Subgroups — #36707559

Order 22784.

520 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028677	C₂	1	1	yes	maximal, characteristic
2	#26022066	C₂	1	4	yes	maximal, complemented
3	#26022062	C₂	1	4	yes	maximal, complemented
4	#26022017	C₂	1	2	yes	maximal, complemented
5	#26021872	C₂	1	2	yes	maximal
6	C₂×#19151376	C₂	1	1	yes	maximal, characteristic, 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	C₂×#15667952	C₂²	1	2	yes	—
12	#19151555	C₄	1	2	yes	—
13	#19151555	C₂²	1	2	yes	—
14	#19151553	C₄	1	1	yes	characteristic
15	#19151553	C₂²	1	1	yes	characteristic
16	C₄×#13083558	C₄	1	1	yes	characteristic
17	#19151550	C₂²	1	2	yes	—
18	#97×D₈₉	C₄	1	2	yes	—
19	#19151418	C₂²	1	8	yes	—
20	#19151379	C₂²	1	8	yes	—
21	#19151376	C₂²	1	2	yes	—
22	#19151376	C₂²	1	2	yes	—
23	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
24	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
25	#347×C₈₉	C₂²	1	2	yes	—
26	#347×C₈₉	C₂²	1	2	yes	—
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	#15668018	—	2	2	no	—
29	#15668018	—	2	2	no	—
30	#15668018	—	2	2	no	—
31	#15668018	—	2	2	no	—
32	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
33	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
34	C₂×#13083557	C₂×C₄	1	2	yes	—
35	C₂×#13083557	C₂³	1	2	yes	—
36	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₇₈	—	2	1	no	—
39	C₄×D₃₅₆	—	2	1	no	—
40	C₄×C₂²⋊D₈₉	—	2	2	no	—
41	#15667971	C₂×C₄	1	1	yes	characteristic
42	#15667969	C₂×C₄	1	2	yes	—
43	C₄²×D₈₉	C₂×C₄	1	2	yes	—
44	#15667952	—	2	2	no	—
45	#15667952	C₂³	1	4	yes	—
46	C₂×#13083549	—	2	2	no	—
47	#15667923	C₂×C₄	1	2	yes	—
48	#15667922	C₂×C₄	1	2	yes	—
49	#15667922	C₂×C₄	1	2	yes	—
50	C₄×#10623494	C₂×C₄	1	2	yes	—
51	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
52	C₈×Dic₈₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₈₉	—	2	2	no	—
57	M₃₂×C₈₉	C₂³	1	4	yes	—
58	C₂×C₁₆×C₈₉	—	2	2	no	—
59	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
60	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
61	#97×C₈₉	C₂³	1	1	yes	characteristic
62	#97×C₈₉	C₂³	1	1	yes	characteristic
63	#13083574	—	2	1	no	—
64	#13083574	—	2	1	no	—
65	#13083574	—	2	1	no	—
66	#13083574	—	2	1	no	—
67	C₂×Dic₁₇₈	M₁₆	1	1	yes	characteristic
68	C₂×D₃₅₆	M₁₆	1	1	yes	characteristic
69	C₂×C₂²⋊D₈₉	M₁₆	1	2	yes	—
70	C₂×C₄×D₈₉	—	2	2	no	—
71	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
72	#13083558	—	2	1	no	—
73	#13083558	—	2	1	no	—
74	#13083558	M₁₆	1	2	yes	—
75	#13083558	M₁₆	1	2	yes	—
76	#13083557	—	2	4	no	—
77	#13083557	—	2	4	no	—
78	C₂×#10623494	C₂²×C₄	1	2	yes	—
79	C₂×#10623494	C₂²×C₄	1	2	yes	—
80	C₈×D₈₉	—	2	4	no	—
81	C₈×D₈₉	—	2	4	no	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	#13083553	—	2	2	no	—
84	C₂²⋊Dic₈₉	—	2	1	no	—
85	#13083551	—	2	2	no	—
86	C₄×Dic₈₉	—	2	2	no	—
87	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
88	#13083549	—	2	4	no	—
89	C₂²×C₄×C₈₉	—	2	1	no	—
90	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	—	2	1	no	—
92	C₂×C₈×C₈₉	—	2	1	no	—
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	#41	1	2	yes	—
97	C₂×C₈×C₈₉	#41	1	2	yes	—
98	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₄²×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₁₆×C₈₉	—	2	4	no	—
103	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂²×D₈₉	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	8	1	no	—
108	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
114	C₂²⋊D₈₉	—	2	2	no	—
