Subgroups — #36520166

Order 22784.

798 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	#26029219	C₂	1	1	yes	maximal, characteristic
2	#26029219	C₂	1	1	yes	maximal, characteristic
3	#26018555	C₂	1	2	yes	maximal, complemented
4	#26018549	C₂	1	2	yes	maximal, complemented
5	#26018544	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018541	C₂	1	2	yes	maximal, complemented
7	C₈×#13083558	C₂	1	2	yes	maximal, complemented
8	M₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
9	M₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#131×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19151729	C₄	1	2	yes	—
13	#19151729	C₄	1	2	yes	—
14	#19151676	C₄	1	2	yes	—
15	#19151676	C₄	1	2	yes	—
16	#19151670	C₄	1	4	yes	—
17	#19151670	C₄	1	4	yes	—
18	#19151651	C₂²	1	1	yes	characteristic
19	#19151651	C₂²	1	1	yes	characteristic
20	#19151651	C₂²	1	2	yes	—
21	#19151651	C₂²	1	2	yes	—
22	C₄×#13083558	C₂²	1	2	yes	—
23	C₄×#13083558	C₂²	1	2	yes	—
24	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	#19151320	C₂²	1	1	yes	characteristic
27	#19151320	C₂²	1	1	yes	characteristic
28	#19151320	C₂²	1	1	yes	characteristic
29	#19151320	C₂²	1	1	yes	characteristic
30	#19151320	C₂²	1	2	yes	—
31	#19151320	C₂²	1	2	yes	—
32	#19151319	C₂²	1	2	yes	—
33	#19151319	C₂²	1	2	yes	—
34	C₈×#10623494	C₂²	1	2	yes	—
35	C₈×#10623494	C₂²	1	2	yes	—
36	C₂×#131×C₈₉	C₄	1	2	yes	—
37	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
38	#405×C₈₉	C₄	1	4	yes	—
39	#405×C₈₉	C₄	1	2	yes	—
40	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
44	C₂×#13083558	C₂×C₄	1	4	yes	—
45	C₂×#13083558	C₂×C₄	1	4	yes	—
46	C₂²×#10623494	C₂×C₄	1	2	yes	—
47	C₂²×#10623494	C₂×C₄	1	2	yes	—
48	C₄⋊#10623494	C₂×C₄	1	4	yes	—
49	C₄⋊#10623494	C₂×C₄	1	4	yes	—
50	C₄⋊#10623494	C₂×C₄	1	2	yes	—
51	C₄⋊#10623494	C₂×C₄	1	2	yes	—
52	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
53	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
54	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
55	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
56	#15667923	C₂×C₄	1	4	yes	—
57	#15667923	C₂×C₄	1	4	yes	—
58	C₄×#10623494	C₂×C₄	1	2	yes	—
59	C₄×#10623494	C₂×C₄	1	2	yes	—
60	C₄×#10623494	C₂³	1	1	yes	characteristic
61	C₄×#10623494	C₂³	1	1	yes	characteristic
62	C₄×#10623494	C₂³	1	2	yes	—
63	C₄×#10623494	C₂³	1	2	yes	—
64	C₄×#10623494	C₂³	1	1	yes	characteristic
65	C₄×#10623494	C₂³	1	1	yes	characteristic
66	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
67	#131×C₈₉	—	2	8	no	—
68	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
69	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
70	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
71	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
72	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
75	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
76	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
78	#117×C₈₉	C₂×C₄	1	2	yes	—
79	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
82	#97×C₈₉	C₂×C₄	1	4	yes	—
83	#97×C₈₉	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₈₉	C₂³	1	2	yes	—
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	#13083558	—	2	8	no	—
90	#13083558	—	4	4	no	—
91	C₂×#10623494	—	2	4	no	—
92	C₂×#10623494	—	4	2	no	—
93	C₂×#10623494	C₂²×C₄	1	2	yes	—
94	C₂×#10623494	C₂²×C₄	1	2	yes	—
95	C₂×#10623494	C₂²×C₄	1	2	yes	—
96	C₂×#10623494	C₂²×C₄	1	4	yes	—
97	C₂×#10623494	C₂²×C₄	1	4	yes	—
98	C₂×#10623494	C₂²×C₄	1	2	yes	—
99	#41×C₈₉	—	2	4	no	—
100	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
101	C₂×D₄×C₈₉	C₄²	1	2	yes	—
102	C₂×D₄×C₈₉	C₄²	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	M₁₆×C₈₉	—	2	8	no	—
110	M₁₆×C₈₉	M₁₆	1	8	yes	complemented
111	C₂×C₈×C₈₉	—	2	4	no	—
112	C₂×C₈×C₈₉	—	2	8	no	—
113	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
116	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
