Subgroups — #36711305

Order 22784.

580 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	#26032601	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022480	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022451	C₂	1	2	yes	maximal, complemented
5	#26022394	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022347	C₂	1	2	yes	maximal, complemented
7	#26022297	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022167	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#358×D₈₉	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	2	yes	—
13	Q₈×D₃₅₆	—	2	1	no	—
14	#19152253	—	2	1	no	—
15	C₂×#15667956	C₂²	1	2	yes	—
16	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
17	#19152003	C₂²	1	1	yes	characteristic
18	#19151978	C₂²	1	1	yes	characteristic
19	#19151910	—	2	2	no	—
20	#19151878	—	2	1	no	—
21	#19151840	C₂²	1	2	yes	—
22	#19151831	C₂²	1	2	yes	—
23	#19151808	—	2	1	no	—
24	#19151755	—	2	2	no	—
25	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
26	#107×D₈₉	C₂²	1	2	yes	—
27	#19151425	C₂²	1	2	yes	—
28	C₄⋊#13083549	C₂²	1	1	yes	characteristic
29	#19151403	C₂²	1	2	yes	—
30	#19151403	C₂²	1	2	yes	—
31	#19151402	C₂²	1	2	yes	—
32	#19151401	C₂²	1	2	yes	—
33	#19151377	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
37	#367×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	1	yes	characteristic
39	#357×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	—	4	1	no	—
42	C₂×Q₈×D₈₉	—	2	2	no	—
43	C₂×Dic₃₅₆	—	2	1	no	—
44	C₂×Dic₃₅₆	C₂³	1	2	yes	—
45	C₂×#13083564	—	2	2	no	—
46	Q₁₆×D₈₉	—	2	4	no	—
47	Q₁₆×D₈₉	—	4	2	no	—
48	C₂×C₈×D₈₉	C₂³	1	2	yes	—
49	#15668012	—	2	1	no	—
50	#15668012	—	2	1	no	—
51	#15668012	—	4	1	no	—
52	#15667994	—	4	2	no	—
53	#15667992	—	2	2	no	—
54	#15667992	—	4	2	no	—
55	#15667991	—	4	1	no	—
56	#15667983	—	2	2	no	—
57	C₄×D₃₅₆	—	4	1	no	—
58	C₄×D₃₅₆	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₈₉	—	2	2	no	—
60	#15667956	—	4	2	no	—
61	C₂×#13083549	C₂³	1	2	yes	—
62	C₁₆×D₈₉	—	4	2	no	—
63	#15667946	—	2	1	no	—
64	#15667946	C₂³	1	2	yes	—
65	#15667945	—	2	2	no	—
66	C₄⋊#10623494	D₄	1	1	yes	characteristic
67	#15667935	—	2	2	no	—
68	#15667933	—	2	2	no	—
69	#15667924	D₄	1	2	yes	—
70	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	SD₃₂×C₈₉	—	2	2	no	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	#107×C₈₉	C₂³	1	1	yes	characteristic
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#103×C₈₉	—	2	1	no	—
80	#102×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	#13083574	—	8	4	no	—
83	C₂×Dic₁₇₈	—	2	2	no	—
84	C₂×Dic₁₇₈	—	4	1	no	—
85	C₂×Dic₁₇₈	—	4	2	no	—
86	Q₈×D₈₉	—	4	4	no	—
87	Q₈×D₈₉	—	8	2	no	—
88	C₂×D₃₅₆	D₈	1	1	yes	characteristic
89	C₂×C₂²⋊D₈₉	—	4	2	no	—
90	C₂×C₄×D₈₉	—	4	2	no	—
91	C₂×C₄×D₈₉	—	4	2	no	—
92	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
93	Dic₃₅₆	—	2	2	no	—
94	Dic₃₅₆	—	2	2	no	—
95	Dic₃₅₆	—	8	1	no	—
96	#13083564	—	4	4	no	—
97	C₂×#10623494	C₂×D₄	1	2	yes	—
98	C₈×D₈₉	—	2	2	no	—
99	C₈×D₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	2	2	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	C₄⋊Dic₈₉	—	4	1	no	—
103	C₄⋊Dic₈₉	—	4	1	no	—
104	C₄⋊Dic₈₉	—	4	2	no	—
105	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
106	#13083553	—	4	2	no	—
107	C₂²⋊Dic₈₉	—	4	2	no	—
108	#13083551	—	4	2	no	—
109	#13083551	D₈	1	2	yes	—
110	#13083549	—	2	2	no	—
111	C₂×Q₈×C₈₉	—	2	1	no	—
112	C₂×D₄×C₈₉	—	2	1	no	—
113	C₂²×C₄×C₈₉	—	4	1	no	—
114	Q₁₆×C₈₉	—	2	1	no	—
115	Q₁₆×C₈₉	#41	1	2	yes	complemented
116	D₈×C₈₉	—	2	1	no	—
