Subgroups — #36444928

Order 22784.

554 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151319	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018079	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018079	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19151651	—	2	2	no	—
9	#19151651	—	2	2	no	—
10	C₂×C₄⋊#10623494	C₄	1	2	yes	—
11	C₂×C₂²⋊#10623494	C₄	1	2	yes	—
12	C₂×C₂²⋊#10623494	C₄	1	2	yes	—
13	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
17	#19151353	C₄	1	2	yes	—
18	#19151320	—	2	2	no	—
19	#19151319	—	2	2	no	—
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#336×C₈₉	C₄	1	2	yes	—
25	C₂²×#10623494	C₂×C₄	1	2	yes	—
26	C₂²×#10623494	C₂×C₄	1	2	yes	—
27	C₂²×#10623494	C₂×C₄	1	2	yes	—
28	C₂²×#10623494	C₂×C₄	1	2	yes	—
29	C₄⋊#10623494	—	2	4	no	—
30	C₂²⋊#10623494	—	2	4	no	—
31	C₂²⋊#10623494	—	2	4	no	—
32	C₄×#10623494	—	2	1	no	—
33	C₄×#10623494	—	2	1	no	—
34	C₄×#10623494	—	2	1	no	—
35	C₄×#10623494	—	2	1	no	—
36	C₄×#10623494	—	2	1	no	—
37	C₄×#10623494	—	2	1	no	—
38	C₄×#10623494	D₄	1	2	yes	—
39	C₄×#10623494	D₄	1	2	yes	—
40	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₈₉	—	2	1	no	—
46	C₂×C₄²×C₈₉	—	2	1	no	—
47	C₂×C₄²×C₈₉	—	2	1	no	—
48	C₂×C₄²×C₈₉	—	2	1	no	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₂²⋊C₈×C₈₉	—	2	4	no	—
54	C₄×C₈×C₈₉	—	2	1	no	—
55	C₄×C₈×C₈₉	—	2	1	no	—
56	C₄×C₈×C₈₉	—	2	1	no	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₂×#10623494	—	2	2	no	—
59	C₂×#10623494	—	2	2	no	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	—	2	2	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	—	2	2	no	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	—	2	2	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	2	2	no	—
69	C₂×#10623494	C₂²⋊C₄	1	4	yes	—
70	C₂⁴×C₈₉	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	—	2	1	no	—
72	C₂²×C₄×C₈₉	—	2	1	no	—
73	C₂²×C₄×C₈₉	—	2	2	no	—
74	C₂²×C₄×C₈₉	—	2	1	no	—
75	C₂²×C₄×C₈₉	—	2	1	no	—
76	C₂²×C₄×C₈₉	—	2	1	no	—
77	C₂²×C₄×C₈₉	—	2	1	no	—
78	C₂²×C₄×C₈₉	—	2	2	no	—
79	C₂²×C₄×C₈₉	—	2	2	no	—
80	C₂²×C₄×C₈₉	—	2	2	no	—
81	C₂²×C₄×C₈₉	—	2	2	no	—
82	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
95	C₄²×C₈₉	—	2	1	no	—
96	C₄²×C₈₉	—	2	1	no	—
97	C₄²×C₈₉	—	2	1	no	—
98	C₄²×C₈₉	—	2	1	no	—
99	C₄²×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	—	2	1	no	—
101	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₈₉	#41	1	1	yes	characteristic
104	C₄²×C₈₉	#41	1	1	yes	characteristic
105	C₂³×C₈₉	—	2	1	no	—
106	C₂³×C₈₉	—	2	1	no	—
107	C₂³×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	—	2	1	no	—
109	C₂³×C₈₉	—	2	1	no	—
110	C₂³×C₈₉	—	2	1	no	—
111	C₂³×C₈₉	#97	1	2	yes	—
112	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	4	no	—
116	C₂×C₄×C₈₉	—	2	4	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	#97	1	2	yes	—
128	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₈₉	#131	1	2	yes	—
137	C₂×C₄×C₈₉	#131	1	2	yes	—
138	C₂×C₄×C₈₉	#131	1	2	yes	—
139	C₂×C₄×C₈₉	#131	1	2	yes	—
140	C₈×C₈₉	—	2	4	no	—
141	C₈×C₈₉	—	4	2	no	—
142	#10623494	—	2	4	no	—
143	#10623494	—	4	2	no	—
144	#10623494	—	4	2	no	—
145	#10623494	—	4	2	no	—
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
155	C₂²×C₈₉	C₄×M₁₆	1	1	yes	characteristic
156	C₂²×C₈₉	#405	1	1	yes	characteristic
157	C₂²×C₈₉	#405	1	1	yes	characteristic
158	C₂²×C₈₉	#405	1	1	yes	characteristic
159	C₂²×C₈₉	#405	1	1	yes	characteristic
160	C₄×C₈₉	—	2	2	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	#408	1	2	yes	—
164	C₄×C₈₉	#433	1	2	yes	—
165	C₄×C₈₉	#435	1	2	yes	—
166	C₄×C₈₉	#435	1	2	yes	—
167	C₄×C₈₉	#435	1	2	yes	—
168	C₄×C₈₉	#436	1	2	yes	—
169	#8149	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	—	4	2	no	—
171	C₂×C₈₉	#1451	1	1	yes	characteristic
172	C₂×C₈₉	#1452	1	1	yes	characteristic
173	C₂×C₈₉	#1453	1	1	yes	characteristic
174	C₂×C₈₉	#1454	1	1	yes	characteristic
175	C₂×C₈₉	#1750	1	1	yes	characteristic
176	C₂×C₈₉	#1752	1	1	yes	characteristic
177	C₂×C₈₉	#1753	1	1	yes	characteristic
178	C₂×#431	—	89	1	no	—
179	C₄×C₂²⋊C₈	—	89	1	no	—
180	C₄×C₂²⋊C₈	D₈₉	1	1	yes	characteristic
