Subgroups — C₄×Q₃₂×D₈₉ #36707699

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₈₉	C₂	1	2	yes	maximal
3	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#15667958	C₂	1	2	yes	maximal, complemented
7	#26022108	C₂	1	2	yes	maximal, complemented
8	#366×D₈₉	C₂	1	1	yes	maximal, characteristic
9	#358×D₈₉	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	2	yes	—
13	C₄×Q₈×D₈₉	—	2	2	no	—
14	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
15	C₂×#15667958	C₂²	1	2	yes	—
16	Q₃₂×D₈₉	C₄	1	8	yes	direct product factor
17	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
21	#19151692	C₂²	1	2	yes	—
22	C₄×#13083564	—	2	2	no	—
23	#19151690	—	2	2	no	—
24	#107×D₈₉	C₂²	1	2	yes	—
25	#103×D₈₉	—	2	2	no	—
26	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
27	#19151424	C₂²	1	1	yes	characteristic
28	#19151423	C₂²	1	2	yes	—
29	#19151403	C₂²	1	2	yes	—
30	#19151402	C₂²	1	2	yes	—
31	#19151401	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
37	#366×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	2	yes	—
39	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₈₉	—	2	2	no	—
41	C₂×Dic₃₅₆	C₂³	1	2	yes	—
42	C₂×#13083564	—	2	2	no	—
43	Q₁₆×D₈₉	—	2	4	no	—
44	Q₁₆×D₈₉	C₂×C₄	1	8	yes	—
45	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
46	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
47	Q₈×Dic₈₉	—	2	2	no	—
48	C₄×Dic₁₇₈	—	2	2	no	—
49	C₄×Dic₁₇₈	—	4	2	no	—
50	#15667977	—	2	2	no	—
51	#15667977	—	4	2	no	—
52	C₄⋊C₄×D₈₉	—	2	2	no	—
53	C₄⋊C₄×D₈₉	—	4	2	no	—
54	C₄²×D₈₉	—	4	2	no	—
55	C₄²×D₈₉	D₄	1	1	yes	characteristic
56	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
57	#15667958	C₂×C₄	1	8	yes	complemented
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₁₆×D₈₉	—	2	2	no	—
61	C₁₆×D₈₉	C₂×C₄	1	4	yes	—
62	#15667946	C₂³	1	2	yes	—
63	#15667945	—	2	2	no	—
64	#15667937	—	2	2	no	—
65	#15667936	—	2	2	no	—
66	#15667935	—	2	2	no	—
67	C₄×#10623494	D₄	1	1	yes	characteristic
68	C₈×Dic₈₉	D₄	1	1	yes	characteristic
69	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
71	C₄×Q₈×C₈₉	—	2	2	no	—
72	Q₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#103×C₈₉	—	2	2	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₇₈	—	2	2	no	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	Q₈×D₈₉	—	2	8	no	—
81	Q₈×D₈₉	—	4	4	no	—
82	C₂×C₄×D₈₉	—	4	2	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
85	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
86	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
87	Dic₃₅₆	—	2	2	no	—
88	Dic₃₅₆	C₂²×C₄	1	4	yes	—
89	#13083564	—	2	8	no	—
90	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
91	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
92	C₈×D₈₉	—	2	2	no	—
93	C₈×D₈₉	C₂²×C₄	1	2	yes	—
94	C₈×D₈₉	#41	1	2	yes	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	C₄⋊Dic₈₉	—	4	2	no	—
97	#13083553	—	4	2	no	—
98	#13083553	—	4	2	no	—
99	#13083553	—	4	2	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	—	4	2	no	—
103	C₄×Dic₈₉	D₈	1	1	yes	characteristic
104	C₄×Dic₈₉	D₈	1	1	yes	characteristic
105	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
106	#13083549	—	2	1	no	—
107	#13083549	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₈₉	—	2	2	no	—
