Subgroups — #36525276

Order 22784.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030807	C₂	1	1	yes	maximal, characteristic
2	#26029346	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019128	C₂	1	2	yes	maximal, complemented
4	#26019109	C₂	1	2	yes	maximal, complemented
5	#26019093	C₂	1	2	yes	maximal, complemented
6	#26019088	C₂	1	2	yes	maximal, complemented
7	#26019013	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018724	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018721	C₂	1	1	yes	maximal, characteristic
10	#26018545	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2981	C₂	1	1	yes	maximal, characteristic
12	#19152155	C₂²	1	1	yes	characteristic
13	#19151994	C₄	1	1	yes	characteristic
14	#19151970	C₄	1	1	yes	characteristic
15	#19151950	C₄	1	1	yes	characteristic
16	#19151944	C₄	1	1	yes	characteristic
17	#19151825	C₄	1	2	yes	—
18	#19151825	C₄	1	2	yes	—
19	#19151728	C₂²	1	2	yes	—
20	#19151727	—	2	2	no	—
21	C₈×Dic₁₇₈	C₂²	1	2	yes	—
22	C₄×#13083562	C₂²	1	2	yes	—
23	C₄×#13083562	C₂²	1	2	yes	—
24	#19151656	—	2	2	no	—
25	#19151655	C₂²	1	1	yes	characteristic
26	#19151651	C₂²	1	1	yes	characteristic
27	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
28	#19151350	C₂²	1	2	yes	—
29	#19151348	C₂²	1	2	yes	—
30	#19151328	C₂²	1	2	yes	—
31	#19151328	C₂²	1	2	yes	—
32	D₄⋊#10623494	C₂²	1	2	yes	—
33	D₄⋊#10623494	C₂²	1	2	yes	—
34	#19151320	C₂²	1	1	yes	characteristic
35	#19151320	C₂²	1	1	yes	characteristic
36	#19151319	C₂²	1	1	yes	characteristic
37	#19151319	C₂²	1	1	yes	characteristic
38	#532×C₈₉	C₂²	1	1	yes	characteristic
39	#436×C₈₉	C₂²	1	2	yes	—
40	#435×C₈₉	—	2	2	no	—
41	C₈×D₄×C₈₉	C₂²	1	2	yes	—
42	#432×C₈₉	C₂²	1	1	yes	characteristic
43	#431×C₈₉	C₂²	1	1	yes	characteristic
44	#408×C₈₉	—	2	2	no	—
45	C₂×#13083562	C₂×C₄	1	2	yes	—
46	C₂×#13083562	C₂×C₄	1	2	yes	—
47	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
48	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
49	#15668012	C₂×C₄	1	1	yes	characteristic
50	#15668012	C₂×C₄	1	1	yes	characteristic
51	#15668004	—	2	1	no	—
52	#15667991	—	2	1	no	—
53	#15667991	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₇₈	—	2	1	no	—
55	C₄×Dic₁₇₈	C₂³	1	2	yes	—
56	#15667971	—	2	1	no	—
57	#15667943	C₂×C₄	1	2	yes	—
58	#15667943	C₂×C₄	1	2	yes	—
59	C₄⋊#10623494	—	2	1	no	—
60	C₄⋊#10623494	C₂³	1	2	yes	—
61	#15667941	—	2	2	no	—
62	#15667941	—	2	2	no	—
63	#15667935	C₂×C₄	1	2	yes	—
64	#15667935	C₂×C₄	1	2	yes	—
65	#15667933	C₂×C₄	1	2	yes	—
66	#15667933	C₂×C₄	1	2	yes	—
67	C₂²⋊#10623494	—	2	1	no	—
68	C₂²⋊#10623494	—	2	1	no	—
69	#15667923	—	2	1	no	—
70	#15667922	—	2	2	no	—
71	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
72	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₄×#10623494	C₂³	1	1	yes	characteristic
75	C₄×#10623494	C₂³	1	1	yes	characteristic
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₈×Dic₈₉	—	2	2	no	—
78	C₂×#41×C₈₉	—	2	1	no	—
79	C₂×M₁₆×C₈₉	—	2	2	no	—
80	C₂²×C₈×C₈₉	—	2	2	no	—
81	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
