Subgroups — #36525944

Order 22784.

786 conjugacy classes of subgroups.

The subgroup lattice has 366 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	M₁₆×D₃₅₆	C₂	1	2	yes	maximal, complemented
3	#26019167	C₂	1	2	yes	maximal, complemented
4	#26019118	C₂	1	4	yes	maximal, complemented
5	#26019077	C₂	1	2	yes	maximal, complemented
6	#26019008	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018678	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1451	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	1	yes	characteristic
16	C₄×D₇₁₂	C₂²	1	2	yes	—
17	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
18	#19151677	C₂²	1	4	yes	complemented
19	#19151671	—	2	4	no	—
20	C₈×D₃₅₆	C₂²	1	4	yes	complemented
21	C₂×#15667946	C₄	1	1	yes	characteristic
22	#19151622	C₂²	1	2	yes	—
23	C₂×#15667934	C₄	1	2	yes	—
24	#19151561	—	2	2	no	—
25	#19151349	C₂²	1	2	yes	complemented
26	#19151349	C₂²	1	2	yes	complemented
27	#19151327	C₂²	1	4	yes	complemented
28	#19151327	C₂²	1	4	yes	complemented
29	#431×C₈₉	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#322×C₈₉	C₂²	1	2	yes	complemented
33	#322×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	—	2	2	no	—
41	C₂×#13083557	—	2	4	no	—
42	M₁₆×D₈₉	—	4	8	no	—
43	C₂×C₈×D₈₉	—	2	4	no	—
44	C₄×D₃₅₆	—	2	2	no	—
45	C₄×D₃₅₆	—	2	2	no	—
46	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
47	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
48	C₄×D₃₅₆	C₂³	1	2	yes	—
49	C₄×D₃₅₆	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
51	C₂×#13083551	—	2	2	no	—
52	#15667946	C₂×C₄	1	2	yes	—
53	#15667946	C₂×C₄	1	1	yes	characteristic
54	#15667946	C₂×C₄	1	1	yes	characteristic
55	C₄⋊#10623494	—	2	2	no	—
56	C₄⋊#10623494	C₂³	1	4	yes	—
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	#15667924	—	2	4	no	—
62	#15667924	—	2	4	no	—
63	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
64	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₈₉	—	2	1	no	—
68	C₂²⋊C₈×C₈₉	—	2	1	no	—
69	#97×C₈₉	D₄	1	2	yes	—
70	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	2	yes	—
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	2	yes	—
76	C₂³×D₈₉	—	2	2	no	—
77	C₂×D₃₅₆	—	2	4	no	—
78	C₂×D₃₅₆	—	2	2	no	—
79	C₂×D₃₅₆	—	4	2	no	—
80	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
81	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₈₉	—	2	2	no	—
83	C₂×C₄×D₈₉	—	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	—	4	4	no	—
88	C₂×C₄×D₈₉	—	4	4	no	—
89	D₇₁₂	—	2	4	no	—
90	D₇₁₂	—	2	2	no	—
91	D₇₁₂	M₁₆	1	4	yes	complemented
92	#13083558	—	4	4	no	—
93	#13083557	—	4	8	no	—
94	C₂×#10623494	—	2	4	no	—
95	C₈×D₈₉	—	4	8	no	—
96	C₄⋊Dic₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
98	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
99	#13083551	—	2	2	no	—
100	#13083551	—	2	2	no	—
101	#13083551	—	2	2	no	—
102	#13083551	—	2	2	no	—
103	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
106	M₁₆×C₈₉	—	2	4	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	4	1	no	—
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
115	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₈₉	#41	1	1	yes	characteristic
118	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
120	C₂²×D₈₉	—	2	2	no	—
121	C₂²×D₈₉	—	2	2	no	—
122	C₂²×D₈₉	—	4	4	no	—
123	C₂²×D₈₉	—	4	4	no	—
124	C₂²×D₈₉	—	8	2	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
132	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₈₉	—	2	2	no	—
136	C₂×Dic₈₉	—	2	2	no	—
137	C₂×Dic₈₉	—	4	4	no	—
