Subgroups — #36526173

Order 22784.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029375	C₂	1	2	yes	maximal, complemented
2	C₄×#15668031	C₂	1	1	yes	maximal, characteristic
3	#26019162	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019160	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019088	C₂	1	2	yes	maximal, complemented
6	#26019085	C₂	1	2	yes	maximal, complemented
7	#26019076	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019006	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018688	C₂	1	2	yes	maximal, complemented
10	#26018538	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1580	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668031	C₄	1	1	yes	characteristic
13	C₄×#13083573	C₂²	1	2	yes	—
14	#19151728	C₂²	1	2	yes	—
15	#19151725	C₂²	1	2	yes	—
16	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
17	#19151692	C₄	1	1	yes	characteristic
18	#19151689	C₄	1	2	yes	—
19	C₄×#13083562	C₂²	1	2	yes	—
20	D₈×Dic₈₉	C₄	1	1	yes	characteristic
21	#19151669	—	2	2	no	—
22	#19151666	—	2	2	no	—
23	D₄×#10623494	C₂²	1	2	yes	—
24	#19151652	—	2	2	no	—
25	#19151649	C₂²	1	1	yes	characteristic
26	#19151635	C₄	1	1	yes	characteristic
27	#19151620	C₂²	1	2	yes	—
28	C₂×#15667941	—	2	2	no	—
29	#19151591	C₄	1	2	yes	—
30	C₄×#13083557	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151349	C₂²	1	1	yes	characteristic
33	#19151348	C₂²	1	2	yes	—
34	#19151330	C₂²	1	2	yes	—
35	#19151328	C₂²	1	2	yes	—
36	D₄⋊#10623494	C₂²	1	2	yes	—
37	#19151320	C₂²	1	1	yes	characteristic
38	#19151319	C₂²	1	1	yes	characteristic
39	#19151319	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₈₉	C₂²	1	2	yes	—
42	#334×C₈₉	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₈₉	C₂²	1	2	yes	—
44	#322×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#13083573	C₂×C₄	1	2	yes	—
46	#15668031	—	2	4	no	—
47	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
48	C₂×#13083562	C₂×C₄	1	2	yes	—
49	C₂×#13083557	—	2	1	no	—
50	C₂×#13083557	—	2	2	no	—
51	C₂²×#10623494	—	2	2	no	—
52	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₇₈	C₂³	1	2	yes	—
55	#15667977	C₂×C₄	1	2	yes	—
56	D₄×Dic₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₂²⋊D₈₉	—	2	2	no	—
58	#15667972	—	2	2	no	—
59	#15667970	C₂×C₄	1	2	yes	—
60	#15667968	—	2	2	no	—
61	C₂×C₄×Dic₈₉	—	2	2	no	—
62	C₄²×D₈₉	C₂³	1	1	yes	characteristic
63	#15667946	C₂×C₄	1	1	yes	characteristic
64	#15667943	C₂×C₄	1	2	yes	—
65	C₄⋊#10623494	C₂³	1	2	yes	—
66	#15667941	—	2	2	no	—
67	#15667941	—	2	2	no	—
68	#15667941	—	2	2	no	—
69	#15667941	D₄	1	2	yes	—
70	#15667937	C₂×C₄	1	2	yes	—
71	#15667935	C₂×C₄	1	2	yes	—
72	#15667933	C₂×C₄	1	2	yes	—
73	C₂²⋊#10623494	—	2	2	no	—
74	#15667924	D₄	1	2	yes	—
75	#15667922	—	2	2	no	—
76	#15667922	—	2	1	no	—
77	#15667922	—	2	2	no	—
78	C₄×#10623494	C₂³	1	1	yes	characteristic
79	C₄×#10623494	C₂³	1	1	yes	characteristic
80	C₄×#10623494	C₂³	1	1	yes	characteristic
