Subgroups — #36526201

Order 22784.

586 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	#26030834	C₂	1	1	yes	maximal, characteristic
2	#26029201	C₂	1	1	yes	maximal, characteristic
3	#26029195	C₂	1	1	yes	maximal, characteristic
4	#26019146	C₂	1	2	yes	maximal, complemented
5	#26019105	C₂	1	2	yes	maximal, complemented
6	#26019099	C₂	1	2	yes	maximal, complemented
7	#26019056	C₂	1	2	yes	maximal, complemented
8	C₂×#19151350	C₂	1	1	yes	maximal, characteristic
9	#26018656	C₂	1	1	yes	maximal, characteristic
10	#26018533	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1481	C₂	1	1	yes	maximal, characteristic
12	#19152155	C₂²	1	1	yes	characteristic
13	#19152001	C₄	1	1	yes	characteristic
14	#19151977	C₄	1	1	yes	characteristic
15	#19151950	C₄	1	1	yes	characteristic
16	#19151948	C₄	1	1	yes	characteristic
17	#19151837	C₄	1	2	yes	—
18	#19151837	C₄	1	2	yes	—
19	#19151729	—	2	2	no	—
20	#19151729	C₂²	1	2	yes	—
21	#19151727	—	2	2	no	—
22	#19151725	C₂²	1	2	yes	—
23	Q₈×#10623494	C₂²	1	2	yes	—
24	#19151722	C₂²	1	2	yes	—
25	C₄×#13083564	C₂²	1	2	yes	—
26	C₄×#13083564	C₂²	1	2	yes	—
27	#19151625	—	2	1	no	—
28	#19151624	—	2	1	no	—
29	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
30	#19151554	—	2	1	no	—
31	#19151553	—	2	1	no	—
32	C₄×#13083558	C₂²	1	1	yes	characteristic
33	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
34	#19151350	C₂²	1	1	yes	characteristic
35	#19151350	C₂²	1	1	yes	characteristic
36	#19151350	C₂²	1	1	yes	characteristic
37	#19151350	C₂²	1	1	yes	characteristic
38	#19151330	C₂²	1	2	yes	—
39	#19151330	C₂²	1	2	yes	—
40	#19151320	C₂²	1	2	yes	—
41	C₈×#10623494	C₂²	1	2	yes	—
42	#533×C₈₉	C₂²	1	1	yes	characteristic
43	#431×C₈₉	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
45	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
46	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
47	C₂×#13083564	C₂×C₄	1	2	yes	—
48	C₂×#13083564	C₂×C₄	1	2	yes	—
49	C₂×#13083558	D₄	1	1	yes	characteristic
50	C₂×#13083558	D₄	1	1	yes	characteristic
51	C₂²×#10623494	—	2	1	no	—
52	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
53	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
54	#15668012	C₂×C₄	1	1	yes	characteristic
55	#15668012	C₂×C₄	1	1	yes	characteristic
56	#15668004	—	2	1	no	—
57	#15667991	—	2	1	no	—
58	#15667991	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₇₈	—	2	1	no	—
60	C₄×Dic₁₇₈	C₂³	1	2	yes	—
61	#15667971	—	2	1	no	—
62	#15667945	C₂×C₄	1	2	yes	—
63	#15667945	C₂×C₄	1	2	yes	—
64	C₄⋊#10623494	—	2	2	no	—
65	C₄⋊#10623494	—	2	2	no	—
66	C₄⋊#10623494	—	2	1	no	—
67	C₄⋊#10623494	C₂³	1	1	yes	characteristic
68	C₄⋊#10623494	C₂³	1	1	yes	characteristic
69	#15667941	—	2	2	no	—
70	#15667941	—	2	2	no	—
71	#15667936	C₂×C₄	1	2	yes	—
72	#15667936	C₂×C₄	1	2	yes	—
73	#15667935	C₂×C₄	1	2	yes	—
74	#15667935	C₂×C₄	1	2	yes	—
75	#15667923	—	2	2	no	—
76	#15667923	D₄	1	2	yes	—
77	#15667922	—	2	2	no	—
78	C₄×#10623494	—	2	2	no	—
79	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
80	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
81	C₄×#10623494	C₂³	1	1	yes	characteristic
82	C₄×#10623494	C₂³	1	1	yes	characteristic
83	C₄×#10623494	C₂³	1	2	yes	—
84	C₈×Dic₈₉	—	2	2	no	—
