Subgroups — #36459642

Order 22784.

842 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151553	C₂	1	2	yes	maximal
2	C₂×#19151369	C₂	1	2	yes	maximal
3	#26019991	C₂	1	1	yes	maximal, characteristic
4	#26019985	C₂	1	1	yes	maximal, characteristic
5	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂×#15667971	C₂²	1	2	yes	—
8	#19151553	—	2	4	no	—
9	#19151553	C₄	1	8	yes	complemented
10	C₂×#15667922	—	2	2	no	—
11	C₂×C₈×Dic₈₉	—	2	2	no	—
12	#19151369	C₄	1	8	yes	complemented
13	#19151367	C₄	1	4	yes	complemented
14	#19151365	C₄	1	4	yes	—
15	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
18	#405×C₈₉	—	2	2	no	—
19	#405×C₈₉	—	2	2	no	—
20	C₂×#97×C₈₉	C₄	1	2	yes	complemented
21	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	complemented
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	C₂×C₄	1	4	yes	—
26	C₂×#13083558	C₂×C₄	1	4	yes	—
27	C₂×#13083558	C₂×C₄	1	4	yes	—
28	C₂²×#10623494	—	2	2	no	—
29	#15667971	—	2	2	no	—
30	#15667971	—	2	2	no	—
31	#15667971	C₂×C₄	1	4	yes	—
32	#15667971	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₈₉	—	2	2	no	—
34	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₈₉	—	2	2	no	—
36	#15667922	—	2	8	no	—
37	C₈×Dic₈₉	—	2	8	no	—
38	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₈₉	—	2	2	no	—
40	C₂×M₁₆×C₈₉	—	2	1	no	—
41	C₂×M₁₆×C₈₉	—	2	1	no	—
42	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	D₄	1	2	yes	—
48	C₂²×C₈×C₈₉	D₄	1	2	yes	—
49	C₂²×C₈×C₈₉	D₄	1	2	yes	—
50	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
51	#117×C₈₉	—	2	2	no	—
52	#117×C₈₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
56	#97×C₈₉	—	2	4	no	—
57	C₄×C₈×C₈₉	—	2	2	no	—
58	C₄×C₈×C₈₉	—	2	2	no	—
59	C₂²×Dic₈₉	—	2	4	no	—
60	#13083558	—	2	4	no	—
61	#13083558	—	2	4	no	—
62	#13083558	—	2	4	no	—
63	#13083558	C₄²	1	8	yes	—
64	C₂×#10623494	—	2	4	no	—
65	C₂×#10623494	—	2	4	no	—
66	C₂×#10623494	—	2	4	no	—
67	C₄⋊Dic₈₉	—	2	4	no	—
68	C₄⋊Dic₈₉	—	2	4	no	—
69	C₂²⋊Dic₈₉	—	2	4	no	—
70	C₂²⋊Dic₈₉	C₄²	1	8	yes	—
71	C₄×Dic₈₉	—	2	4	no	—
72	C₄×Dic₈₉	—	2	4	no	—
73	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	#41	1	2	yes	—
85	M₁₆×C₈₉	—	2	4	no	—
86	M₁₆×C₈₉	—	2	4	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	C₄²	1	4	yes	—
90	C₂×C₈×C₈₉	C₄²	1	2	yes	—
91	C₂×C₈×C₈₉	C₄²	1	2	yes	—
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
94	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
98	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
99	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
100	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
101	C₄⋊C₄×C₈₉	—	2	4	no	—
102	C₂²⋊C₄×C₈₉	—	2	4	no	—
103	C₄²×C₈₉	—	2	2	no	—
104	C₄²×C₈₉	—	2	2	no	—
105	C₂³×C₈₉	—	4	2	no	—
106	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
107	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
108	C₂³×C₈₉	C₄×D₄	1	2	yes	—
109	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
110	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	4	4	no	—
113	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#117	1	2	yes	—
121	C₂×C₄×C₈₉	#117	1	2	yes	—
122	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
131	C₂×Dic₈₉	—	2	8	no	—
132	C₂×Dic₈₉	—	4	8	no	—
133	C₈×C₈₉	—	2	4	no	—
134	C₈×C₈₉	#95	1	4	yes	—
135	C₈×C₈₉	#95	1	2	yes	—
136	C₈×C₈₉	#95	1	2	yes	—
137	#10623494	—	2	8	no	—
138	C₂²×C₈₉	—	2	4	no	—
139	C₂²×C₈₉	—	2	2	no	—
140	C₂²×C₈₉	—	4	2	no	—
141	C₂²×C₈₉	—	4	1	no	—
142	C₂²×C₈₉	—	4	1	no	—
143	C₂²×C₈₉	C₄³	1	1	yes	characteristic
144	C₂²×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
145	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
148	C₄×C₈₉	—	4	4	no	—
149	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
150	C₄×C₈₉	#376	1	1	yes	characteristic
151	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
153	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
155	Dic₈₉	—	4	8	no	—
156	#10711	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	—	2	1	no	—
159	C₂×C₈₉	—	2	1	no	—
160	C₂×C₈₉	—	2	2	no	—
161	C₂×C₈₉	C₄×#95	1	1	yes	characteristic
162	C₂×C₈₉	#1757	1	2	yes	—
163	C₂×#405	—	89	2	no	—
164	C₂×#405	D₈₉	1	1	yes	characteristic
165	C₂×#344	—	89	2	no	—
166	#1742	—	89	1	no	—
167	#1740	—	89	1	no	—
168	C₈₉	#10711	1	1	yes	characteristic, complemented, Sylow
169	C₂²×M₁₆	—	89	2	no	—
170	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
171	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
172	C₂×#117	—	89	2	no	—
173	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
174	#405	—	2	2	no	—
175	#405	—	2	2	no	—
176	#405	—	89	8	no	—
177	#405	—	178	4	no	—
178	C₂×#97	—	178	2	no	—
179	C₂×#97	Dic₈₉	1	2	yes	complemented
180	C₂×C₄×C₈	—	178	2	no	—
181	C₂×C₄×C₈	Dic₈₉	1	2	yes	complemented
182	#344	—	89	8	no	—
183	#343	—	89	4	no	—
184	#342	—	89	4	no	—
185	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	89	4	no	—
190	C₂×M₁₆	—	89	4	no	—
191	C₂×M₁₆	—	89	4	no	—
