Subgroups — #36526124

Order 22784.

656 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029380	C₂	1	2	yes	maximal, complemented
2	C₄×#15668031	C₂	1	1	yes	maximal, characteristic
3	#26019137	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019133	C₂	1	2	yes	maximal, complemented
5	#26019128	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019058	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019009	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018688	C₂	1	2	yes	maximal, complemented
10	C₈×#13083557	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1589	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668031	C₄	1	1	yes	characteristic
13	C₄×#13083573	C₂²	1	2	yes	—
14	C₈×Dic₁₇₈	C₂²	1	2	yes	—
15	#19151722	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	#19151676	C₂²	1	2	yes	—
22	#19151674	—	2	2	no	—
23	#19151673	—	2	2	no	—
24	#19151649	C₂²	1	1	yes	characteristic
25	#19151648	—	2	2	no	—
26	#19151635	C₄	1	1	yes	characteristic
27	#19151621	—	2	2	no	—
28	#19151620	C₂²	1	2	yes	—
29	#19151591	C₄	1	2	yes	—
30	C₄×#13083557	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151351	C₂²	1	1	yes	characteristic
33	#19151350	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	C₈×#10623494	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
41	#436×C₈₉	C₂²	1	2	yes	—
42	#335×C₈₉	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₈₉	C₂²	1	2	yes	—
44	C₈²×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×#13083558	—	2	2	no	—
50	C₂×#13083557	—	2	1	no	—
51	C₂×#13083557	—	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	1	yes	characteristic
70	#15667941	D₄	1	1	yes	characteristic
71	#15667937	C₂×C₄	1	2	yes	—
72	#15667935	C₂×C₄	1	2	yes	—
73	#15667933	C₂×C₄	1	2	yes	—
74	C₂²⋊#10623494	—	2	2	no	—
75	#15667924	D₄	1	1	yes	characteristic
76	#15667924	D₄	1	1	yes	characteristic
77	#15667922	—	2	1	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	—	2	2	no	—
82	C₈×Dic₈₉	—	2	2	no	—
83	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
85	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
86	C₂×M₁₆×C₈₉	—	2	2	no	—
87	C₄×D₄×C₈₉	C₂³	1	2	yes	—
88	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
89	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
90	#102×C₈₉	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈×C₈₉	—	2	2	no	—
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
95	#13083573	—	2	4	no	—
96	#13083573	—	4	2	no	—
97	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
98	C₂×C₂²⋊D₈₉	—	2	2	no	—
99	C₂²×Dic₈₉	—	2	2	no	—
100	C₂²×Dic₈₉	—	2	2	no	—
101	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
104	Dic₃₅₆	M₁₆	1	4	yes	complemented
105	#13083562	—	2	4	no	—
106	#13083558	—	8	2	no	—
107	#13083557	—	2	4	no	—
108	#13083557	—	8	2	no	—
109	C₂×#10623494	—	2	1	no	—
110	C₂×#10623494	—	2	2	no	—
