Subgroups — #36469552

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033354	C₂	1	1	yes	maximal, characteristic
2	#26021881	C₂	1	1	yes	maximal, characteristic
3	#26021856	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151376	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020096	C₂	1	1	yes	maximal, characteristic
6	#26018339	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	#19152016	C₄	1	4	yes	complemented
10	#19152014	—	2	2	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	M₃₂×D₈₉	—	2	2	no	—
14	C₂×#15667940	C₄	1	1	yes	characteristic
15	#19151595	C₄	1	1	yes	characteristic
16	#19151553	C₂²	1	1	yes	characteristic
17	#19151383	C₄	1	4	yes	complemented
18	#19151376	—	2	2	no	—
19	#19151375	—	2	2	no	—
20	#19151363	C₄	1	1	yes	characteristic
21	#19151358	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
26	#15668018	—	2	1	no	—
27	#15668018	—	2	1	no	—
28	#15668018	C₂×C₄	1	2	yes	—
29	#15668018	C₂×C₄	1	2	yes	—
30	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
32	C₂×#13083557	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	1	no	—
34	#15667971	C₂×C₄	1	1	yes	characteristic
35	#15667952	—	2	1	no	—
36	#15667952	—	2	1	no	—
37	#15667952	D₄	1	2	yes	—
38	#15667952	D₄	1	2	yes	—
39	#15667951	—	2	4	no	—
40	C₂×#13083549	C₂×C₄	1	2	yes	—
41	C₂×#13083549	C₂×C₄	1	2	yes	—
42	C₁₆×D₈₉	—	2	4	no	—
43	#15667940	—	2	2	no	—
44	#15667937	—	2	1	no	—
45	#15667934	—	2	1	no	—
46	#15667922	—	2	1	no	—
47	C₈×Dic₈₉	—	2	1	no	—
48	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₈₉	—	2	1	no	—
52	M₃₂×C₈₉	—	2	1	no	—
53	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
54	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₁₆×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	—	2	1	no	—
57	#97×C₈₉	—	2	1	no	—
58	#97×C₈₉	—	2	1	no	—
59	#13083574	—	2	1	no	—
60	#13083574	—	2	1	no	—
61	#13083574	C₄²	1	2	yes	—
62	#13083574	M₁₆	1	2	yes	—
63	C₂×Dic₁₇₈	M₁₆	1	1	yes	characteristic
64	C₂×D₃₅₆	M₁₆	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	#13083558	—	2	1	no	—
68	#13083558	—	2	1	no	—
69	#13083558	—	2	1	no	—
70	#13083558	C₄²	1	2	yes	—
71	#13083557	—	2	2	no	—
72	#13083557	—	2	2	no	—
73	C₂×#10623494	—	2	1	no	—
74	C₂×#10623494	—	2	1	no	—
75	C₈×D₈₉	—	2	2	no	—
76	C₈×D₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₂²⋊Dic₈₉	—	2	1	no	—
79	C₄×Dic₈₉	—	2	1	no	—
80	#13083549	—	2	2	no	—
81	#13083549	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₈₉	—	2	1	no	—
83	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	—	2	1	no	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
92	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
93	C₄²×C₈₉	—	2	1	no	—
94	C₄²×C₈₉	—	2	1	no	—
95	C₁₆×C₈₉	—	2	2	no	—
96	C₁₆×C₈₉	—	2	2	no	—
97	C₂³×C₈₉	#131	1	1	yes	characteristic
98	C₂²×D₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	4	1	no	—
100	C₂×C₄×C₈₉	—	8	1	no	—
101	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
107	C₂²⋊D₈₉	—	2	2	no	—
108	C₂²⋊D₈₉	—	2	2	no	—
109	C₂×Dic₈₉	—	2	1	no	—
110	C₂×Dic₈₉	—	4	1	no	—
111	D₃₅₆	—	2	1	no	—
112	D₃₅₆	#97	1	2	yes	—
113	C₄×D₈₉	—	2	2	no	—
114	C₄×D₈₉	—	2	2	no	—
115	Dic₁₇₈	—	2	1	no	—
116	Dic₁₇₈	#97	1	2	yes	—
117	C₈×C₈₉	—	4	1	no	—
118	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₈×C₈₉	#117	1	1	yes	characteristic
120	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
121	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
122	#10623494	—	2	2	no	—
123	#10623494	—	4	1	no	—
124	C₂²×C₈₉	—	4	1	no	—
125	C₂²×C₈₉	#405	1	1	yes	characteristic
126	C₂²×C₈₉	#435	1	1	yes	characteristic
127	C₂²×C₈₉	#435	1	1	yes	characteristic
128	C₂×D₈₉	—	2	2	no	—
129	C₂×D₈₉	—	8	1	no	—
130	C₄×C₈₉	—	8	1	no	—
131	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
133	C₄×C₈₉	#435	1	1	yes	characteristic
134	C₄×C₈₉	#435	1	1	yes	characteristic
135	Dic₈₉	—	2	2	no	—
136	Dic₈₉	—	8	1	no	—
137	#12618	—	89	1	no	maximal, Sylow
138	C₂×C₈₉	—	2	1	no	—
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	#1752	1	1	yes	characteristic
142	D₈₉	—	4	2	no	—
143	#3407	—	89	1	no	—
144	#2117	—	89	1	no	—
145	#2110	—	89	1	no	—
146	C₂×#347	—	89	1	no	—
147	#1765	—	89	1	no	—
148	#1378	—	89	1	no	—
149	#1377	D₈₉	1	1	yes	characteristic, complemented
150	C₈₉	#12618	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	89	1	no	—
152	#504	—	89	4	no	—
153	#504	—	178	2	no	—
