Subgroups — #36525849

Order 22784.

794 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#26028987	C₂	1	2	yes	maximal, complemented
3	#26019136	C₂	1	2	yes	maximal, complemented
4	#26019080	C₂	1	4	yes	maximal, complemented
5	#26019046	C₂	1	2	yes	maximal, complemented
6	#26018676	C₂	1	2	yes	maximal, complemented
7	C₈×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₁₂	C₄	1	1	yes	characteristic
10	C₂×M₁₆×D₈₉	—	2	2	no	—
11	C₂×C₄×D₃₅₆	C₂²	1	2	yes	—
12	C₄×D₇₁₂	C₄	1	1	yes	characteristic
13	C₄×D₇₁₂	C₄	1	2	yes	—
14	C₄×D₇₁₂	C₄	1	1	yes	characteristic
15	C₄×D₇₁₂	C₂²	1	2	yes	—
16	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
17	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
18	#19151675	C₂²	1	4	yes	complemented
19	#19151668	—	2	4	no	—
20	#19151663	C₂²	1	4	yes	complemented
21	#19151651	C₂²	1	1	yes	characteristic
22	C₂×#15667946	C₄	1	1	yes	characteristic
23	C₂×#15667934	C₄	1	2	yes	—
24	#19151561	—	2	2	no	—
25	C₄×#13083558	C₂²	1	1	yes	characteristic
26	#19151325	C₂²	1	4	yes	complemented
27	#19151325	C₂²	1	4	yes	complemented
28	#19151320	C₂²	1	2	yes	complemented
29	C₈×#10623494	C₂²	1	2	yes	complemented
30	#423×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#335×C₈₉	C₂²	1	2	yes	complemented
33	#335×C₈₉	C₂²	1	2	yes	complemented
34	C₂²×D₃₅₆	C₂×C₄	1	2	yes	—
35	C₂²×C₄×D₈₉	—	2	2	no	—
36	C₂×D₇₁₂	—	2	4	no	—
37	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
38	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
40	C₂×#13083558	D₄	1	1	yes	characteristic
41	C₂×#13083558	D₄	1	1	yes	characteristic
42	C₂×#13083557	—	2	4	no	—
43	M₁₆×D₈₉	—	4	8	no	—
44	C₂×C₈×D₈₉	—	2	4	no	—
45	C₄×D₃₅₆	—	2	2	no	—
46	C₄×D₃₅₆	—	2	2	no	—
47	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
48	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
49	C₄×D₃₅₆	C₂³	1	2	yes	—
50	C₄×D₃₅₆	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
52	C₂×#13083551	—	2	2	no	—
53	#15667946	C₂×C₄	1	2	yes	—
54	#15667946	C₂×C₄	1	1	yes	characteristic
55	#15667946	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#10623494	—	2	1	no	—
57	#15667934	C₂×C₄	1	2	yes	—
58	#15667934	C₂×C₄	1	2	yes	—
59	#15667934	C₂×C₄	1	2	yes	—
60	#15667934	C₂×C₄	1	2	yes	—
61	C₂²⋊#10623494	—	2	1	no	—
62	#15667924	—	2	4	no	—
63	#15667924	—	2	4	no	—
64	#15667923	D₄	1	2	yes	—
65	C₄×#10623494	C₂³	1	2	yes	—
66	C₄×#10623494	C₂³	1	2	yes	—
67	C₂×M₁₆×C₈₉	—	2	2	no	—
68	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₈₉	—	2	2	no	—
72	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
73	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₂³×D₈₉	—	2	2	no	—
78	C₂×D₃₅₆	—	2	4	no	—
79	C₂×D₃₅₆	—	2	2	no	—
80	C₂×D₃₅₆	—	4	2	no	—
81	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
82	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₈₉	—	2	2	no	—
84	C₂×C₄×D₈₉	—	2	2	no	—
85	C₂×C₄×D₈₉	—	2	2	no	—