81	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
83	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₈₉	—	2	2	no	—
85	C₄×D₄×C₈₉	C₂³	1	2	yes	—
86	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
88	#102×C₈₉	C₂×C₄	1	2	yes	—
89	C₂²⋊C₈×C₈₉	—	2	2	no	—
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	#13083573	—	2	4	no	—
94	#13083573	—	4	2	no	—
95	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
96	C₂×C₂²⋊D₈₉	—	2	2	no	—
97	C₂²×Dic₈₉	—	2	2	no	—
98	C₂²×Dic₈₉	—	2	2	no	—
99	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
102	Dic₃₅₆	—	2	2	no	—
103	#13083562	—	2	4	no	—
104	#13083557	—	4	2	no	—
105	#13083557	—	8	2	no	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	—	2	1	no	—
108	C₂×#10623494	—	8	2	no	—
109	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
110	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
111	C₂×#10623494	C₂×D₄	1	2	yes	—
112	C₈×D₈₉	—	2	2	no	—
113	C₈×D₈₉	—	4	1	no	—
114	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
115	#13083553	—	2	2	no	—
116	#13083553	—	2	2	no	—
117	C₂²⋊Dic₈₉	—	2	2	no	—
118	#13083551	—	2	2	no	—
119	C₄×Dic₈₉	—	2	2	no	—
120	C₄×Dic₈₉	—	2	2	no	—
121	C₄×Dic₈₉	—	2	2	no	—
122	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₈₉	#41	1	1	yes	characteristic
125	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₈₉	—	2	2	no	—
127	D₈×C₈₉	M₁₆	1	4	yes	complemented
128	C₂×C₈×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	—	8	2	no	—
131	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
135	C₂²⋊C₄×C₈₉	—	2	2	no	—
136	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₂³×C₈₉	—	2	2	no	—
138	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
139	D₄×C₈₉	—	2	2	no	—
140	D₄×C₈₉	C₂×M₁₆	1	4	yes	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₈₉	—	4	4	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	2	2	no	—
150	C₂×Dic₈₉	—	4	4	no	—
151	C₂×Dic₈₉	—	4	4	no	—
152	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
155	C₄×D₈₉	—	2	1	no	—
156	C₄×D₈₉	—	2	1	no	—
157	C₄×D₈₉	—	2	2	no	—
158	C₄×D₈₉	—	2	2	no	—
159	Dic₁₇₈	—	2	2	no	—
160	Dic₁₇₈	—	2	2	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	2	2	no	—
163	C₈×C₈₉	—	4	1	no	—
164	C₈×C₈₉	—	4	2	no	—
165	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
166	#10623494	—	2	1	no	—
167	#10623494	—	2	1	no	—
168	#10623494	—	2	2	no	—
169	#10623494	—	4	1	no	—
170	#10623494	—	4	2	no	—
171	C₂²×C₈₉	—	2	4	no	—
172	C₂²×C₈₉	—	4	2	no	—
173	C₂²×C₈₉	—	4	2	no	—
174	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₈₉	—	2	1	no	—
176	C₂×D₈₉	—	2	1	no	—
177	C₂×D₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	4	2	no	—
180	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₈₉	#568	1	1	yes	characteristic
182	C₄×C₈₉	#577	1	2	yes	—
183	Dic₈₉	—	2	1	no	—
184	Dic₈₉	—	2	1	no	—
185	Dic₈₉	—	4	2	no	—
186	Dic₈₉	—	4	2	no	—
187	Dic₈₉	#444	1	2	yes	—
188	#18065	—	89	1	no	maximal, Sylow
189	C₂×C₈₉	—	4	4	no	—
