Subgroups — #36444640

Order 22784.

698 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028885	C₂	1	1	yes	maximal, characteristic
2	#26028885	C₂	1	1	yes	maximal, characteristic
3	#26018145	C₂	1	4	yes	maximal, complemented
4	C₂×#376×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151655	—	2	1	no	—
6	#19151655	C₄	1	2	yes	complemented
7	#19151655	C₄	1	2	yes	complemented
8	#19151655	C₂²	1	2	yes	—
9	#19151651	—	2	1	no	—
10	#19151651	—	2	1	no	—
11	#19151651	—	2	1	no	—
12	#19151651	C₄	1	2	yes	complemented
13	#19151622	—	2	1	no	—
14	#19151622	C₄	1	2	yes	complemented
15	#19151562	—	2	1	no	—
16	#19151562	—	2	1	no	—
17	C₄×#13083558	—	2	1	no	—
18	C₄×#13083558	C₂²	1	2	yes	—
19	C₂²×C₄²×C₈₉	—	2	1	no	—
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	#376×C₈₉	—	2	1	no	—
22	#376×C₈₉	—	2	1	no	—
23	#376×C₈₉	C₄	1	2	yes	—
24	#376×C₈₉	C₂²	1	2	yes	—
25	C₂×#95×C₈₉	—	2	1	no	—
26	C₂×#95×C₈₉	C₄	1	2	yes	—
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083558	—	2	2	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	C₄⋊#10623494	—	2	2	no	—
31	C₄⋊#10623494	—	2	2	no	—
32	C₄⋊#10623494	C₂×C₄	1	4	yes	complemented
33	C₂²⋊#10623494	—	2	2	no	—
34	C₂²⋊#10623494	—	2	2	no	—
35	C₂²⋊#10623494	—	2	2	no	—
36	C₂²⋊#10623494	—	2	2	no	—
37	#15667923	—	2	2	no	—
38	#15667923	C₂×C₄	1	4	yes	complemented
39	C₄×#10623494	—	2	2	no	—
40	C₄×#10623494	—	2	2	no	—
41	C₂³×C₄×C₈₉	—	2	2	no	—
42	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	—	2	8	no	—
46	C₂×C₄²×C₈₉	—	2	2	no	—
47	C₂×C₄²×C₈₉	—	2	2	no	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	#95×C₈₉	—	2	2	no	—
61	#95×C₈₉	—	2	2	no	—
62	#95×C₈₉	—	2	2	no	—
63	#95×C₈₉	C₂×C₄	1	4	yes	—
64	#13083558	—	4	4	no	—
65	#13083558	—	4	4	no	—
66	C₂×#10623494	—	2	4	no	—
67	C₂×#10623494	—	2	4	no	—
68	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	—	2	8	no	—
71	C₂²×C₄×C₈₉	—	2	4	no	—
72	C₂²×C₄×C₈₉	—	4	2	no	—
73	C₂²×C₄×C₈₉	—	4	2	no	—
74	C₂²×C₄×C₈₉	—	4	4	no	—
75	C₂²×C₄×C₈₉	—	4	4	no	—
76	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₉	—	2	8	no	—
95	C₄²×C₈₉	—	2	8	no	—
96	C₄²×C₈₉	C₄²	1	1	yes	characteristic
97	C₄²×C₈₉	C₄²	1	1	yes	characteristic
98	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
112	C₂³×C₈₉	—	2	2	no	—
113	C₂³×C₈₉	—	2	2	no	—
114	C₂³×C₈₉	—	2	2	no	—
115	C₂³×C₈₉	#95	1	1	yes	characteristic
116	C₂³×C₈₉	#95	1	1	yes	characteristic
117	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	8	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	4	16	no	—
137	C₂×C₄×C₈₉	—	4	4	no	—
138	C₂×C₄×C₈₉	—	4	4	no	—
139	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
140	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
144	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#104	1	4	yes	—
146	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
152	#10623494	—	4	4	no	—
153	#10623494	—	4	4	no	—
154	C₂²×C₈₉	—	2	8	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	#339	1	2	yes	—
162	C₂²×C₈₉	#344	1	2	yes	—
163	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
164	C₂²×C₈₉	#421	1	2	yes	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	4	8	no	—
176	C₄×C₈₉	#339	1	2	yes	—
177	C₄×C₈₉	C₂×#104	1	2	yes	—
178	#8072	—	89	1	no	maximal, Sylow
179	C₂×C₈₉	—	2	2	no	—
180	C₂×C₈₉	—	2	2	no	—
181	C₂×C₈₉	—	2	2	no	—
182	C₂×C₈₉	C₂×#339	1	1	yes	characteristic
183	C₂×C₈₉	#1733	1	1	yes	characteristic
184	C₂×C₈₉	C₂×#344	1	1	yes	characteristic
185	#2920	—	89	1	no	—
186	#2920	—	89	1	no	—
187	C₂×#376	D₈₉	1	1	yes	characteristic
188	#1300	—	89	4	no	—
189	C₈₉	#8072	1	1	yes	characteristic, complemented, Sylow
190	C₂²×C₄²	—	2	1	no	—
191	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
192	#432	—	89	2	no	—
193	#432	—	89	2	no	—
194	#432	—	89	2	no	—
195	#432	—	178	1	no	—
196	#431	—	89	2	no	—
197	#431	—	178	1	no	—
198	#431	—	178	1	no	—
199	#431	—	178	1	no	—
200	#423	—	89	2	no	—
201	#423	—	178	1	no	—
202	#407	—	178	1	no	—
203	#407	—	178	1	no	—
204	C₄×M₁₆	—	89	2	no	—
205	C₄×M₁₆	—	178	1	no	—
206	#376	—	2	1	no	—
207	#376	—	2	1	no	—
208	#376	Dic₈₉	1	2	yes	—
209	#376	C₂×D₈₉	1	2	yes	—
210	C₂×#95	—	2	1	no	—
211	C₂×#95	Dic₈₉	1	2	yes	—
212	C₂³×C₄	—	2	2	no	—
213	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
214	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
215	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
216	C₂×M₁₆	—	178	2	no	—
217	C₂×M₁₆	—	178	2	no	—
218	C₂×C₄²	—	2	8	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
