Subgroups — C₂×SD₃₂×Dic₈₉ #36707696

Order 22784.

742 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	SD₃₂×Dic₈₉	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151402	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151399	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₈₉	—	2	1	no	—
10	C₂×D₄×Dic₈₉	—	2	1	no	—
11	C₂²×#13083549	C₂²	1	1	yes	characteristic
12	Q₁₆×Dic₈₉	—	2	2	no	—
13	Q₁₆×Dic₈₉	C₂²	1	4	yes	complemented
14	D₈×Dic₈₉	—	2	2	no	—
15	D₈×Dic₈₉	C₂²	1	4	yes	—
16	C₂×#15667946	C₂²	1	1	yes	characteristic
17	C₂×#15667946	C₂²	1	1	yes	characteristic
18	C₂×#15667936	—	2	1	no	—
19	C₂×#15667933	—	2	1	no	—
20	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
21	#19151426	C₂²	1	4	yes	complemented
22	#19151402	C₂²	1	4	yes	complemented
23	#19151399	C₂²	1	4	yes	—
24	C₁₆×Dic₈₉	C₂²	1	4	yes	complemented
25	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×SD₃₂×C₈₉	C₄	1	8	yes	complemented
28	C₂×SD₃₂×C₈₉	C₂²	1	4	yes	—
29	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₂³×Dic₈₉	—	4	1	no	—
31	C₂²×#10623494	D₄	1	1	yes	characteristic
32	Q₈×Dic₈₉	—	2	4	no	—
33	Q₈×Dic₈₉	—	4	2	no	—
34	D₄×Dic₈₉	—	2	4	no	—
35	D₄×Dic₈₉	—	4	2	no	—
36	C₂×C₄⋊Dic₈₉	—	2	1	no	—
37	C₂×C₄⋊Dic₈₉	—	2	1	no	—
38	C₂×C₄⋊Dic₈₉	—	4	1	no	—
39	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
40	C₂×C₄×Dic₈₉	—	4	1	no	—
41	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
42	C₂×#13083549	—	2	2	no	—
43	C₂×#13083549	C₂³	1	2	yes	—
44	#15667946	—	2	1	no	—
45	#15667946	—	2	1	no	—
46	#15667946	C₂³	1	2	yes	—
47	#15667946	C₂³	1	2	yes	—
48	#15667936	—	2	4	no	—
49	#15667933	—	2	4	no	—
50	C₈×Dic₈₉	D₄	1	2	yes	—
51	C₈×Dic₈₉	C₂³	1	2	yes	—
52	C₂²×Q₈×C₈₉	—	2	1	no	—
53	C₂²×D₄×C₈₉	—	2	1	no	—
54	C₂×Q₁₆×C₈₉	—	2	1	no	—
55	C₂×Q₁₆×C₈₉	—	2	2	no	—
56	C₂×Q₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
57	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
58	C₂×D₈×C₈₉	—	2	1	no	—
59	C₂×D₈×C₈₉	—	2	2	no	—
60	C₂×D₈×C₈₉	C₂×C₄	1	4	yes	—
61	C₂×D₈×C₈₉	C₂³	1	2	yes	—
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	SD₃₂×C₈₉	C₂×C₄	1	16	yes	complemented
64	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
65	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
66	C₂²×Dic₈₉	—	4	1	no	—
67	C₂²×Dic₈₉	—	4	1	no	—
68	C₂²×Dic₈₉	—	4	4	no	—
69	C₂²×Dic₈₉	—	8	4	no	—
70	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
71	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	C₂×D₄	1	2	yes	—
74	C₄⋊Dic₈₉	—	2	2	no	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	4	1	no	—
77	C₄⋊Dic₈₉	—	4	1	no	—
78	C₄⋊Dic₈₉	—	4	4	no	—
79	C₂²⋊Dic₈₉	—	4	4	no	—
80	C₄×Dic₈₉	—	4	4	no	—
81	C₄×Dic₈₉	D₈	1	2	yes	—