117	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
118	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
120	C₂²⋊C₄×C₈₉	C₄²	1	2	yes	—
121	C₂²⋊C₄×C₈₉	C₄²	1	2	yes	—
122	C₂²⋊C₄×C₈₉	C₄²	1	2	yes	—
123	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
126	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
127	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
129	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
131	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
132	Q₈×C₈₉	—	2	2	no	—
133	D₄×C₈₉	—	2	4	no	—
134	D₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	4	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	4	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	4	no	—
140	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
144	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
145	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
147	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
150	C₈×C₈₉	—	2	2	no	—
151	C₈×C₈₉	—	2	2	no	—
152	C₈×C₈₉	C₂×M₁₆	1	8	yes	—
153	#10623494	—	2	2	no	—
154	#10623494	—	2	4	no	—
155	#10623494	—	2	2	no	—
156	#10623494	—	2	2	no	—
157	#10623494	—	2	4	no	—
158	#10623494	#131	1	4	yes	complemented
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	—	2	2	no	—
162	C₂²×C₈₉	—	2	2	no	—
163	C₂²×C₈₉	—	2	2	no	—
164	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
165	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	2	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	2	no	—
171	C₄×C₈₉	C₄×M₁₆	1	2	yes	—
172	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
173	C₄×C₈₉	C₂×#131	1	2	yes	—
174	C₄×C₈₉	#568	1	2	yes	—
175	#17337	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	2	no	—
177	C₂×C₈₉	—	4	2	no	—
178	C₂×C₈₉	C₂×C₄×M₁₆	1	1	yes	characteristic
179	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
180	C₂×C₈₉	#2875	1	1	yes	characteristic
181	#2965	—	89	1	no	—
182	#2965	—	89	1	no	—
183	C₄×#131	D₈₉	1	1	yes	characteristic
184	#1455	—	89	2	no	—
185	#1453	—	89	2	no	—
186	#1452	—	89	1	no	—
187	#1451	—	89	2	no	—
188	C₈×M₁₆	—	89	1	no	—
189	C₈×M₁₆	—	89	1	no	—
190	C₈×M₁₆	—	89	2	no	—
191	C₂×#322	—	89	1	no	—
192	C₈₉	#17337	1	1	yes	characteristic, complemented, Sylow
193	C₂×#131	Dic₈₉	1	2	yes	—
194	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
195	#446	—	89	2	no	—
196	#446	—	89	2	no	—
197	#436	—	89	2	no	—
198	#436	—	89	2	no	—
199	#435	—	89	4	no	—
200	#435	—	89	4	no	—
201	#431	—	89	1	no	—
202	#431	—	89	1	no	—
203	#431	—	89	2	no	—
204	#431	—	89	2	no	—
205	#405	Dic₈₉	1	4	yes	—
206	#405	Dic₈₉	1	2	yes	—
207	C₄×M₁₆	—	89	2	no	—
208	C₄×M₁₆	—	89	2	no	—
209	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
210	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
211	C₂×C₄×C₈	—	89	1	no	—
212	C₂×C₄×C₈	—	89	1	no	—
213	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
215	#322	—	89	1	no	—
216	#322	—	89	1	no	—
217	#322	—	89	1	no	—
218	#322	—	89	1	no	—
219	#322	—	89	2	no	—
220	#322	—	89	2	no	—
221	#322	—	89	2	no	—
222	#322	—	89	2	no	—
223	C₈²	—	89	2	no	—
224	C₈²	—	89	2	no	—
225	C₂×#41	C₂×Dic₈₉	1	1	yes	characteristic
226	#131	—	2	8	no	—
227	C₂×M₁₆	—	89	4	no	—
228	C₂×M₁₆	—	89	4	no	—
229	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
230	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
231	C₂²×C₈	—	89	2	no	—
232	C₂²×C₈	—	89	2	no	—
233	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
234	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
235	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
236	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
237	C₄×D₄	C₄×D₈₉	1	2	yes	—
238	C₄×D₄	C₄×D₈₉	1	2	yes	—
239	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
240	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
241	#117	C₂×Dic₈₉	1	2	yes	—
242	#117	C₂×Dic₈₉	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
245	C₄⋊C₈	—	89	4	no	—