109	Q₁₆×C₈₉	—	2	2	no	—
110	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	4	2	no	—
115	C₄²×C₈₉	—	4	2	no	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	1	no	—
118	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
119	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₈₉	—	2	4	no	—
121	Q₈×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
132	C₄×D₈₉	—	2	2	no	—
133	C₄×D₈₉	—	4	8	no	—
134	C₄×D₈₉	—	8	4	no	—
135	C₄×D₈₉	Q₃₂	1	4	yes	direct product factor
136	C₄×D₈₉	C₄×D₄	1	2	yes	—
137	C₄×D₈₉	#135	1	2	yes	—
138	Dic₁₇₈	—	2	4	no	—
139	Dic₁₇₈	—	4	2	no	—
140	Dic₁₇₈	—	4	8	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
144	#10623494	—	2	1	no	—
145	#10623494	C₄×D₄	1	2	yes	—
146	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₈₉	C₄×D₈	1	2	yes	—
148	C₂×D₈₉	C₂×Q₃₂	1	2	yes	—
149	C₂×D₈₉	#508	1	2	yes	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	4	4	no	—
152	C₄×C₈₉	—	8	2	no	—
153	C₄×C₈₉	C₂×Q₃₂	1	2	yes	complemented
154	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
156	Dic₈₉	—	2	1	no	—
157	Dic₈₉	—	4	4	no	—
158	Dic₈₉	—	8	2	no	—
159	Dic₈₉	C₄×D₈	1	2	yes	—
160	Dic₈₉	C₂×Q₃₂	1	2	yes	complemented
161	C₂×C₄×Q₃₂	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
165	D₈₉	C₄×Q₃₂	1	4	yes	direct product factor
166	C₂²×Q₃₂	—	89	1	no	—
167	C₂×C₄×Q₁₆	—	89	2	no	—
168	C₄×Q₃₂	—	89	1	no	—
169	C₄×Q₃₂	—	89	2	no	—
170	C₄×Q₃₂	—	89	1	no	—
171	C₄×Q₃₂	—	89	2	no	—
172	C₄×Q₃₂	—	89	1	no	—
173	C₄×Q₃₂	D₈₉	1	1	yes	characteristic, direct product factor
174	C₂×#366	—	89	1	no	—
175	C₂×#358	—	89	2	no	—
176	C₂×C₄×C₁₆	—	89	1	no	—
177	C₈₉	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	89	2	no	—
179	C₂×C₄×Q₈	—	178	2	no	—
180	C₂×Q₃₂	—	89	2	no	—
181	C₂×Q₃₂	—	89	8	no	—
182	C₂×Q₃₂	—	89	1	no	—
183	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
184	C₂²×C₁₆	—	89	1	no	—
185	C₂²×C₁₆	—	89	1	no	—
186	C₄×Q₁₆	—	89	2	no	—
187	C₄×Q₁₆	—	89	2	no	—
188	C₄×Q₁₆	—	89	2	no	—
189	C₄×Q₁₆	—	178	2	no	—
190	C₄×Q₁₆	—	178	2	no	—
191	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
192	C₂×#107	—	89	2	no	—
193	C₂×#103	—	178	2	no	—
194	C₂×C₄×C₈	—	89	1	no	—
195	#366	—	89	1	no	—
196	#366	—	89	2	no	—
197	#366	C₂×D₈₉	1	1	yes	characteristic
198	#358	—	89	2	no	—
199	#358	—	89	2	no	—
200	#358	—	89	2	no	—
201	#358	C₂×D₈₉	1	2	yes	—
202	C₄×C₁₆	—	89	1	no	—
203	C₄×C₁₆	—	89	1	no	—
204	C₄×C₁₆	—	89	1	no	—
205	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
206	C₂²×Q₈	—	178	2	no	—
207	C₂×Q₁₆	—	89	8	no	—
208	C₂×Q₁₆	—	89	2	no	—
209	C₂×Q₁₆	—	178	4	no	—
210	C₂×Q₁₆	—	178	2	no	—
211	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
212	C₂²×C₈	—	89	1	no	—
213	C₂²×C₈	—	89	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	356	2	no	—
216	C₄×Q₈	—	356	2	no	—
217	C₄×Q₈	—	178	2	no	—
218	C₄×Q₈	—	178	2	no	—
219	C₄×Q₈	—	178	2	no	—
220	C₂×C₄⋊C₄	—	356	2	no	—
221	C₂×C₄⋊C₄	—	178	2	no	—
222	C₂×C₄²	—	89	1	no	—
223	C₂×C₄²	—	356	2	no	—
224	Q₃₂	—	89	8	no	—
225	Q₃₂	—	89	4	no	—
226	Q₃₂	C₄×D₈₉	1	4	yes	direct product factor
227	C₂×C₁₆	—	89	1	no	—
228	C₂×C₁₆	—	89	4	no	—
229	C₂×C₁₆	—	89	1	no	—
230	C₂×C₁₆	—	178	2	no	—
231	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
233	#107	—	89	2	no	—
234	#107	—	178	2	no	—
235	#107	C₂²×D₈₉	1	2	yes	—
