Subgroups — #36520101

Order 22784.

718 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	#26029219	C₂	1	1	yes	maximal, characteristic
2	#41×#10623494	C₂	1	1	yes	maximal, characteristic
3	#26018552	C₂	1	2	yes	maximal, complemented
4	#26018547	C₂	1	2	yes	maximal, complemented
5	#26018545	C₂	1	2	yes	maximal, complemented
6	#26018544	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018542	C₂	1	2	yes	maximal, complemented
8	#26018539	C₂	1	1	yes	maximal, characteristic, complemented
9	M₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151319	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2922	C₂	1	1	yes	maximal, characteristic
12	#19151729	C₄	1	2	yes	—
13	Q₈×#10623494	C₄	1	2	yes	—
14	#19151676	C₄	1	2	yes	—
15	#19151670	C₄	1	4	yes	—
16	D₄×#10623494	C₄	1	4	yes	—
17	D₄×#10623494	C₄	1	2	yes	—
18	#19151651	C₂²	1	1	yes	characteristic
19	#19151651	C₂²	1	2	yes	—
20	#19151651	C₂²	1	1	yes	characteristic
21	#19151651	C₂²	1	2	yes	—
22	C₄×#13083558	C₂²	1	2	yes	—
23	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
24	C₂×C₄×#10623494	C₂²	1	2	yes	—
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	#19151320	C₂²	1	2	yes	—
27	#19151320	C₂²	1	2	yes	—
28	#19151320	C₂²	1	2	yes	—
29	#19151320	C₂²	1	2	yes	—
30	#19151319	C₂²	1	1	yes	characteristic
31	#19151319	C₂²	1	1	yes	characteristic
32	#19151319	C₂²	1	1	yes	characteristic
33	#19151319	C₂²	1	1	yes	characteristic
34	#19151319	C₂²	1	2	yes	—
35	C₈×#10623494	C₂²	1	2	yes	—
36	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
37	#433×C₈₉	C₄	1	4	yes	—
38	#431×C₈₉	C₂²	1	2	yes	—
39	#431×C₈₉	C₂²	1	2	yes	—
40	#424×C₈₉	C₄	1	2	yes	—
41	#408×C₈₉	C₄	1	2	yes	—
42	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
44	C₂×#13083558	C₂×C₄	1	4	yes	—
45	C₂²×#10623494	C₂×C₄	1	2	yes	—
46	C₂²×#10623494	C₂×C₄	1	4	yes	—
47	C₂²×#10623494	C₂×C₄	1	2	yes	—
48	C₄⋊#10623494	C₂×C₄	1	2	yes	—
49	C₄⋊#10623494	C₂×C₄	1	4	yes	—
50	C₄⋊#10623494	C₂×C₄	1	4	yes	—
51	C₄⋊#10623494	C₂×C₄	1	2	yes	—
52	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
53	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
54	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
55	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
56	#15667923	C₂×C₄	1	4	yes	—
57	C₄×#10623494	C₂×C₄	1	2	yes	—
58	C₄×#10623494	C₂×C₄	1	4	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	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
68	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
69	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
72	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
74	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
75	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
76	#117×C₈₉	C₂×C₄	1	2	yes	—
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
79	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
80	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
81	C₂²⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
82	C₂²⋊C₈×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	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₈₉	C₂³	1	2	yes	—
88	#13083558	—	4	4	no	—
89	C₂×#10623494	—	4	2	no	—
90	C₂×#10623494	—	4	4	no	—
91	C₂×#10623494	—	4	2	no	—
92	C₂×#10623494	C₂²×C₄	1	2	yes	—
93	C₂×#10623494	C₂²×C₄	1	2	yes	—
94	C₂×#10623494	C₂²×C₄	1	2	yes	—
95	C₂×#10623494	C₂²×C₄	1	4	yes	—
96	C₂×#10623494	C₂²×C₄	1	2	yes	—
97	C₂×#10623494	C₂²×C₄	1	4	yes	—
98	#41×C₈₉	—	2	4	no	—
99	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
100	C₂×D₄×C₈₉	C₄²	1	2	yes	—
101	C₂×D₄×C₈₉	C₄²	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
108	M₁₆×C₈₉	—	2	4	no	—
109	C₂×C₈×C₈₉	—	2	4	no	—
110	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