82	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
83	#13083549	—	2	2	no	—
84	C₂⁴×C₈₉	—	4	1	no	—
85	C₂×Q₈×C₈₉	—	2	4	no	—
86	C₂×Q₈×C₈₉	—	2	2	no	—
87	C₂×Q₈×C₈₉	—	4	1	no	—
88	C₂×Q₈×C₈₉	—	4	2	no	—
89	C₂×D₄×C₈₉	—	2	4	no	—
90	C₂×D₄×C₈₉	—	2	2	no	—
91	C₂×D₄×C₈₉	—	4	1	no	—
92	C₂×D₄×C₈₉	—	4	2	no	—
93	C₂²×C₄×C₈₉	—	4	1	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₈₉	—	2	4	no	—
96	Q₁₆×C₈₉	—	2	2	no	—
97	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	complemented
98	D₈×C₈₉	—	2	4	no	—
99	D₈×C₈₉	—	2	2	no	—
100	D₈×C₈₉	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	#41	1	2	yes	—
104	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
105	C₁₆×C₈₉	C₂²×C₄	1	4	yes	complemented
106	C₂³×C₈₉	—	4	2	no	—
107	C₂³×C₈₉	—	4	4	no	—
108	C₂³×C₈₉	—	8	2	no	—
109	C₂³×C₈₉	—	8	2	no	—
110	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
111	Q₈×C₈₉	—	2	4	no	—
112	Q₈×C₈₉	—	4	2	no	—
113	Q₈×C₈₉	—	4	4	no	—
114	D₄×C₈₉	—	2	4	no	—
115	D₄×C₈₉	—	4	2	no	—
116	D₄×C₈₉	—	4	4	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	4	4	no	—
119	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#135	1	2	yes	—
122	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₈₉	—	2	2	no	—
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	8	2	no	—
127	C₂×Dic₈₉	—	8	2	no	—
128	C₂×Dic₈₉	—	8	8	no	—
129	C₂×Dic₈₉	SD₃₂	1	4	yes	direct product factor
130	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
131	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
132	C₈×C₈₉	C₄×D₄	1	2	yes	—
133	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
135	#10623494	—	2	2	no	—
136	C₂²×C₈₉	—	4	4	no	—
137	C₂²×C₈₉	—	8	2	no	—
138	C₂²×C₈₉	—	8	2	no	—
139	C₂²×C₈₉	—	8	8	no	—
140	C₂²×C₈₉	C₄×D₈	1	2	yes	—
141	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
142	C₂²×C₈₉	#508	1	2	yes	—
143	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₈₉	—	4	4	no	—
145	C₄×C₈₉	C₄×D₈	1	2	yes	—
146	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
148	Dic₈₉	—	2	2	no	—
149	Dic₈₉	—	8	2	no	—
150	Dic₈₉	—	8	2	no	—
151	Dic₈₉	C₂×SD₃₂	1	4	yes	direct product factor
152	C₂×C₄×SD₃₂	—	89	1	no	maximal, Sylow
153	C₂×C₈₉	—	8	4	no	—
154	C₂×C₈₉	C₄×SD₃₂	1	4	yes	complemented
155	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
156	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
158	C₂²×SD₃₂	D₈₉	1	1	yes	characteristic
159	C₂×C₄×Q₁₆	—	89	1	no	—
160	C₂×C₄×D₈	—	89	1	no	—
161	C₄×SD₃₂	—	89	8	no	—
162	C₂×#367	—	89	1	no	—
163	C₂×#358	—	89	1	no	—
164	C₂×#357	—	89	1	no	—
165	C₂×C₄×C₁₆	—	89	1	no	—
166	C₈₉	C₂×C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
168	C₂²×D₈	C₂×D₈₉	1	1	yes	characteristic