85	C₂²×C₈×C₈₉	—	2	1	no	—
86	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
88	#125×C₈₉	—	2	1	no	—
89	C₂²⋊Q₈×C₈₉	—	2	1	no	—
90	C₂²⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
91	C₄×Q₈×C₈₉	—	2	1	no	—
92	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
93	#117×C₈₉	—	2	1	no	—
94	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₈₉	—	2	1	no	—
96	C₄⋊C₈×C₈₉	—	2	1	no	—
97	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²⋊C₈×C₈₉	—	2	1	no	—
100	C₄×C₈×C₈₉	C₂³	1	2	yes	—
101	C₂×Dic₁₇₈	—	2	1	no	—
102	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
103	#13083564	—	2	4	no	—
104	#13083564	—	4	2	no	—
105	#13083558	—	2	2	no	—
106	#13083558	Q₁₆	1	4	yes	complemented
107	C₂×#10623494	—	2	2	no	—
108	C₂×#10623494	—	2	1	no	—
109	C₂×#10623494	—	2	1	no	—
110	C₂×#10623494	—	4	1	no	—
111	C₂×#10623494	—	4	2	no	—
112	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
113	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
114	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
115	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
116	C₂×#10623494	C₂×D₄	1	2	yes	—
117	C₂×#10623494	C₂×D₄	1	2	yes	—
118	C₄⋊Dic₈₉	—	2	1	no	—
119	C₄⋊Dic₈₉	—	2	1	no	—
120	C₄⋊Dic₈₉	—	2	1	no	—
121	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
122	#13083553	—	2	2	no	—
123	#13083553	—	2	2	no	—
124	C₂²⋊Dic₈₉	—	2	2	no	—
125	C₄×Dic₈₉	—	2	2	no	—
126	C₂×Q₈×C₈₉	—	2	1	no	—
127	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
131	C₂×C₈×C₈₉	—	2	1	no	—
132	C₂×C₈×C₈₉	—	2	1	no	—
133	C₂×C₈×C₈₉	—	2	2	no	—
134	C₂×C₈×C₈₉	—	4	2	no	—
135	C₄⋊C₄×C₈₉	—	2	1	no	—
136	C₄⋊C₄×C₈₉	—	2	1	no	—
137	C₄⋊C₄×C₈₉	—	2	1	no	—
138	C₄⋊C₄×C₈₉	—	2	2	no	—
139	C₄⋊C₄×C₈₉	—	2	2	no	—
140	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
141	C₂²⋊C₄×C₈₉	—	2	2	no	—
142	C₄²×C₈₉	—	2	2	no	—
143	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
144	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₈₉	#41	1	1	yes	characteristic
146	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
147	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
148	Q₈×C₈₉	—	2	2	no	—
149	Q₈×C₈₉	—	2	2	no	—
150	Q₈×C₈₉	—	8	1	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	1	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
160	C₂×C₄×C₈₉	#135	1	2	yes	—
161	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
164	C₂×Dic₈₉	—	2	2	no	—
165	C₂×Dic₈₉	—	2	2	no	—
166	Dic₁₇₈	—	2	2	no	—
167	Dic₁₇₈	—	2	2	no	—
168	Dic₁₇₈	—	8	1	no	—
169	C₈×C₈₉	—	4	2	no	—
170	C₈×C₈₉	—	4	1	no	—
171	C₈×C₈₉	—	4	1	no	—
172	#10623494	—	2	1	no	—
173	#10623494	—	2	1	no	—
174	#10623494	—	2	1	no	—
175	#10623494	—	2	2	no	—
176	#10623494	—	4	1	no	—
177	#10623494	—	4	1	no	—
178	#10623494	#131	1	2	yes	—
179	#10623494	C₂×Q₁₆	1	4	yes	—
180	C₂²×C₈₉	—	2	1	no	—
181	C₂²×C₈₉	—	2	1	no	—
182	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
183	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₈₉	—	2	1	no	—
185	C₄×C₈₉	—	2	1	no	—
186	C₄×C₈₉	—	2	1	no	—
187	C₄×C₈₉	—	4	2	no	—
188	C₄×C₈₉	—	4	2	no	—
189	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
190	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
191	C₄×C₈₉	#568	1	1	yes	characteristic
