Subgroups — #36538331

Order 22784.

702 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151978	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030068	C₂	1	2	yes	maximal
3	C₈×#13083574	C₂	1	1	yes	maximal, characteristic
4	#26019623	C₂	1	4	yes	maximal, complemented
5	#26019618	C₂	1	4	yes	maximal, complemented
6	#26018938	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₅₆	—	2	2	no	—
9	C₂×#15668012	C₂²	1	2	yes	—
10	C₄×#13083574	C₂²	1	1	yes	characteristic
11	#19151978	—	2	1	no	—
12	#19151978	—	2	1	no	—
13	#19151978	C₂²	1	4	yes	—
14	#19151894	—	2	2	no	—
15	#19151889	—	2	2	no	—
16	#19151884	C₂²	1	4	yes	—
17	#19151879	C₂²	1	4	yes	—
18	C₈×Dic₁₇₈	C₂²	1	2	yes	—
19	C₈×D₃₅₆	C₂²	1	2	yes	—
20	C₈×C₂²⋊D₈₉	—	2	2	no	—
21	#19151651	C₂²	1	1	yes	characteristic
22	#19151649	—	2	1	no	—
23	#19151640	—	2	2	no	—
24	#19151600	—	2	2	no	—
25	C₄×C₈×D₈₉	—	2	1	no	—
26	#19151343	C₂²	1	8	yes	—
27	#423×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	#335×C₈₉	C₂²	1	4	yes	—
30	C₂×#13083574	D₄	1	1	yes	characteristic
31	C₂²×Dic₁₇₈	—	2	2	no	—
32	C₂×Dic₃₅₆	—	2	4	no	—
33	C₂×Dic₃₅₆	—	2	4	no	—
34	C₂×Dic₃₅₆	—	4	8	no	—
35	C₂×C₈×D₈₉	—	2	2	no	—
36	#15668012	—	2	2	no	—
37	#15668012	—	2	2	no	—
38	#15668012	—	2	2	no	—
39	#15668012	C₂³	1	4	yes	—
40	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
41	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
42	C₄×D₃₅₆	D₄	1	1	yes	characteristic
43	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
44	C₄×C₂²⋊D₈₉	—	2	1	no	—
45	C₄×C₂²⋊D₈₉	—	2	1	no	—
46	#15667971	D₄	1	1	yes	characteristic
47	#15667969	—	2	1	no	—
48	C₂×C₄⋊Dic₈₉	—	2	2	no	—
49	C₂×C₄⋊Dic₈₉	—	2	2	no	—
50	C₄²×D₈₉	—	2	1	no	—
51	#15667946	—	2	4	no	—
52	#15667944	—	2	4	no	—
53	C₄⋊#10623494	C₂³	1	2	yes	—
54	#15667941	—	2	2	no	—
55	#15667937	—	2	4	no	—
56	#15667934	—	2	4	no	—
57	C₂²⋊#10623494	—	2	1	no	—
58	#15667924	—	2	2	no	—
59	C₄×#10623494	—	2	1	no	—
60	C₈×Dic₈₉	—	2	2	no	—
61	C₂×M₁₆×C₈₉	—	2	2	no	—
62	C₂²×C₈×C₈₉	D₄	1	2	yes	—
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₈₉	—	2	2	no	—
65	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
66	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
67	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₈₉	C₂³	1	2	yes	—
69	#13083574	—	2	2	no	—
70	#13083574	Q₁₆	1	4	yes	complemented
71	C₂×Dic₁₇₈	—	2	4	no	—
72	C₂×Dic₁₇₈	—	4	8	no	—
73	C₂×Dic₁₇₈	—	4	2	no	—
74	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
75	C₂×D₃₅₆	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₈₉	—	2	1	no	—
77	C₂²×Dic₈₉	—	4	2	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	C₂×C₄×D₈₉	—	2	1	no	—
80	C₂×C₄×D₈₉	—	2	1	no	—
81	Dic₃₅₆	—	4	8	no	—
82	Dic₃₅₆	—	4	8	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₈×D₈₉	—	4	4	no	—
85	C₄⋊Dic₈₉	—	2	4	no	—
86	C₄⋊Dic₈₉	—	2	4	no	—
87	C₄⋊Dic₈₉	—	4	2	no	—
88	C₄⋊Dic₈₉	—	4	2	no	—
89	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
90	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
91	#13083553	—	2	1	no	—
92	#13083553	—	2	1	no	—
93	C₂²⋊Dic₈₉	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	—	2	1	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
102	M₁₆×C₈₉	—	4	4	no	—
103	C₂×C₈×C₈₉	—	2	4	no	—
104	C₂×C₈×C₈₉	—	2	4	no	—
105	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	#41	1	1	yes	characteristic