190	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
191	C₂×C₈₉	#2952	1	1	yes	characteristic
192	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
193	D₈₉	—	4	1	no	—
194	#2990	—	89	2	no	—
195	C₄×#135	—	89	1	no	—
196	#1593	—	89	1	no	—
197	#1593	—	89	1	no	—
198	#1582	—	89	2	no	—
199	#1582	—	89	2	no	—
200	#1580	—	89	1	no	—
201	#1580	D₈₉	1	1	yes	characteristic, complemented
202	#1566	—	89	1	no	—
203	#1489	—	89	2	no	—
204	#1451	—	89	1	no	—
205	C₈₉	#18065	1	1	yes	characteristic, complemented, Sylow
206	C₂×#135	—	89	1	no	—
207	C₄×#41	—	89	2	no	—
208	#446	—	89	2	no	—
209	#446	—	89	2	no	—
210	C₄×Q₁₆	—	89	1	no	—
211	C₄×Q₁₆	—	89	1	no	—
212	C₄×SD₁₆	—	89	2	no	—
213	C₄×SD₁₆	—	89	2	no	—
214	C₄×D₈	—	89	1	no	—
215	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
216	#435	—	178	2	no	—
217	#435	—	178	2	no	—
218	C₈×D₄	—	89	2	no	—
219	C₈×D₄	C₂×D₈₉	1	2	yes	—
220	#432	—	178	2	no	—
221	#431	—	89	1	no	—
222	#427	—	89	1	no	—
223	#423	—	89	2	no	—
224	C₂×C₄⋊C₈	—	178	2	no	—
225	#416	—	89	2	no	—
226	C₄×M₁₆	—	89	1	no	—
227	C₂×C₄×C₈	—	89	1	no	—
228	#334	—	89	2	no	—
229	#334	—	89	1	no	—
230	#334	C₂×D₈₉	1	1	yes	characteristic
231	Q₈⋊C₈	—	89	2	no	—
232	Q₈⋊C₈	—	89	2	no	—
233	D₄⋊C₈	—	89	2	no	—
234	D₄⋊C₈	C₂×D₈₉	1	2	yes	—
235	#322	—	89	1	no	—
236	#322	—	89	1	no	—
237	#322	—	89	1	no	—
238	#322	C₂×D₈₉	1	1	yes	characteristic
239	C₂×#41	—	89	2	no	—
240	#135	—	178	4	no	—
241	C₂×Q₁₆	—	89	1	no	—
242	C₂×SD₁₆	—	89	2	no	—
243	C₂×D₈	C₄×D₈₉	1	1	yes	characteristic
244	C₂×M₁₆	—	178	1	no	—
245	C₂×M₁₆	—	178	2	no	—
246	C₂²×C₈	—	2	2	no	—
247	C₂²×C₈	—	89	1	no	—
248	C₂²×C₈	—	89	1	no	—
249	C₂²×C₈	—	178	2	no	—
250	C₄×Q₈	—	89	2	no	—
251	C₄×Q₈	—	89	2	no	—
252	C₄×D₄	—	89	2	no	—
253	C₄×D₄	—	178	2	no	—
254	C₄×D₄	—	178	2	no	—
255	C₄×D₄	C₂²×D₈₉	1	2	yes	—
256	#117	—	89	2	no	—
257	#117	—	178	2	no	—
258	C₂×C₄²	—	89	1	no	—
259	C₂×C₄²	—	178	2	no	—
260	#107	—	89	1	no	—
261	#107	C₄×D₈₉	1	1	yes	characteristic
262	#106	—	89	2	no	—
263	C₄⋊C₈	—	89	2	no	—
264	C₄⋊C₈	—	89	2	no	—
265	C₄⋊C₈	—	178	2	no	—
266	C₄⋊C₈	—	178	2	no	—
267	C₄⋊C₈	—	178	2	no	—
268	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
269	#103	—	89	2	no	—
270	#103	—	89	2	no	—
271	#102	—	89	2	no	—
272	#102	C₄×D₈₉	1	2	yes	—
273	C₂²⋊C₈	—	2	2	no	—
274	C₂²⋊C₈	—	89	2	no	—
275	C₂²⋊C₈	—	178	2	no	—
276	#97	—	178	1	no	—
277	#97	—	178	2	no	—
278	#97	—	178	2	no	—
279	C₄×C₈	—	89	1	no	—
280	C₄×C₈	—	89	1	no	—
281	C₄×C₈	—	89	1	no	—
282	C₄×C₈	—	89	1	no	—
283	C₄×C₈	—	89	1	no	—
284	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	#41	—	356	2	no	—
288	#41	—	178	4	no	—
289	C₂×Q₈	—	89	2	no	—
290	C₂×D₄	—	178	2	no	—
291	C₂×D₄	C₂×C₄×D₈₉	1	2	yes	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	89	1	no	—
294	C₂²×C₄	—	89	1	no	—
295	C₂²×C₄	—	89	1	no	—
296	C₂²×C₄	—	178	2	no	—
297	C₂²×C₄	—	178	2	no	—
298	Q₁₆	—	178	2	no	—
299	SD₁₆	—	178	4	no	—
300	D₈	#13083557	1	4	yes	complemented
301	M₁₆	—	356	2	no	—