154	C₂×M₃₂	—	89	1	no	—
155	C₂×M₃₂	—	89	1	no	—
156	C₂×M₃₂	—	178	2	no	—
157	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
158	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
159	C₂×#104	—	89	1	no	—
160	#416	—	89	1	no	—
161	#405	—	89	1	no	—
162	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
163	#350	—	89	4	no	—
164	#347	—	178	2	no	—
165	#347	—	178	2	no	—
166	#341	—	89	1	no	—
167	#339	—	89	1	no	—
168	C₂×#41	—	89	1	no	—
169	#131	—	89	2	no	—
170	#131	—	89	2	no	—
171	#131	—	178	1	no	—
172	#131	—	178	1	no	—
173	C₂×M₁₆	—	89	1	no	—
174	C₂×M₁₆	—	89	1	no	—
175	C₂×M₁₆	—	178	1	no	—
176	C₂²×C₈	—	178	1	no	—
177	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
179	#117	—	89	1	no	—
180	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	89	2	no	—
184	M₃₂	—	89	2	no	—
185	M₃₂	—	178	4	no	—
186	M₃₂	—	178	1	no	—
187	M₃₂	—	178	1	no	—
188	M₃₂	C₄×D₈₉	1	2	yes	—
189	M₃₂	C₄×D₈₉	1	2	yes	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	89	2	no	—
193	C₂×C₁₆	—	89	2	no	—
194	C₂×C₁₆	—	178	4	no	—
195	#104	—	178	2	no	—
196	#103	—	178	1	no	—
197	#102	—	178	1	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	178	1	no	—
201	C₄×C₈	—	178	1	no	—
202	#41	—	89	2	no	—
203	#41	—	89	2	no	—
204	#41	—	178	1	no	—
205	#41	—	178	1	no	—
206	C₂×Q₈	—	89	1	no	—
207	C₂×D₄	—	89	1	no	—
208	C₂×D₄	—	178	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	178	1	no	—
211	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
212	M₁₆	—	89	2	no	—
213	M₁₆	—	178	1	no	—
214	M₁₆	—	178	1	no	—
215	M₁₆	—	178	1	no	—
216	M₁₆	—	178	2	no	—
217	M₁₆	—	178	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	178	1	no	—
222	C₂×C₈	—	178	1	no	—
223	C₂×C₈	—	178	2	no	—
224	C₂×C₈	—	178	2	no	—
225	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
227	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
228	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
229	C₂×C₈	#13083573	1	1	yes	characteristic
230	C₂×C₈	#13083575	1	1	yes	characteristic
231	C₄⋊C₄	—	178	1	no	—
232	C₂²⋊C₄	—	178	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	178	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	2	2	no	—
238	C₁₆	—	89	4	no	—
239	C₁₆	—	178	2	no	—
240	C₂³	—	178	1	no	—
241	C₂³	#15668018	1	1	yes	characteristic
242	Q₈	—	89	2	no	—
243	Q₈	—	178	1	no	—
244	D₄	—	89	2	no	—
245	D₄	—	178	2	no	—
246	D₄	—	178	2	no	—
247	D₄	—	178	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	356	1	no	—
251	C₂×C₄	—	178	1	no	—
252	C₂×C₄	—	178	2	no	—
253	C₂×C₄	—	178	2	no	—
254	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
255	C₂×C₄	#15667972	1	1	yes	characteristic
256	C₂×C₄	#15667974	1	1	yes	characteristic
257	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
258	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
259	C₂×C₄	#15668018	1	1	yes	characteristic
260	C₈	—	4	1	no	—
261	C₈	—	356	1	no	—
262	C₈	—	178	2	no	—
263	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
264	C₈	#15667970	1	1	yes	characteristic
265	C₈	#15667972	1	1	yes	characteristic
266	C₈	#15667974	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	178	2	no	—
269	C₂²	—	712	1	no	—
270	C₂²	#19151552	1	1	yes	characteristic
271	C₂²	#19151668	1	1	yes	characteristic
272	C₂²	#19151672	1	1	yes	characteristic
273	C₄	—	8	1	no	—
274	C₄	—	178	2	no	—
275	C₄	—	712	1	no	—
276	C₄	#19151445	1	1	yes	characteristic
277	C₄	#97×D₈₉	1	1	yes	characteristic
278	C₄	#19151668	1	1	yes	characteristic
279	C₄	#19151672	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	356	2	no	—
284	C₂	#26020027	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151445
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1377	C₂
socle	C₂×C₈₉	#1752
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1377	D₈₉
hypocenter	C₈₉	#12618
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1377

Complements

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

N	H ≅ G/N	type
3. #26021856	C₂	semidirect
4. C₂×#19151376	C₂	semidirect
6. #26018339	C₂	semidirect
7. C₈₉×#1377	C₂	semidirect
9. #19152016	C₄	semidirect
17. #19151383	C₄	semidirect
149. #1377	D₈₉	semidirect
150. C₈₉	#12618	semidirect

Maximal subgroups

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

subgroup	index
#26033354	2
#26021881	2
#26021856	2
C₂×#19151376	2
#26020096	2
#26018339	2
C₈₉×#1377	2
#12618	89