Subgroups — #36524323

Order 22784.

618 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030291	C₂	1	1	yes	maximal, characteristic
2	#26029186	C₂	1	1	yes	maximal, characteristic
3	#26019102	C₂	1	4	yes	maximal, complemented
4	C₈×#13083564	C₂	1	4	yes	maximal, complemented
5	#26019026	C₂	1	1	yes	maximal, characteristic
6	#26018681	C₂	1	1	yes	maximal, characteristic
7	#26018654	C₂	1	1	yes	maximal, characteristic
8	#26018535	C₂	1	1	yes	maximal, characteristic
9	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
10	#19152155	C₂²	1	1	yes	characteristic
11	#19151933	C₄	1	2	yes	—
12	#19151893	C₄	1	2	yes	—
13	#19151808	C₄	1	2	yes	—
14	#19151754	C₄	1	2	yes	—
15	#19151728	—	2	1	no	—
16	#19151728	C₂²	1	2	yes	—
17	C₈×Dic₁₇₈	—	2	1	no	—
18	C₈×Dic₁₇₈	C₂²	1	2	yes	—
19	C₄×#13083564	C₂²	1	4	yes	—
20	#19151651	C₂²	1	1	yes	characteristic
21	#19151651	C₂²	1	1	yes	characteristic
22	#19151624	—	2	1	no	—
23	#19151622	C₂²	1	1	yes	characteristic
24	#19151553	—	2	1	no	—
25	#19151350	C₂²	1	4	yes	—
26	Q₈⋊#10623494	C₂²	1	4	yes	—
27	#19151328	C₂²	1	4	yes	—
28	#19151319	C₂²	1	2	yes	—
29	C₈×#10623494	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₈₉	—	2	1	no	—
32	C₈×Q₈×C₈₉	—	2	1	no	—
33	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
34	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
36	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	—	2	1	no	—
38	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#13083564	C₂×C₄	1	4	yes	—
40	C₂×#13083558	—	2	1	no	—
41	#15668012	C₂×C₄	1	1	yes	characteristic
42	#15668012	C₂×C₄	1	1	yes	characteristic
43	#15668004	—	2	1	no	—
44	#15667991	—	2	1	no	—
45	#15667991	C₂×C₄	1	2	yes	—
46	C₄×Dic₁₇₈	—	2	1	no	—
47	C₄×Dic₁₇₈	C₂³	1	2	yes	—
48	#15667971	—	2	1	no	—
49	#15667945	C₂×C₄	1	4	yes	—
50	C₄⋊#10623494	—	2	1	no	—
51	C₄⋊#10623494	—	2	1	no	—
52	C₄⋊#10623494	C₂×C₄	1	2	yes	—
53	C₄⋊#10623494	C₂³	1	2	yes	—
54	#15667941	—	2	2	no	—
55	#15667941	—	2	2	no	—
56	#15667936	C₂×C₄	1	4	yes	—
57	#15667935	C₂×C₄	1	4	yes	—
58	C₂²⋊#10623494	—	2	1	no	—
59	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
60	#15667922	—	2	2	no	—
61	C₄×#10623494	C₂³	1	2	yes	—
62	C₄×#10623494	C₂³	1	2	yes	—
63	C₈×Dic₈₉	—	2	2	no	—
64	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	—	2	1	no	—
66	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
68	C₄×Q₈×C₈₉	—	2	1	no	—
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₈₉	—	2	1	no	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	—	2	2	no	—
77	C₄⋊C₈×C₈₉	—	2	2	no	—
78	C₄⋊C₈×C₈₉	—	2	1	no	—
79	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
80	C₄×C₈×C₈₉	—	2	2	no	—
81	C₄×C₈×C₈₉	—	2	2	no	—
82	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
83	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
84	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₂×Dic₁₇₈	—	2	1	no	—
87	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
88	#13083564	—	2	8	no	—
89	#13083558	—	4	2	no	—
90	C₂×#10623494	—	2	2	no	—
91	C₂×#10623494	—	2	2	no	—
92	C₂×#10623494	C₂²×C₄	1	4	yes	—
93	C₄⋊Dic₈₉	—	2	1	no	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
97	#13083553	—	2	2	no	—
98	#13083553	—	2	2	no	—
99	C₂²⋊Dic₈₉	—	2	2	no	—
100	C₄×Dic₈₉	—	2	2	no	—
101	C₂×Q₈×C₈₉	—	2	4	no	—
102	C₂×Q₈×C₈₉	—	2	1	no	—
103	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₈₉	—	2	1	no	—