236	#103	—	2	2	no	—
237	#103	—	178	2	no	—
238	#103	—	178	2	no	—
239	#103	—	178	2	no	—
240	C₄×C₈	—	89	1	no	—
241	C₄×C₈	—	89	1	no	—
242	C₄×C₈	—	89	1	no	—
243	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	356	2	no	—
246	C₂×Q₈	—	356	4	no	—
247	C₂×Q₈	—	178	8	no	—
248	C₂×Q₈	—	178	2	no	—
249	C₂²×C₄	—	89	1	no	—
250	C₂²×C₄	—	89	1	no	—
251	C₂²×C₄	—	89	1	no	—
252	C₂²×C₄	—	356	2	no	—
253	C₂²×C₄	—	356	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	89	4	no	—
256	Q₁₆	—	178	8	no	—
257	Q₁₆	—	178	2	no	—
258	Q₁₆	C₂×C₄×D₈₉	1	4	yes	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	2	no	—
261	C₂×C₈	—	89	2	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	356	2	no	—
269	C₄⋊C₄	—	356	2	no	—
270	C₄⋊C₄	—	356	2	no	—
271	C₄⋊C₄	—	356	2	no	—
272	C₄⋊C₄	—	178	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	89	1	no	—
275	C₄²	—	89	1	no	—
276	C₄²	—	89	1	no	—
277	C₄²	—	356	2	no	—
278	C₄²	—	356	2	no	—
279	C₄²	—	356	2	no	—
280	C₄²	D₄×D₈₉	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	89	2	no	—
283	C₁₆	—	178	1	no	—
284	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
285	C₂³	—	89	1	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	356	8	no	—
289	Q₈	—	356	2	no	—
290	Q₈	—	178	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	89	4	no	—
296	C₂×C₄	—	89	2	no	—
297	C₂×C₄	—	89	2	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	356	8	no	—
300	C₂×C₄	—	356	2	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	—	712	4	no	—
304	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
305	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	89	2	no	—
309	C₈	—	178	1	no	—
310	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
311	C₈	#41×D₈₉	1	1	yes	characteristic
312	C₂²	—	89	2	no	—
313	C₂²	—	89	2	no	—
314	C₂²	—	89	2	no	—
315	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	89	2	no	—
320	C₄	—	89	2	no	—
321	C₄	—	356	4	no	—
322	C₄	—	178	1	no	—
323	C₄	—	712	2	no	—
324	C₄	Q₃₂×D₈₉	1	2	yes	direct product factor
325	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
326	C₄	#135×D₈₉	1	1	yes	characteristic
327	C₂	—	89	4	no	—
328	C₂	C₄×D₈×D₈₉	1	1	yes	characteristic
329	C₂	C₂×Q₃₂×D₈₉	1	1	yes	characteristic
330	C₂	#508×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×Q₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₄×Q₃₂	D₈₉
hypocenter	C₈₉	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₇₁₂	C₂	semidirect
4. Q₃₂×Dic₈₉	C₂	semidirect
5. #26022110	C₂	semidirect
6. C₄×#15667958	C₂	semidirect
7. #26022108	C₂	semidirect
11. C₄×Q₃₂×C₈₉	C₂	semidirect
16. Q₃₂×D₈₉	C₄	direct (N × H)
56. Dic₇₁₂	C₂×C₄	semidirect
57. #15667958	C₂×C₄	semidirect
72. Q₃₂×C₈₉	C₂×C₄	semidirect
135. C₄×D₈₉	Q₃₂	direct (N × H)
153. C₄×C₈₉	C₂×Q₃₂	semidirect
160. Dic₈₉	C₂×Q₃₂	semidirect
165. D₈₉	C₄×Q₃₂	direct (N × H)
173. C₄×Q₃₂	D₈₉	direct (N × H)
177. C₈₉	C₂×C₄×Q₃₂	semidirect
226. Q₃₂	C₄×D₈₉	direct (N × H)
324. C₄	Q₃₂×D₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₈₉	2
C₄×Q₁₆×D₈₉	2
C₄×Dic₇₁₂	2
Q₃₂×Dic₈₉	2
#26022110	2
C₄×#15667958	2
#26022108	2
#366×D₈₉	2
#358×D₈₉	2
C₄×C₁₆×D₈₉	2
C₄×Q₃₂×C₈₉	2
C₂×C₄×Q₃₂	89

Subgroups — C4×Q32×D89 #36707699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₃₂×D₈₉ #36707699