82	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
83	#124×C₈₉	—	2	1	no	—
84	#121×C₈₉	—	2	1	no	—
85	#121×C₈₉	C₂×C₄	1	2	yes	—
86	C₄×D₄×C₈₉	—	2	1	no	—
87	C₄×D₄×C₈₉	C₂³	1	2	yes	—
88	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
90	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
91	C₂²⋊C₈×C₈₉	—	2	1	no	—
92	C₂²⋊C₈×C₈₉	—	2	2	no	—
93	C₂²⋊C₈×C₈₉	—	2	2	no	—
94	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
95	#97×C₈₉	—	2	1	no	—
96	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×Dic₁₇₈	—	2	1	no	—
99	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
100	#13083562	—	4	2	no	—
101	#13083562	—	4	2	no	—
102	C₂×#10623494	—	2	2	no	—
103	C₂×#10623494	—	2	1	no	—
104	C₂×#10623494	—	2	1	no	—
105	C₂×#10623494	—	4	1	no	—
106	C₂×#10623494	—	4	1	no	—
107	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
108	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
109	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
110	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
111	C₄⋊Dic₈₉	—	2	1	no	—
112	C₄⋊Dic₈₉	—	2	1	no	—
113	C₄⋊Dic₈₉	—	2	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	C₄×Dic₈₉	—	2	2	no	—
119	#41×C₈₉	—	8	2	no	—
120	C₂×Q₈×C₈₉	—	2	1	no	—
121	C₂×D₄×C₈₉	—	2	2	no	—
122	C₂×D₄×C₈₉	—	2	1	no	—
123	C₂×D₄×C₈₉	—	2	1	no	—
124	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₈₉	—	2	2	no	—
126	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
129	M₁₆×C₈₉	—	8	2	no	—
130	C₂×C₈×C₈₉	—	2	2	no	—
131	C₂×C₈×C₈₉	—	8	2	no	—
132	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₈₉	—	2	1	no	—
135	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
136	C₂²⋊C₄×C₈₉	—	2	2	no	—
137	C₂²⋊C₄×C₈₉	—	2	2	no	—
138	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₈₉	#41	1	1	yes	characteristic
141	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
142	C₂³×C₈₉	—	2	2	no	—
143	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
144	Q₈×C₈₉	—	8	1	no	—
145	D₄×C₈₉	—	2	2	no	—
146	D₄×C₈₉	—	2	2	no	—
147	D₄×C₈₉	—	8	1	no	—
148	D₄×C₈₉	—	8	1	no	—
149	D₄×C₈₉	—	8	2	no	—
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	4	1	no	—
155	C₂×C₄×C₈₉	—	4	2	no	—
156	C₂×C₄×C₈₉	—	4	2	no	—
157	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₂×Dic₈₉	—	2	2	no	—
164	C₂×Dic₈₉	—	2	2	no	—
165	Dic₁₇₈	—	2	2	no	—
166	Dic₁₇₈	—	2	2	no	—
167	Dic₁₇₈	—	8	1	no	—
168	C₈×C₈₉	—	2	2	no	—
169	C₈×C₈₉	—	2	2	no	—
170	C₈×C₈₉	—	4	2	no	—
171	#10623494	—	2	1	no	—
172	#10623494	—	2	1	no	—
173	#10623494	—	2	1	no	—
174	#10623494	—	2	1	no	—
175	#10623494	—	4	1	no	—
176	#10623494	—	4	1	no	—
177	#10623494	—	4	2	no	—
178	C₂²×C₈₉	—	2	1	no	—
179	C₂²×C₈₉	—	2	1	no	—
180	C₂²×C₈₉	—	2	1	no	—
181	C₂²×C₈₉	—	4	2	no	—
182	C₂²×C₈₉	—	4	2	no	—
183	C₂²×C₈₉	—	4	2	no	—
184	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₈₉	—	2	1	no	—
186	C₄×C₈₉	—	2	1	no	—
187	C₄×C₈₉	—	2	1	no	—
188	C₄×C₈₉	—	2	1	no	—
189	C₄×C₈₉	—	4	2	no	—
190	C₄×C₈₉	#568	1	1	yes	characteristic
191	C₄×C₈₉	#568	1	1	yes	characteristic
192	C₄×C₈₉	#577	1	2	yes	—
193	Dic₈₉	—	4	2	no	—
194	Dic₈₉	—	4	2	no	—
195	#17990	—	89	1	no	maximal, Sylow