138	D₃₅₆	—	2	2	no	—
139	D₃₅₆	—	2	2	no	—
140	D₃₅₆	—	2	2	no	—
141	D₃₅₆	—	4	4	no	—
142	D₃₅₆	C₂×M₁₆	1	4	yes	complemented
143	C₄×D₈₉	—	4	2	no	—
144	C₄×D₈₉	—	4	2	no	—
145	C₄×D₈₉	—	4	2	no	—
146	C₄×D₈₉	—	4	2	no	—
147	C₄×D₈₉	—	4	4	no	—
148	C₄×D₈₉	—	4	4	no	—
149	C₈×C₈₉	—	2	1	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	—	2	4	no	—
153	C₈×C₈₉	—	4	2	no	—
154	C₈×C₈₉	C₂×M₁₆	1	2	yes	complemented
155	#10623494	—	4	4	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	#442	1	2	yes	—
159	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂×D₈₉	—	2	4	no	—
161	C₂×D₈₉	—	4	2	no	—
162	C₂×D₈₉	—	4	2	no	—
163	C₂×D₈₉	—	4	2	no	—
164	C₂×D₈₉	—	4	2	no	—
165	C₂×D₈₉	—	4	2	no	—
166	C₂×D₈₉	—	8	8	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	#442	1	2	yes	—
171	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
172	C₄×C₈₉	#568	1	1	yes	characteristic
173	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
174	Dic₈₉	—	4	2	no	—
175	Dic₈₉	—	4	2	no	—
176	#18044	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	2	no	—
178	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
179	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
180	C₂×C₈₉	#2957	1	1	yes	characteristic
181	D₈₉	—	4	4	no	—
182	D₈₉	—	8	2	no	—
183	C₂×C₄×D₈	—	89	1	no	—
184	D₄×M₁₆	—	89	2	no	—
185	#1594	—	89	2	no	—
186	#1586	—	89	4	no	—
187	#1580	—	89	2	no	—
188	#1566	—	89	1	no	—
189	#1486	—	89	2	no	—
190	#1451	D₈₉	1	1	yes	characteristic, complemented
191	C₈₉	#18044	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₈	—	89	1	no	—
193	C₂²×M₁₆	—	178	2	no	—
194	C₂×C₄×D₄	—	89	2	no	—
195	C₄×D₈	—	89	1	no	—
196	C₄×D₈	—	89	2	no	—
197	C₄×D₈	—	89	1	no	—
198	C₄×D₈	—	89	1	no	—
199	C₄×D₈	—	89	2	no	—
200	C₄×D₈	—	89	1	no	—
201	#436	—	89	4	no	—
202	#435	—	178	4	no	—
203	C₈×D₄	—	89	4	no	—
204	#431	C₂×D₈₉	1	1	yes	characteristic
205	C₂×#107	—	89	1	no	—
206	#423	—	89	2	no	—
207	C₂×#102	—	89	2	no	—
208	#407	—	178	2	no	—
209	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
211	#334	—	89	2	no	—
212	#334	—	89	2	no	—
213	D₄⋊C₈	—	89	4	no	—
214	D₄⋊C₈	—	89	4	no	—
215	#322	C₂×D₈₉	1	2	yes	complemented
216	#322	C₂×D₈₉	1	2	yes	complemented
217	C₂²×D₄	—	89	2	no	—
218	C₂³×C₄	—	178	2	no	—
219	C₂×D₈	—	89	1	no	—
220	C₂×D₈	—	89	2	no	—
221	C₂×D₈	—	89	1	no	—
222	C₂×D₈	—	178	4	no	—
223	C₂×M₁₆	—	356	8	no	—
224	C₂×M₁₆	—	178	2	no	—
225	C₂×M₁₆	—	178	4	no	—
226	C₂×M₁₆	D₃₅₆	1	2	yes	—
227	C₂²×C₈	—	178	4	no	—
228	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
230	C₄×D₄	—	89	2	no	—
231	C₄×D₄	—	89	2	no	—
232	C₄×D₄	—	89	2	no	—
233	C₄×D₄	—	89	2	no	—
234	C₄×D₄	—	178	2	no	—
235	C₄×D₄	—	178	2	no	—
236	C₂×C₄⋊C₄	—	89	2	no	—
237	C₂×C₂²⋊C₄	—	178	2	no	—
238	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
239	#107	—	89	2	no	—
240	#107	—	89	1	no	—
241	#107	—	89	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	89	4	no	—
244	C₄⋊C₈	—	178	2	no	—
245	#102	—	89	2	no	—
246	#102	—	89	2	no	—
247	#102	—	89	2	no	—
248	#102	—	89	2	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	178	4	no	—
251	C₂²⋊C₈	—	178	4	no	—
252	#97	D₃₅₆	1	2	yes	—
253	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
254	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₈₉	1	2	yes	—
256	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₈₉	1	2	yes	—
259	C₂⁴	—	178	2	no	—
260	C₂×D₄	—	89	2	no	—
261	C₂×D₄	—	89	2	no	—
262	C₂×D₄	—	356	2	no	—
263	C₂×D₄	—	178	4	no	—
264	C₂×D₄	—	178	2	no	—
265	C₂²×C₄	—	356	4	no	—
266	C₂²×C₄	—	356	4	no	—
267	C₂²×C₄	—	178	2	no	—