192	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
193	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
194	Dic₈₉	—	4	2	no	—
195	Dic₈₉	—	4	2	no	—
196	#18068	—	89	1	no	maximal, Sylow
197	C₂×C₈₉	—	2	2	no	—
198	C₂×C₈₉	#2935	1	1	yes	characteristic
199	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
200	C₂×C₈₉	#2957	1	1	yes	characteristic
201	#3170	—	89	1	no	—
202	#2962	—	89	1	no	—
203	#2962	—	89	1	no	—
204	#1591	—	89	2	no	—
205	#1584	—	89	2	no	—
206	#1584	—	89	2	no	—
207	C₈×Q₁₆	—	89	2	no	—
208	C₂×#335	—	89	1	no	—
209	#1481	—	89	1	no	—
210	#1481	D₈₉	1	1	yes	characteristic
211	C₈×M₁₆	—	89	1	no	—
212	C₈₉	#18068	1	1	yes	characteristic, complemented, Sylow
213	#533	—	89	1	no	—
214	#533	C₂×D₈₉	1	1	yes	characteristic
215	#500	—	89	1	no	—
216	#494	—	89	1	no	—
217	#487	—	89	1	no	—
218	#487	—	89	1	no	—
219	#467	—	89	2	no	—
220	#467	—	89	2	no	—
221	#446	—	89	2	no	—
222	#446	—	89	2	no	—
223	#446	—	178	2	no	—
224	#446	—	178	2	no	—
225	C₈×Q₈	—	89	2	no	—
226	C₈×Q₈	—	89	2	no	—
227	C₄×Q₁₆	—	89	2	no	—
228	C₄×Q₁₆	—	89	2	no	—
229	#431	C₂×D₈₉	1	1	yes	characteristic
230	#424	—	178	1	no	—
231	#424	—	178	1	no	—
232	C₂×C₄⋊C₈	—	89	1	no	—
233	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
234	#405	—	178	1	no	—
235	#405	—	178	1	no	—
236	C₄×M₁₆	—	89	1	no	—
237	C₂×C₄×C₈	—	89	1	no	—
238	#335	—	89	1	no	—
239	#335	—	89	1	no	—
240	#335	—	89	1	no	—
241	#335	—	89	1	no	—
242	Q₈⋊C₈	—	89	2	no	—
243	Q₈⋊C₈	—	89	2	no	—
244	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
245	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
246	#322	—	89	2	no	—
247	C₈²	—	89	2	no	—
248	C₂×Q₁₆	—	89	2	no	—
249	C₂×Q₁₆	—	89	2	no	—
250	C₂×M₁₆	—	89	1	no	—
251	C₂×M₁₆	—	89	1	no	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	89	1	no	—
254	C₂²×C₈	—	89	1	no	—
255	C₂²×C₈	—	178	1	no	—
256	C₄⋊Q₈	—	89	1	no	—
257	C₄⋊Q₈	—	89	1	no	—
258	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
259	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
260	#125	—	2	1	no	—
261	#125	—	178	1	no	—
262	C₂²⋊Q₈	—	2	1	no	—
263	C₂²⋊Q₈	—	89	2	no	—
264	C₂²⋊Q₈	—	178	1	no	—
265	C₂²⋊Q₈	C₄×D₈₉	1	2	yes	—
266	C₄×Q₈	—	2	1	no	—
267	C₄×Q₈	—	89	2	no	—
268	C₄×Q₈	—	178	1	no	—
269	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
270	#117	—	2	1	no	—
271	#117	—	178	1	no	—
272	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
273	#107	—	89	2	no	—
274	#107	—	89	2	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	—	89	1	no	—
278	C₄⋊C₈	—	89	1	no	—
279	C₄⋊C₈	—	178	2	no	—
280	C₄⋊C₈	—	178	2	no	—
281	C₄⋊C₈	—	178	2	no	—
282	C₄⋊C₈	—	178	2	no	—
283	C₄⋊C₈	—	178	1	no	—
284	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
285	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
286	#103	—	89	2	no	—
287	#103	—	89	2	no	—
288	#103	—	89	2	no	—
289	#103	—	89	2	no	—
290	C₂²⋊C₈	—	2	1	no	—
291	#97	—	89	2	no	—
292	#97	—	178	2	no	—
293	#97	—	178	2	no	—
294	C₄×C₈	—	89	1	no	—
295	C₄×C₈	—	89	1	no	—
296	C₄×C₈	—	89	1	no	—
297	C₄×C₈	—	89	2	no	—
298	C₄×C₈	—	89	1	no	—
299	C₄×C₈	—	178	2	no	—
300	C₄×C₈	—	178	2	no	—
301	C₄×C₈	C₂²×D₈₉	1	2	yes	—
302	C₂×Q₈	—	2	1	no	—
303	C₂×Q₈	—	89	2	no	—
304	C₂×Q₈	—	178	1	no	—
305	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
