Subgroups — #36707087

Order 22784.

716 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	#26033354	C₂	1	2	yes	maximal
2	#26028677	C₂	1	1	yes	maximal, characteristic
3	#26021856	C₂	1	2	yes	maximal, complemented
4	#26021855	C₂	1	4	yes	maximal, complemented
5	C₂×#19151376	C₂	1	1	yes	maximal, characteristic, complemented
6	#347×D₈₉	C₂	1	4	yes	maximal, complemented
7	C₂×#347×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₄	1	1	yes	characteristic
9	C₂×#15668018	C₂²	1	1	yes	characteristic
10	C₄×#13083574	C₄	1	1	yes	characteristic
11	#19152016	C₄	1	8	yes	complemented
12	#19152014	C₄	1	8	yes	complemented
13	C₂×#15667952	C₂²	1	2	yes	—
14	M₃₂×D₈₉	C₂²	1	8	yes	—
15	#19151555	C₄	1	2	yes	—
16	#19151555	C₂²	1	2	yes	—
17	#19151553	C₄	1	1	yes	characteristic
18	#19151553	C₂²	1	1	yes	characteristic
19	C₄×#13083558	C₄	1	1	yes	characteristic
20	#19151550	C₄	1	2	yes	—
21	#97×D₈₉	C₂²	1	2	yes	—
22	#19151376	C₂²	1	2	yes	—
23	#19151376	C₂²	1	2	yes	—
24	#19151375	C₂²	1	8	yes	—
25	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
26	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
27	#347×C₈₉	C₂²	1	2	yes	—
28	#347×C₈₉	C₂²	1	2	yes	—
29	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
30	#15668018	—	2	2	no	—
31	#15668018	—	2	2	no	—
32	#15668018	C₂×C₄	1	4	yes	—
33	#15668018	C₂×C₄	1	4	yes	—
34	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
35	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
36	C₂×#13083557	C₂×C₄	1	2	yes	—
37	C₂×#13083557	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₈₉	C₂³	1	2	yes	—
40	C₄×Dic₁₇₈	—	2	1	no	—
41	C₄×D₃₅₆	—	2	1	no	—
42	C₄×C₂²⋊D₈₉	—	2	2	no	—
43	#15667971	C₂×C₄	1	1	yes	characteristic
44	#15667969	C₂×C₄	1	2	yes	—
45	C₄²×D₈₉	C₂×C₄	1	2	yes	—
46	#15667952	C₂×C₄	1	4	yes	complemented
47	#15667952	C₂³	1	4	yes	—
48	#15667951	C₂×C₄	1	16	yes	complemented
49	C₂×#13083549	C₂×C₄	1	4	yes	complemented
50	C₁₆×D₈₉	C₂×C₄	1	16	yes	complemented
51	#15667923	C₂×C₄	1	2	yes	—
52	#15667922	C₂×C₄	1	2	yes	—
53	#15667922	C₂³	1	2	yes	—
54	C₄×#10623494	C₂×C₄	1	2	yes	—
55	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
56	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
60	M₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
61	M₃₂×C₈₉	C₂³	1	4	yes	—
62	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
63	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
64	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
65	#97×C₈₉	C₂³	1	1	yes	characteristic
66	#97×C₈₉	C₂³	1	1	yes	characteristic
67	#13083574	C₄²	1	2	yes	—
68	#13083574	C₄²	1	2	yes	—
69	#13083574	M₁₆	1	2	yes	—
70	#13083574	M₁₆	1	2	yes	—
71	C₂×Dic₁₇₈	M₁₆	1	1	yes	characteristic
72	C₂×D₃₅₆	M₁₆	1	1	yes	characteristic
73	C₂×C₂²⋊D₈₉	M₁₆	1	2	yes	—
74	C₂×C₄×D₈₉	—	2	2	no	—
75	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
76	#13083558	—	2	1	no	—
77	#13083558	—	2	1	no	—
78	#13083558	C₄²	1	2	yes	—
79	#13083558	C₄²	1	2	yes	—
80	#13083557	—	2	4	no	—
81	#13083557	C₄²	1	8	yes	—
82	C₂×#10623494	C₂²×C₄	1	2	yes	—
83	C₂×#10623494	C₂²×C₄	1	2	yes	—
84	C₈×D₈₉	—	2	4	no	—
85	C₈×D₈₉	C₂²×C₄	1	8	yes	—
86	C₄⋊Dic₈₉	—	2	1	no	—
87	#13083553	—	2	2	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	#13083551	—	2	2	no	—
90	C₄×Dic₈₉	—	2	2	no	—
