Subgroups — #36524128

Order 22784.

626 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029335	C₂	1	1	yes	maximal, characteristic
2	#26029332	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
4	#26019125	C₂	1	4	yes	maximal, complemented
5	#26019104	C₂	1	4	yes	maximal, complemented
6	#26019000	C₂	1	1	yes	maximal, characteristic
7	#26018648	C₂	1	1	yes	maximal, characteristic
8	#26018648	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1452	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	#19151728	C₂²	1	4	yes	—
14	#19151725	—	2	2	no	—
15	C₈×Dic₁₇₈	C₂²	1	4	yes	—
16	#19151722	—	2	2	no	—
17	C₄×Dic₃₅₆	C₄	1	2	yes	—
18	C₄×Dic₃₅₆	C₄	1	2	yes	—
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	C₄×Dic₃₅₆	C₂²	1	2	yes	—
21	#19151652	—	2	2	no	—
22	#19151651	C₂²	1	1	yes	characteristic
23	#19151651	C₂²	1	1	yes	characteristic
24	#19151648	—	2	2	no	—
25	C₂×#15667946	C₄	1	1	yes	characteristic
26	C₂×#15667937	C₄	1	1	yes	characteristic
27	C₂×#15667937	C₄	1	1	yes	characteristic
28	#19151351	C₂²	1	2	yes	—
29	#19151349	C₂²	1	2	yes	—
30	#19151330	C₂²	1	4	yes	—
31	#19151330	C₂²	1	4	yes	—
32	#431×C₈₉	C₂²	1	1	yes	characteristic
33	#431×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	#322×C₈₉	C₂²	1	2	yes	—
36	#322×C₈₉	C₂²	1	2	yes	—
37	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
39	C₂×Dic₃₅₆	—	2	4	no	—
40	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
41	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₇₈	—	2	1	no	—
43	C₄×Dic₁₇₈	—	2	1	no	—
44	C₄×Dic₁₇₈	—	2	1	no	—
45	C₄×Dic₁₇₈	—	2	1	no	—
46	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₇₈	C₂³	1	2	yes	—
49	C₄×Dic₁₇₈	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×#13083553	—	2	1	no	—
53	C₂×#13083553	—	2	1	no	—
54	C₂×C₄×Dic₈₉	—	2	1	no	—
55	C₂×C₄×Dic₈₉	—	2	1	no	—
56	#15667946	C₂×C₄	1	2	yes	—
57	#15667946	C₂×C₄	1	2	yes	—
58	C₄⋊#10623494	C₂³	1	2	yes	—
59	C₄⋊#10623494	C₂³	1	2	yes	—
60	#15667941	—	2	4	no	—
61	#15667941	—	2	4	no	—
62	#15667937	C₂×C₄	1	2	yes	—
63	#15667937	C₂×C₄	1	2	yes	—
64	#15667937	C₂×C₄	1	2	yes	—
65	#15667937	C₂×C₄	1	2	yes	—
66	C₂²⋊#10623494	—	2	1	no	—
67	C₂²⋊#10623494	—	2	1	no	—
68	#15667922	—	2	4	no	—
69	C₄×#10623494	—	2	1	no	—
70	C₄×#10623494	—	2	1	no	—
71	C₈×Dic₈₉	—	2	4	no	—
72	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₈₉	—	2	1	no	—
76	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
77	C₂²⋊C₈×C₈₉	—	2	1	no	—
78	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
79	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₈₉	C₂³	1	2	yes	—
82	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₈₉	C₂³	1	2	yes	—
85	C₂×Dic₁₇₈	—	2	1	no	—
86	C₂×Dic₁₇₈	—	2	2	no	—
87	C₂×Dic₁₇₈	—	2	2	no	—
88	C₂×Dic₁₇₈	—	2	2	no	—
89	C₂×Dic₁₇₈	—	2	1	no	—
90	C₂×Dic₁₇₈	M₁₆	1	4	yes	—
91	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₈₉	—	2	1	no	—
94	C₂²×Dic₈₉	—	2	1	no	—
95	C₂²×Dic₈₉	—	2	1	no	—
96	C₂²×Dic₈₉	—	2	1	no	—
97	Dic₃₅₆	—	2	4	no	—
98	Dic₃₅₆	—	2	4	no	—
99	C₂×#10623494	—	2	2	no	—
100	C₂×#10623494	—	2	2	no	—
101	C₄⋊Dic₈₉	—	2	1	no	—
102	C₄⋊Dic₈₉	—	2	1	no	—
103	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
105	#13083553	—	2	2	no	—
106	#13083553	—	2	2	no	—
107	#13083553	—	2	2	no	—
108	#13083553	—	2	2	no	—
109	C₄×Dic₈₉	—	2	2	no	—
110	C₄×Dic₈₉	—	2	2	no	—