117	D₈×C₈₉	—	2	1	no	—
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₂²⋊C₄×C₈₉	—	4	1	no	—
124	C₄²×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₈₉	—	2	1	no	—
127	C₁₆×C₈₉	—	2	1	no	—
128	C₂³×C₈₉	—	4	1	no	—
129	C₂²×D₈₉	C₂×D₈	1	2	yes	—
130	Q₈×C₈₉	—	2	2	no	—
131	Q₈×C₈₉	—	4	1	no	—
132	D₄×C₈₉	—	4	1	no	—
133	D₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	4	1	no	—
137	C₂×C₄×C₈₉	—	8	2	no	—
138	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₈₉	—	8	4	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	4	2	no	—
144	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
145	D₃₅₆	SD₃₂	1	4	yes	complemented
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	8	4	no	—
149	C₄×D₈₉	—	8	4	no	—
150	Dic₁₇₈	—	4	2	no	—
151	Dic₁₇₈	—	4	2	no	—
152	Dic₁₇₈	—	8	4	no	—
153	Dic₁₇₈	—	8	2	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
156	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
157	#10623494	—	2	2	no	—
158	C₂²×C₈₉	—	8	1	no	—
159	C₂²×C₈₉	—	8	1	no	—
160	C₂²×C₈₉	—	8	1	no	—
161	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₈₉	—	2	2	no	—
163	C₂×D₈₉	—	2	2	no	—
164	C₂×D₈₉	C₂×SD₃₂	1	4	yes	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	4	2	no	—
167	C₄×C₈₉	—	8	1	no	—
168	C₄×C₈₉	—	8	1	no	—
169	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
170	C₄×C₈₉	#546	1	1	yes	characteristic
171	C₄×C₈₉	#577	1	1	yes	characteristic
172	Dic₈₉	—	2	2	no	—
173	Dic₈₉	—	8	2	no	—
174	Dic₈₉	—	8	2	no	—
175	#33865	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	16	1	no	—
177	C₂×C₈₉	#3294	1	1	yes	characteristic
178	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
179	C₂×C₈₉	#3417	1	1	yes	characteristic
180	D₈₉	—	2	4	no	—
181	#3334	—	89	1	no	—
182	D₄×Q₁₆	—	89	1	no	—
183	#2241	—	89	1	no	—
184	#2235	—	89	2	no	—
185	#2225	—	89	1	no	—
186	#2217	—	89	2	no	—
187	#2208	—	89	1	no	—
188	#2185	—	89	2	no	—
189	C₄×SD₃₂	D₈₉	1	1	yes	characteristic, complemented
190	D₄×C₁₆	—	89	1	no	—
191	C₂×#358	—	89	2	no	—
192	C₈₉	#33865	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₁₆	—	89	2	no	—
194	D₄×Q₈	—	178	1	no	—
195	#547	—	178	1	no	—
196	C₂×SD₃₂	—	89	2	no	—
197	C₂×SD₃₂	C₂×D₈₉	1	1	yes	characteristic
198	C₂²×C₁₆	—	89	2	no	—
199	#500	—	89	1	no	—
200	#494	—	89	1	no	—
201	#480	—	178	2	no	—
202	#474	—	178	1	no	—
203	#467	—	89	2	no	—
204	#466	—	89	2	no	—
205	#462	—	178	1	no	—
206	#451	—	178	2	no	—
207	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
209	C₈×D₄	—	89	1	no	—
210	C₂×#107	—	89	2	no	—
211	#367	—	89	2	no	—
212	#367	C₂×D₈₉	1	1	yes	characteristic
213	C₄⋊C₁₆	—	89	1	no	—
214	#358	—	89	2	no	—
215	#358	—	89	2	no	—
216	#358	—	89	2	no	—
217	#358	—	89	2	no	—
218	#358	C₂×D₈₉	1	1	yes	characteristic
219	#357	C₂×D₈₉	1	1	yes	characteristic
220	C₂²⋊C₁₆	—	89	2	no	—
221	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
222	C₂×#41	—	356	1	no	—
223	C₂²×Q₈	—	178	2	no	—
224	C₂×Q₁₆	—	89	2	no	—
225	C₂×Q₁₆	—	356	2	no	—
226	C₂×Q₁₆	—	178	2	no	—
227	C₂×Q₁₆	—	178	4	no	—
228	C₂×Q₁₆	—	178	1	no	—
229	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₂²×C₈	—	89	2	no	—
232	C₄⋊Q₈	—	356	1	no	—
233	C₄⋊Q₈	—	178	1	no	—
234	C₄⋊Q₈	—	178	1	no	—
235	#123	—	356	2	no	—
236	C₂²⋊Q₈	—	356	2	no	—
237	C₂²⋊Q₈	—	356	1	no	—
238	C₂²⋊Q₈	—	178	2	no	—
239	#121	—	178	2	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	89	1	no	—
243	C₄×D₄	—	356	1	no	—
244	C₂×C₄⋊C₄	—	178	2	no	—
245	SD₃₂	—	2	2	no	—
246	SD₃₂	—	356	2	no	—
247	C₂×C₁₆	—	89	2	no	—
248	C₂×C₁₆	—	356	2	no	—
249	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