196	C₂×C₈₉	—	4	1	no	—
197	C₂×C₈₉	—	8	2	no	—
198	C₂×C₈₉	#2932	1	1	yes	characteristic
199	C₂×C₈₉	#2949	1	1	yes	characteristic
200	C₂×C₈₉	#2957	1	1	yes	characteristic
201	#3167	—	89	1	no	—
202	#2985	—	89	1	no	—
203	#2981	D₈₉	1	1	yes	characteristic
204	#1588	—	89	2	no	—
205	#1585	—	89	2	no	—
206	#1583	—	89	2	no	—
207	#1582	—	89	2	no	—
208	#1568	—	89	1	no	—
209	#1497	—	89	1	no	—
210	#1496	—	89	1	no	—
211	#1452	—	89	1	no	—
212	C₈₉	#17990	1	1	yes	characteristic, complemented, Sylow
213	#533	—	89	1	no	—
214	#532	C₂×D₈₉	1	1	yes	characteristic
215	#498	—	89	1	no	—
216	#492	—	89	1	no	—
217	#487	—	89	1	no	—
218	#486	—	89	1	no	—
219	#465	—	89	2	no	—
220	#465	—	89	2	no	—
221	#446	—	89	2	no	—
222	#446	—	178	2	no	—
223	C₈×Q₈	—	89	2	no	—
224	C₄×SD₁₆	—	89	2	no	—
225	C₄×SD₁₆	—	89	2	no	—
226	#436	C₂×D₈₉	1	2	yes	—
227	#435	—	2	2	no	—
228	C₈×D₄	C₂×D₈₉	1	2	yes	—
229	#433	—	178	2	no	—
230	#432	—	89	1	no	—
231	#432	C₂×D₈₉	1	1	yes	characteristic
232	#431	—	89	1	no	—
233	#431	C₂×D₈₉	1	1	yes	characteristic
234	#408	—	2	2	no	—
235	C₂×C₄×C₈	—	89	1	no	—
236	#335	—	89	2	no	—
237	#334	—	89	2	no	—
238	Q₈⋊C₈	—	89	2	no	—
239	Q₈⋊C₈	—	89	2	no	—
240	D₄⋊C₈	—	89	2	no	—
241	D₄⋊C₈	—	89	2	no	—
242	#322	—	89	1	no	—
243	#322	—	89	1	no	—
244	#322	—	89	1	no	—
245	#322	—	89	1	no	—
246	C₂×#41	—	2	1	no	—
247	C₂×SD₁₆	—	89	2	no	—
248	C₂×SD₁₆	—	89	2	no	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂²×C₈	—	2	2	no	—
251	C₂²×C₈	—	89	1	no	—
252	C₂²×C₈	—	89	1	no	—
253	C₄⋊Q₈	—	89	1	no	—
254	C₄⋊Q₈	—	89	1	no	—
255	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
256	#127	C₄×D₈₉	1	1	yes	characteristic
257	#125	—	178	1	no	—
258	#124	—	2	1	no	—
259	C₂²⋊Q₈	—	89	2	no	—
260	C₂²⋊Q₈	—	178	1	no	—
261	#121	—	2	1	no	—
262	#121	C₄×D₈₉	1	2	yes	—
263	C₄×Q₈	—	89	2	no	—
264	C₄×Q₈	—	178	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	C₂²×D₈₉	1	2	yes	—
267	#117	—	178	1	no	—
268	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
269	#106	—	89	2	no	—
270	#106	—	89	2	no	—
271	C₄⋊C₈	—	89	2	no	—
272	C₄⋊C₈	—	178	2	no	—
273	C₄⋊C₈	—	178	2	no	—
274	C₄⋊C₈	—	178	1	no	—
275	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
276	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
277	#103	—	89	2	no	—
278	#103	—	89	2	no	—
279	#102	—	89	2	no	—
280	#102	—	89	2	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	2	no	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₂²⋊C₈	—	178	1	no	—
285	C₂²⋊C₈	—	178	1	no	—
286	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
287	#97	—	2	1	no	—
288	#97	—	178	2	no	—
289	#97	—	178	1	no	—
290	C₄×C₈	—	89	1	no	—
291	C₄×C₈	—	89	1	no	—
292	C₄×C₈	—	89	1	no	—
293	C₄×C₈	—	89	1	no	—
294	C₄×C₈	—	89	1	no	—
295	C₄×C₈	—	89	1	no	—
296	C₄×C₈	—	178	2	no	—
297	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
298	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
299	#41	—	8	2	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×Q₈	—	89	2	no	—
302	C₂×Q₈	—	178	1	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	C₂×C₄×D₈₉	1	2	yes	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₂²×C₄	#13083574	1	1	yes	characteristic
