Subgroups — #36572031

Order 22784.

692 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028910	C₂	1	2	yes	maximal
2	#26021006	C₂	1	2	yes	maximal, complemented
3	#26020853	C₂	1	4	yes	maximal, complemented
4	#26020351	C₂	1	2	yes	maximal
5	#26020290	C₂	1	1	yes	maximal, characteristic
6	#26020138	C₂	1	2	yes	maximal, complemented
7	#26020028	C₂	1	1	yes	maximal, characteristic
8	C₂×#427×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19151659	—	2	4	no	—
10	#19151655	—	2	2	no	—
11	#19151651	—	2	2	no	—
12	C₂×#15667948	C₂²	1	2	yes	—
13	#19151625	C₂²	1	4	yes	—
14	#19151562	C₂²	1	2	yes	—
15	C₂×C₂²⋊#10623494	C₂²	1	1	yes	characteristic
16	#19151554	C₂²	1	4	yes	—
17	C₂×#15667922	C₂²	1	2	yes	—
18	C₄×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	#19151363	C₂²	1	2	yes	—
20	#19151357	C₂²	1	4	yes	—
21	#19151357	C₂²	1	4	yes	—
22	#19151352	C₂²	1	2	yes	—
23	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#117×C₈₉	C₂²	1	2	yes	—
25	#427×C₈₉	C₄	1	1	yes	characteristic, complemented
26	#427×C₈₉	C₄	1	1	yes	characteristic, complemented
27	#427×C₈₉	C₄	1	2	yes	complemented
28	#427×C₈₉	C₂²	1	1	yes	characteristic
29	#427×C₈₉	C₂²	1	1	yes	characteristic
30	#427×C₈₉	C₂²	1	2	yes	—
31	C₂×#107×C₈₉	C₄	1	1	yes	characteristic, complemented
32	C₂×#107×C₈₉	C₄	1	1	yes	characteristic, complemented
33	C₂×#106×C₈₉	C₄	1	2	yes	complemented
34	C₂×#13083558	C₂³	1	4	yes	—
35	C₂²×#10623494	C₂³	1	2	yes	—
36	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
37	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
38	#15667948	—	2	4	no	—
39	C₄⋊#10623494	—	2	4	no	—
40	C₄⋊#10623494	—	2	4	no	—
41	C₂²⋊#10623494	—	2	2	no	—
42	C₂²⋊#10623494	—	2	2	no	—
43	C₂²⋊#10623494	D₄	1	2	yes	—
44	C₂²⋊#10623494	D₄	1	2	yes	—
45	#15667923	—	2	4	no	—
46	#15667922	—	2	4	no	—
47	C₄×#10623494	—	2	4	no	—
48	#15667919	—	2	1	no	—
49	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	—	2	4	no	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	#117×C₈₉	—	2	2	no	—
58	#117×C₈₉	—	2	2	no	—
59	#117×C₈₉	C₂×C₄	1	2	yes	—
60	#117×C₈₉	C₂×C₄	1	2	yes	—
61	#117×C₈₉	C₂³	1	2	yes	—
62	#117×C₈₉	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
66	C₂×C₄²×C₈₉	—	2	2	no	—
67	#107×C₈₉	C₂×C₄	1	2	yes	—
68	#107×C₈₉	C₂×C₄	1	2	yes	—
69	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
70	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
71	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
72	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
73	#106×C₈₉	C₂×C₄	1	2	yes	—
74	#106×C₈₉	C₂×C₄	1	2	yes	—
75	#106×C₈₉	C₂×C₄	1	2	yes	—
76	#106×C₈₉	C₂×C₄	1	2	yes	—
77	C₂²×Dic₈₉	—	2	2	no	—
78	#13083558	—	2	4	no	—
79	#13083558	—	2	4	no	—
80	C₂×#10623494	—	2	2	no	—
81	C₂×#10623494	—	2	2	no	—
82	C₂×#10623494	—	2	4	no	—
83	C₂×#10623494	C₂×D₄	1	4	yes	—
84	C₂²⋊Dic₈₉	—	4	2	no	—
85	C₄×Dic₈₉	—	2	2	no	—
86	C₄×Dic₈₉	—	2	2	no	—
87	C₂⁴×C₈₉	#41	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	—	2	2	no	—
89	C₂²×C₄×C₈₉	—	2	2	no	—
90	C₂²×C₄×C₈₉	—	2	2	no	—
91	C₂²×C₄×C₈₉	—	2	2	no	—
92	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	#41	1	4	yes	—
108	C₄⋊C₄×C₈₉	—	2	2	no	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