251	#107	—	89	2	no	—
252	#107	—	178	2	no	—
253	#107	—	178	1	no	—
254	#107	C₂²×D₈₉	1	1	yes	characteristic
255	#107	C₂²×D₈₉	1	1	yes	characteristic
256	C₄⋊C₈	—	89	1	no	—
257	#103	—	2	1	no	—
258	#103	—	178	2	no	—
259	#102	—	2	1	no	—
260	#102	—	178	2	no	—
261	C₂²⋊C₈	—	89	2	no	—
262	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
263	#41	—	712	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	356	2	no	—
266	C₂×Q₈	—	356	1	no	—
267	C₂×Q₈	—	356	4	no	—
268	C₂×Q₈	—	178	2	no	—
269	C₂×Q₈	—	712	2	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	89	1	no	—
272	C₂×D₄	—	356	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	89	2	no	—
275	C₂²×C₄	—	356	2	no	—
276	C₂²×C₄	—	356	2	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	356	4	no	—
279	Q₁₆	—	178	2	no	—
280	Q₁₆	—	178	2	no	—
281	Q₁₆	—	712	1	no	—
282	Q₁₆	#13083574	1	2	yes	complemented
283	D₈	—	2	1	no	—
284	D₈	—	2	1	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	178	2	no	—
287	C₂×C₈	—	178	2	no	—
288	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	89	1	no	—
294	C₄⋊C₄	—	356	2	no	—
295	C₄⋊C₄	—	356	2	no	—
296	C₄⋊C₄	—	356	1	no	—
297	C₄⋊C₄	—	356	1	no	—
298	C₄⋊C₄	—	356	1	no	—
299	C₄⋊C₄	—	178	2	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	89	2	no	—
302	C₂²⋊C₄	—	356	2	no	—
303	C₂²⋊C₄	—	356	2	no	—
304	C₄²	—	4	1	no	—
305	C₄²	D₄×D₈₉	1	1	yes	characteristic
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	178	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	89	2	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	356	2	no	—
314	Q₈	—	356	2	no	—
315	Q₈	—	712	4	no	—
316	Q₈	—	712	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	89	4	no	—
320	D₄	—	712	4	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	89	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	—	178	2	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	712	4	no	—
331	C₂×C₄	—	712	4	no	—
332	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
333	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	178	2	no	—
337	C₈	C₂×#13083574	1	1	yes	characteristic
338	C₈	#15668058	1	1	yes	characteristic
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	89	4	no	—
343	C₂²	—	178	2	no	—
344	C₂²	—	178	2	no	—
345	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	178	2	no	—
351	C₄	—	712	2	no	—
352	C₄	—	712	2	no	—
353	C₄	SD₃₂×D₈₉	1	2	yes	—
354	C₄	#19152244	1	1	yes	characteristic
355	C₄	#19152405	1	1	yes	characteristic
356	C₂	—	16	1	no	—
357	C₂	—	178	4	no	—
358	C₂	#26032067	1	1	yes	characteristic
359	C₂	C₂×SD₃₂×D₈₉	1	1	yes	characteristic
360	C₂	#26033402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×SD₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26032601	C₂
hypercenter	C₄×SD₃₂	D₈₉
hypocenter	C₈₉	#33865
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₃₅₆	C₂	semidirect
3. #26022480	C₂	semidirect
4. #26022451	C₂	semidirect
5. #26022394	C₂	semidirect
6. #26022347	C₂	semidirect
7. #26022297	C₂	semidirect
8. #26022167	C₂	semidirect
9. C₁₆×D₃₅₆	C₂	semidirect
10. #358×D₈₉	C₂	semidirect
11. C₄×SD₃₂×C₈₉	C₂	semidirect
115. Q₁₆×C₈₉	#41	semidirect
145. D₃₅₆	SD₃₂	semidirect
189. C₄×SD₃₂	D₈₉	semidirect
192. C₈₉	#33865	semidirect
282. Q₁₆	#13083574	semidirect

Maximal subgroups

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

subgroup	index
#26032601	2
Q₁₆×D₃₅₆	2
#26022480	2
#26022451	2
#26022394	2
#26022347	2
#26022297	2
#26022167	2
C₁₆×D₃₅₆	2
#358×D₈₉	2
C₄×SD₃₂×C₈₉	2
#33865	89