Subgroups — #36447221

Order 22784.

516 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020404	C₂	1	1	yes	maximal, characteristic
2	#26020404	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151344	C₂	1	2	yes	maximal
4	#26018135	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	—	2	1	no	—
7	C₂×C₄⋊#10623494	—	2	1	no	—
8	C₂×C₄⋊#10623494	—	2	1	no	—
9	C₂×C₄⋊#10623494	—	2	1	no	—
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	#19151344	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
15	#391×C₈₉	—	2	1	no	—
16	#389×C₈₉	—	2	1	no	—
17	#389×C₈₉	C₄	1	2	yes	—
18	#384×C₈₉	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
20	C₄×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
21	C₄³×C₈₉	C₂²	1	1	yes	characteristic
22	C₂²×#10623494	—	2	2	no	—
23	C₂²×#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	C₄⋊#10623494	—	2	2	no	—
27	C₄⋊#10623494	—	2	2	no	—
28	C₄⋊#10623494	—	4	2	no	—
29	C₄⋊#10623494	—	4	2	no	—
30	C₄⋊#10623494	—	4	2	no	—
31	C₄⋊#10623494	—	4	2	no	—
32	C₂²×Q₈×C₈₉	—	2	1	no	—
33	C₄⋊Q₈×C₈₉	—	2	1	no	—
34	C₄⋊Q₈×C₈₉	—	2	1	no	—
35	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
36	C₄⋊Q₈×C₈₉	Q₈	1	2	yes	—
37	C₄×Q₈×C₈₉	—	4	2	no	—
38	C₄×Q₈×C₈₉	—	4	2	no	—
39	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₈₉	—	2	2	no	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	#95×C₈₉	—	2	2	no	—
54	#95×C₈₉	—	2	2	no	—
55	C₂×#10623494	—	4	2	no	—
56	C₂×#10623494	—	4	2	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₂×#10623494	—	4	2	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₂×Q₈×C₈₉	—	2	2	no	—
62	C₂×Q₈×C₈₉	—	2	2	no	—
63	C₂×Q₈×C₈₉	—	4	1	no	—
64	C₂×Q₈×C₈₉	—	4	1	no	—
65	C₂×Q₈×C₈₉	—	4	1	no	—
66	C₂×Q₈×C₈₉	—	4	1	no	—
67	C₂²×C₄×C₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	—	2	2	no	—
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₈₉	—	2	2	no	—
77	C₄⋊C₄×C₈₉	—	2	2	no	—
78	C₄⋊C₄×C₈₉	—	2	2	no	—
79	C₄⋊C₄×C₈₉	—	4	2	no	—
80	C₄⋊C₄×C₈₉	—	4	4	no	—
81	C₄⋊C₄×C₈₉	—	4	1	no	—
82	C₄⋊C₄×C₈₉	—	4	1	no	—
83	C₄⋊C₄×C₈₉	—	4	1	no	—
84	C₄⋊C₄×C₈₉	—	4	1	no	—
85	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
86	C₄²×C₈₉	—	2	2	no	—
87	C₄²×C₈₉	—	2	1	no	—
88	C₄²×C₈₉	—	2	1	no	—
89	C₄²×C₈₉	—	2	1	no	—
90	C₄²×C₈₉	—	2	1	no	—
91	C₄²×C₈₉	—	4	1	no	—
92	C₄²×C₈₉	—	4	4	no	—
93	C₄²×C₈₉	D₈	1	2	yes	—
94	C₄²×C₈₉	SD₁₆	1	2	yes	—
95	C₄²×C₈₉	SD₁₆	1	2	yes	—
96	C₄²×C₈₉	Q₁₆	1	2	yes	—
97	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
99	C₄²×C₈₉	#41	1	1	yes	characteristic
100	C₄²×C₈₉	#41	1	1	yes	characteristic
101	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₈₉	—	4	2	no	—
103	Q₈×C₈₉	—	4	2	no	—
104	Q₈×C₈₉	—	4	2	no	—
105	Q₈×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	#101	1	2	yes	—
122	C₂×C₄×C₈₉	#102	1	2	yes	—
123	C₂×C₄×C₈₉	#102	1	2	yes	—
124	C₂×C₄×C₈₉	#103	1	2	yes	—
125	C₂×C₄×C₈₉	#103	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
136	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
140	#10623494	—	8	2	no	—
141	#10623494	—	8	2	no	—
142	C₂²×C₈₉	#380	1	1	yes	characteristic
143	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
144	C₂²×C₈₉	#413	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
146	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
147	C₂²×C₈₉	#417	1	1	yes	characteristic
148	C₂²×C₈₉	#417	1	1	yes	characteristic
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	2	no	—
153	C₄×C₈₉	—	4	4	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	8	2	no	—
156	C₄×C₈₉	#332	1	4	yes	complemented
157	C₄×C₈₉	#459	1	1	yes	characteristic
158	C₄×C₈₉	#460	1	1	yes	characteristic
159	C₄×C₈₉	#461	1	1	yes	characteristic
160	C₄×C₈₉	#462	1	1	yes	characteristic
161	C₄×C₈₉	#474	1	1	yes	characteristic
162	C₄×C₈₉	#475	1	1	yes	characteristic
163	C₄×C₈₉	#476	1	1	yes	characteristic
164	C₄×C₈₉	#477	1	1	yes	characteristic
165	#8586	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	C₂×#332	1	1	yes	characteristic
167	C₂×C₈₉	#1547	1	1	yes	characteristic
168	C₂×C₈₉	#1550	1	1	yes	characteristic
169	C₂×C₈₉	#1561	1	1	yes	characteristic
170	C₂×C₈₉	#1801	1	1	yes	characteristic
171	C₂×C₈₉	#1802	1	1	yes	characteristic