111	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
112	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₈₉	—	2	2	no	—
115	C₂²⋊C₄×C₈₉	—	2	2	no	—
116	C₂²⋊C₄×C₈₉	—	2	2	no	—
117	C₂²⋊C₄×C₈₉	—	2	2	no	—
118	C₄²×C₈₉	—	2	2	no	—
119	C₄²×C₈₉	—	2	2	no	—
120	C₄²×C₈₉	—	2	2	no	—
121	C₄²×C₈₉	—	2	2	no	—
122	C₂³×C₈₉	—	2	2	no	—
123	C₂³×C₈₉	—	2	2	no	—
124	C₂³×C₈₉	—	4	1	no	—
125	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	4	4	no	—
140	C₂×C₄×C₈₉	—	8	2	no	—
141	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
146	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
147	C₂×Dic₈₉	—	4	2	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	#117	1	2	yes	—
153	C₈×C₈₉	#117	1	2	yes	—
154	#10623494	—	2	4	no	—
155	#10623494	—	4	4	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	—	4	4	no	—
164	C₂²×C₈₉	#444	1	2	yes	—
165	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₈₉	#552	1	2	yes	—
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	—	4	2	no	—
173	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
174	C₄×C₈₉	#519	1	1	yes	characteristic
175	C₄×C₈₉	#547	1	2	yes	—
176	Dic₈₉	—	8	2	no	—
177	#22607	—	89	1	no	maximal, Sylow
178	C₂×C₈₉	—	2	2	no	—
179	C₂×C₈₉	—	2	2	no	—
180	C₂×C₈₉	—	4	1	no	—
181	C₂×C₈₉	#2344	1	1	yes	characteristic
182	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
183	C₂×C₈₉	#2957	1	1	yes	characteristic
184	#2924	—	89	2	no	—
185	C₂×#427	D₈₉	1	1	yes	characteristic
186	#1946	—	89	2	no	—
187	#1922	—	89	4	no	—
188	#1826	—	89	2	no	—
189	#1810	—	89	1	no	—
190	#1773	—	89	2	no	—
191	#1752	—	89	1	no	—
192	C₈₉	#22607	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
194	C₂×#117	C₂×D₈₉	1	2	yes	—
195	#433	—	178	4	no	—
196	#432	—	178	2	no	—
197	#431	—	178	2	no	—
198	C₂×#108	—	89	2	no	—
199	#427	Dic₈₉	1	1	yes	characteristic, complemented
200	#427	Dic₈₉	1	1	yes	characteristic, complemented
201	#427	Dic₈₉	1	2	yes	complemented
202	#427	C₂×D₈₉	1	2	yes	—
203	#427	C₂×D₈₉	1	1	yes	characteristic
204	#427	C₂×D₈₉	1	1	yes	characteristic
205	C₂×#107	Dic₈₉	1	1	yes	characteristic, complemented
206	C₂×#107	Dic₈₉	1	1	yes	characteristic, complemented
207	C₂×#106	Dic₈₉	1	2	yes	complemented
208	#424	—	89	4	no	—
209	#407	—	89	2	no	—
210	C₂×C₂²⋊C₈	—	89	1	no	—
211	#405	—	89	4	no	—
212	C₂×#97	—	89	2	no	—
213	C₄×C₂²⋊C₄	—	89	1	no	—
214	#341	—	89	2	no	—
215	#339	—	89	4	no	—
216	#339	—	89	4	no	—
217	#336	—	89	2	no	—
218	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
219	C₂×M₁₆	—	89	4	no	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	89	2	no	—
222	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
223	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
224	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
225	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	#117	C₂×Dic₈₉	1	2	yes	—
231	#117	C₂×Dic₈₉	1	2	yes	—
232	#117	C₂²×D₈₉	1	2	yes	—
233	#117	C₂²×D₈₉	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
235	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	—	178	1	no	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	—	89	2	no	—
240	#108	—	178	4	no	—
241	#107	C₂×Dic₈₉	1	2	yes	—
242	#107	C₂×Dic₈₉	1	2	yes	—
243	#107	C₂×Dic₈₉	1	1	yes	characteristic
244	#107	C₂×Dic₈₉	1	1	yes	characteristic
