	—	4	12	no	—
302	D₄	—	4	6	no	—
303	D₄	—	4	1	no	—
304	D₄	C₂×D₈×C₈₉	1	12	yes	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	6	no	—
307	C₂×C₄	—	4	12	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	12	no	—
310	C₂×C₄	—	4	12	no	—
311	C₂×C₄	C₂×D₈×C₈₉	1	3	yes	—
312	C₂×C₄	C₂×D₈×C₈₉	1	6	yes	—
313	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₈₉	1	6	yes	—
315	C₈	C₂²×D₄×C₈₉	1	1	yes	characteristic
316	C₈	C₂²×D₄×C₈₉	1	6	yes	—
317	C₈	C₂⁵×C₈₉	1	1	yes	characteristic
318	C₂²	—	4	3	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	16	6	no	—
323	C₂²	C₂²×D₈×C₈₉	1	1	yes	characteristic
324	C₂²	C₂²×D₈×C₈₉	1	6	yes	—
325	C₄	—	4	4	no	—
326	C₄	—	4	6	no	—
327	C₄	C₂²×D₈×C₈₉	1	1	yes	characteristic
328	C₄	C₂²×D₈×C₈₉	1	6	yes	—
329	C₄	C₂³×D₄×C₈₉	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	6	no	—
332	C₂	—	8	2	no	—
333	C₂	—	8	4	no	—
334	C₂	C₂³×D₈×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	C₂³×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₈₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂×#41×C₈₉ ◁ C₂×#131×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3584	C₂	semidirect
2. C₂×#578×C₈₉	C₂	semidirect
3. C₈₉×#3417	C₂	semidirect
4. C₈₉×#3416	C₂	semidirect
5. C₈₉×#3414	C₂	semidirect
6. C₂×#510×C₈₉	C₂	semidirect
7. C₈₉×#3407	C₂	semidirect
11. #578×C₈₉	C₂²	semidirect
12. #578×C₈₉	C₂²	semidirect
14. #576×C₈₉	C₂²	semidirect
18. C₂×#135×C₈₉	C₂²	semidirect
19. C₂²×Q₁₆×C₈₉	C₂²	semidirect
21. #510×C₈₉	C₂²	semidirect
22. #510×C₈₉	C₂²	semidirect
23. #510×C₈₉	C₂²	semidirect
24. #509×C₈₉	C₂²	semidirect
25. #509×C₈₉	C₂²	semidirect
26. #508×C₈₉	C₂²	semidirect
27. C₂×Q₃₂×C₈₉	C₂²	semidirect
28. C₂×SD₃₂×C₈₉	C₂²	semidirect
29. #504×C₈₉	C₂²	semidirect
30. C₂×M₃₂×C₈₉	C₂²	semidirect
162. #62637	C₈₉	direct (N × H)
168. #3584	C₂×C₈₉	semidirect
169. C₂×#578	C₂×C₈₉	semidirect
170. #3417	C₂×C₈₉	semidirect
171. #3416	C₂×C₈₉	semidirect
172. #3414	C₂×C₈₉	semidirect
173. C₂×#510	C₂×C₈₉	semidirect
174. #3407	C₂×C₈₉	semidirect
175. C₈₉	#62637	direct (N × H)
179. #578	C₂²×C₈₉	semidirect
180. #578	C₂²×C₈₉	semidirect
182. #576	C₂²×C₈₉	semidirect
186. C₂×#135	C₂²×C₈₉	semidirect
187. C₂²×Q₁₆	C₂²×C₈₉	semidirect
189. #510	C₂²×C₈₉	semidirect
190. #510	C₂²×C₈₉	semidirect
191. #510	C₂²×C₈₉	semidirect
192. #509	C₂²×C₈₉	semidirect
193. #509	C₂²×C₈₉	semidirect
194. #508	C₂²×C₈₉	semidirect
195. C₂×Q₃₂	C₂²×C₈₉	semidirect
196. C₂×SD₃₂	C₂²×C₈₉	semidirect
197. #504	C₂²×C₈₉	semidirect
198. C₂×M₃₂	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×#3584	2
C₂×#578×C₈₉	2
C₈₉×#3417	2
C₈₉×#3416	2
C₈₉×#3414	2
C₂×#510×C₈₉	2
C₈₉×#3407	2
#62637	89

Subgroups — C89×#62637 #36440485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#62637 #36440485