111	C₄×Dic₈₉	—	2	2	no	—
112	C₄×Dic₈₉	—	2	2	no	—
113	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
116	C₂×C₈×C₈₉	—	2	2	no	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
124	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	#41	1	1	yes	characteristic
127	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₈₉	#131	1	2	yes	—
136	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	—	2	4	no	—
141	C₂×Dic₈₉	—	2	2	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	8	1	no	—
146	C₂×Dic₈₉	—	8	1	no	—
147	Dic₁₇₈	—	2	2	no	—
148	Dic₁₇₈	—	2	4	no	—
149	Dic₁₇₈	—	2	2	no	—
150	Dic₁₇₈	—	2	2	no	—
151	Dic₁₇₈	—	4	2	no	—
152	Dic₁₇₈	C₂×M₁₆	1	4	yes	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	—	2	4	no	—
158	C₈×C₈₉	—	4	2	no	—
159	#10623494	—	4	2	no	—
160	#10623494	—	4	2	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	#435	1	2	yes	—
164	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	#435	1	2	yes	—
169	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
170	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
171	C₄×C₈₉	#578	1	2	yes	—
172	Dic₈₉	—	2	4	no	—
173	Dic₈₉	—	4	2	no	—
174	Dic₈₉	—	4	2	no	—
175	Dic₈₉	—	4	2	no	—
176	#17880	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	2	no	—
178	C₂×C₈₉	C₂×#435	1	1	yes	characteristic
179	C₂×C₈₉	#2950	1	1	yes	characteristic
180	C₂×C₈₉	#2957	1	1	yes	characteristic
181	#2983	—	89	1	no	—
182	#2982	—	89	1	no	—
183	C₂×C₄×Q₁₆	—	89	1	no	—
184	#1587	—	89	4	no	—
185	#1584	—	89	4	no	—
186	#1564	—	89	1	no	—
187	#1479	—	89	1	no	—
188	#1479	—	89	1	no	—
189	#1452	D₈₉	1	1	yes	characteristic
190	C₈₉	#17880	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₁₆	—	89	1	no	—
192	C₂×C₄×Q₈	—	89	1	no	—
193	C₂×C₄×Q₈	—	89	1	no	—
194	#446	—	89	4	no	—
195	#446	—	178	2	no	—
196	C₈×Q₈	—	89	4	no	—
197	C₈×Q₈	—	178	2	no	—
198	C₄×Q₁₆	—	89	2	no	—
199	C₄×Q₁₆	—	89	2	no	—
200	C₄×Q₁₆	—	89	2	no	—
201	C₄×Q₁₆	—	89	2	no	—
202	#432	—	178	2	no	—
203	#431	—	89	1	no	—
204	#431	—	89	1	no	—
205	#431	—	178	2	no	—
206	#431	C₂×D₈₉	1	1	yes	characteristic
207	#431	C₂×D₈₉	1	1	yes	characteristic
208	C₂×#107	—	89	1	no	—
209	C₂×#103	—	89	1	no	—
210	C₂×#103	—	89	1	no	—
211	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
212	#335	—	89	2	no	—
213	#334	—	89	2	no	—
214	Q₈⋊C₈	—	89	4	no	—
215	Q₈⋊C₈	—	89	4	no	—
216	#322	C₂×D₈₉	1	2	yes	—
217	#322	C₂×D₈₉	1	2	yes	—
218	C₂²×Q₈	—	89	1	no	—
219	C₂²×Q₈	—	89	1	no	—
220	C₂×Q₁₆	—	89	2	no	—
221	C₂×Q₁₆	—	89	2	no	—
222	C₂×Q₁₆	—	178	4	no	—
223	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
225	C₄×Q₈	—	89	2	no	—
226	C₄×Q₈	—	89	2	no	—
227	C₄×Q₈	—	89	2	no	—
228	C₄×Q₈	—	89	2	no	—
229	C₄×Q₈	—	178	1	no	—
230	C₄×Q₈	—	178	1	no	—
231	C₄×Q₈	—	178	1	no	—
232	C₄×Q₈	—	178	1	no	—
233	C₂×C₄⋊C₄	—	89	1	no	—
234	C₂×C₄⋊C₄	—	89	1	no	—
235	C₂×C₄⋊C₄	—	178	1	no	—
236	C₂×C₄⋊C₄	—	178	1	no	—
237	C₂×C₄²	—	178	1	no	—
238	C₂×C₄²	—	178	1	no	—
239	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
240	#107	—	89	2	no	—
241	#107	—	89	2	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	89	2	no	—
244	C₄⋊C₈	—	89	2	no	—
245	C₄⋊C₈	—	178	4	no	—
246	C₄⋊C₈	—	178	4	no	—
247	C₄⋊C₈	D₃₅₆	1	2	yes	—
248	#103	—	89	2	no	—
249	#103	—	89	2	no	—
250	#103	—	89	2	no	—
251	#103	—	89	2	no	—
252	C₂²⋊C₈	—	2	1	no	—
253	C₂²⋊C₈	—	178	1	no	—
