Subgroups — #36460243

Order 22784.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
2	#26022659	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151358	C₂	1	4	yes	maximal
4	C₂×#376×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	2	yes	—
6	C₂×#15667971	C₂²	1	2	yes	—
7	#19151622	—	2	2	no	—
8	#19151622	C₄	1	4	yes	complemented
9	C₂×C₄⋊#10623494	—	2	2	no	—
10	#19151468	C₄	1	2	yes	complemented
11	#19151464	C₄	1	2	yes	complemented
12	#19151458	—	2	2	no	—
13	C₄×C₄⋊Dic₈₉	—	2	2	no	—
14	#19151358	—	2	8	no	—
15	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	#376×C₈₉	—	2	2	no	—
18	#376×C₈₉	C₄	1	4	yes	complemented
19	C₂×#95×C₈₉	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	4	no	—
22	C₂×#13083558	—	2	4	no	—
23	C₂×#13083558	C₂×C₄	1	4	yes	—
24	C₂²×#10623494	—	2	2	no	—
25	#15667971	—	2	4	no	—
26	#15667971	—	2	4	no	—
27	C₂×C₄⋊Dic₈₉	—	2	4	no	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄×Dic₈₉	—	2	2	no	—
31	C₄⋊#10623494	—	2	8	no	—
32	#15667919	—	2	4	no	—
33	C₂³×C₄×C₈₉	—	2	2	no	—
34	C₂³×C₄×C₈₉	D₄	1	2	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	—	2	2	no	—
37	C₂×C₄²×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	—	2	2	no	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₈₉	D₄	1	4	yes	—
45	C₂×C₄²×C₈₉	D₄	1	2	yes	—
46	C₂×C₄²×C₈₉	D₄	1	2	yes	—
47	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
48	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
49	#95×C₈₉	—	2	8	no	—
50	C₂²×Dic₈₉	—	2	4	no	—
51	#13083558	—	2	8	no	—
52	#13083558	—	4	4	no	—
53	C₂×#10623494	—	2	4	no	—
54	C₂×#10623494	—	4	2	no	—
55	C₂×#10623494	—	4	2	no	—
56	C₄⋊Dic₈₉	—	2	8	no	—
57	C₄⋊Dic₈₉	—	4	8	no	—
58	C₂²⋊Dic₈₉	—	4	4	no	—
59	C₄×Dic₈₉	—	4	2	no	—
60	C₄×Dic₈₉	—	4	2	no	—
61	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	—	2	4	no	—
63	C₂²×C₄×C₈₉	—	2	4	no	—
64	C₂²×C₄×C₈₉	—	2	4	no	—
65	C₂²×C₄×C₈₉	—	4	4	no	—
66	C₂²×C₄×C₈₉	—	4	8	no	—
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
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	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₈₉	#41	1	2	yes	—
80	C₂²×C₄×C₈₉	#41	1	2	yes	—
81	C₄²×C₈₉	—	2	4	no	—
82	C₄²×C₈₉	—	2	4	no	—
83	C₄²×C₈₉	—	2	4	no	—
84	C₄²×C₈₉	C₄²	1	2	yes	—
85	C₄²×C₈₉	C₄²	1	2	yes	—
86	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
87	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
88	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
89	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
90	C₄²×C₈₉	C₄⋊C₄	1	8	yes	complemented
91	C₂³×C₈₉	—	2	4	no	—
92	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₈₉	#121	1	2	yes	—
96	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	2	4	no	—
99	C₂×C₄×C₈₉	—	2	8	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	—	4	4	no	—
102	C₂×C₄×C₈₉	—	4	8	no	—
103	C₂×C₄×C₈₉	—	4	8	no	—
104	C₂×C₄×C₈₉	—	4	8	no	—
105	C₂×C₄×C₈₉	#95	1	4	yes	—
106	C₂×C₄×C₈₉	#95	1	4	yes	—
107	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#117	1	2	yes	—
113	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₈₉	#121	1	2	yes	—
118	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#125	1	2	yes	—
122	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×Dic₈₉	—	4	4	no	—
126	C₂×Dic₈₉	—	4	8	no	—
127	#10623494	—	4	4	no	—
128	C₂²×C₈₉	—	2	4	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	4	no	—
132	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
133	C₂²×C₈₉	#380	1	2	yes	—
134	C₂²×C₈₉	#389	1	2	yes	—
135	C₂²×C₈₉	#390	1	1	yes	characteristic
136	C₂²×C₈₉	#391	1	1	yes	characteristic
137	C₂²×C₈₉	#421	1	4	yes	—
138	C₄×C₈₉	—	2	4	no	—
139	C₄×C₈₉	—	2	4	no	—
140	C₄×C₈₉	—	4	8	no	—
141	C₄×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
142	C₄×C₈₉	#382	1	1	yes	characteristic
143	C₄×C₈₉	#384	1	1	yes	characteristic
144	C₄×C₈₉	#389	1	2	yes	—
145	C₄×C₈₉	#390	1	1	yes	characteristic
146	C₄×C₈₉	#391	1	1	yes	characteristic
147	Dic₈₉	—	8	4	no	—
148	#10840	—	89	1	no	maximal, Sylow
149	C₂×C₈₉	—	2	2	no	—
150	C₂×C₈₉	—	2	2	no	—
151	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
152	C₂×C₈₉	#1733	1	2	yes	—
153	C₂×C₈₉	#1963	1	2	yes	—
154	C₂×C₈₉	#1995	1	2	yes	—
155	C₂×#423	—	89	1	no	—
156	#2285	—	89	1	no	—
157	C₂×#376	D₈₉	1	1	yes	characteristic
158	C₂×#339	—	89	4	no	—
159	C₈₉	#10840	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	89	2	no	—
161	C₂×#117	—	89	2	no	—
162	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
163	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
164	#423	—	89	4	no	—