192	C₂×M₁₆	—	178	2	no	—
193	C₂×M₁₆	—	178	2	no	—
194	C₂×M₁₆	—	178	2	no	—
195	C₂×M₁₆	C₄×D₈₉	1	4	yes	—
196	C₂²×C₈	—	178	2	no	—
197	C₂²×C₈	Dic₁₇₈	1	2	yes	—
198	C₂²×C₈	C₄×D₈₉	1	4	yes	—
199	C₂²×C₈	D₃₅₆	1	2	yes	—
200	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
201	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
202	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
204	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	#117	—	89	4	no	—
208	#117	—	89	4	no	—
209	#117	—	178	2	no	—
210	#117	—	178	2	no	—
211	C₂×C₄⋊C₄	—	178	2	no	—
212	C₂×C₄⋊C₄	C₄×D₈₉	1	2	yes	—
213	C₂×C₂²⋊C₄	—	89	4	no	—
214	C₂×C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
215	C₂×C₄²	—	178	2	no	—
216	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
217	#97	—	2	4	no	—
218	#97	—	178	8	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	178	8	no	—
222	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
223	C₂²×C₄	—	178	4	no	—
224	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
225	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
230	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
231	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
232	C₂²×C₄	#13083574	1	1	yes	characteristic
233	C₂²×C₄	#13083574	1	1	yes	characteristic
234	C₂²×C₄	#13083574	1	2	yes	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	89	8	no	—
238	M₁₆	—	178	4	no	—
239	M₁₆	—	178	4	no	—
240	M₁₆	—	178	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	178	4	no	—
244	C₂×C₈	—	178	4	no	—
245	C₂×C₈	—	178	4	no	—
246	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
247	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
248	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
249	C₂×C₈	#13083551	1	4	yes	—
250	C₂×C₈	#13083551	1	4	yes	—
251	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
253	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
254	C₂×C₈	#13083553	1	4	yes	—
255	C₂×C₈	#13083553	1	4	yes	—
256	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
257	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	178	4	no	—
260	C₄⋊C₄	—	178	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	89	8	no	—
263	C₂²⋊C₄	—	178	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	178	4	no	—
267	C₄²	—	178	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	C₄²×D₈₉	1	2	yes	—
270	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
271	C₂³	C₄×C₂²⋊D₈₉	1	2	yes	—
272	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
273	C₂³	C₄×Dic₁₇₈	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	356	8	no	—
277	C₂×C₄	—	178	8	no	—
278	C₂×C₄	C₄²×D₈₉	1	2	yes	—
279	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
281	C₂×C₄	C₂×#13083551	1	2	yes	—
282	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
283	C₂×C₄	C₂×#13083553	1	2	yes	—
284	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
285	C₂×C₄	#15667969	1	2	yes	—
286	C₂×C₄	#15667969	1	2	yes	—
287	C₂×C₄	#15667971	1	1	yes	characteristic
288	C₂×C₄	#15667971	1	1	yes	characteristic
289	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
290	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
291	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
292	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
293	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
294	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
295	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
296	C₈	—	2	4	no	—
297	C₈	—	178	8	no	—
298	C₈	#15667919	1	4	yes	—
299	C₈	#15667919	1	2	yes	—
300	C₈	#15667919	1	2	yes	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₄²×Dic₈₉	1	1	yes	characteristic
307	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
308	C₂²	C₄×#13083551	1	2	yes	—
309	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
310	C₂²	C₄×#13083553	1	2	yes	—
311	C₄	—	4	4	no	—
312	C₄	—	356	8	no	—
313	C₄	C₂×#15667919	1	1	yes	characteristic
314	C₄	#19151444	1	1	yes	characteristic
315	C₄	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
316	C₄	C₄×#13083551	1	2	yes	—
317	C₄	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
318	C₄	C₄×#13083553	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	C₄×#15667919	1	1	yes	characteristic
324	C₂	#26020052	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₈₉
commutator	C₂²×C₈₉	C₄³
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#405×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#405	D₈₉
hypocenter	C₈₉	#10711
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
9. #19151553	C₄	semidirect
12. #19151369	C₄	semidirect
13. #19151367	C₄	semidirect
20. C₂×#97×C₈₉	C₄	semidirect
21. C₂×C₄×C₈×C₈₉	C₄	semidirect
168. C₈₉	#10711	semidirect
179. C₂×#97	Dic₈₉	semidirect
181. C₂×C₄×C₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151553	2
C₂×#19151369	2
#26019991	2
#26019985	2
C₂×#405×C₈₉	2
#10711	89