169	C₂×C₄×Q₈	—	178	1	no	—
170	C₂×C₄×D₄	—	178	1	no	—
171	C₂×SD₃₂	Dic₈₉	1	8	yes	direct product factor
172	C₂×SD₃₂	C₂×D₈₉	1	4	yes	—
173	C₂²×C₁₆	—	89	1	no	—
174	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
175	C₄×Q₁₆	—	89	4	no	—
176	C₄×Q₁₆	—	178	2	no	—
177	C₄×D₈	—	89	4	no	—
178	C₄×D₈	—	178	2	no	—
179	C₂×#107	—	89	1	no	—
180	C₂×#107	—	89	1	no	—
181	C₂×#103	—	178	1	no	—
182	C₂×#102	—	178	1	no	—
183	C₂×C₄×C₈	—	89	1	no	—
184	#367	—	89	4	no	—
185	#358	—	89	4	no	—
186	#357	—	89	4	no	—
187	C₄×C₁₆	—	89	4	no	—
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂³×C₄	—	356	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	C₂×Dic₈₉	1	4	yes	complemented
194	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	C₂×Dic₈₉	1	4	yes	—
198	C₂×D₈	C₂²×D₈₉	1	2	yes	—
199	C₂²×C₈	—	89	1	no	—
200	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
201	C₄×Q₈	—	356	2	no	—
202	C₄×Q₈	—	178	4	no	—
203	C₄×D₄	—	356	2	no	—
204	C₄×D₄	—	178	4	no	—
205	C₂×C₄⋊C₄	—	356	1	no	—
206	C₂×C₄⋊C₄	—	178	1	no	—
207	C₂×C₄⋊C₄	—	178	1	no	—
208	C₂×C₂²⋊C₄	—	356	1	no	—
209	C₂×C₄²	—	89	1	no	—
210	C₂×C₄²	—	356	1	no	—
211	SD₃₂	C₂×Dic₈₉	1	16	yes	direct product factor
212	C₂×C₁₆	—	89	2	no	—
213	C₂×C₁₆	—	178	2	no	—
214	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	complemented
215	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
216	#107	—	89	2	no	—
217	#107	—	89	2	no	—
218	#107	—	178	1	no	—
219	#107	—	178	1	no	—
220	#103	—	178	4	no	—
221	#102	—	178	4	no	—
222	C₄×C₈	—	89	2	no	—
223	C₄×C₈	—	89	2	no	—
224	C₂⁴	—	4	1	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	89	1	no	—
235	C₂²×C₄	—	89	1	no	—
236	C₂²×C₄	—	356	4	no	—
237	C₂²×C₄	—	356	1	no	—
238	C₂²×C₄	—	356	1	no	—
239	C₂²×C₄	—	712	4	no	—
240	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	C₂²×Dic₈₉	1	4	yes	complemented
244	D₈	—	2	4	no	—
245	D₈	—	2	2	no	—
246	D₈	C₂²×Dic₈₉	1	4	yes	—
247	C₂×C₈	—	89	2	no	—
248	C₂×C₈	—	178	2	no	—
249	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
250	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
251	C₂×C₈	#13083573	1	2	yes	—
252	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	356	4	no	—
254	C₄⋊C₄	—	356	1	no	—
255	C₄⋊C₄	—	356	1	no	—
256	C₄⋊C₄	—	178	2	no	—
257	C₄⋊C₄	—	178	2	no	—
258	C₂²⋊C₄	—	356	4	no	—
259	C₄²	—	89	2	no	—
260	C₄²	—	89	2	no	—
261	C₄²	—	356	4	no	—
262	C₁₆	—	178	2	no	—
263	C₁₆	C₂²×Dic₈₉	1	4	yes	complemented
264	C₂³	—	4	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	8	2	no	—
267	C₂³	—	8	2	no	—
268	C₂³	D₈×D₈₉	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	4	2	no	—
272	D₄	—	2	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	89	4	no	—