246	C₄⋊C₈	—	89	4	no	—
247	C₄⋊C₈	—	89	2	no	—
248	C₄⋊C₈	—	89	2	no	—
249	C₂²⋊C₈	—	89	2	no	—
250	C₂²⋊C₈	—	89	2	no	—
251	C₂²⋊C₈	—	89	4	no	—
252	C₂²⋊C₈	—	89	4	no	—
253	#97	—	89	4	no	—
254	#97	—	89	4	no	—
255	#97	C₂×Dic₈₉	1	4	yes	—
256	#97	C₂×Dic₈₉	1	2	yes	—
257	C₄×C₈	—	89	2	no	—
258	C₄×C₈	—	89	2	no	—
259	C₄×C₈	—	89	1	no	—
260	C₄×C₈	—	89	1	no	—
261	C₄×C₈	—	89	2	no	—
262	C₄×C₈	—	89	2	no	—
263	C₄×C₈	—	89	1	no	—
264	C₄×C₈	—	89	1	no	—
265	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
266	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
267	C₄×C₈	C₂²×D₈₉	1	2	yes	—
268	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	#41	—	2	4	no	—
271	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
272	C₂×D₄	C₄×Dic₈₉	1	2	yes	—
273	C₂×D₄	C₄×Dic₈₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
277	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
278	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
279	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
280	M₁₆	—	2	8	no	—
281	M₁₆	—	356	4	no	—
282	M₁₆	—	178	8	no	—
283	M₁₆	#13083558	1	8	yes	complemented
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	8	no	—
286	C₂×C₈	—	89	2	no	—
287	C₂×C₈	—	89	2	no	—
288	C₂×C₈	—	89	2	no	—
289	C₂×C₈	—	89	2	no	—
290	C₂×C₈	—	89	4	no	—
291	C₂×C₈	—	89	4	no	—
292	C₂×C₈	—	356	2	no	—
293	C₂×C₈	—	178	4	no	—
294	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
295	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
296	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
297	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
298	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
299	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
300	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
301	C₂²⋊C₄	C₄×Dic₈₉	1	2	yes	—
302	C₂²⋊C₄	C₄×Dic₈₉	1	2	yes	—
303	C₂²⋊C₄	C₄×Dic₈₉	1	2	yes	—
304	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
305	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
306	C₄²	C₂×C₄×D₈₉	1	2	yes	—
307	C₄²	C₂×C₄×D₈₉	1	2	yes	—
308	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
309	C₄²	C₂²×Dic₈₉	1	2	yes	—
310	C₄²	C₂³×D₈₉	1	1	yes	characteristic
311	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
312	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
322	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
324	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
326	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
327	C₂×C₄	C₂×#13083558	1	4	yes	—
328	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	89	4	no	—
334	C₈	—	178	2	no	—
335	C₈	—	178	4	no	—
336	C₈	—	178	2	no	—
337	C₈	—	178	2	no	—
338	C₈	—	178	4	no	—
339	C₈	C₂×#13083558	1	8	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	C₄×#13083558	1	2	yes	—
346	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	C₄×#13083558	1	2	yes	—
353	C₄	C₂²×#13083558	1	2	yes	—
354	C₄	#131×D₈₉	1	2	yes	—
355	C₄	#19152363	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×C₄×#13083558	1	1	yes	characteristic
359	C₂	#131×Dic₈₉	1	1	yes	characteristic
360	C₂	#26028678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₄²
Fitting	C₄×#131×C₈₉	C₂
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₄×#131	D₈₉
hypocenter	C₈₉	#17337
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #26018555	C₂	semidirect
4. #26018549	C₂	semidirect
5. #26018544	C₂	semidirect
6. #26018541	C₂	semidirect
7. C₈×#13083558	C₂	semidirect
8. M₁₆×#10623494	C₂	semidirect
9. M₁₆×#10623494	C₂	semidirect
10. C₂×#19151320	C₂	semidirect
110. M₁₆×C₈₉	M₁₆	semidirect
158. #10623494	#131	semidirect
192. C₈₉	#17337	semidirect
283. M₁₆	#13083558	semidirect

Maximal subgroups

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

subgroup	index
#26029219	2
#26018555	2
#26018549	2
#26018544	2
#26018541	2
C₈×#13083558	2
M₁₆×#10623494	2
C₂×#19151320	2
C₄×#131×C₈₉	2
#17337	89