311	SD₁₆	—	356	2	no	—
312	SD₁₆	—	356	2	no	—
313	M₁₆	—	8	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	8	2	no	—
316	C₂×C₈	—	89	1	no	—
317	C₂×C₈	—	89	1	no	—
318	C₂×C₈	—	89	1	no	—
319	C₂×C₈	—	89	1	no	—
320	C₂×C₈	—	356	1	no	—
321	C₂×C₈	—	356	1	no	—
322	C₂×C₈	—	178	2	no	—
323	C₂×C₈	—	178	1	no	—
324	C₂×C₈	—	178	1	no	—
325	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
326	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	89	2	no	—
329	C₄⋊C₄	—	178	1	no	—
330	C₄⋊C₄	—	178	1	no	—
331	C₄⋊C₄	—	178	2	no	—
332	C₄⋊C₄	—	178	1	no	—
333	C₄⋊C₄	—	178	2	no	—
334	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
335	C₂²⋊C₄	—	2	2	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	178	2	no	—
338	C₄²	—	178	2	no	—
339	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
340	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
341	C₄²	#13083574	1	1	yes	characteristic
342	C₄²	C₂³×D₈₉	1	1	yes	characteristic
343	C₂³	—	2	2	no	—
344	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
345	Q₈	—	8	1	no	—
346	Q₈	—	178	2	no	—
347	Q₈	—	178	2	no	—
348	Q₈	—	712	1	no	—
349	D₄	—	2	2	no	—
350	D₄	—	2	2	no	—
351	D₄	—	8	1	no	—
352	D₄	—	8	1	no	—
353	D₄	—	8	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	178	2	no	—
362	C₂×C₄	—	178	2	no	—
363	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
364	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
365	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
367	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
368	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
369	C₈	—	2	2	no	—
370	C₈	—	2	2	no	—
371	C₈	—	4	2	no	—
372	C₈	—	356	1	no	—
373	C₈	—	356	1	no	—
374	C₈	—	356	2	no	—
375	C₈	—	178	1	no	—
376	C₈	—	178	1	no	—
377	C₈	—	178	1	no	—
378	C₈	—	178	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	356	2	no	—
392	C₄	—	356	2	no	—
393	C₄	#19152363	1	1	yes	characteristic
394	C₄	#19152363	1	1	yes	characteristic
395	C₄	#19152409	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	8	2	no	—
398	C₂	#26028968	1	1	yes	characteristic
399	C₂	#26029109	1	1	yes	characteristic
400	C₂	#26029167	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2981	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#532×C₈₉	C₂²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2981	D₈₉
hypocenter	C₈₉	#17990
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2981

Complements

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

N	H ≅ G/N	type
2. #26029346	C₂	semidirect
3. #26019128	C₂	semidirect
4. #26019109	C₂	semidirect
5. #26019093	C₂	semidirect
6. #26019088	C₂	semidirect
7. #26019013	C₂	semidirect
8. #26018724	C₂	semidirect
10. #26018545	C₂	semidirect
212. C₈₉	#17990	semidirect

Maximal subgroups

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

subgroup	index
#26030807	2
#26029346	2
#26019128	2
#26019109	2
#26019093	2
#26019088	2
#26019013	2
#26018724	2
#26018721	2
#26018545	2
C₈₉×#2981	2
#17990	89