Subgroups — #36524127

Order 22784.

626 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029337	C₂	1	1	yes	maximal, characteristic
2	#26029333	C₂	1	1	yes	maximal, characteristic
3	#26019124	C₂	1	4	yes	maximal, complemented
4	#26019103	C₂	1	4	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic
6	#26018647	C₂	1	1	yes	maximal, characteristic
7	#26018647	C₂	1	1	yes	maximal, characteristic
8	#26018544	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
10	#19151729	—	2	2	no	—
11	#19151729	C₂²	1	4	yes	—
12	Q₈×#10623494	—	2	2	no	—
13	Q₈×#10623494	C₂²	1	4	yes	—
14	#19151655	—	2	2	no	—
15	#19151651	—	2	2	no	—
16	#19151651	C₂²	1	1	yes	characteristic
17	#19151651	C₂²	1	1	yes	characteristic
18	#19151651	C₂²	1	1	yes	characteristic
19	#19151651	C₂²	1	1	yes	characteristic
20	#19151351	C₂²	1	2	yes	—
21	#19151349	C₂²	1	2	yes	—
22	Q₈⋊#10623494	C₂²	1	4	yes	—
23	Q₈⋊#10623494	C₂²	1	4	yes	—
24	#19151320	C₂²	1	2	yes	—
25	#19151320	C₂²	1	2	yes	—
26	C₂²×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₈₉	C₄	1	2	yes	—
30	C₄×Q₁₆×C₈₉	C₄	1	2	yes	—
31	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
32	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
33	C₂×#107×C₈₉	C₄	1	1	yes	characteristic
34	C₂×#103×C₈₉	C₄	1	1	yes	characteristic
35	C₂×#103×C₈₉	C₄	1	1	yes	characteristic
36	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊#10623494	—	2	4	no	—
38	C₄⋊#10623494	—	2	4	no	—
39	C₄⋊#10623494	—	2	1	no	—
40	C₄⋊#10623494	D₄	1	2	yes	—
41	C₄⋊#10623494	C₂³	1	2	yes	—
42	C₄⋊#10623494	C₂³	1	2	yes	—
43	C₂²⋊#10623494	—	2	1	no	—
44	C₂²⋊#10623494	—	2	1	no	—
45	C₂²⋊#10623494	—	2	1	no	—
46	C₂²⋊#10623494	D₄	1	2	yes	—
47	#15667923	—	2	4	no	—
48	C₄×#10623494	—	2	4	no	—
49	C₄×#10623494	—	2	1	no	—
50	C₄×#10623494	—	2	1	no	—
51	C₄×#10623494	C₂³	1	2	yes	—
52	C₄×#10623494	C₂³	1	2	yes	—
53	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂×Q₁₆×C₈₉	—	2	4	no	—
56	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
57	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₄×Q₈×C₈₉	—	2	1	no	—
61	C₄×Q₈×C₈₉	—	2	1	no	—
62	C₄×Q₈×C₈₉	—	2	1	no	—
63	C₄×Q₈×C₈₉	—	2	1	no	—
64	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
65	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
66	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
67	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
69	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
70	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₈₉	—	2	1	no	—
73	C₂×C₄²×C₈₉	—	2	1	no	—
74	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂×C₄	1	2	yes	—
76	#107×C₈₉	C₂×C₄	1	2	yes	—
77	#103×C₈₉	C₂×C₄	1	2	yes	—
78	#103×C₈₉	C₂×C₄	1	2	yes	—
79	#103×C₈₉	C₂×C₄	1	2	yes	—
80	#103×C₈₉	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₂×#10623494	—	2	2	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	—	2	2	no	—
88	C₂×#10623494	C₂×D₄	1	4	yes	—
89	C₂×Q₈×C₈₉	—	2	1	no	—
90	C₂×Q₈×C₈₉	—	2	1	no	—
91	C₂×Q₈×C₈₉	—	2	2	no	—
92	C₂×Q₈×C₈₉	—	2	2	no	—
93	C₂×Q₈×C₈₉	—	2	2	no	—
94	C₂×Q₈×C₈₉	M₁₆	1	4	yes	—
95	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
96	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₈₉	—	2	1	no	—
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	C₂²×C₄×C₈₉	—	2	1	no	—
101	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
104	Q₁₆×C₈₉	—	2	4	no	—
105	Q₁₆×C₈₉	—	2	4	no	—
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	2	1	no	—
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
119	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
120	C₄²×C₈₉	—	2	2	no	—
121	C₄²×C₈₉	—	2	2	no	—
122	C₄²×C₈₉	—	2	2	no	—
123	C₄²×C₈₉	—	2	2	no	—
124	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	#41	1	1	yes	characteristic
127	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
129	Q₈×C₈₉	—	2	2	no	—
130	Q₈×C₈₉	—	2	4	no	—
131	Q₈×C₈₉	—	2	2	no	—
132	Q₈×C₈₉	—	2	2	no	—
133	Q₈×C₈₉	—	4	2	no	—
134	Q₈×C₈₉	C₂×M₁₆	1	4	yes	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	4	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	8	1	no	—
144	C₂×C₄×C₈₉	—	8	1	no	—
145	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
149	C₂×C₄×C₈₉	#131	1	2	yes	—
150	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	#10623494	—	2	4	no	—
158	#10623494	—	4	2	no	—
159	#10623494	—	4	2	no	—
160	#10623494	—	4	2	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	#435	1	2	yes	—
164	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₈₉	—	2	4	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	4	2	no	—
170	C₄×C₈₉	—	4	2	no	—
171	C₄×C₈₉	—	4	2	no	—
172	C₄×C₈₉	#435	1	2	yes	—
173	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
174	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
175	C₄×C₈₉	#578	1	2	yes	—
176	#17880	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	2	no	—