172	C₂×C₈₉	#1803	1	1	yes	characteristic
173	C₄×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
174	#1842	—	89	1	no	—
175	#1842	—	89	1	no	—
176	C₂×#332	—	89	2	no	—
177	#1296	—	89	2	no	—
178	C₈₉	#8586	1	1	yes	characteristic, complemented, Sylow
179	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
180	C₂×C₄×Q₈	—	2	1	no	—
181	C₂×C₄⋊C₈	—	89	2	no	—
182	C₂×C₄⋊C₈	—	89	2	no	—
183	C₂×C₄⋊C₈	—	178	1	no	—
184	C₂×C₄⋊C₈	—	178	1	no	—
185	C₂×C₄⋊C₈	—	178	1	no	—
186	C₂×C₄⋊C₈	—	178	1	no	—
187	#391	—	2	1	no	—
188	#389	—	2	1	no	—
189	#389	Dic₈₉	1	2	yes	—
190	#384	C₂×D₈₉	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	2	1	no	—
192	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
193	C₄³	C₂×D₈₉	1	1	yes	characteristic
194	#332	—	178	4	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×C₈	—	178	2	no	—
197	C₂²×C₈	—	178	2	no	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	Dic₁₇₈	1	2	yes	—
201	C₄⋊Q₈	D₃₅₆	1	2	yes	—
202	C₄×Q₈	—	4	2	no	—
203	C₄×Q₈	—	4	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
218	C₄⋊C₈	—	356	2	no	—
219	C₄⋊C₈	—	356	2	no	—
220	C₄⋊C₈	—	356	2	no	—
221	C₄⋊C₈	—	356	2	no	—
222	C₄⋊C₈	—	178	2	no	—
223	C₄⋊C₈	—	178	2	no	—
224	C₄⋊C₈	—	178	2	no	—
225	C₄⋊C₈	—	178	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
243	C₂×C₈	—	356	2	no	—
244	C₂×C₈	—	356	2	no	—
245	C₂×C₈	—	356	2	no	—
246	C₂×C₈	—	356	2	no	—
247	C₂×C₈	—	356	2	no	—
248	C₂×C₈	—	356	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	C₄⋊Dic₈₉	1	4	yes	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	4	no	—
266	C₄²	#13083559	1	2	yes	—
267	C₄²	#13083561	1	2	yes	—
268	C₄²	#13083562	1	2	yes	—
269	C₄²	#13083564	1	2	yes	—
270	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
271	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
272	C₄²	#13083574	1	1	yes	characteristic
273	C₄²	#13083574	1	1	yes	characteristic
274	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	#15667931	1	2	yes	—
295	C₂×C₄	#15667933	1	2	yes	—
296	C₂×C₄	#15667933	1	2	yes	—
297	C₂×C₄	#15667936	1	2	yes	—
298	C₂×C₄	#15667936	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
300	C₂×C₄	#15667971	1	1	yes	characteristic
301	C₂×C₄	#15667986	1	1	yes	characteristic
302	C₂×C₄	#15667986	1	1	yes	characteristic
303	C₂×C₄	#15667991	1	1	yes	characteristic
304	C₂×C₄	#15667991	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
306	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
307	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
308	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
309	C₂×C₄	#15668037	1	1	yes	characteristic
310	C₂×C₄	#15668037	1	1	yes	characteristic
311	C₂×C₄	#15668042	1	1	yes	characteristic
312	C₂×C₄	#15668042	1	1	yes	characteristic
313	C₈	—	712	2	no	—
314	C₈	—	712	2	no	—
315	C₂²	#19151458	1	1	yes	characteristic
316	C₂²	C₂×#15667931	1	1	yes	characteristic
317	C₂²	#19151582	1	1	yes	characteristic
318	C₂²	C₂×#15667933	1	1	yes	characteristic
319	C₂²	C₂×#15667936	1	1	yes	characteristic
320	C₂²	#19151599	1	1	yes	characteristic
321	C₂²	#19151599	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	2	no	—
329	C₄	#19151344	1	4	yes	complemented
330	C₄	#19151793	1	1	yes	characteristic
331	C₄	#19151798	1	1	yes	characteristic
332	C₄	#19151803	1	1	yes	characteristic
333	C₄	#19151808	1	1	yes	characteristic
334	C₄	#19151878	1	1	yes	characteristic
335	C₄	#19151883	1	1	yes	characteristic
336	C₄	#19151888	1	1	yes	characteristic
337	C₄	#19151893	1	1	yes	characteristic
338	C₂	C₂×#19151344	1	1	yes	characteristic
339	C₂	#26018929	1	1	yes	characteristic
340	C₂	#26018942	1	1	yes	characteristic
341	C₂	#26018990	1	1	yes	characteristic
342	C₂	#26020248	1	1	yes	characteristic
343	C₂	#26020252	1	1	yes	characteristic
344	C₂	#26020257	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#8586
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
156. C₄×C₈₉	#332	semidirect
178. C₈₉	#8586	semidirect
329. C₄	#19151344	semidirect

Maximal subgroups

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

subgroup	index
#26020404	2
C₂×#19151344	2
#26018135	2
C₄×C₄⋊Q₈×C₈₉	2
#8586	89