114	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
115	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
117	C₂²⋊C₄×C₈₉	C₄²	1	2	yes	—
118	C₂²⋊C₄×C₈₉	C₄²	1	2	yes	—
119	C₂²⋊C₄×C₈₉	C₄²	1	2	yes	—
120	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
121	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
124	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
126	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
128	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
129	Q₈×C₈₉	—	2	2	no	—
130	D₄×C₈₉	—	2	4	no	—
131	D₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	4	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	4	no	—
136	C₂×C₄×C₈₉	—	2	4	no	—
137	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#131	1	4	yes	—
144	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
147	C₈×C₈₉	—	2	2	no	—
148	C₈×C₈₉	—	2	2	no	—
149	C₈×C₈₉	—	2	4	no	—
150	#10623494	—	2	2	no	—
151	#10623494	—	2	2	no	—
152	#10623494	—	2	4	no	—
153	#10623494	—	2	2	no	—
154	#10623494	—	2	2	no	—
155	#10623494	—	2	4	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	2	no	—
159	C₂²×C₈₉	—	2	2	no	—
160	C₂²×C₈₉	—	2	2	no	—
161	C₂²×C₈₉	#405	1	2	yes	—
162	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
163	C₄×C₈₉	—	2	2	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	#405	1	2	yes	—
169	C₄×C₈₉	C₂×#131	1	2	yes	—
170	C₄×C₈₉	#568	1	2	yes	—
171	C₄×C₈₉	#568	1	2	yes	—
172	#17329	—	89	1	no	maximal, Sylow
173	C₂×C₈₉	—	2	2	no	—
174	C₂×C₈₉	—	4	2	no	—
175	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
176	C₂×C₈₉	#2875	1	1	yes	characteristic
177	C₂×C₈₉	#2875	1	1	yes	characteristic
178	#2965	—	89	1	no	—
179	C₈×#41	—	89	1	no	—
180	#2922	D₈₉	1	1	yes	characteristic
181	#1454	—	89	2	no	—
182	#1453	—	89	2	no	—
183	#1452	—	89	2	no	—
184	#1452	—	89	1	no	—
185	#1451	—	89	2	no	—
186	#1451	—	89	1	no	—
187	C₈×M₁₆	—	89	1	no	—
188	C₂×#322	—	89	1	no	—
189	C₈₉	#17329	1	1	yes	characteristic, complemented, Sylow
190	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
191	#446	—	89	2	no	—
192	C₈×Q₈	—	89	2	no	—
193	#436	—	89	2	no	—
194	#435	—	89	4	no	—
195	C₈×D₄	—	89	2	no	—
196	C₈×D₄	—	89	4	no	—
197	#433	Dic₈₉	1	4	yes	—
198	#431	—	89	1	no	—
199	#431	—	89	2	no	—
200	#431	—	89	1	no	—
201	#431	—	89	2	no	—
202	#431	C₂×D₈₉	1	2	yes	—
203	#431	C₂×D₈₉	1	2	yes	—
204	#424	Dic₈₉	1	2	yes	—
205	#408	Dic₈₉	1	2	yes	—
206	C₄×M₁₆	—	89	2	no	—
207	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₂×C₄×C₈	—	89	1	no	—
209	C₂×C₄×C₈	—	89	2	no	—
210	C₂×C₄×C₈	—	89	1	no	—
211	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
212	#322	—	89	1	no	—
213	#322	—	89	1	no	—
214	#322	—	89	1	no	—
215	#322	—	89	1	no	—
216	#322	—	89	2	no	—
217	#322	—	89	2	no	—
218	#322	—	89	2	no	—
219	#322	—	89	2	no	—
220	#322	—	89	2	no	—
221	C₈²	—	89	2	no	—
222	C₂×#41	C₂×Dic₈₉	1	1	yes	characteristic
223	C₂×M₁₆	—	89	4	no	—
224	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
225	C₂²×C₈	—	89	2	no	—
226	C₂²×C₈	—	89	4	no	—
227	C₂²×C₈	—	89	2	no	—
228	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
229	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
230	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
231	C₄×D₄	C₄×D₈₉	1	2	yes	—
232	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
233	C₄×D₄	C₄×D₈₉	1	2	yes	—
234	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
235	#117	C₂×Dic₈₉	1	1	yes	characteristic
236	#117	C₂×Dic₈₉	1	2	yes	—
237	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
239	C₄⋊C₈	—	89	2	no	—
240	C₄⋊C₈	—	89	4	no	—
241	C₄⋊C₈	—	89	4	no	—
242	C₄⋊C₈	—	89	2	no	—
243	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
244	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
245	C₂²⋊C₈	—	89	2	no	—