110	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₂³×C₈₉	#121	1	1	yes	characteristic
112	C₂²×D₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₈₉	#135	1	2	yes	—
120	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
123	C₂²⋊D₈₉	—	2	4	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	4	4	no	—
128	C₂×Dic₈₉	—	8	4	no	—
129	D₃₅₆	—	2	1	no	—
130	D₃₅₆	C₂×Q₁₆	1	2	yes	complemented
131	C₄×D₈₉	—	2	4	no	—
132	C₄×D₈₉	—	4	2	no	—
133	C₄×D₈₉	—	4	2	no	—
134	Dic₁₇₈	—	2	1	no	—
135	Dic₁₇₈	—	4	8	no	—
136	Dic₁₇₈	—	8	4	no	—
137	Dic₁₇₈	C₂×Q₁₆	1	2	yes	complemented
138	C₈×C₈₉	—	2	2	no	—
139	C₈×C₈₉	—	4	4	no	—
140	C₈×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
141	#10623494	—	4	2	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	#467	1	2	yes	—
145	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
146	C₂×D₈₉	—	2	2	no	—
147	C₂×D₈₉	—	4	1	no	—
148	C₂×D₈₉	—	4	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	#467	1	2	yes	—
152	C₄×C₈₉	#473	1	2	yes	—
153	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
155	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
156	Dic₈₉	—	2	2	no	—
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	8	4	no	—
160	#19268	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	2	2	no	—
162	C₂×C₈₉	#3038	1	1	yes	characteristic
163	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
165	D₈₉	—	4	2	no	—
166	C₂×#494	—	89	1	no	—
167	#3079	—	89	2	no	—
168	C₈×#41	—	89	1	no	—
169	#1670	—	89	4	no	—
170	#1669	—	89	4	no	—
171	#1567	D₈₉	1	1	yes	characteristic, complemented
172	#1549	—	89	2	no	—
173	C₈₉	#19268	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₁₆	—	178	2	no	—
175	C₂×C₄⋊Q₈	—	89	2	no	—
176	C₄×#41	—	89	1	no	—
177	#494	—	89	4	no	—
178	#494	—	178	1	no	—
179	#494	—	178	1	no	—
180	#477	—	178	2	no	—
181	#476	—	178	2	no	—
182	#475	—	89	4	no	—
183	#474	—	89	4	no	—
184	C₈×Q₈	—	89	2	no	—
185	C₈×D₄	—	89	2	no	—
186	C₈×D₄	—	178	2	no	—
187	#431	—	89	1	no	—
188	#431	—	178	1	no	—
189	#428	—	178	2	no	—
190	#423	C₂×D₈₉	1	2	yes	—
191	#417	—	178	2	no	—
192	C₂×C₄×C₈	—	178	1	no	—
193	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
194	#335	C₂×D₈₉	1	4	yes	—
195	#332	—	89	8	no	—
196	C₂×#41	—	89	1	no	—
197	C₂²×Q₈	—	178	2	no	—
198	C₂×Q₁₆	—	356	8	no	—
199	C₂×Q₁₆	—	178	4	no	—
200	C₂×Q₁₆	—	178	4	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂²×C₈	—	178	2	no	—
203	C₂²×C₈	D₃₅₆	1	2	yes	—
204	C₄⋊Q₈	—	89	4	no	—
205	C₄⋊Q₈	—	178	2	no	—
206	C₄⋊Q₈	—	178	2	no	—
207	C₄⋊Q₈	—	178	2	no	—
208	C₄×Q₈	—	89	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×D₄	—	89	1	no	—
211	C₄×D₄	—	89	1	no	—
212	C₄×D₄	—	178	1	no	—
213	C₄×D₄	—	178	1	no	—
214	#117	—	89	1	no	—
215	#117	—	178	1	no	—
216	C₂×C₄⋊C₄	—	178	2	no	—
217	C₂×C₄⋊C₄	—	178	2	no	—
218	C₂×C₄²	—	178	1	no	—
219	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
220	#107	—	178	4	no	—
221	#106	—	178	4	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	89	2	no	—
224	C₄⋊C₈	—	178	2	no	—
225	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
226	#103	—	178	4	no	—
227	#102	—	178	4	no	—
228	C₂²⋊C₈	—	178	1	no	—
229	C₂²⋊C₈	—	178	2	no	—
230	C₄×C₈	—	178	2	no	—
231	C₄×C₈	—	178	1	no	—
232	C₄×C₈	D₃₅₆	1	1	yes	characteristic
233	C₄×C₈	D₃₅₆	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₈₉	1	2	yes	—