306	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
307	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
308	C₂²×C₄	#13083573	1	1	yes	characteristic
309	Q₁₆	—	356	2	no	—
310	Q₁₆	—	178	4	no	—
311	M₁₆	—	89	4	no	—
312	M₁₆	—	178	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	89	1	no	—
318	C₂×C₈	—	89	2	no	—
319	C₂×C₈	—	89	1	no	—
320	C₂×C₈	—	89	1	no	—
321	C₂×C₈	—	89	1	no	—
322	C₂×C₈	—	89	2	no	—
323	C₂×C₈	—	356	2	no	—
324	C₂×C₈	—	356	1	no	—
325	C₂×C₈	—	178	2	no	—
326	C₂×C₈	—	178	1	no	—
327	C₂×C₈	—	178	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	89	2	no	—
334	C₄⋊C₄	—	178	1	no	—
335	C₄⋊C₄	—	178	1	no	—
336	C₄⋊C₄	—	178	2	no	—
337	C₄⋊C₄	—	178	1	no	—
338	C₄⋊C₄	—	178	2	no	—
339	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
340	C₂²⋊C₄	—	2	2	no	—
341	C₂²⋊C₄	—	178	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	178	2	no	—
344	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
345	C₄²	D₄×D₈₉	1	1	yes	characteristic
346	C₄²	#13083573	1	1	yes	characteristic
347	C₄²	C₂³×D₈₉	1	1	yes	characteristic
348	C₂³	#15667972	1	1	yes	characteristic
349	Q₈	—	2	2	no	—
350	Q₈	—	2	2	no	—
351	Q₈	—	8	1	no	—
352	Q₈	—	178	2	no	—
353	Q₈	—	178	2	no	—
354	Q₈	—	712	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	178	2	no	—
361	C₂×C₄	—	178	2	no	—
362	C₂×C₄	#15667972	1	1	yes	characteristic
363	C₂×C₄	#15667972	1	1	yes	characteristic
364	C₂×C₄	#15667972	1	1	yes	characteristic
365	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
366	C₂×C₄	#15668033	1	2	yes	—
367	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
368	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
369	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
370	C₈	—	4	2	no	—
371	C₈	—	4	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	89	2	no	—
374	C₈	—	89	4	no	—
375	C₈	—	356	1	no	—
376	C₈	—	356	1	no	—
377	C₈	—	178	1	no	—
378	C₈	—	178	1	no	—
379	C₈	—	178	1	no	—
380	C₈	—	178	2	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	#19151690	1	2	yes	—
384	C₂²	C₂×#15667972	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	4	2	no	—
389	C₄	—	4	2	no	—
390	C₄	—	356	2	no	—
391	C₄	—	356	2	no	—
392	C₄	#19151690	1	2	yes	—
393	C₄	#131×D₈₉	1	1	yes	characteristic
394	C₄	#19152363	1	1	yes	characteristic
395	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
396	C₄	C₂×#15668033	1	1	yes	characteristic
397	C₂	—	2	2	no	—
398	C₂	#26028992	1	1	yes	characteristic
399	C₂	C₂×#19151690	1	1	yes	characteristic
400	C₂	#26029165	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×#1481	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1481	D₈₉
hypocenter	C₈₉	#18068
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1481

Complements

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

N	H ≅ G/N	type
4. #26019146	C₂	semidirect
5. #26019105	C₂	semidirect
6. #26019099	C₂	semidirect
7. #26019056	C₂	semidirect
106. #13083558	Q₁₆	semidirect
212. C₈₉	#18068	semidirect

Maximal subgroups

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

subgroup	index
#26030834	2
#26029201	2
#26029195	2
#26019146	2
#26019105	2
#26019099	2
#26019056	2
C₂×#19151350	2
#26018656	2
#26018533	2
C₈₉×#1481	2
#18068	89