111	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
112	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
113	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
114	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
115	C₈×D₈₉	—	2	2	no	—
116	C₈×D₈₉	—	4	1	no	—
117	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
118	#13083553	—	2	2	no	—
119	#13083553	—	2	2	no	—
120	C₂²⋊Dic₈₉	—	2	2	no	—
121	#13083551	—	2	2	no	—
122	C₄×Dic₈₉	—	2	2	no	—
123	C₄×Dic₈₉	—	2	2	no	—
124	C₄×Dic₈₉	—	2	2	no	—
125	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₈₉	#41	1	1	yes	characteristic
128	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₈₉	—	2	2	no	—
130	D₈×C₈₉	—	2	2	no	—
131	M₁₆×C₈₉	—	8	2	no	—
132	C₂×C₈×C₈₉	—	2	1	no	—
133	C₂×C₈×C₈₉	—	2	2	no	—
134	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
139	C₂²⋊C₄×C₈₉	—	2	2	no	—
140	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
141	C₂³×C₈₉	—	2	2	no	—
142	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
143	D₄×C₈₉	—	2	2	no	—
144	D₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
151	C₂²⋊D₈₉	—	4	4	no	—
152	C₂×Dic₈₉	—	2	2	no	—
153	C₂×Dic₈₉	—	2	2	no	—
154	C₂×Dic₈₉	—	4	4	no	—
155	C₂×Dic₈₉	—	4	4	no	—
156	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
157	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
158	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
159	C₄×D₈₉	—	2	1	no	—
160	C₄×D₈₉	—	2	1	no	—
161	C₄×D₈₉	—	2	1	no	—
162	C₄×D₈₉	—	2	1	no	—
163	C₄×D₈₉	—	2	2	no	—
164	Dic₁₇₈	—	2	2	no	—
165	Dic₁₇₈	C₂×M₁₆	1	4	yes	—
166	C₈×C₈₉	—	2	2	no	—
167	C₈×C₈₉	—	2	1	no	—
168	C₈×C₈₉	—	2	1	no	—
169	C₈×C₈₉	—	2	1	no	—
170	C₈×C₈₉	—	4	2	no	—
171	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
172	#10623494	—	2	2	no	—
173	#10623494	—	2	1	no	—
174	#10623494	—	2	1	no	—
175	#10623494	—	2	1	no	—
176	#10623494	—	2	1	no	—
177	#10623494	—	4	2	no	—
178	C₂²×C₈₉	—	4	2	no	—
179	C₂²×C₈₉	—	4	2	no	—
180	C₂²×C₈₉	—	4	2	no	—
181	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂×D₈₉	—	2	1	no	—
183	C₂×D₈₉	—	2	1	no	—
184	C₂×D₈₉	—	2	1	no	—
185	C₄×C₈₉	—	2	1	no	—
186	C₄×C₈₉	—	4	2	no	—
187	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
188	C₄×C₈₉	#568	1	1	yes	characteristic
189	C₄×C₈₉	#576	1	1	yes	characteristic
190	C₄×C₈₉	#578	1	1	yes	characteristic
191	Dic₈₉	—	2	4	no	—
192	Dic₈₉	—	2	1	no	—
193	Dic₈₉	—	2	1	no	—
194	Dic₈₉	—	4	2	no	—
195	Dic₈₉	#443	1	2	yes	—
196	#18059	—	89	1	no	maximal, Sylow
197	C₂×C₈₉	—	8	2	no	—
198	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
199	C₂×C₈₉	#2951	1	1	yes	characteristic
200	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
201	D₈₉	—	4	1	no	—
202	#2991	—	89	2	no	—
203	C₄×#135	—	89	1	no	—
204	#1589	—	89	1	no	—
205	#1589	D₈₉	1	1	yes	characteristic, complemented
206	#1588	—	89	2	no	—
207	#1588	—	89	2	no	—
208	C₈×Q₁₆	—	89	1	no	—
209	C₈×Q₁₆	—	89	1	no	—