105	C₂²×C₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	4	2	no	—
112	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
121	C₄²×C₈₉	—	2	2	no	—
122	C₄²×C₈₉	—	2	2	no	—
123	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₉	#41	1	1	yes	characteristic
126	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
128	Q₈×C₈₉	—	2	4	no	—
129	Q₈×C₈₉	—	2	2	no	—
130	Q₈×C₈₉	#131	1	4	yes	—
131	C₂×C₄×C₈₉	—	2	4	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
141	C₂×C₄×C₈₉	#135	1	2	yes	—
142	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	Dic₁₇₈	—	2	4	no	—
148	Dic₁₇₈	—	4	2	no	—
149	C₈×C₈₉	—	2	1	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	4	2	no	—
152	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
153	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
154	#10623494	—	2	4	no	—
155	#10623494	—	4	2	no	—
156	#10623494	—	4	2	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
160	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₈₉	—	2	4	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	4	2	no	—
166	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
167	C₄×C₈₉	C₂×#131	1	2	yes	—
168	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
169	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
170	Dic₈₉	—	4	2	no	—
171	Dic₈₉	—	4	2	no	—
172	#17903	—	89	1	no	maximal, Sylow
173	C₂×C₈₉	—	2	2	no	—
174	C₂×C₈₉	#2929	1	1	yes	characteristic
175	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
176	C₂×C₈₉	#2955	1	1	yes	characteristic
177	#3104	—	89	1	no	—
178	#2960	—	89	1	no	—
179	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
180	#1584	—	89	4	no	—
181	C₈×Q₁₆	—	89	4	no	—
182	#1571	—	89	1	no	—
183	#1487	—	89	1	no	—
184	#1481	—	89	1	no	—
185	#1450	—	89	1	no	—
186	C₈₉	#17903	1	1	yes	characteristic, complemented, Sylow
187	#533	—	89	1	no	—
188	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
189	#484	—	89	2	no	—
190	#477	—	89	2	no	—
191	#462	—	89	2	no	—
192	#451	—	89	2	no	—
193	#446	—	89	2	no	—
194	#446	—	178	1	no	—
195	C₈×Q₈	—	2	1	no	—
196	C₈×Q₈	—	2	1	no	—
197	C₈×Q₈	—	89	2	no	—
198	C₈×Q₈	—	178	1	no	—
199	C₈×Q₈	C₂×D₈₉	1	2	yes	—
200	C₈×Q₈	C₂×D₈₉	1	2	yes	—
201	C₄×Q₁₆	—	89	4	no	—
202	#431	—	89	1	no	—
203	#431	—	89	1	no	—
204	#424	—	178	1	no	—
205	#423	—	89	1	no	—
206	C₂×C₄⋊C₈	—	2	1	no	—
207	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
208	#405	—	178	1	no	—
209	C₂×C₄×C₈	—	2	1	no	—
210	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
211	#335	—	89	4	no	—
212	Q₈⋊C₈	—	89	4	no	—
213	Q₈⋊C₈	—	89	4	no	—
214	#322	—	89	2	no	—
215	C₈²	—	89	2	no	—
216	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
217	C₂×Q₁₆	—	89	4	no	—
218	C₂×M₁₆	—	178	1	no	—
219	C₂²×C₈	—	2	1	no	—
220	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
221	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
222	C₄⋊Q₈	—	89	1	no	—
223	C₄⋊Q₈	—	89	1	no	—
224	#125	—	178	1	no	—
225	C₂²⋊Q₈	—	89	2	no	—
226	C₂²⋊Q₈	—	178	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	89	2	no	—
230	C₄×Q₈	—	178	1	no	—
231	C₄×Q₈	C₄×D₈₉	1	2	yes	—
232	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
233	#117	—	178	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
236	C₂×C₄²	—	2	1	no	—
237	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
238	#107	—	89	4	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	89	2	no	—
243	C₄⋊C₈	—	89	2	no	—
