Subgroups — #36708016

Order 22784.

618 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#26022079	C₂	1	4	yes	maximal, complemented
3	#26022076	C₂	1	4	yes	maximal, complemented
4	#26022055	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021861	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₁₆×D₈₉	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2163	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃₅₆	C₄	1	1	yes	characteristic
11	C₂×#15667951	C₂²	1	4	yes	complemented
12	C₂×C₁₆×D₈₉	C₂²	1	4	yes	complemented
13	C₈×D₃₅₆	C₄	1	2	yes	—
14	C₈×D₃₅₆	C₄	1	2	yes	—
15	C₈×D₃₅₆	C₂²	1	2	yes	—
16	C₈×D₃₅₆	C₂²	1	2	yes	—
17	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
18	C₂×#15667924	C₄	1	1	yes	characteristic
19	C₂×#15667924	C₄	1	1	yes	characteristic
20	C₄⋊#13083549	C₂²	1	2	yes	complemented
21	C₂²⋊#13083549	C₂²	1	2	yes	—
22	#19151377	C₂²	1	4	yes	complemented
23	#19151377	C₂²	1	4	yes	complemented
24	#19151374	C₂²	1	2	yes	complemented
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
27	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
28	#346×C₈₉	C₂²	1	2	yes	complemented
29	C₂²×D₃₅₆	C₈	1	1	yes	characteristic
30	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
33	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₈₉	—	4	2	no	—
35	C₂×C₈×D₈₉	—	4	2	no	—
36	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	C₂³	1	2	yes	—
39	C₂×C₈×D₈₉	C₂³	1	2	yes	—
40	C₄×D₃₅₆	C₈	1	2	yes	—
41	C₄×D₃₅₆	C₈	1	2	yes	—
42	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
43	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
45	C₂×#13083551	C₈	1	1	yes	characteristic
46	C₂×#13083551	C₈	1	1	yes	characteristic
47	#15667951	—	4	4	no	—
48	C₂×#13083549	C₂³	1	4	yes	—
49	C₁₆×D₈₉	—	4	4	no	—
50	C₄⋊#10623494	C₂×C₄	1	2	yes	—
51	C₄⋊#10623494	C₂×C₄	1	2	yes	—
52	#15667924	C₂×C₄	1	2	yes	—
53	#15667924	C₂×C₄	1	2	yes	—
54	#15667924	C₂×C₄	1	2	yes	—
55	#15667924	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
60	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
65	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
66	C₂×D₃₅₆	—	2	4	no	—
67	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
68	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
69	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
70	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₈₉	—	4	1	no	—
72	C₂×C₄×D₈₉	—	4	1	no	—
73	C₂×C₄×D₈₉	—	4	1	no	—
74	C₂×C₄×D₈₉	—	4	1	no	—
75	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
79	C₂×#10623494	—	4	1	no	—
80	C₂×#10623494	—	4	1	no	—
81	C₂×#10623494	C₂²×C₄	1	2	yes	—
82	C₂×#10623494	C₂²×C₄	1	2	yes	—
83	C₈×D₈₉	—	2	4	no	—
84	C₈×D₈₉	—	2	4	no	—
85	C₈×D₈₉	—	4	2	no	—
86	C₈×D₈₉	—	4	2	no	—
87	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
89	#13083551	C₂×C₈	1	2	yes	—
90	#13083551	C₂×C₈	1	2	yes	—
91	#13083551	C₂×C₈	1	2	yes	—
92	#13083551	C₂×C₈	1	2	yes	—
93	#13083549	—	2	4	no	—
94	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	—	4	1	no	—
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
103	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
105	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₁₆×C₈₉	—	2	4	no	—