210	#1567	—	89	1	no	—
211	#1489	—	89	2	no	—
212	C₈×M₁₆	—	89	1	no	—
213	C₈₉	#18059	1	1	yes	characteristic, complemented, Sylow
214	C₂×#135	—	89	1	no	—
215	C₄×#41	—	89	2	no	—
216	C₈×Q₈	—	89	2	no	—
217	C₈×Q₈	—	89	2	no	—
218	C₄×Q₁₆	—	89	1	no	—
219	C₄×Q₁₆	—	89	1	no	—
220	C₄×SD₁₆	—	89	2	no	—
221	C₄×SD₁₆	—	89	2	no	—
222	C₄×D₈	—	89	1	no	—
223	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
224	#436	—	89	2	no	—
225	#436	—	178	2	no	—
226	#436	—	178	2	no	—
227	#436	C₂×D₈₉	1	2	yes	—
228	#431	—	89	1	no	—
229	#431	—	178	2	no	—
230	#427	—	89	1	no	—
231	#423	—	89	2	no	—
232	#423	—	178	2	no	—
233	#416	—	89	2	no	—
234	C₄×M₁₆	—	89	1	no	—
235	C₂×C₄×C₈	—	89	1	no	—
236	#335	—	89	2	no	—
237	#335	—	89	1	no	—
238	#335	C₂×D₈₉	1	1	yes	characteristic
239	Q₈⋊C₈	—	89	2	no	—
240	Q₈⋊C₈	—	89	2	no	—
241	D₄⋊C₈	—	89	2	no	—
242	D₄⋊C₈	C₂×D₈₉	1	2	yes	—
243	#322	—	89	1	no	—
244	#322	—	89	1	no	—
245	C₈²	—	89	1	no	—
246	C₈²	C₂×D₈₉	1	1	yes	characteristic
247	C₂×#41	—	89	2	no	—
248	#135	—	178	4	no	—
249	C₂×Q₁₆	—	89	1	no	—
250	C₂×SD₁₆	—	89	2	no	—
251	C₂×D₈	C₄×D₈₉	1	1	yes	characteristic
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	178	1	no	—
254	C₂×M₁₆	—	178	2	no	—
255	C₂×M₁₆	—	178	2	no	—
256	C₂²×C₈	—	89	1	no	—
257	C₂²×C₈	—	89	1	no	—
258	C₄×Q₈	—	89	2	no	—
259	C₄×Q₈	—	89	2	no	—
260	C₄×D₄	—	89	2	no	—
261	C₄×D₄	—	178	2	no	—
262	C₄×D₄	—	178	2	no	—
263	C₄×D₄	C₂²×D₈₉	1	2	yes	—
264	#117	—	89	2	no	—
265	#117	—	178	2	no	—
266	C₂×C₄²	—	89	1	no	—
267	C₂×C₄²	—	178	2	no	—
268	#107	—	89	1	no	—
269	#107	C₄×D₈₉	1	1	yes	characteristic
270	#106	—	89	2	no	—
271	C₄⋊C₈	—	89	1	no	—
272	C₄⋊C₈	—	89	1	no	—
273	C₄⋊C₈	—	89	2	no	—
274	C₄⋊C₈	—	178	2	no	—
275	C₄⋊C₈	—	178	2	no	—
276	C₄⋊C₈	—	178	2	no	—
277	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
278	#103	—	89	2	no	—
279	#103	—	89	2	no	—
280	#102	—	89	2	no	—
281	#102	C₄×D₈₉	1	2	yes	—
282	C₂²⋊C₈	—	2	2	no	—
283	C₂²⋊C₈	—	89	1	no	—
284	C₂²⋊C₈	—	89	1	no	—
285	C₂²⋊C₈	—	178	2	no	—
286	#97	—	178	1	no	—
287	C₄×C₈	—	89	1	no	—
288	C₄×C₈	—	89	1	no	—
289	C₄×C₈	—	89	1	no	—
290	C₄×C₈	—	89	1	no	—
291	C₄×C₈	—	89	1	no	—
292	C₄×C₈	—	178	2	no	—
293	C₄×C₈	—	178	2	no	—
294	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
297	#41	—	356	2	no	—
298	#41	—	178	4	no	—
299	C₂×Q₈	—	89	2	no	—
300	C₂×D₄	—	178	2	no	—
301	C₂×D₄	C₂×C₄×D₈₉	1	2	yes	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	89	1	no	—
304	C₂²×C₄	—	89	1	no	—
305	C₂²×C₄	—	89	1	no	—
306	C₂²×C₄	—	178	2	no	—
307	C₂²×C₄	—	178	2	no	—
308	Q₁₆	—	89	4	no	—
309	SD₁₆	—	178	4	no	—
310	D₈	—	2	2	no	—
311	M₁₆	—	8	2	no	—
312	M₁₆	—	178	4	no	—
313	M₁₆	—	712	2	no	—
314	M₁₆	—	712	2	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	89	1	no	—
318	C₂×C₈	—	89	1	no	—
319	C₂×C₈	—	89	1	no	—
320	C₂×C₈	—	89	1	no	—
321	C₂×C₈	—	356	1	no	—
322	C₂×C₈	—	178	2	no	—
323	C₂×C₈	—	178	1	no	—
324	C₂×C₈	—	178	2	no	—