235	#41	—	89	4	no	—
236	#41	—	178	2	no	—
237	C₂×Q₈	—	89	1	no	—
238	C₂×Q₈	—	356	8	no	—
239	C₂×Q₈	—	356	2	no	—
240	C₂×Q₈	—	178	4	no	—
241	C₂×D₄	—	89	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂²×C₄	—	356	2	no	—
244	C₂²×C₄	—	178	1	no	—
245	C₂²×C₄	—	178	1	no	—
246	C₂²×C₄	—	178	1	no	—
247	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
249	C₂²×C₄	#13083575	1	1	yes	characteristic
250	Q₁₆	—	356	8	no	—
251	Q₁₆	—	356	8	no	—
252	M₁₆	—	4	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	356	4	no	—
256	C₂×C₈	—	178	2	no	—
257	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
258	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
259	C₄⋊C₄	—	89	1	no	—
260	C₄⋊C₄	—	89	1	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₄⋊C₄	—	356	2	no	—
263	C₄⋊C₄	—	178	4	no	—
264	C₄⋊C₄	—	178	4	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₂²⋊C₄	—	178	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₄²	—	178	1	no	—
271	C₄²	—	178	1	no	—
272	C₄²	—	178	1	no	—
273	C₄²	D₄×D₈₉	1	1	yes	characteristic
274	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
275	C₄²	#13083575	1	1	yes	characteristic
276	C₄²	C₂³×D₈₉	1	1	yes	characteristic
277	C₂³	—	178	1	no	—
278	C₂³	#15667987	1	1	yes	characteristic
279	Q₈	—	89	2	no	—
280	Q₈	—	356	8	no	—
281	Q₈	—	178	1	no	—
282	Q₈	—	712	4	no	—
283	D₄	—	89	2	no	—
284	D₄	—	178	4	no	—
285	D₄	—	178	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	356	2	no	—
289	C₂×C₄	—	356	2	no	—
290	C₂×C₄	—	356	4	no	—
291	C₂×C₄	—	178	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	178	1	no	—
294	C₂×C₄	—	178	4	no	—
295	C₂×C₄	—	712	4	no	—
296	C₂×C₄	#15667987	1	1	yes	characteristic
297	C₂×C₄	#15667987	1	1	yes	characteristic
298	C₂×C₄	#15667987	1	1	yes	characteristic
299	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
300	C₂×C₄	#15668031	1	2	yes	—
301	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
303	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	4	4	no	—
306	C₈	—	356	2	no	—
307	C₈	#15668040	1	4	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	356	1	no	—
311	C₂²	—	356	1	no	—
312	C₂²	—	178	2	no	—
313	C₂²	#19151840	1	2	yes	—
314	C₂²	C₂×#15667987	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	356	1	no	—
318	C₄	—	356	1	no	—
319	C₄	—	178	2	no	—
320	C₄	—	712	4	no	—
321	C₄	#19151840	1	2	yes	—
322	C₄	#19151874	1	2	yes	—
323	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
324	C₄	C₂×#15668031	1	1	yes	characteristic
325	C₄	C₂×#15668040	1	2	yes	—
326	C₂	—	2	2	no	—
327	C₂	—	356	2	no	—
328	C₂	#26029733	1	1	yes	characteristic
329	C₂	C₂×#19151840	1	1	yes	characteristic
330	C₂	C₂×#19151874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1567	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂×#13083574	D₄
hypercenter	#1567	D₈₉
hypocenter	C₈₉	#19268
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
1. C₂×#19151978	C₂	semidirect
4. #26019623	C₂	semidirect
5. #26019618	C₂	semidirect
6. #26018938	C₂	semidirect
7. C₈₉×#1567	C₂	semidirect
70. #13083574	Q₁₆	semidirect
130. D₃₅₆	C₂×Q₁₆	semidirect
137. Dic₁₇₈	C₂×Q₁₆	semidirect
171. #1567	D₈₉	semidirect
173. C₈₉	#19268	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151978	2
#26030068	2
C₈×#13083574	2
#26019623	2
#26019618	2
#26018938	2
C₈₉×#1567	2
#19268	89