108	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₈₉	—	2	2	no	—
110	C₂²×D₈₉	—	4	1	no	—
111	C₂²×D₈₉	—	4	1	no	—
112	C₂²×D₈₉	—	4	1	no	—
113	C₂²×D₈₉	—	4	1	no	—
114	C₂²×D₈₉	M₃₂	1	4	yes	complemented
115	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
116	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	1	no	—
129	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
130	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
131	D₃₅₆	—	2	4	no	—
132	D₃₅₆	—	2	4	no	—
133	C₄×D₈₉	—	2	2	no	—
134	C₄×D₈₉	—	2	2	no	—
135	C₄×D₈₉	—	2	2	no	—
136	C₄×D₈₉	—	2	2	no	—
137	C₄×D₈₉	—	4	2	no	—
138	C₄×D₈₉	—	4	2	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	H(5,𝔽₂)	1	2	yes	—
143	#10623494	—	2	2	no	—
144	#10623494	—	2	2	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	C₂×M₃₂	1	2	yes	—
148	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₈₉	—	2	2	no	—
150	C₂×D₈₉	—	2	2	no	—
151	C₂×D₈₉	—	2	2	no	—
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₂×D₈₉	—	4	4	no	—
155	C₂×D₈₉	—	4	4	no	—
156	C₂×D₈₉	C₂×M₃₂	1	4	yes	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	#518	1	1	yes	characteristic
163	C₄×C₈₉	#568	1	1	yes	characteristic
164	Dic₈₉	—	2	2	no	—
165	Dic₈₉	—	2	2	no	—
166	#33487	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	2	no	—
168	C₂×C₈₉	#2972	1	1	yes	characteristic
169	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
170	C₂×C₈₉	#3407	1	1	yes	characteristic
171	D₈₉	—	2	4	no	—
172	D₈₉	—	4	2	no	—
173	C₂×C₈×D₄	—	89	1	no	—
174	#2168	—	89	4	no	—
175	#2168	—	89	4	no	—
176	#2163	—	89	1	no	—
177	#2163	D₈₉	1	1	yes	characteristic, complemented
178	#2112	—	89	2	no	—
179	C₂×C₂²⋊C₁₆	—	89	2	no	—
180	C₈₉	#33487	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₈	—	89	1	no	—
182	C₂³×C₈	—	89	1	no	—
183	C₂×C₄×D₄	—	89	1	no	—
184	C₂×M₃₂	—	89	4	no	—
185	C₂²×C₁₆	—	89	4	no	—
186	C₈×D₄	—	89	2	no	—
187	C₈×D₄	—	89	2	no	—
188	C₈×D₄	—	89	2	no	—
189	C₈×D₄	—	89	2	no	—
190	C₂×C₄⋊C₈	—	89	1	no	—
191	C₂×C₂²⋊C₈	—	89	1	no	—
192	C₂×C₂²⋊C₈	—	89	1	no	—
193	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
194	C₄⋊C₁₆	—	89	2	no	—
195	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
196	C₂²⋊C₁₆	—	89	2	no	—
197	C₂²⋊C₁₆	—	89	4	no	—
198	C₂²⋊C₁₆	—	89	4	no	—
199	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
200	#346	—	89	2	no	—
201	#346	C₂×D₈₉	1	2	yes	complemented
202	C₂²×D₄	—	89	1	no	—
203	C₂³×C₄	—	89	1	no	—
204	C₂³×C₄	—	89	1	no	—
205	C₂²×C₈	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₂²×C₈	—	89	2	no	—
208	C₂²×C₈	—	89	2	no	—
209	C₂²×C₈	—	89	2	no	—
210	C₂²×C₈	—	89	2	no	—
211	C₂²×C₈	—	356	2	no	—
212	C₂²×C₈	—	356	2	no	—
213	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₄×D₄	—	89	2	no	—
216	C₄×D₄	—	89	2	no	—
217	C₄×D₄	—	89	2	no	—
218	C₄×D₄	—	89	2	no	—
219	C₂×C₄⋊C₄	—	89	1	no	—
220	C₂×C₂²⋊C₄	—	89	1	no	—
221	C₂×C₂²⋊C₄	—	89	1	no	—
222	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
223	M₃₂	—	356	4	no	—
224	C₂×C₁₆	—	89	4	no	—
225	C₂×C₁₆	—	356	4	no	—
226	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
227	C₄⋊C₈	—	89	2	no	—
228	C₄⋊C₈	—	89	2	no	—
229	C₂²⋊C₈	—	89	2	no	—
230	C₂²⋊C₈	—	89	2	no	—
231	C₂²⋊C₈	—	89	2	no	—
232	C₂²⋊C₈	—	89	2	no	—
233	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