325	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
326	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
327	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
328	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
329	C₄⋊C₄	—	89	2	no	—
330	C₄⋊C₄	—	178	2	no	—
331	C₄⋊C₄	—	178	2	no	—
332	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	178	2	no	—
335	C₂²⋊C₄	—	178	2	no	—
336	C₄²	—	89	1	no	—
337	C₄²	—	89	1	no	—
338	C₄²	—	89	1	no	—
339	C₄²	—	178	2	no	—
340	C₄²	—	178	2	no	—
341	C₄²	—	178	2	no	—
342	C₄²	C₂³×D₈₉	1	1	yes	characteristic
343	C₂³	—	2	2	no	—
344	C₂³	—	89	1	no	—
345	Q₈	—	89	4	no	—
346	Q₈	—	178	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	2	2	no	—
349	D₄	—	356	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	89	1	no	—
354	C₂×C₄	—	89	1	no	—
355	C₂×C₄	—	89	1	no	—
356	C₂×C₄	—	356	4	no	—
357	C₂×C₄	—	356	4	no	—
358	C₂×C₄	—	178	2	no	—
359	C₂×C₄	—	178	2	no	—
360	C₂×C₄	—	178	2	no	—
361	C₂×C₄	—	178	1	no	—
362	C₂×C₄	—	178	1	no	—
363	C₂×C₄	—	178	1	no	—
364	C₂×C₄	—	178	1	no	—
365	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
366	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
367	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	2	no	—
373	C₈	—	356	2	no	—
374	C₈	—	178	2	no	—
375	C₈	—	178	1	no	—
376	C₈	—	178	1	no	—
377	C₈	—	178	1	no	—
378	C₈	—	178	1	no	—
379	C₈	M₁₆×D₈₉	1	2	yes	—
380	C₂²	—	4	2	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	178	1	no	—
384	C₂²	—	178	1	no	—
385	C₂²	—	178	1	no	—
386	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	4	2	no	—
389	C₄	—	89	2	no	—
390	C₄	—	356	2	no	—
391	C₄	—	178	1	no	—
392	C₄	—	178	1	no	—
393	C₄	—	178	4	no	—
394	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
395	C₄	#19152363	1	1	yes	characteristic
396	C₄	#19152399	1	1	yes	characteristic
397	C₄	#19152412	1	1	yes	characteristic
398	C₂	—	8	2	no	—
399	C₂	—	356	1	no	—
400	C₂	#26028988	1	1	yes	characteristic
401	C₂	#26029122	1	1	yes	characteristic
402	C₂	#443×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×#1589	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668031	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1589	D₈₉
hypocenter	C₈₉	#18059
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
1. #26029380	C₂	semidirect
3. #26019137	C₂	semidirect
4. #26019133	C₂	semidirect
5. #26019128	C₂	semidirect
6. C₈×Dic₃₅₆	C₂	semidirect
7. #26019058	C₂	semidirect
8. #26019009	C₂	semidirect
9. #26018688	C₂	semidirect
10. C₈×#13083557	C₂	semidirect
11. C₈₉×#1589	C₂	semidirect
104. Dic₃₅₆	M₁₆	semidirect
205. #1589	D₈₉	semidirect
213. C₈₉	#18059	semidirect

Maximal subgroups

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

subgroup	index
#26029380	2
C₄×#15668031	2
#26019137	2
#26019133	2
#26019128	2
C₈×Dic₃₅₆	2
#26019058	2
#26019009	2
#26018688	2
C₈×#13083557	2
C₈₉×#1589	2
#18059	89