178	C₂×C₈₉	C₂×#435	1	1	yes	characteristic
179	C₂×C₈₉	#2950	1	1	yes	characteristic
180	C₂×C₈₉	#2957	1	1	yes	characteristic
181	#2983	—	89	1	no	—
182	#2982	—	89	1	no	—
183	C₂×C₄×Q₁₆	D₈₉	1	1	yes	characteristic
184	#1587	—	89	4	no	—
185	#1584	—	89	4	no	—
186	#1564	—	89	1	no	—
187	#1479	—	89	1	no	—
188	#1479	—	89	1	no	—
189	#1452	—	89	1	no	—
190	C₈₉	#17880	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₁₆	Dic₈₉	1	1	yes	characteristic
192	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
193	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
194	#446	—	89	4	no	—
195	#446	—	178	2	no	—
196	C₈×Q₈	—	89	4	no	—
197	C₈×Q₈	—	178	2	no	—
198	C₄×Q₁₆	Dic₈₉	1	2	yes	—
199	C₄×Q₁₆	Dic₈₉	1	2	yes	—
200	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
201	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
202	#432	—	178	2	no	—
203	#431	—	89	1	no	—
204	#431	—	89	1	no	—
205	#431	—	89	1	no	—
206	#431	—	89	1	no	—
207	#431	—	178	2	no	—
208	C₂×#107	Dic₈₉	1	1	yes	characteristic
209	C₂×#103	Dic₈₉	1	1	yes	characteristic
210	C₂×#103	Dic₈₉	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
212	#335	—	89	2	no	—
213	#334	—	89	2	no	—
214	Q₈⋊C₈	—	89	4	no	—
215	Q₈⋊C₈	—	89	4	no	—
216	#322	—	89	2	no	—
217	#322	—	89	2	no	—
218	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
219	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
220	C₂×Q₁₆	—	2	4	no	—
221	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
222	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
223	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
224	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
230	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
231	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
232	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
237	C₂×C₄²	—	2	1	no	—
238	C₂×C₄²	—	2	1	no	—
239	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
240	#107	C₂×Dic₈₉	1	2	yes	—
241	#107	C₂×Dic₈₉	1	2	yes	—
242	C₄⋊C₈	—	89	2	no	—
243	C₄⋊C₈	—	89	2	no	—
244	C₄⋊C₈	—	89	2	no	—
245	C₄⋊C₈	—	178	4	no	—
246	C₄⋊C₈	—	178	4	no	—
247	C₄⋊C₈	—	178	1	no	—
248	#103	C₂×Dic₈₉	1	2	yes	—
249	#103	C₂×Dic₈₉	1	2	yes	—
250	#103	C₂×Dic₈₉	1	2	yes	—
251	#103	C₂×Dic₈₉	1	2	yes	—
252	C₂²⋊C₈	—	89	2	no	—
253	C₂²⋊C₈	—	178	1	no	—
254	C₂²⋊C₈	—	178	1	no	—
255	C₂²⋊C₈	—	178	1	no	—
256	#97	—	178	4	no	—
257	C₄×C₈	—	89	2	no	—
258	C₄×C₈	—	89	2	no	—
259	C₄×C₈	—	178	4	no	—
260	C₄×C₈	—	178	1	no	—
261	C₄×C₈	—	178	1	no	—
262	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
263	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	#13083558	1	4	yes	—
272	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
273	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
280	C₂²×C₄	#13083573	1	1	yes	characteristic
281	Q₁₆	—	2	4	no	—
282	Q₁₆	—	2	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	89	4	no	—
286	C₂×C₈	—	178	2	no	—
287	C₂×C₈	—	178	2	no	—
288	C₂×C₈	—	178	2	no	—
289	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
290	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
291	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
300	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
306	C₄²	D₄×D₈₉	1	1	yes	characteristic
307	C₄²	#13083573	1	1	yes	characteristic
308	C₄²	C₂³×D₈₉	1	1	yes	characteristic
309	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	C₂×#13083558	1	4	yes	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
327	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
328	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×#13083558	1	2	yes	—
330	C₂×C₄	#15668019	1	2	yes	—
331	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
333	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	356	2	no	—
339	C₈	—	356	2	no	—
340	C₈	—	356	2	no	—
341	C₈	—	178	4	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	#19151670	1	2	yes	—
345	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	4	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	#19151670	1	2	yes	—
354	C₄	C₂²×#13083558	1	1	yes	characteristic
355	C₄	C₂×#15668019	1	1	yes	characteristic
356	C₄	#19152411	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	C₂×#19151670	1	1	yes	characteristic
359	C₂	#26029120	1	1	yes	characteristic
360	C₂	#26029169	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₄×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₈₉
hypocenter	C₈₉	#17880
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #26019124	C₂	semidirect
4. #26019103	C₂	semidirect
190. C₈₉	#17880	semidirect

Maximal subgroups

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

subgroup	index
#26029337	2
#26029333	2
#26019124	2
#26019103	2
#26019000	2
#26018647	2
#26018544	2
C₂×C₄×Q₁₆×C₈₉	2
#17880	89