254	C₂²⋊C₈	—	178	1	no	—
255	C₂²⋊C₈	D₃₅₆	1	2	yes	—
256	#97	—	178	4	no	—
257	C₄×C₈	—	178	4	no	—
258	C₄×C₈	—	178	1	no	—
259	C₄×C₈	—	178	1	no	—
260	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
261	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₈₉	1	2	yes	—
263	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₈₉	1	2	yes	—
266	C₂×Q₈	—	89	4	no	—
267	C₂×Q₈	—	89	2	no	—
268	C₂×Q₈	—	89	2	no	—
269	C₂×Q₈	—	178	2	no	—
270	C₂×Q₈	—	178	2	no	—
271	C₂×Q₈	—	178	1	no	—
272	C₂×Q₈	—	178	1	no	—
273	C₂×Q₈	—	178	2	no	—
274	C₂²×C₄	—	178	1	no	—
275	C₂²×C₄	—	178	1	no	—
276	C₂²×C₄	—	178	1	no	—
277	C₂²×C₄	—	178	1	no	—
278	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
280	C₂²×C₄	#13083574	1	1	yes	characteristic
281	Q₁₆	—	178	4	no	—
282	Q₁₆	—	178	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	178	2	no	—
287	C₂×C₈	—	178	2	no	—
288	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
293	C₄⋊C₄	—	89	2	no	—
294	C₄⋊C₄	—	89	2	no	—
295	C₄⋊C₄	—	178	1	no	—
296	C₄⋊C₄	—	178	1	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	2	no	—
299	C₄⋊C₄	—	178	2	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄²	—	178	2	no	—
302	C₄²	—	178	2	no	—
303	C₄²	—	178	2	no	—
304	C₄²	—	178	2	no	—
305	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
306	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
307	C₄²	#13083574	1	1	yes	characteristic
308	C₄²	C₂³×D₈₉	1	1	yes	characteristic
309	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
310	Q₈	—	89	4	no	—
311	Q₈	—	356	2	no	—
312	Q₈	—	178	2	no	—
313	Q₈	—	178	2	no	—
314	Q₈	—	178	4	no	—
315	Q₈	—	178	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	178	2	no	—
320	C₂×C₄	—	178	2	no	—
321	C₂×C₄	—	178	2	no	—
322	C₂×C₄	—	178	4	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	712	1	no	—
325	C₂×C₄	—	712	1	no	—
326	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
327	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
328	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
329	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
330	C₂×C₄	#15668020	1	2	yes	—
331	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
333	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
334	C₈	—	2	4	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	356	2	no	—
341	C₈	—	356	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	#19151671	1	2	yes	—
345	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	356	2	no	—
350	C₄	—	356	2	no	—
351	C₄	—	356	2	no	—
352	C₄	—	178	4	no	—
353	C₄	#19151671	1	2	yes	—
354	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
355	C₄	C₂×#15668020	1	1	yes	characteristic
356	C₄	#19152415	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	C₂×#19151671	1	1	yes	characteristic
359	C₂	#26029121	1	1	yes	characteristic
360	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1452	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1452	D₈₉
hypocenter	C₈₉	#17880
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
4. #26019125	C₂	semidirect
5. #26019104	C₂	semidirect
190. C₈₉	#17880	semidirect

Maximal subgroups

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

subgroup	index
#26029335	2
#26029332	2
C₂×C₄×Dic₃₅₆	2
#26019125	2
#26019104	2
#26019000	2
#26018648	2
C₈₉×#1452	2
#17880	89