91	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
92	#13083549	C₂²×C₄	1	8	yes	—
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
103	C₄²×C₈₉	—	2	1	no	—
104	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₁₆×C₈₉	C₂²×C₄	1	8	yes	—
107	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
108	C₂²×D₈₉	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	8	1	no	—
112	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
118	C₂²⋊D₈₉	#97	1	4	yes	—
119	C₂²⋊D₈₉	#97	1	4	yes	—
120	C₂×Dic₈₉	—	2	2	no	—
121	C₂×Dic₈₉	C₂×M₁₆	1	2	yes	—
122	D₃₅₆	#97	1	2	yes	—
123	D₃₅₆	#97	1	2	yes	—
124	C₄×D₈₉	—	8	2	no	—
125	C₄×D₈₉	C₂×C₄²	1	4	yes	—
126	C₄×D₈₉	C₂×M₁₆	1	4	yes	—
127	Dic₁₇₈	#97	1	2	yes	—
128	Dic₁₇₈	#97	1	2	yes	—
129	C₈×C₈₉	—	2	1	no	—
130	C₈×C₈₉	—	2	1	no	—
131	C₈×C₈₉	C₂×C₄²	1	2	yes	—
132	C₈×C₈₉	C₂³×C₄	1	2	yes	—
133	#10623494	—	2	2	no	—
134	#10623494	C₂×C₄²	1	4	yes	—
135	C₂²×C₈₉	—	4	1	no	—
136	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
137	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
138	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
139	C₂×D₈₉	—	4	2	no	—
140	C₂×D₈₉	C₂×#97	1	4	yes	—
141	C₄×C₈₉	—	4	1	no	—
142	C₄×C₈₉	—	4	1	no	—
143	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
144	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
145	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
147	Dic₈₉	—	4	2	no	—
148	Dic₈₉	C₂×#97	1	4	yes	—
149	#33329	—	89	1	no	maximal, Sylow
150	C₂×C₈₉	—	2	1	no	—
151	C₂×C₈₉	—	2	1	no	—
152	C₂×C₈₉	—	2	1	no	—
153	C₂×C₈₉	C₂²×#97	1	1	yes	characteristic
154	D₈₉	—	2	4	no	—
155	#3407	—	89	2	no	—
156	#2875	—	89	1	no	—
157	#2110	—	89	2	no	—
158	#2110	—	89	4	no	—
159	C₂×#347	—	89	1	no	—
160	C₂×#347	—	89	4	no	—
161	C₂×#347	D₈₉	1	1	yes	characteristic, complemented
162	C₈₉	#33329	1	1	yes	characteristic, complemented, Sylow
163	C₂×#131	—	89	1	no	—
164	C₂×#131	—	89	1	no	—
165	C₄×#41	—	89	1	no	—
166	#504	—	89	8	no	—
167	#504	—	89	8	no	—
168	C₂×M₃₂	—	89	2	no	—
169	C₂×M₃₂	—	89	8	no	—
170	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
171	#405	—	89	1	no	—
172	#405	—	89	1	no	—
173	#405	—	89	2	no	—
174	#405	—	89	2	no	—
175	C₄×M₁₆	—	89	1	no	—
176	C₄×M₁₆	—	89	2	no	—
177	C₂×#97	—	89	2	no	—
178	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
179	#347	—	89	8	no	—
180	#347	—	89	2	no	—
181	#347	—	89	2	no	—
182	#347	C₂×D₈₉	1	2	yes	—
183	#347	C₂×D₈₉	1	2	yes	—
184	C₂×#41	—	89	1	no	—
185	#131	—	89	4	no	—
186	#131	—	89	4	no	—
187	#131	—	178	2	no	—
188	#131	—	178	2	no	—
189	C₂×M₁₆	—	89	1	no	—
190	C₂×M₁₆	—	89	1	no	—
191	C₂×M₁₆	—	89	2	no	—
192	C₂×M₁₆	—	89	2	no	—
193	C₂²×C₈	—	89	2	no	—
194	C₂²×C₈	—	89	2	no	—
195	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
197	C₄×Q₈	—	178	1	no	—
198	C₄×D₄	—	178	1	no	—
199	C₄×D₄	—	178	2	no	—
200	#117	—	89	1	no	—
201	#117	—	89	2	no	—
202	C₂×C₄²	—	89	2	no	—
203	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
204	M₃₂	—	89	4	no	—
205	M₃₂	—	89	4	no	—
206	M₃₂	—	89	16	no	—
207	M₃₂	C₄×D₈₉	1	4	yes	complemented
208	M₃₂	C₂²×D₈₉	1	4	yes	—
209	C₂×C₁₆	—	89	4	no	—
210	C₂×C₁₆	—	89	16	no	—
211	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
212	#97	—	89	2	no	—
213	#97	—	89	2	no	—
214	#97	—	89	2	no	—
215	#97	C₄×D₈₉	1	1	yes	characteristic