181	C₂×#322	—	89	1	no	—
182	C₂×#322	—	89	1	no	—
183	#1273	—	89	1	no	—
184	#1273	—	89	1	no	—
185	C₈₉	#8149	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
187	#431	—	178	2	no	—
188	#431	—	178	2	no	—
189	C₂×C₄⋊C₈	—	89	2	no	—
190	C₂×C₂²⋊C₈	—	89	2	no	—
191	C₂×C₂²⋊C₈	—	89	2	no	—
192	C₂×C₂²⋊C₈	Dic₈₉	1	2	yes	—
193	C₂×C₄×C₈	—	89	1	no	—
194	C₂×C₄×C₈	—	89	1	no	—
195	C₂×C₄×C₈	—	89	1	no	—
196	C₂×C₄×C₈	—	89	1	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
199	#336	—	89	2	no	—
200	#336	Dic₈₉	1	2	yes	—
201	#322	—	178	2	no	—
202	#322	—	178	2	no	—
203	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
204	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
205	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
206	C₂²×C₈	—	89	2	no	—
207	C₂²×C₈	—	89	2	no	—
208	C₂²×C₈	—	89	2	no	—
209	C₂²×C₈	—	89	2	no	—
210	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
211	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
220	C₄⋊C₈	—	178	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₂²⋊C₈	—	178	4	no	—
223	C₂²⋊C₈	—	178	4	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	89	2	no	—
229	C₄×C₈	—	89	2	no	—
230	C₄×C₈	—	178	1	no	—
231	C₄×C₈	—	178	1	no	—
232	C₄×C₈	—	178	1	no	—
233	C₄×C₈	—	178	1	no	—
234	C₄×C₈	—	178	1	no	—
235	C₄×C₈	—	178	1	no	—
236	C₂⁴	C₄×Dic₈₉	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
249	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
250	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
251	C₂²×C₄	#13083557	1	4	yes	—
252	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	89	4	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	—	178	2	no	—
266	C₂×C₈	—	178	2	no	—
267	C₂×C₈	—	178	2	no	—
268	C₂×C₈	—	178	2	no	—
269	C₂×C₈	—	178	2	no	—
270	C₂×C₈	—	178	2	no	—
271	C₂×C₈	—	178	2	no	—
272	C₂×C₈	#13083558	1	4	yes	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	D₄×D₈₉	1	1	yes	characteristic
280	C₄²	D₄×D₈₉	1	1	yes	characteristic
281	C₄²	#13083573	1	1	yes	characteristic
282	C₄²	#13083573	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	#15667922	1	2	yes	—
290	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	#15667922	1	2	yes	—
306	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	#15667968	1	1	yes	characteristic
308	C₂×C₄	#15667972	1	1	yes	characteristic
309	C₂×C₄	#15667972	1	1	yes	characteristic
310	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
311	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
312	C₂×C₄	C₂×#13083557	1	2	yes	—
313	C₂×C₄	C₂×#13083558	1	2	yes	—
314	C₂×C₄	#15668018	1	2	yes	—
315	C₂×C₄	#15668019	1	2	yes	—
316	C₂×C₄	#15668020	1	2	yes	—
317	C₂×C₄	#15668020	1	2	yes	—
318	C₈	—	2	4	no	—
319	C₈	—	4	2	no	—
320	C₈	—	356	2	no	—
321	C₈	—	356	2	no	—
322	C₈	—	356	2	no	—
323	C₈	—	178	4	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
332	C₂²	C₂×#15667922	1	1	yes	characteristic
333	C₂²	C₄×#13083558	1	1	yes	characteristic
334	C₂²	#19151553	1	1	yes	characteristic
335	C₂²	#19151554	1	1	yes	characteristic
336	C₂²	#19151556	1	1	yes	characteristic
337	C₂²	#19151556	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	#19151563	1	2	yes	—
342	C₄	#19151656	1	2	yes	—
343	C₄	#19151666	1	2	yes	—
344	C₄	#19151666	1	2	yes	—
345	C₄	#19151670	1	2	yes	—
346	C₄	#19151676	1	2	yes	—
347	C₂	—	4	2	no	—
348	C₂	#26018541	1	1	yes	characteristic
349	C₂	#26018545	1	1	yes	characteristic
350	C₂	#26018549	1	1	yes	characteristic
351	C₂	#26018552	1	1	yes	characteristic
352	C₂	#26020020	1	1	yes	characteristic
353	C₂	#26020031	1	1	yes	characteristic
354	C₂	#26020036	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₈₉
commutator, socle	C₂³×C₈₉	C₂×C₄²
Fitting	C₄×C₂²⋊C₈×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₂²⋊C₈	D₈₉
hypocenter	C₈₉	#8149
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#19151320	C₂	semidirect
4. C₂×#19151319	C₂	semidirect
5. #26018079	C₂	semidirect
6. #26018079	C₂	semidirect
185. C₈₉	#8149	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151651	2
C₄×C₂²⋊#10623494	2
C₂×#19151320	2
C₂×#19151319	2
#26018079	2
C₄×C₂²⋊C₈×C₈₉	2
#8149	89