278	C₂×C₄	—	89	2	no	—
279	C₂×C₄	—	89	2	no	—
280	C₂×C₄	—	356	2	no	—
281	C₂×C₄	—	356	2	no	—
282	C₂×C₄	—	178	2	no	—
283	C₂×C₄	—	712	8	no	—
284	C₂×C₄	—	712	2	no	—
285	C₂×C₄	—	712	2	no	—
286	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
287	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
288	C₂×C₄	#15668031	1	2	yes	—
289	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
290	C₈	—	178	2	no	—
291	C₈	D₄×Dic₈₉	1	2	yes	—
292	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
293	C₈	C₂×#13083573	1	1	yes	characteristic
294	C₂²	—	4	4	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	8	no	—
297	C₂²	—	8	2	no	—
298	C₂²	D₈×Dic₈₉	1	2	yes	—
299	C₂²	SD₃₂×D₈₉	1	2	yes	—
300	C₂²	#19152029	1	2	yes	—
301	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	89	4	no	—
304	C₄	—	178	2	no	—
305	C₄	—	712	2	no	—
306	C₄	—	712	2	no	—
307	C₄	D₈×Dic₈₉	1	2	yes	—
308	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
309	C₄	C₂×#15668031	1	1	yes	characteristic
310	C₂	—	8	4	no	—
311	C₂	SD₃₂×Dic₈₉	1	4	yes	direct product factor
312	C₂	C₂×D₈×Dic₈₉	1	1	yes	characteristic
313	C₂	C₂×SD₃₂×D₈₉	1	1	yes	characteristic
314	C₂	C₂×#19152029	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂²×SD₃₂×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₈×C₈₉	C₂²
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂²×SD₃₂	D₈₉
hypocenter	C₈₉	C₂×C₄×SD₃₂
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×SD₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×Dic₈₉	C₂	semidirect
3. SD₃₂×Dic₈₉	C₂	direct (N × H)
4. C₂×#19151426	C₂	semidirect
5. C₂×#19151402	C₂	semidirect
7. C₂×C₁₆×Dic₈₉	C₂	semidirect
13. Q₁₆×Dic₈₉	C₂²	semidirect
21. #19151426	C₂²	semidirect
22. #19151402	C₂²	semidirect
24. C₁₆×Dic₈₉	C₂²	semidirect
27. C₂×SD₃₂×C₈₉	C₄	semidirect
56. C₂×Q₁₆×C₈₉	C₂×C₄	semidirect
63. SD₃₂×C₈₉	C₂×C₄	semidirect
64. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
97. Q₁₆×C₈₉	C₂²×C₄	semidirect
105. C₁₆×C₈₉	C₂²×C₄	semidirect
129. C₂×Dic₈₉	SD₃₂	direct (N × H)
151. Dic₈₉	C₂×SD₃₂	direct (N × H)
154. C₂×C₈₉	C₄×SD₃₂	semidirect
166. C₈₉	C₂×C₄×SD₃₂	semidirect
171. C₂×SD₃₂	Dic₈₉	direct (N × H)
193. C₂×Q₁₆	C₂×Dic₈₉	semidirect
211. SD₃₂	C₂×Dic₈₉	direct (N × H)
214. C₂×C₁₆	C₂×Dic₈₉	semidirect
243. Q₁₆	C₂²×Dic₈₉	semidirect
263. C₁₆	C₂²×Dic₈₉	semidirect
311. C₂	SD₃₂×Dic₈₉	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×Q₁₆×Dic₈₉	2
C₂×D₈×Dic₈₉	2
SD₃₂×Dic₈₉	2
C₂×#19151426	2
C₂×#19151402	2
C₂×#19151399	2
C₂×C₁₆×Dic₈₉	2
C₂²×SD₃₂×C₈₉	2
C₂×C₄×SD₃₂	89

Subgroups — C2×SD32×Dic89 #36707696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×SD₃₂×Dic₈₉ #36707696