Subgroups — C₈₉×#29154 #36407002

Order 22784.

612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#443×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2031	C₂	1	4	yes	maximal, complemented
4	C₈₉×#1976	C₂	1	2	yes	maximal
5	C₈₉×#1833	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1814	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1812	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#131×C₈₉	C₂²	1	2	yes	—
11	#533×C₈₉	C₂²	1	2	yes	—
12	#500×C₈₉	—	2	1	no	—
13	#499×C₈₉	—	2	1	no	—
14	#499×C₈₉	—	2	1	no	—
15	#498×C₈₉	—	2	1	no	—
16	#443×C₈₉	—	2	1	no	—
17	#443×C₈₉	—	2	1	no	—
18	#443×C₈₉	—	2	1	no	—
19	#443×C₈₉	—	2	1	no	—
20	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
22	#427×C₈₉	C₂²	1	1	yes	characteristic
23	#427×C₈₉	C₂²	1	2	yes	—
24	#427×C₈₉	C₂²	1	2	yes	—
25	#421×C₈₉	C₂²	1	4	yes	—
26	C₂×#104×C₈₉	C₂²	1	2	yes	—
27	#410×C₈₉	C₂²	1	4	yes	—
28	#405×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	#344×C₈₉	C₂²	1	4	yes	—
31	#339×C₈₉	C₂²	1	2	yes	—
32	H(3,ℤ/4)×C₈₉	C₂²	1	4	yes	—
33	C₂×#41×C₈₉	C₂³	1	2	yes	—
34	#135×C₈₉	—	2	1	no	—
35	#135×C₈₉	—	2	1	no	—
36	#135×C₈₉	—	2	1	no	—
37	#135×C₈₉	—	2	1	no	—
38	C₂×Q₁₆×C₈₉	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₈₉	Q₈	1	2	yes	complemented
40	C₂×D₈×C₈₉	Q₈	1	1	yes	characteristic, complemented
41	#131×C₈₉	—	2	2	no	—
42	#131×C₈₉	—	2	2	no	—
43	#131×C₈₉	—	2	2	no	—
44	#131×C₈₉	—	2	2	no	—
45	C₂×M₁₆×C₈₉	—	2	2	no	—
46	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
47	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
48	C₂²×C₈×C₈₉	—	2	2	no	—
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₈₉	—	2	2	no	—
52	#125×C₈₉	—	2	2	no	—
53	C₂²⋊Q₈×C₈₉	—	4	2	no	—
54	C₄×Q₈×C₈₉	—	4	2	no	—
55	#117×C₈₉	C₂³	1	2	yes	—
56	#117×C₈₉	C₂³	1	4	yes	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	#108×C₈₉	—	2	1	no	—
59	#108×C₈₉	—	2	1	no	—
60	#108×C₈₉	—	2	1	no	—
61	#108×C₈₉	—	2	1	no	—
62	#107×C₈₉	—	2	2	no	—
63	#107×C₈₉	—	2	2	no	—
64	#107×C₈₉	—	2	1	no	—
65	#106×C₈₉	—	2	2	no	—
66	#106×C₈₉	—	2	2	no	—
67	#106×C₈₉	—	2	1	no	—
68	#104×C₈₉	—	2	2	no	—
69	#104×C₈₉	—	2	2	no	—
70	#104×C₈₉	—	4	4	no	—
71	#99×C₈₉	—	4	4	no	—
72	#97×C₈₉	—	2	2	no	—
73	C₄×C₈×C₈₉	—	2	2	no	—
74	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
75	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
78	#41×C₈₉	—	2	2	no	—
79	#41×C₈₉	—	2	2	no	—
80	#41×C₈₉	—	2	2	no	—
81	#41×C₈₉	—	2	2	no	—
82	C₂×Q₈×C₈₉	—	4	2	no	—
83	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
84	C₂×D₄×C₈₉	—	2	2	no	—
85	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₈₉	—	2	1	no	—
87	C₂²×C₄×C₈₉	—	2	2	no	—
88	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
89	Q₁₆×C₈₉	—	2	1	no	—
90	Q₁₆×C₈₉	—	2	1	no	—
91	SD₁₆×C₈₉	—	2	2	no	—
92	SD₁₆×C₈₉	—	2	2	no	—
93	D₈×C₈₉	—	2	1	no	—
94	D₈×C₈₉	—	2	1	no	—
95	M₁₆×C₈₉	—	2	2	no	—
96	M₁₆×C₈₉	—	2	2	no	—
97	M₁₆×C₈₉	—	2	2	no	—
98	M₁₆×C₈₉	—	2	2	no	—
99	M₁₆×C₈₉	—	4	2	no	—
100	M₁₆×C₈₉	—	4	2	no	—
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	4	2	no	—
104	C₂×C₈×C₈₉	—	4	2	no	—
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
115	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
