Subgroups — #36523653

Order 22784.

702 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083574	C₂	1	1	yes	maximal, characteristic
2	#26028821	C₂	1	2	yes	maximal
3	#26019024	C₂	1	2	yes	maximal, complemented
4	#26019021	C₂	1	4	yes	maximal, complemented
5	#26019006	C₂	1	2	yes	maximal, complemented
6	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
7	#334×D₈₉	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668020	—	2	2	no	—
10	C₄×#13083574	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
12	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
13	C₈×D₃₅₆	C₄	1	1	yes	characteristic
14	C₈×D₃₅₆	C₄	1	1	yes	characteristic
15	C₈×C₂²⋊D₈₉	C₄	1	2	yes	—
16	C₈×C₂²⋊D₈₉	C₄	1	2	yes	—
17	#19151651	C₂²	1	1	yes	characteristic
18	#19151649	C₂²	1	2	yes	—
19	#19151626	—	2	2	no	—
20	#19151623	—	2	4	no	—
21	#19151620	C₂²	1	4	yes	—
22	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
23	C₄⋊C₈×D₈₉	C₂²	1	4	yes	—
24	C₄×C₈×D₈₉	C₂²	1	2	yes	—
25	#19151349	C₂²	1	1	yes	characteristic
26	#19151349	C₂²	1	1	yes	characteristic
27	#19151349	C₂²	1	2	yes	—
28	#19151348	C₂²	1	4	yes	—
29	#19151348	C₂²	1	4	yes	—
30	#423×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#334×C₈₉	C₂²	1	2	yes	—
33	#334×C₈₉	C₂²	1	2	yes	—
34	C₂×#13083574	D₄	1	1	yes	characteristic
35	#15668020	—	4	8	no	—
36	C₂×#13083557	—	2	4	no	—
37	C₂²×#10623494	—	2	2	no	—
38	C₂×C₈×D₈₉	—	2	4	no	—
39	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
43	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
44	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
46	C₄×C₂²⋊D₈₉	Q₈	1	2	yes	—
47	#15667971	D₄	1	1	yes	characteristic
48	#15667969	D₄	1	2	yes	—
49	C₄²×D₈₉	C₂³	1	2	yes	—
50	C₄⋊#10623494	—	2	2	no	—
51	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
52	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
53	C₄⋊#10623494	C₂³	1	2	yes	—
54	C₄⋊#10623494	C₂³	1	2	yes	—
55	#15667941	—	2	4	no	—
56	#15667941	—	2	4	no	—
57	#15667941	C₂×C₄	1	2	yes	—
58	#15667941	C₂×C₄	1	2	yes	—
59	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
60	#15667924	C₂×C₄	1	2	yes	—
61	#15667924	C₂×C₄	1	2	yes	—
62	C₄×#10623494	C₂³	1	2	yes	—
63	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
64	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₈₉	—	2	2	no	—
66	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₉	—	2	2	no	—
70	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	#13083574	SD₁₆	1	8	yes	—
76	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
77	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₈₉	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
82	#13083557	—	4	8	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₂×#10623494	—	2	2	no	—
85	C₂×#10623494	—	4	4	no	—
86	C₂×#10623494	C₂²×C₄	1	2	yes	—
87	C₂×#10623494	C₂²×C₄	1	2	yes	—
88	C₈×D₈₉	—	2	4	no	—
89	C₈×D₈₉	—	4	2	no	—
90	C₈×D₈₉	—	4	8	no	—
91	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
92	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
93	#13083553	C₄⋊C₄	1	2	yes	—
94	#13083553	C₄⋊C₄	1	2	yes	—
95	C₂²⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
96	#13083551	C₄⋊C₄	1	2	yes	—
97	#13083551	C₄⋊C₄	1	2	yes	—
98	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
99	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