216	#97	C₄×D₈₉	1	1	yes	characteristic
217	#97	C₂²×D₈₉	1	1	yes	characteristic
218	#97	C₂²×D₈₉	1	1	yes	characteristic
219	C₄×C₈	—	89	2	no	—
220	C₄×C₈	—	89	2	no	—
221	C₄×C₈	—	89	2	no	—
222	#41	—	89	2	no	—
223	#41	—	89	2	no	—
224	#41	—	89	2	no	—
225	#41	—	89	2	no	—
226	C₂×Q₈	—	89	1	no	—
227	C₂×D₄	—	89	2	no	—
228	C₂×D₄	—	89	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	89	2	no	—
231	C₂²×C₄	—	178	2	no	—
232	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
233	M₁₆	—	89	2	no	—
234	M₁₆	—	89	2	no	—
235	M₁₆	—	89	8	no	—
236	M₁₆	—	178	4	no	—
237	M₁₆	—	178	1	no	—
238	M₁₆	—	178	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	89	2	no	—
242	C₂×C₈	—	89	8	no	—
243	C₂×C₈	—	89	2	no	—
244	C₂×C₈	—	178	4	no	—
245	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
251	C₄⋊C₄	—	178	2	no	—
252	C₄⋊C₄	—	178	1	no	—
253	C₂²⋊C₄	—	178	2	no	—
254	C₂²⋊C₄	—	178	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	89	2	no	—
257	C₄²	—	178	2	no	—
258	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
260	C₁₆	—	89	8	no	—
261	C₁₆	C₂×C₄×D₈₉	1	8	yes	—
262	C₂³	—	89	2	no	—
263	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
264	Q₈	—	89	2	no	—
265	Q₈	—	89	2	no	—
266	D₄	—	89	4	no	—
267	D₄	—	89	4	no	—
268	D₄	—	89	2	no	—
269	D₄	—	89	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	89	2	no	—
274	C₂×C₄	—	89	4	no	—
275	C₂×C₄	—	89	4	no	—
276	C₂×C₄	—	178	2	no	—
277	C₂×C₄	—	712	2	no	—
278	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
279	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
280	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
281	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
282	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	89	4	no	—
287	C₈	—	178	2	no	—
288	C₈	C₄²×D₈₉	1	2	yes	—
289	C₈	C₂²×C₄×D₈₉	1	2	yes	—
290	C₂²	—	4	1	no	—
291	C₂²	—	89	4	no	—
292	C₂²	—	356	2	no	—
293	C₂²	#97×D₈₉	1	1	yes	characteristic
294	C₂²	#97×D₈₉	1	1	yes	characteristic
295	C₂²	C₂×M₁₆×D₈₉	1	1	yes	characteristic
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	89	4	no	—
299	C₄	—	356	2	no	—
300	C₄	#97×D₈₉	1	1	yes	characteristic
301	C₄	#97×D₈₉	1	1	yes	characteristic
302	C₄	C₂×C₄²×D₈₉	1	1	yes	characteristic
303	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	178	4	no	—
308	C₂	C₂×#97×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄²×D₈₉
commutator	C₄×C₈₉	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#347×C₈₉	C₂
socle	C₂×C₈₉	C₂²×#97
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#347	D₈₉
hypocenter	C₈₉	#33329
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. #26021856	C₂	semidirect
4. #26021855	C₂	semidirect
5. C₂×#19151376	C₂	semidirect
6. #347×D₈₉	C₂	semidirect
7. C₂×#347×C₈₉	C₂	semidirect
11. #19152016	C₄	semidirect
12. #19152014	C₄	semidirect
46. #15667952	C₂×C₄	semidirect
48. #15667951	C₂×C₄	semidirect
49. C₂×#13083549	C₂×C₄	semidirect
50. C₁₆×D₈₉	C₂×C₄	semidirect
60. M₃₂×C₈₉	C₂×C₄	semidirect
62. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
161. C₂×#347	D₈₉	semidirect
162. C₈₉	#33329	semidirect
207. M₃₂	C₄×D₈₉	semidirect
211. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033354	2
#26028677	2
#26021856	2
#26021855	2
C₂×#19151376	2
#347×D₈₉	2
C₂×#347×C₈₉	2
#33329	89