302	M₁₆	—	712	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	8	2	no	—
306	C₂×C₈	—	89	1	no	—
307	C₂×C₈	—	89	1	no	—
308	C₂×C₈	—	89	2	no	—
309	C₂×C₈	—	356	1	no	—
310	C₂×C₈	—	178	2	no	—
311	C₂×C₈	—	178	1	no	—
312	C₂×C₈	—	178	2	no	—
313	C₂×C₈	—	712	2	no	—
314	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
316	C₂×C₈	D₄×D₈₉	1	2	yes	—
317	C₄⋊C₄	—	89	2	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₄⋊C₄	—	178	2	no	—
320	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	178	2	no	—
323	C₂²⋊C₄	—	178	2	no	—
324	C₄²	—	89	1	no	—
325	C₄²	—	89	1	no	—
326	C₄²	—	89	1	no	—
327	C₄²	—	178	2	no	—
328	C₄²	—	178	2	no	—
329	C₄²	—	178	2	no	—
330	C₄²	C₂³×D₈₉	1	1	yes	characteristic
331	C₂³	—	2	2	no	—
332	C₂³	—	89	1	no	—
333	Q₈	—	178	2	no	—
334	Q₈	—	178	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	356	4	no	—
337	D₄	C₂×#13083557	1	4	yes	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	89	1	no	—
342	C₂×C₄	—	89	1	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	356	4	no	—
345	C₂×C₄	—	356	4	no	—
346	C₂×C₄	—	178	2	no	—
347	C₂×C₄	—	178	2	no	—
348	C₂×C₄	—	178	2	no	—
349	C₂×C₄	—	178	2	no	—
350	C₂×C₄	—	178	1	no	—
351	C₂×C₄	—	178	1	no	—
352	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
354	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	356	1	no	—
360	C₈	—	356	2	no	—
361	C₈	—	178	2	no	—
362	C₈	—	178	1	no	—
363	C₈	—	178	1	no	—
364	C₈	C₂×#13083557	1	2	yes	—
365	C₂²	—	2	4	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	178	1	no	—
369	C₂²	—	178	1	no	—
370	C₂²	—	178	1	no	—
371	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	89	2	no	—
375	C₄	—	356	2	no	—
376	C₄	—	356	2	no	—
377	C₄	—	178	1	no	—
378	C₄	—	178	1	no	—
379	C₄	C₂²×#13083557	1	1	yes	characteristic
380	C₄	#19152363	1	1	yes	characteristic
381	C₄	#19152408	1	2	yes	—
382	C₂	—	4	4	no	—
383	C₂	—	356	1	no	—
384	C₂	D₄×#13083557	1	1	yes	characteristic
385	C₂	#26029134	1	1	yes	characteristic
386	C₂	#444×D₈₉	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×#1580	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668031	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1580	D₈₉
hypocenter	C₈₉	#18065
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1580

Complements

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

N	H ≅ G/N	type
1. #26029375	C₂	semidirect
3. #26019162	C₂	semidirect
4. #26019160	C₂	semidirect
5. #26019088	C₂	semidirect
6. #26019085	C₂	semidirect
7. #26019076	C₂	semidirect
8. #26019006	C₂	semidirect
9. #26018688	C₂	semidirect
10. #26018538	C₂	semidirect
11. C₈₉×#1580	C₂	semidirect
127. D₈×C₈₉	M₁₆	semidirect
201. #1580	D₈₉	semidirect
205. C₈₉	#18065	semidirect
300. D₈	#13083557	semidirect

Maximal subgroups

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

subgroup	index
#26029375	2
C₄×#15668031	2
#26019162	2
#26019160	2
#26019088	2
#26019085	2
#26019076	2
#26019006	2
#26018688	2
#26018538	2
C₈₉×#1580	2
#18065	89