245	#107	C₂×Dic₈₉	1	1	yes	characteristic
246	#107	C₂×Dic₈₉	1	1	yes	characteristic
247	#106	C₂×Dic₈₉	1	2	yes	—
248	#106	C₂×Dic₈₉	1	2	yes	—
249	#106	C₂×Dic₈₉	1	2	yes	—
250	#106	C₂×Dic₈₉	1	2	yes	—
251	C₄⋊C₈	—	178	4	no	—
252	C₄⋊C₈	—	178	4	no	—
253	C₂²⋊C₈	—	89	2	no	—
254	C₂²⋊C₈	—	89	2	no	—
255	C₂²⋊C₈	—	178	2	no	—
256	C₂²⋊C₈	—	178	2	no	—
257	#97	—	178	4	no	—
258	#97	—	178	4	no	—
259	C₄×C₈	—	178	4	no	—
260	#95	—	178	1	no	—
261	C₂⁴	#13083573	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	178	2	no	—
267	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
271	C₂²×C₄	#13083573	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
273	M₁₆	—	178	4	no	—
274	M₁₆	—	178	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	89	4	no	—
281	C₂×C₈	—	178	4	no	—
282	C₂×C₈	—	178	2	no	—
283	C₂×C₈	—	178	2	no	—
284	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
285	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
286	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
287	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
288	C₂×C₈	#13083574	1	4	yes	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
292	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
293	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
294	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	356	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	178	2	no	—
305	C₄²	—	178	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
310	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
311	C₂³	C₂×#13083573	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
329	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂×#13083574	1	2	yes	—
333	C₂×C₄	#15668060	1	2	yes	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	356	4	no	—
337	C₈	—	178	4	no	—
338	C₈	#15667971	1	2	yes	—
339	C₈	#15667971	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	#19151720	1	2	yes	—
349	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
350	C₂²	#19152283	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	712	2	no	—
358	C₄	C₂×#15667971	1	1	yes	characteristic
359	C₄	#19152076	1	1	yes	characteristic
360	C₄	#19152249	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	#26023121	1	1	yes	characteristic
365	C₂	C₂×#19151720	1	1	yes	characteristic
366	C₂	#26029169	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#427×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂⁴×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#427	D₈₉
hypocenter	C₈₉	#22607
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #26021006	C₂	semidirect
3. #26020853	C₂	semidirect
6. #26020138	C₂	semidirect
25. #427×C₈₉	C₄	semidirect
26. #427×C₈₉	C₄	semidirect
27. #427×C₈₉	C₄	semidirect
31. C₂×#107×C₈₉	C₄	semidirect
32. C₂×#107×C₈₉	C₄	semidirect
33. C₂×#106×C₈₉	C₄	semidirect
192. C₈₉	#22607	semidirect
199. #427	Dic₈₉	semidirect
200. #427	Dic₈₉	semidirect
201. #427	Dic₈₉	semidirect
205. C₂×#107	Dic₈₉	semidirect
206. C₂×#107	Dic₈₉	semidirect
207. C₂×#106	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26028910	2
#26021006	2
#26020853	2
#26020351	2
#26020290	2
#26020138	2
#26020028	2
C₂×#427×C₈₉	2
#22607	89