100	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
101	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
104	M₁₆×C₈₉	—	4	4	no	—
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	—	2	4	no	—
107	C₂×C₈×C₈₉	—	4	1	no	—
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
115	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
125	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
128	C₂²⋊D₈₉	#106	1	8	yes	—
129	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
132	D₃₅₆	#106	1	4	yes	—
133	C₄×D₈₉	—	2	2	no	—
134	C₄×D₈₉	—	2	2	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	C₂×SD₁₆	1	8	yes	—
137	Dic₁₇₈	#106	1	4	yes	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	4	4	no	—
142	C₈×C₈₉	#131	1	2	yes	complemented
143	#10623494	—	2	2	no	—
144	#10623494	—	2	2	no	—
145	#10623494	—	4	2	no	—
146	#10623494	—	4	2	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	C₂×#106	1	2	yes	—
150	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂×D₈₉	—	2	2	no	—
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	C₂×#106	1	4	yes	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	C₂×#106	1	2	yes	—
158	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
159	C₄×C₈₉	#568	1	1	yes	characteristic
160	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
161	Dic₈₉	—	2	2	no	—
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	C₂×#106	1	4	yes	—
164	#17826	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	2	no	—
166	C₂×C₈₉	#2906	1	1	yes	characteristic
167	C₂×C₈₉	C₂²×#106	1	1	yes	characteristic
168	C₂×C₈₉	#2916	1	1	yes	characteristic
169	D₈₉	—	2	4	no	—
170	C₈×#41	—	89	1	no	—
171	#2906	—	89	2	no	—
172	#1570	—	89	2	no	—
173	#1570	—	89	4	no	—
174	#1566	—	89	2	no	—
175	#1566	D₈₉	1	1	yes	characteristic, complemented
176	C₂×#334	—	89	1	no	—
177	C₂×#334	—	89	2	no	—
178	C₈₉	#17826	1	1	yes	characteristic, complemented, Sylow
179	C₂×#131	—	178	2	no	—
180	C₄×#41	—	89	1	no	—
181	C₈×Q₈	—	89	1	no	—
182	C₈×Q₈	—	89	1	no	—
183	C₈×D₄	—	89	2	no	—
184	C₈×D₄	—	89	2	no	—
185	C₈×D₄	—	89	1	no	—
186	C₈×D₄	—	89	1	no	—
187	#431	—	89	1	no	—
188	#431	—	89	2	no	—
189	#424	—	178	4	no	—
190	#424	—	178	2	no	—
191	#423	—	89	4	no	—
192	#423	C₂×D₈₉	1	2	yes	—
193	C₂×C₄⋊C₈	—	89	2	no	—
194	C₂×C₄⋊C₈	—	89	4	no	—
195	C₂×C₄×C₈	—	89	2	no	—
196	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
197	#334	—	89	4	no	—
198	#334	—	89	4	no	—
199	#334	—	89	1	no	—
200	#334	—	89	1	no	—
201	#334	—	89	2	no	—
202	#334	C₂×D₈₉	1	2	yes	—
203	#334	C₂×D₈₉	1	2	yes	—
204	C₂×#41	—	89	1	no	—
205	#131	—	356	8	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	178	4	no	—
208	C₂²×C₈	—	89	2	no	—
209	C₂²×C₈	—	89	2	no	—
210	C₂²×C₈	—	178	4	no	—
211	C₂²×C₈	—	178	2	no	—
212	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
213	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
214	C₄×Q₈	—	89	1	no	—
215	C₄×Q₈	—	89	1	no	—
216	C₄×D₄	—	89	2	no	—
217	C₄×D₄	—	89	2	no	—
218	C₄×D₄	—	89	1	no	—
219	C₄×D₄	—	89	1	no	—
220	#117	—	89	1	no	—
221	#117	—	89	2	no	—
222	C₂×C₄²	—	89	2	no	—
223	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	89	2	no	—
226	C₄⋊C₈	—	89	2	no	—
227	C₄⋊C₈	—	89	1	no	—
228	C₄⋊C₈	—	89	1	no	—
229	C₄⋊C₈	—	89	2	no	—
230	C₄⋊C₈	—	89	2	no	—
231	C₄⋊C₈	—	178	4	no	—