86	C₂×C₄×D₈₉	—	2	2	no	—
87	C₂×C₄×D₈₉	—	2	2	no	—
88	C₂×C₄×D₈₉	—	4	4	no	—
89	C₂×C₄×D₈₉	—	4	4	no	—
90	D₇₁₂	—	2	4	no	—
91	D₇₁₂	—	2	2	no	—
92	D₇₁₂	M₁₆	1	4	yes	complemented
93	#13083558	—	2	2	no	—
94	#13083558	D₈	1	4	yes	complemented
95	#13083557	—	4	8	no	—
96	C₂×#10623494	—	2	2	no	—
97	C₂×#10623494	C₂×D₄	1	2	yes	—
98	C₂×#10623494	C₂×D₄	1	2	yes	—
99	C₈×D₈₉	—	4	8	no	—
100	C₄⋊Dic₈₉	—	2	2	no	—
101	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
103	#13083551	—	2	2	no	—
104	#13083551	—	2	2	no	—
105	#13083551	—	2	2	no	—
106	#13083551	—	2	2	no	—
107	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
110	M₁₆×C₈₉	—	4	4	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	4	no	—
113	C₂×C₈×C₈₉	—	4	1	no	—
114	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₈₉	#41	1	1	yes	characteristic
121	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₂²×D₈₉	—	2	2	no	—
124	C₂²×D₈₉	—	2	2	no	—
125	C₂²×D₈₉	—	4	4	no	—
126	C₂²×D₈₉	—	4	4	no	—
127	C₂²×D₈₉	—	8	2	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
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×D₈	1	2	yes	—
135	C₂×C₄×C₈₉	#135	1	2	yes	—
136	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	—	2	2	no	—
141	C₂×Dic₈₉	—	4	4	no	—
142	D₃₅₆	—	2	2	no	—
143	D₃₅₆	—	2	2	no	—
144	D₃₅₆	—	2	2	no	—
145	D₃₅₆	—	4	4	no	—
146	D₃₅₆	C₂×M₁₆	1	4	yes	complemented
147	C₄×D₈₉	—	4	2	no	—
148	C₄×D₈₉	—	4	2	no	—
149	C₄×D₈₉	—	4	2	no	—
150	C₄×D₈₉	—	4	2	no	—
151	C₄×D₈₉	—	4	4	no	—
152	C₄×D₈₉	—	4	4	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	4	4	no	—
157	C₈×C₈₉	C₂×M₁₆	1	2	yes	complemented
158	#10623494	—	2	2	no	—
159	#10623494	—	4	2	no	—
160	#10623494	C₂×D₈	1	4	yes	complemented
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	C₄×D₈	1	2	yes	—
164	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×D₈₉	—	2	4	no	—
166	C₂×D₈₉	—	4	2	no	—
167	C₂×D₈₉	—	4	2	no	—
168	C₂×D₈₉	—	4	2	no	—
169	C₂×D₈₉	—	4	2	no	—
170	C₂×D₈₉	—	4	2	no	—
171	C₂×D₈₉	—	8	8	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	C₄×D₈	1	2	yes	—
176	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
177	C₄×C₈₉	#568	1	1	yes	characteristic
178	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
179	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
180	Dic₈₉	—	4	2	no	—
181	Dic₈₉	—	4	2	no	—
182	M₁₆×D₈	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	2	2	no	—
184	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
185	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
186	C₂×C₈₉	#2957	1	1	yes	characteristic
187	D₈₉	—	4	4	no	—
188	D₈₉	—	8	2	no	—
189	C₂×C₄×D₈	—	89	1	no	—
190	D₄×M₁₆	—	89	2	no	—
191	#1589	—	89	2	no	—
192	#1581	—	89	4	no	—
193	C₈×D₈	—	89	2	no	—
194	#1567	D₈₉	1	1	yes	characteristic, complemented
195	#1486	—	89	2	no	—
196	C₈×M₁₆	—	89	1	no	—
197	C₈₉	M₁₆×D₈	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₈	—	89	1	no	—