234	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂⁴	—	89	1	no	—
238	C₂⁴	—	89	1	no	—
239	C₂×D₄	—	89	2	no	—
240	C₂×D₄	—	89	2	no	—
241	C₂×D₄	—	178	4	no	—
242	C₂²×C₄	—	89	1	no	—
243	C₂²×C₄	—	89	1	no	—
244	C₂²×C₄	—	89	2	no	—
245	C₂²×C₄	—	89	2	no	—
246	C₂²×C₄	—	89	2	no	—
247	C₂²×C₄	—	89	2	no	—
248	C₂²×C₄	—	356	1	no	—
249	C₂²×C₄	—	356	1	no	—
250	C₂²×C₄	—	356	1	no	—
251	C₂²×C₄	—	356	1	no	—
252	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	89	2	no	—
258	C₂×C₈	—	89	2	no	—
259	C₂×C₈	—	356	2	no	—
260	C₂×C₈	—	356	2	no	—
261	C₂×C₈	—	356	1	no	—
262	C₂×C₈	—	356	1	no	—
263	C₂×C₈	—	178	4	no	—
264	C₂×C₈	—	178	4	no	—
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	—	89	2	no	—
270	C₄⋊C₄	—	89	2	no	—
271	C₂²⋊C₄	—	89	2	no	—
272	C₂²⋊C₄	—	89	2	no	—
273	C₂²⋊C₄	—	89	2	no	—
274	C₂²⋊C₄	—	89	2	no	—
275	C₄²	C₈×D₈₉	1	1	yes	characteristic
276	C₄²	C₈×D₈₉	1	1	yes	characteristic
277	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
278	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
279	C₁₆	—	2	4	no	—
280	C₁₆	—	178	4	no	—
281	C₂³	—	89	4	no	—
282	C₂³	—	89	2	no	—
283	C₂³	—	89	2	no	—
284	C₂³	—	356	1	no	—
285	C₂³	—	356	1	no	—
286	C₂³	—	356	1	no	—
287	C₂³	—	356	1	no	—
288	C₂³	—	178	2	no	—
289	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
290	D₄	—	178	4	no	—
291	D₄	—	178	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	89	2	no	—
297	C₂×C₄	—	89	2	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	356	2	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	—	178	2	no	—
304	C₂×C₄	—	178	2	no	—
305	C₂×C₄	—	178	2	no	—
306	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	178	2	no	—
316	C₈	—	178	2	no	—
317	C₈	#15668058	1	2	yes	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	89	4	no	—
321	C₂²	—	356	4	no	—
322	C₂²	—	356	4	no	—
323	C₂²	—	178	2	no	—
324	C₂²	—	178	2	no	—
325	C₂²	—	178	2	no	—
326	C₂²	—	178	2	no	—
327	C₂²	—	178	2	no	—
328	C₂²	M₃₂×D₈₉	1	2	yes	—
329	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	178	2	no	—
336	C₄	—	178	2	no	—
337	C₄	#19152072	1	1	yes	characteristic
338	C₄	#19152364	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	356	2	no	—
341	C₂	—	178	4	no	—
342	C₂	#26029275	1	1	yes	characteristic
343	C₂	C₂×M₃₂×D₈₉	1	1	yes	characteristic
344	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2163	C₂
prime radical	C₂²×D₃₅₆	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2163	D₈₉
hypocenter	C₈₉	#33487
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #26022079	C₂	semidirect
3. #26022076	C₂	semidirect
4. #26022055	C₂	semidirect
5. #26021861	C₂	semidirect
6. C₂²⋊C₁₆×D₈₉	C₂	semidirect
7. C₈₉×#2163	C₂	semidirect
11. C₂×#15667951	C₂²	semidirect
12. C₂×C₁₆×D₈₉	C₂²	semidirect
20. C₄⋊#13083549	C₂²	semidirect
22. #19151377	C₂²	semidirect
23. #19151377	C₂²	semidirect
24. #19151374	C₂²	semidirect
26. C₄⋊C₁₆×C₈₉	C₂²	semidirect
28. #346×C₈₉	C₂²	semidirect
114. C₂²×D₈₉	M₃₂	semidirect
177. #2163	D₈₉	semidirect
180. C₈₉	#33487	semidirect
195. C₄⋊C₁₆	C₂×D₈₉	semidirect
201. #346	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₅₆	2
#26022079	2
#26022076	2
#26022055	2
#26021861	2
C₂²⋊C₁₆×D₈₉	2
C₈₉×#2163	2
#33487	89