232	C₄⋊C₈	—	178	4	no	—
233	C₄⋊C₈	—	178	2	no	—
234	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
235	C₂²⋊C₈	—	89	2	no	—
236	C₂²⋊C₈	—	89	2	no	—
237	C₂²⋊C₈	—	89	2	no	—
238	C₄×C₈	—	89	2	no	—
239	C₄×C₈	—	89	2	no	—
240	C₄×C₈	—	89	2	no	—
241	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
242	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	#41	—	89	8	no	—
246	C₂×Q₈	—	89	1	no	—
247	C₂×D₄	—	89	2	no	—
248	C₂×D₄	—	89	1	no	—
249	C₂²×C₄	—	89	2	no	—
250	C₂²×C₄	—	89	2	no	—
251	C₂²×C₄	—	89	2	no	—
252	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
255	M₁₆	—	4	4	no	—
256	M₁₆	—	356	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	89	2	no	—
261	C₂×C₈	—	89	2	no	—
262	C₂×C₈	—	356	2	no	—
263	C₂×C₈	—	356	8	no	—
264	C₂×C₈	—	356	4	no	—
265	C₂×C₈	—	178	2	no	—
266	C₂×C₈	—	178	2	no	—
267	C₂×C₈	—	178	4	no	—
268	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
272	C₄⋊C₄	—	89	2	no	—
273	C₄⋊C₄	—	89	2	no	—
274	C₄⋊C₄	—	89	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₂²⋊C₄	—	89	2	no	—
277	C₂²⋊C₄	—	89	2	no	—
278	C₂²⋊C₄	—	89	2	no	—
279	C₄²	—	89	2	no	—
280	C₄²	—	89	2	no	—
281	C₄²	—	89	2	no	—
282	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
283	C₄²	D₄×D₈₉	1	1	yes	characteristic
284	C₄²	Q₈×D₈₉	1	1	yes	characteristic
285	C₄²	C₂³×D₈₉	1	1	yes	characteristic
286	C₂³	—	89	2	no	—
287	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
288	Q₈	—	89	4	no	—
289	D₄	—	89	8	no	—
290	D₄	—	89	4	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	89	2	no	—
295	C₂×C₄	—	89	8	no	—
296	C₂×C₄	—	89	2	no	—
297	C₂×C₄	—	89	2	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	178	2	no	—
300	C₂×C₄	—	178	2	no	—
301	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
302	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
303	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	SD₁₆×D₈₉	1	4	yes	—
305	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	4	no	—
312	C₈	—	356	2	no	—
313	C₈	—	356	2	no	—
314	C₈	—	178	2	no	—
315	C₈	—	178	2	no	—
316	C₈	#15668020	1	2	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	89	4	no	—
320	C₂²	—	178	2	no	—
321	C₂²	—	178	2	no	—
322	C₂²	#106×D₈₉	1	2	yes	—
323	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	89	4	no	—
328	C₄	—	178	2	no	—
329	C₄	—	178	2	no	—
330	C₄	#106×D₈₉	1	2	yes	—
331	C₄	C₂×#15668020	1	1	yes	characteristic
332	C₄	#19152364	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₈₉	1	2	yes	—
334	C₂	—	2	2	no	—
335	C₂	—	178	4	no	—
336	C₂	#26028821	1	1	yes	characteristic
337	C₂	C₂×#106×D₈₉	1	1	yes	characteristic
338	C₂	#26028860	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×#1566	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#13083574	D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#17826
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #26019024	C₂	semidirect
4. #26019021	C₂	semidirect
5. #26019006	C₂	semidirect
6. C₂×#19151349	C₂	semidirect
7. #334×D₈₉	C₂	semidirect
8. C₈₉×#1566	C₂	semidirect
142. C₈×C₈₉	#131	semidirect
175. #1566	D₈₉	semidirect
178. C₈₉	#17826	semidirect
316. C₈	#15668020	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083574	2
#26028821	2
#26019024	2
#26019021	2
#26019006	2
C₂×#19151349	2
#334×D₈₉	2
C₈₉×#1566	2
#17826	89