116	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	4	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	4	4	no	—
120	C₄⋊C₄×C₈₉	—	4	2	no	—
121	C₂²⋊C₄×C₈₉	—	2	2	no	—
122	C₂²⋊C₄×C₈₉	—	2	4	no	—
123	C₄²×C₈₉	—	2	1	no	—
124	C₄²×C₈₉	—	2	2	no	—
125	C₄²×C₈₉	—	2	4	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₈₉	—	2	2	no	—
129	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
130	Q₈×C₈₉	—	2	2	no	—
131	Q₈×C₈₉	—	2	2	no	—
132	Q₈×C₈₉	—	8	4	no	—
133	D₄×C₈₉	—	2	2	no	—
134	D₄×C₈₉	—	2	2	no	—
135	D₄×C₈₉	—	4	2	no	—
136	D₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	2	1	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	4	no	—
144	C₂×C₄×C₈₉	—	4	2	no	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
151	C₈×C₈₉	—	4	2	no	—
152	C₈×C₈₉	—	4	2	no	—
153	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
157	C₈×C₈₉	#124	1	1	yes	characteristic
158	C₈×C₈₉	#124	1	1	yes	characteristic
159	C₈×C₈₉	#124	1	1	yes	characteristic
160	C₈×C₈₉	#124	1	1	yes	characteristic
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	2	no	—
163	C₂²×C₈₉	—	8	2	no	—
164	C₂²×C₈₉	#535	1	1	yes	characteristic
165	C₂²×C₈₉	#554	1	1	yes	characteristic
166	C₂²×C₈₉	#555	1	1	yes	characteristic
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	—	4	2	no	—
170	C₄×C₈₉	—	8	2	no	—
171	C₄×C₈₉	—	8	4	no	—
172	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
174	C₄×C₈₉	#554	1	1	yes	characteristic
175	C₄×C₈₉	#555	1	1	yes	characteristic
176	#29154	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂×C₈₉	—	2	1	no	—
178	C₂×C₈₉	—	2	1	no	—
179	C₂×C₈₉	—	2	1	no	—
180	C₂×C₈₉	—	8	2	no	—
181	C₂×C₈₉	#2563	1	1	yes	characteristic
182	#3160	C₂×C₈₉	1	1	yes	characteristic, complemented
183	C₂×#443	C₂×C₈₉	1	1	yes	characteristic
184	#2031	C₂×C₈₉	1	4	yes	complemented
185	#1976	C₂×C₈₉	1	2	yes	—
186	#1833	C₂×C₈₉	1	2	yes	complemented
187	#1814	C₂×C₈₉	1	2	yes	complemented
188	#1812	C₂×C₈₉	1	2	yes	complemented
189	#1772	C₂×C₈₉	1	1	yes	characteristic, complemented
190	C₈₉	#29154	1	1	yes	characteristic, direct product factor, Sylow
191	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
192	C₂×#131	C₂²×C₈₉	1	2	yes	—
193	#533	C₂²×C₈₉	1	2	yes	—
194	#500	—	2	1	no	—
195	#499	—	2	1	no	—
196	#499	—	2	1	no	—
197	#498	—	2	1	no	—
198	#443	—	2	1	no	—
199	#443	—	2	1	no	—
200	#443	—	2	1	no	—
201	#443	—	2	1	no	—
202	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
203	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
204	#427	C₂²×C₈₉	1	1	yes	characteristic
205	#427	C₂²×C₈₉	1	2	yes	—
206	#427	C₂²×C₈₉	1	2	yes	—
207	#421	C₂²×C₈₉	1	4	yes	—
208	C₂×#104	C₂²×C₈₉	1	2	yes	—
209	#410	C₂²×C₈₉	1	4	yes	—
210	#405	C₂²×C₈₉	1	2	yes	—
211	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
212	#344	C₂²×C₈₉	1	4	yes	—
213	#339	C₂²×C₈₉	1	2	yes	—
214	H(3,ℤ/4)	C₂²×C₈₉	1	4	yes	—
215	C₂×#41	C₂³×C₈₉	1	2	yes	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	C₂×Q₁₆	Q₈×C₈₉	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	Q₈×C₈₉	1	2	yes	complemented
222	C₂×D₈	Q₈×C₈₉	1	1	yes	characteristic, complemented
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
229	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
233	C₄⋊Q₈	—	2	2	no	—
234	#125	—	2	2	no	—