268	C₂²×C₄	—	178	2	no	—
269	C₂²×C₄	—	178	2	no	—
270	C₂²×C₄	—	178	2	no	—
271	C₂²×C₄	—	178	2	no	—
272	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
274	C₂²×C₄	#13083574	1	1	yes	characteristic
275	D₈	—	89	4	no	—
276	D₈	—	178	4	no	—
277	D₈	—	178	2	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	356	8	no	—
280	M₁₆	—	356	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	356	8	no	—
285	C₂×C₈	—	178	4	no	—
286	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
290	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
291	C₄⋊C₄	—	89	2	no	—
292	C₄⋊C₄	—	89	2	no	—
293	C₄⋊C₄	—	178	2	no	—
294	C₂²⋊C₄	—	178	2	no	—
295	C₂²⋊C₄	—	178	2	no	—
296	C₂²⋊C₄	—	178	2	no	—
297	C₂²⋊C₄	—	178	2	no	—
298	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
299	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
300	C₄²	#13083574	1	1	yes	characteristic
301	C₄²	C₂³×D₈₉	1	1	yes	characteristic
302	C₂³	—	356	4	no	—
303	C₂³	—	356	4	no	—
304	C₂³	—	178	2	no	—
305	C₂³	—	178	2	no	—
306	C₂³	—	712	2	no	—
307	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
308	D₄	—	89	4	no	—
309	D₄	—	356	4	no	—
310	D₄	—	178	2	no	—
311	D₄	—	178	2	no	—
312	D₄	—	178	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	356	4	no	—
317	C₂×C₄	—	356	2	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	356	2	no	—
320	C₂×C₄	—	356	2	no	—
321	C₂×C₄	—	356	4	no	—
322	C₂×C₄	—	356	4	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
326	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
327	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
328	C₂×C₄	#15668040	1	4	yes	—
329	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
331	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
332	C₈	—	2	4	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	356	4	no	—
338	C₈	M₁₆×D₈₉	1	2	yes	complemented
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	356	2	no	—
342	C₂²	—	356	2	no	—
343	C₂²	—	356	2	no	—
344	C₂²	—	356	2	no	—
345	C₂²	—	356	2	no	—
346	C₂²	—	178	4	no	—
347	C₂²	—	712	8	no	—
348	C₂²	#19151711	1	2	yes	—
349	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	356	2	no	—
354	C₄	—	356	2	no	—
355	C₄	#19151711	1	2	yes	—
356	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
357	C₄	#19152364	1	1	yes	characteristic
358	C₄	C₂×#15668040	1	2	yes	—
359	C₂	—	2	2	no	—
360	C₂	—	356	4	no	—
361	C₂	—	712	2	no	—
362	C₂	M₁₆×D₃₅₆	1	1	yes	characteristic
363	C₂	C₂×#19151711	1	1	yes	characteristic
364	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1451	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₇₁₂	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1451	D₈₉
hypocenter	C₈₉	#18044
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₃₅₆	C₂	semidirect
3. #26019167	C₂	semidirect
4. #26019118	C₂	semidirect
5. #26019077	C₂	semidirect
6. #26019008	C₂	semidirect
7. #26018678	C₂	semidirect
8. C₈₉×#1451	C₂	semidirect
18. #19151677	C₂²	semidirect
20. C₈×D₃₅₆	C₂²	semidirect
25. #19151349	C₂²	semidirect
26. #19151349	C₂²	semidirect
27. #19151327	C₂²	semidirect
28. #19151327	C₂²	semidirect
32. #322×C₈₉	C₂²	semidirect
33. #322×C₈₉	C₂²	semidirect
91. D₇₁₂	M₁₆	semidirect
142. D₃₅₆	C₂×M₁₆	semidirect
154. C₈×C₈₉	C₂×M₁₆	semidirect
190. #1451	D₈₉	semidirect
191. C₈₉	#18044	semidirect
215. #322	C₂×D₈₉	semidirect
216. #322	C₂×D₈₉	semidirect
338. 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
M₁₆×D₃₅₆	2
#26019167	2
#26019118	2
#26019077	2
#26019008	2
#26018678	2
C₈₉×#1451	2
#18044	89