244	C₄⋊C₈	—	178	2	no	—
245	C₄⋊C₈	—	178	2	no	—
246	C₄⋊C₈	—	178	1	no	—
247	C₄⋊C₈	—	178	1	no	—
248	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
249	#103	—	89	4	no	—
250	#103	—	89	4	no	—
251	C₂²⋊C₈	—	89	2	no	—
252	C₂²⋊C₈	—	178	1	no	—
253	#97	—	178	2	no	—
254	C₄×C₈	—	2	2	no	—
255	C₄×C₈	—	2	2	no	—
256	C₄×C₈	—	89	2	no	—
257	C₄×C₈	—	89	2	no	—
258	C₄×C₈	—	178	2	no	—
259	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
260	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
263	C₂×Q₈	—	2	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	89	2	no	—
266	C₂×Q₈	—	178	1	no	—
267	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
272	C₂²×C₄	#13083574	1	1	yes	characteristic
273	Q₁₆	—	178	8	no	—
274	M₁₆	—	356	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	89	4	no	—
279	C₂×C₈	—	178	2	no	—
280	C₂×C₈	—	178	2	no	—
281	C₂×C₈	#13083564	1	4	yes	complemented
282	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
283	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
284	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
285	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	89	2	no	—
290	C₄⋊C₄	—	178	1	no	—
291	C₄⋊C₄	—	178	1	no	—
292	C₄⋊C₄	—	178	2	no	—
293	C₄⋊C₄	—	178	1	no	—
294	C₄⋊C₄	—	178	2	no	—
295	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
296	C₂²⋊C₄	—	178	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	178	2	no	—
300	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
301	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
302	C₄²	#13083574	1	1	yes	characteristic
303	C₄²	C₂³×D₈₉	1	1	yes	characteristic
304	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
305	Q₈	—	2	4	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	356	2	no	—
308	Q₈	—	178	4	no	—
309	Q₈	#15668018	1	4	yes	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	178	2	no	—
317	C₂×C₄	—	178	2	no	—
318	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
319	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
320	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
321	C₂×C₄	C₂×#13083564	1	2	yes	—
322	C₂×C₄	#15668032	1	2	yes	—
323	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
325	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	4	2	no	—
329	C₈	—	356	2	no	—
330	C₈	—	356	2	no	—
331	C₈	—	178	4	no	—
332	C₈	C₂×#13083564	1	2	yes	—
333	C₈	C₂×#13083564	1	2	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	C₄×#13083564	1	2	yes	—
337	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	4	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	2	no	—
343	C₄	—	356	2	no	—
344	C₄	—	356	2	no	—
345	C₄	C₄×#13083564	1	2	yes	—
346	C₄	C₂×#15668018	1	2	yes	—
347	C₄	C₂²×#13083564	1	1	yes	characteristic
348	C₄	C₂×#15668032	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	#26028946	1	1	yes	characteristic
351	C₂	C₂×C₄×#13083564	1	1	yes	characteristic
352	C₂	#26029153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	#17903
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #26019102	C₂	semidirect
4. C₈×#13083564	C₂	semidirect
112. C₂×C₈×C₈₉	Q₁₆	semidirect
186. C₈₉	#17903	semidirect
281. C₂×C₈	#13083564	semidirect

Maximal subgroups

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

subgroup	index
#26030291	2
#26029186	2
#26019102	2
C₈×#13083564	2
#26019026	2
#26018681	2
#26018654	2
#26018535	2
C₂×C₈×Q₈×C₈₉	2
#17903	89