235	C₂²⋊Q₈	—	4	2	no	—
236	C₄×Q₈	—	4	2	no	—
237	#117	C₂³×C₈₉	1	4	yes	—
238	#117	C₂³×C₈₉	1	2	yes	—
239	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
240	#108	—	2	1	no	—
241	#108	—	2	1	no	—
242	#108	—	2	1	no	—
243	#108	—	2	1	no	—
244	#107	—	2	2	no	—
245	#107	—	2	2	no	—
246	#107	—	2	1	no	—
247	#106	—	2	2	no	—
248	#106	—	2	2	no	—
249	#106	—	2	1	no	—
250	#104	—	2	2	no	—
251	#104	—	2	2	no	—
252	#104	—	4	4	no	—
253	#99	—	4	4	no	—
254	#97	—	2	2	no	—
255	C₄×C₈	—	2	2	no	—
256	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
257	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
258	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
259	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	4	2	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
291	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
293	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
294	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
295	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
296	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
297	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
298	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	4	no	—
308	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	8	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
328	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
332	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
336	C₈	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
337	C₈	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
338	C₈	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
339	C₈	#124×C₈₉	1	1	yes	characteristic
340	C₈	#124×C₈₉	1	1	yes	characteristic
341	C₈	#124×C₈₉	1	1	yes	characteristic
342	C₈	#124×C₈₉	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#535×C₈₉	1	1	yes	characteristic
347	C₂²	#554×C₈₉	1	1	yes	characteristic
348	C₂²	#555×C₈₉	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	4	no	—
353	C₄	—	8	2	no	—
354	C₄	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
355	C₄	C₂×#124×C₈₉	1	1	yes	characteristic
356	C₄	#554×C₈₉	1	1	yes	characteristic
357	C₄	#555×C₈₉	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₈₉×#2563	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
socle	C₂×C₈₉	#2563
prime radical	C₂×#135×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3160	C₂	semidirect
3. C₈₉×#2031	C₂	semidirect
5. C₈₉×#1833	C₂	semidirect
6. C₈₉×#1814	C₂	semidirect
7. C₈₉×#1812	C₂	semidirect
8. C₈₉×#1772	C₂	semidirect
38. C₂×Q₁₆×C₈₉	Q₈	semidirect
39. C₂×SD₁₆×C₈₉	Q₈	semidirect
40. C₂×D₈×C₈₉	Q₈	semidirect
176. #29154	C₈₉	direct (N × H)
182. #3160	C₂×C₈₉	semidirect
184. #2031	C₂×C₈₉	semidirect
186. #1833	C₂×C₈₉	semidirect
187. #1814	C₂×C₈₉	semidirect
188. #1812	C₂×C₈₉	semidirect
189. #1772	C₂×C₈₉	semidirect
190. C₈₉	#29154	direct (N × H)
220. C₂×Q₁₆	Q₈×C₈₉	semidirect
221. C₂×SD₁₆	Q₈×C₈₉	semidirect
222. C₂×D₈	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3160	2
C₂×#443×C₈₉	2
C₈₉×#2031	2
C₈₉×#1976	2
C₈₉×#1833	2
C₈₉×#1814	2
C₈₉×#1812	2
C₈₉×#1772	2
#29154	89

Subgroups — C89×#29154 #36407002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29154 #36407002