165	#423	—	178	2	no	—
166	C₂×C₄⋊C₈	—	178	2	no	—
167	#384	—	89	2	no	—
168	#382	—	89	2	no	—
169	#380	—	178	2	no	—
170	C₄×C₄⋊C₄	—	178	2	no	—
171	#376	—	2	2	no	—
172	#376	Dic₈₉	1	4	yes	complemented
173	C₂×#95	—	2	2	no	—
174	#339	—	178	8	no	—
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	C₂²⋊D₈₉	1	2	yes	—
177	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
178	C₂×M₁₆	—	89	4	no	—
179	C₂×M₁₆	—	178	4	no	—
180	C₂×M₁₆	—	178	4	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂²×C₈	—	178	2	no	—
183	#117	—	178	4	no	—
184	#117	—	178	4	no	—
185	C₂×C₄⋊C₄	—	89	4	no	—
186	C₂×C₄⋊C₄	—	178	4	no	—
187	C₂×C₂²⋊C₄	—	178	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	178	2	no	—
192	C₂×C₄²	Dic₁₇₈	1	4	yes	—
193	C₂×C₄²	Dic₁₇₈	1	2	yes	—
194	C₂×C₄²	C₄×D₈₉	1	2	yes	—
195	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
197	C₂×C₄²	D₃₅₆	1	4	yes	—
198	C₂×C₄²	D₃₅₆	1	2	yes	—
199	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
200	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
201	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
202	C₄⋊C₈	—	178	8	no	—
203	#95	—	2	8	no	—
204	#95	—	178	4	no	—
205	C₂⁴	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	178	4	no	—
212	C₂²×C₄	#13083551	1	2	yes	—
213	C₂²×C₄	#13083551	1	2	yes	—
214	C₂²×C₄	C₂²⋊Dic₈₉	1	4	yes	—
215	C₂²×C₄	#13083553	1	4	yes	—
216	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
221	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
222	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
223	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
224	C₂²×C₄	#13083574	1	2	yes	—
225	C₂²×C₄	#13083574	1	2	yes	—
226	M₁₆	—	356	4	no	—
227	M₁₆	—	178	8	no	—
228	C₂×C₈	—	356	2	no	—
229	C₂×C₈	—	356	2	no	—
230	C₂×C₈	—	178	4	no	—
231	C₄⋊C₄	—	356	8	no	—
232	C₄⋊C₄	—	178	8	no	—
233	C₂²⋊C₄	—	356	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	356	2	no	—
238	C₄²	—	356	2	no	—
239	C₄²	C₄×Dic₈₉	1	2	yes	—
240	C₄²	C₄×Dic₈₉	1	2	yes	—
241	C₄²	#13083551	1	4	yes	—
242	C₄²	#13083553	1	2	yes	—
243	C₄²	#13083553	1	2	yes	—
244	C₄²	C₄⋊Dic₈₉	1	4	yes	—
245	C₄²	C₄⋊Dic₈₉	1	8	yes	complemented
246	C₂³	—	2	4	no	—
247	C₂³	C₂×#13083551	1	1	yes	characteristic
248	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
249	C₂³	C₂×#13083553	1	1	yes	characteristic
250	C₂³	#15667986	1	2	yes	—
251	C₂³	#15667991	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	356	8	no	—
261	C₂×C₄	—	356	4	no	—
262	C₂×C₄	#15667919	1	4	yes	—
263	C₂×C₄	#15667919	1	4	yes	—
264	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
266	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
269	C₂×C₄	#15667971	1	2	yes	—
270	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
271	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
272	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
273	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
274	C₂×C₄	#15667986	1	2	yes	—
275	C₂×C₄	#15667991	1	2	yes	—
276	C₂×C₄	#15668000	1	1	yes	characteristic
277	C₂×C₄	#15668000	1	1	yes	characteristic
278	C₂×C₄	#15668004	1	2	yes	—
279	C₂×C₄	#15668009	1	1	yes	characteristic
280	C₂×C₄	#15668012	1	2	yes	—
281	C₂×C₄	#15668012	1	1	yes	characteristic
282	C₈	—	356	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	C₂×#15667919	1	1	yes	characteristic
288	C₂²	#19151458	1	2	yes	—
289	C₂²	#19151496	1	2	yes	—
290	C₂²	#19151500	1	1	yes	characteristic
291	C₂²	#19151504	1	1	yes	characteristic
292	C₂²	#19151616	1	4	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	8	no	—
296	C₄	—	712	4	no	—
297	C₄	C₄×C₄⋊Dic₈₉	1	2	yes	—
298	C₄	#19151464	1	1	yes	characteristic
299	C₄	#19151468	1	1	yes	characteristic
300	C₄	#19151496	1	2	yes	—
301	C₄	#19151500	1	1	yes	characteristic
302	C₄	#19151504	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#26018482	1	1	yes	characteristic
306	C₂	#26019962	1	2	yes	—
307	C₂	#26021084	1	2	yes	—
308	C₂	#26021233	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#376×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂×D₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#376	D₈₉
hypocenter	C₈₉	#10840
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
8. #19151622	C₄	semidirect
10. #19151468	C₄	semidirect
11. #19151464	C₄	semidirect
18. #376×C₈₉	C₄	semidirect
90. C₄²×C₈₉	C₄⋊C₄	semidirect
159. C₈₉	#10840	semidirect
172. #376	Dic₈₉	semidirect
245. C₄²	C₄⋊Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151622	2
#26022659	2
C₂×#19151358	2
C₂×#376×C₈₉	2
#10840	89