199	C₂²×M₁₆	—	178	2	no	—
200	C₂×C₄×D₄	—	89	2	no	—
201	C₄×D₈	—	89	1	no	—
202	C₄×D₈	—	89	2	no	—
203	C₄×D₈	—	89	1	no	—
204	C₄×D₈	—	89	1	no	—
205	C₄×D₈	—	89	2	no	—
206	C₄×D₈	—	89	1	no	—
207	#436	—	89	4	no	—
208	#435	—	178	4	no	—
209	C₈×D₄	—	89	4	no	—
210	#431	—	89	1	no	—
211	C₂×#107	—	89	1	no	—
212	#423	C₂×D₈₉	1	2	yes	—
213	C₂×#102	—	89	2	no	—
214	#407	—	178	2	no	—
215	C₄×M₁₆	—	89	1	no	—
216	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
217	#335	C₂×D₈₉	1	2	yes	complemented
218	#335	C₂×D₈₉	1	2	yes	complemented
219	D₄⋊C₈	—	89	4	no	—
220	D₄⋊C₈	—	89	4	no	—
221	#322	—	89	2	no	—
222	C₈²	—	89	2	no	—
223	C₂²×D₄	—	89	2	no	—
224	C₂³×C₄	—	178	2	no	—
225	C₂×D₈	—	89	1	no	—
226	C₂×D₈	—	89	2	no	—
227	C₂×D₈	—	89	1	no	—
228	C₂×D₈	—	178	4	no	—
229	C₂×M₁₆	—	2	2	no	—
230	C₂×M₁₆	—	89	1	no	—
231	C₂×M₁₆	—	89	1	no	—
232	C₂×M₁₆	—	356	8	no	—
233	C₂×M₁₆	—	178	4	no	—
234	C₂²×C₈	—	178	4	no	—
235	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
236	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
237	C₄×D₄	—	89	2	no	—
238	C₄×D₄	—	89	2	no	—
239	C₄×D₄	—	89	2	no	—
240	C₄×D₄	—	89	2	no	—
241	C₄×D₄	—	178	2	no	—
242	C₄×D₄	—	178	2	no	—
243	C₂×C₄⋊C₄	—	89	2	no	—
244	C₂×C₂²⋊C₄	—	178	2	no	—
245	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
246	#107	—	89	2	no	—
247	#107	—	89	1	no	—
248	#107	—	89	1	no	—
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	178	1	no	—
251	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
252	#102	—	89	2	no	—
253	#102	—	89	2	no	—
254	#102	—	89	2	no	—
255	#102	—	89	2	no	—
256	C₂²⋊C₈	—	178	4	no	—
257	C₂²⋊C₈	—	178	4	no	—
258	C₂²⋊C₈	—	178	1	no	—
259	#97	—	89	2	no	—
260	C₄×C₈	—	89	2	no	—
261	C₄×C₈	—	89	2	no	—
262	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
263	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₂⁴	—	178	2	no	—
267	C₂×D₄	—	89	2	no	—
268	C₂×D₄	—	89	2	no	—
269	C₂×D₄	—	356	2	no	—
270	C₂×D₄	—	178	4	no	—
271	C₂×D₄	—	178	2	no	—
272	C₂²×C₄	—	356	4	no	—
273	C₂²×C₄	—	356	4	no	—
274	C₂²×C₄	—	178	2	no	—
275	C₂²×C₄	—	178	2	no	—
276	C₂²×C₄	—	178	2	no	—
277	C₂²×C₄	—	178	2	no	—
278	C₂²×C₄	—	178	2	no	—
279	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
281	C₂²×C₄	#13083575	1	1	yes	characteristic
282	D₈	—	89	4	no	—
283	D₈	—	178	4	no	—
284	D₈	—	178	2	no	—
285	M₁₆	—	4	4	no	—
286	M₁₆	—	89	4	no	—
287	M₁₆	—	356	8	no	—
288	M₁₆	—	178	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	89	2	no	—
293	C₂×C₈	—	89	2	no	—
294	C₂×C₈	—	356	8	no	—
295	C₂×C₈	—	178	2	no	—
296	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	89	2	no	—
301	C₄⋊C₄	—	89	2	no	—
302	C₄⋊C₄	—	178	2	no	—
303	C₂²⋊C₄	—	178	2	no	—
304	C₂²⋊C₄	—	178	2	no	—
305	C₂²⋊C₄	—	178	2	no	—
306	C₂²⋊C₄	—	178	2	no	—
307	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
308	C₄²	D₄×D₈₉	1	1	yes	characteristic