222	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
223	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
225	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
226	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
227	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
228	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
233	C₄⋊C₈	—	89	4	no	—
234	C₄⋊C₈	—	178	2	no	—
235	C₄⋊C₈	—	178	2	no	—
236	C₄⋊C₈	—	178	2	no	—
237	C₂²⋊C₈	—	178	2	no	—
238	C₂²⋊C₈	—	178	2	no	—
239	C₂²⋊C₈	—	178	2	no	—
240	C₂²⋊C₈	—	178	2	no	—
241	#97	—	89	4	no	—
242	#97	—	178	2	no	—
243	C₄×C₈	—	178	2	no	—
244	C₄×C₈	—	178	2	no	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	C₂×Dic₈₉	1	4	yes	—
249	C₂⁴	C₂²⋊Dic₈₉	1	1	yes	characteristic
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	8	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	4	no	—
256	C₂²×C₄	—	4	4	no	—
257	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
258	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
259	C₂²×C₄	#13083551	1	1	yes	characteristic
260	C₂²×C₄	#13083551	1	1	yes	characteristic
261	C₂²×C₄	#13083551	1	1	yes	characteristic
262	C₂²×C₄	#13083551	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
265	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
266	C₂²×C₄	#13083553	1	1	yes	characteristic
267	C₂²×C₄	#13083553	1	1	yes	characteristic
268	C₂²×C₄	#13083553	1	1	yes	characteristic
269	C₂²×C₄	#13083553	1	1	yes	characteristic
270	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
271	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
272	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
275	M₁₆	—	356	4	no	—
276	M₁₆	—	356	4	no	—
277	C₂×C₈	—	178	4	no	—
278	C₂×C₈	—	178	4	no	—
279	C₄²	—	2	8	no	—
280	C₄²	—	2	8	no	—
281	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
282	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
283	C₄²	#13083551	1	1	yes	characteristic
284	C₄²	#13083551	1	1	yes	characteristic
285	C₄²	#13083551	1	1	yes	characteristic
286	C₄²	#13083551	1	1	yes	characteristic
287	C₄²	#13083553	1	1	yes	characteristic
288	C₄²	#13083553	1	1	yes	characteristic
289	C₄²	#13083553	1	1	yes	characteristic
290	C₄²	#13083553	1	1	yes	characteristic
291	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
292	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
293	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
294	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
295	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
296	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	#15667919	1	1	yes	characteristic
301	C₂³	#15667919	1	1	yes	characteristic
302	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	8	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	16	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	#15667919	1	1	yes	characteristic
325	C₂×C₄	#15667919	1	1	yes	characteristic
326	C₂×C₄	#15667919	1	1	yes	characteristic
327	C₂×C₄	#15667919	1	1	yes	characteristic
328	C₂×C₄	#15667919	1	1	yes	characteristic
329	C₂×C₄	#15667919	1	1	yes	characteristic
330	C₂×C₄	#15667940	1	4	yes	—
331	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
333	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
334	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
335	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
337	C₈	—	356	4	no	—
338	C₈	—	356	4	no	—
339	C₂²	—	2	8	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	#19151358	1	2	yes	—
347	C₂²	#19151368	1	2	yes	—
348	C₂²	C₂×#15667919	1	1	yes	characteristic
349	C₂²	#19151615	1	2	yes	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	8	no	—
361	C₄	#19151358	1	2	yes	—
362	C₄	C₂×#15667940	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	2	2	no	—
366	C₂	C₂×#19151358	1	1	yes	characteristic
367	C₂	#26019961	1	1	yes	characteristic
368	C₂	C₂×#19151368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#376×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#376	D₈₉
hypocenter	C₈₉	#8072
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
3. #26018145	C₂	semidirect
6. #19151655	C₄	semidirect
7. #19151655	C₄	semidirect
12. #19151651	C₄	semidirect
14. #19151622	C₄	semidirect
32. C₄⋊#10623494	C₂×C₄	semidirect
38. #15667923	C₂×C₄	semidirect
189. C₈₉	#8072	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26018145	2
C₂×#376×C₈₉	2
#8072	89