246	C₂²⋊C₈	—	89	4	no	—
247	C₂²⋊C₈	—	89	2	no	—
248	C₂²⋊C₈	—	89	4	no	—
249	C₂²⋊C₈	C₂×Dic₈₉	1	4	yes	—
250	C₂²⋊C₈	C₂×Dic₈₉	1	4	yes	—
251	#97	—	89	4	no	—
252	#97	C₂×Dic₈₉	1	2	yes	—
253	C₄×C₈	—	89	2	no	—
254	C₄×C₈	—	89	4	no	—
255	C₄×C₈	—	89	2	no	—
256	C₄×C₈	—	89	1	no	—
257	C₄×C₈	—	89	2	no	—
258	C₄×C₈	—	89	1	no	—
259	C₄×C₈	—	89	1	no	—
260	C₄×C₈	—	89	1	no	—
261	C₄×C₈	—	89	2	no	—
262	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
263	C₄×C₈	C₂²×D₈₉	1	2	yes	—
264	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	#41	—	2	4	no	—
267	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
268	C₂×D₄	C₄×Dic₈₉	1	2	yes	—
269	C₂×D₄	C₄×Dic₈₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
271	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
274	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
275	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
276	M₁₆	—	2	4	no	—
277	M₁₆	—	356	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	—	89	2	no	—
281	C₂×C₈	—	89	2	no	—
282	C₂×C₈	—	89	4	no	—
283	C₂×C₈	—	89	2	no	—
284	C₂×C₈	—	89	4	no	—
285	C₂×C₈	—	356	2	no	—
286	C₂×C₈	—	356	4	no	—
287	C₂×C₈	—	356	2	no	—
288	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
289	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
290	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
291	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
292	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
293	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
294	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
295	C₂²⋊C₄	C₄×Dic₈₉	1	2	yes	—
296	C₂²⋊C₄	C₄×Dic₈₉	1	2	yes	—
297	C₂²⋊C₄	C₄×Dic₈₉	1	2	yes	—
298	C₄²	C₂×C₄×D₈₉	1	2	yes	—
299	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
300	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₈₉	1	2	yes	—
302	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
303	C₄²	C₂²×Dic₈₉	1	2	yes	—
304	C₄²	C₂³×D₈₉	1	1	yes	characteristic
305	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
306	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	2	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
316	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
319	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
320	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
321	C₂×C₄	#15668019	1	4	yes	—
322	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	4	no	—
328	C₈	—	178	2	no	—
329	C₈	—	178	4	no	—
330	C₈	—	178	2	no	—
331	C₈	—	178	2	no	—
332	C₈	—	178	2	no	—
333	C₈	—	178	4	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	#19151554	1	2	yes	—
340	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	#19151554	1	2	yes	—
347	C₄	C₂×#15668019	1	2	yes	—
348	C₄	#19152363	1	2	yes	—
349	C₄	#19152363	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×#19151554	1	1	yes	characteristic
353	C₂	#26028678	1	1	yes	characteristic
354	C₂	#26028678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₄²
Fitting	C₈₉×#2922	C₂
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2922	D₈₉
hypocenter	C₈₉	#17329
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ #2922

Complements

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

N	H ≅ G/N	type
3. #26018552	C₂	semidirect
4. #26018547	C₂	semidirect
5. #26018545	C₂	semidirect
6. #26018544	C₂	semidirect
7. #26018542	C₂	semidirect
8. #26018539	C₂	semidirect
9. M₁₆×#10623494	C₂	semidirect
10. C₂×#19151319	C₂	semidirect
189. C₈₉	#17329	semidirect

Maximal subgroups

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

subgroup	index
#26029219	2
#41×#10623494	2
#26018552	2
#26018547	2
#26018545	2
#26018544	2
#26018542	2
#26018539	2
M₁₆×#10623494	2
C₂×#19151319	2
C₈₉×#2922	2
#17329	89