309	C₄²	#13083575	1	1	yes	characteristic
310	C₄²	C₂³×D₈₉	1	1	yes	characteristic
311	C₂³	—	356	4	no	—
312	C₂³	—	356	4	no	—
313	C₂³	—	178	2	no	—
314	C₂³	—	178	2	no	—
315	C₂³	—	712	2	no	—
316	C₂³	#15667974	1	1	yes	characteristic
317	D₄	—	89	4	no	—
318	D₄	—	356	4	no	—
319	D₄	—	178	2	no	—
320	D₄	—	178	2	no	—
321	D₄	—	178	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	356	4	no	—
326	C₂×C₄	—	356	4	no	—
327	C₂×C₄	—	356	4	no	—
328	C₂×C₄	—	356	2	no	—
329	C₂×C₄	—	356	2	no	—
330	C₂×C₄	—	356	2	no	—
331	C₂×C₄	—	356	2	no	—
332	C₂×C₄	—	178	2	no	—
333	C₂×C₄	—	178	2	no	—
334	C₂×C₄	#15667974	1	1	yes	characteristic
335	C₂×C₄	#15667974	1	1	yes	characteristic
336	C₂×C₄	#15667974	1	1	yes	characteristic
337	C₂×C₄	D₈×D₈₉	1	2	yes	—
338	C₂×C₄	#15668035	1	2	yes	—
339	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
341	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	4	4	no	—
346	C₈	—	89	4	no	—
347	C₈	—	356	2	no	—
348	C₈	—	178	2	no	—
349	C₈	M₁₆×D₈₉	1	2	yes	complemented
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	356	2	no	—
353	C₂²	—	356	2	no	—
354	C₂²	—	356	2	no	—
355	C₂²	—	356	2	no	—
356	C₂²	—	356	2	no	—
357	C₂²	—	178	4	no	—
358	C₂²	—	712	8	no	—
359	C₂²	#19151680	1	2	yes	—
360	C₂²	C₂×#15667974	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	356	2	no	—
365	C₄	—	356	2	no	—
366	C₄	#19151680	1	2	yes	—
367	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
368	C₄	#19152364	1	1	yes	characteristic
369	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
370	C₄	C₂×#15668035	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	356	4	no	—
373	C₂	—	712	2	no	—
374	C₂	#26028987	1	1	yes	characteristic
375	C₂	C₂×#19151680	1	1	yes	characteristic
376	C₂	#26029168	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1567	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₇₁₂	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1567	D₈₉
hypocenter	C₈₉	M₁₆×D₈
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
2. #26028987	C₂	semidirect
3. #26019136	C₂	semidirect
4. #26019080	C₂	semidirect
5. #26019046	C₂	semidirect
6. #26018676	C₂	semidirect
7. C₈×#13083558	C₂	semidirect
8. C₈₉×#1567	C₂	semidirect
18. #19151675	C₂²	semidirect
20. #19151663	C₂²	semidirect
26. #19151325	C₂²	semidirect
27. #19151325	C₂²	semidirect
28. #19151320	C₂²	semidirect
29. C₈×#10623494	C₂²	semidirect
32. #335×C₈₉	C₂²	semidirect
33. #335×C₈₉	C₂²	semidirect
92. D₇₁₂	M₁₆	semidirect
94. #13083558	D₈	semidirect
146. D₃₅₆	C₂×M₁₆	semidirect
157. C₈×C₈₉	C₂×M₁₆	semidirect
160. #10623494	C₂×D₈	semidirect
194. #1567	D₈₉	semidirect
197. C₈₉	M₁₆×D₈	semidirect
217. #335	C₂×D₈₉	semidirect
218. #335	C₂×D₈₉	semidirect
349. C₈	M₁₆×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₁₂	2
#26028987	2
#26019136	2
#26019080	2
#26019046	2
#26018676	2
C₈×#13083558	2
C₈₉×#1567	2
M₁₆×D₈	89