115	C₂²⋊D₈₉	—	2	2	no	—
116	C₂×Dic₈₉	—	2	2	no	—
117	C₂×Dic₈₉	C₂×M₁₆	1	2	yes	—
118	D₃₅₆	—	2	1	no	—
119	D₃₅₆	—	2	1	no	—
120	C₄×D₈₉	—	2	2	no	—
121	C₄×D₈₉	—	2	2	no	—
122	C₄×D₈₉	—	8	2	no	—
123	Dic₁₇₈	—	2	1	no	—
124	Dic₁₇₈	—	2	1	no	—
125	C₈×C₈₉	—	2	1	no	—
126	C₈×C₈₉	—	2	1	no	—
127	C₈×C₈₉	#117	1	2	yes	—
128	C₈×C₈₉	C₂×#41	1	2	yes	—
129	#10623494	—	2	2	no	—
130	#10623494	C₂×M₁₆	1	4	yes	—
131	C₂²×C₈₉	—	4	1	no	—
132	C₂²×C₈₉	#432	1	1	yes	characteristic
133	C₂²×C₈₉	#432	1	1	yes	characteristic
134	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
135	C₂×D₈₉	—	2	2	no	—
136	C₂×D₈₉	—	4	2	no	—
137	C₄×C₈₉	—	4	1	no	—
138	C₄×C₈₉	—	4	1	no	—
139	C₄×C₈₉	#432	1	1	yes	characteristic
140	C₄×C₈₉	#432	1	1	yes	characteristic
141	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
142	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
143	Dic₈₉	—	2	2	no	—
144	Dic₈₉	—	4	2	no	—
145	#33426	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	—	2	1	no	—
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	C₂×#432	1	1	yes	characteristic
150	D₈₉	—	4	2	no	—
151	#2875	—	89	1	no	—
152	#2166	—	89	4	no	—
153	#2165	—	89	4	no	—
154	#2153	—	89	2	no	—
155	#2115	—	89	2	no	—
156	C₂×#347	—	89	1	no	—
157	C₂×#347	D₈₉	1	1	yes	characteristic, complemented
158	C₈₉	#33426	1	1	yes	characteristic, complemented, Sylow
159	C₂×#131	—	89	1	no	—
160	C₂×#131	—	89	1	no	—
161	C₄×#41	—	89	1	no	—
162	C₂×M₃₂	—	89	2	no	—
163	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
164	#405	—	89	1	no	—
165	#405	—	89	1	no	—
166	#405	—	89	2	no	—
167	#405	—	89	2	no	—
168	C₄×M₁₆	—	89	1	no	—
169	C₄×M₁₆	—	89	2	no	—
170	C₂×#97	—	89	2	no	—
171	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
172	#364	—	89	8	no	—
173	#349	—	89	8	no	—
174	#347	—	89	2	no	—
175	#347	—	89	2	no	—
176	#347	C₂×D₈₉	1	2	yes	—
177	#347	C₂×D₈₉	1	2	yes	—
178	C₂×#41	—	89	1	no	—
179	#131	—	178	2	no	—
180	#131	—	178	2	no	—
181	#131	—	178	2	no	—
182	#131	—	178	2	no	—
183	C₂×M₁₆	—	89	1	no	—
184	C₂×M₁₆	—	89	1	no	—
185	C₂×M₁₆	—	89	2	no	—
186	C₂×M₁₆	—	89	2	no	—
187	C₂²×C₈	—	89	2	no	—
188	C₂²×C₈	—	89	2	no	—
189	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
191	C₄×Q₈	—	178	1	no	—
192	C₄×D₄	—	178	1	no	—
193	C₄×D₄	—	178	2	no	—
194	#117	—	89	1	no	—
195	#117	—	89	2	no	—
196	C₂×C₄²	—	89	2	no	—
197	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
198	M₃₂	—	2	2	no	—
199	M₃₂	—	89	4	no	—
200	M₃₂	—	178	2	no	—
201	M₃₂	C₂²×D₈₉	1	4	yes	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	178	2	no	—
204	#97	—	89	2	no	—
205	#97	—	89	2	no	—
206	#97	—	89	2	no	—
207	#97	C₄×D₈₉	1	1	yes	characteristic
208	#97	C₄×D₈₉	1	1	yes	characteristic
209	#97	C₂²×D₈₉	1	1	yes	characteristic
210	#97	C₂²×D₈₉	1	1	yes	characteristic
211	C₄×C₈	—	89	2	no	—
212	C₄×C₈	—	89	2	no	—
213	C₄×C₈	—	89	2	no	—
214	#41	—	178	1	no	—
215	#41	—	178	1	no	—
216	#41	—	178	1	no	—
217	#41	—	178	1	no	—
218	C₂×Q₈	—	89	1	no	—
219	C₂×D₄	—	89	2	no	—
220	C₂×D₄	—	89	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	89	2	no	—
223	C₂²×C₄	—	178	2	no	—
224	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
225	M₁₆	—	89	2	no	—
226	M₁₆	—	89	2	no	—
227	M₁₆	—	178	4	no	—
228	M₁₆	—	178	1	no	—
229	M₁₆	—	178	1	no	—
230	M₁₆	—	178	4	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	89	2	no	—
234	C₂×C₈	—	89	2	no	—
235	C₂×C₈	—	178	4	no	—
236	C₂×C₈	—	178	4	no	—
237	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
240	C₂×C₈	#13083574	1	2	yes	—
241	C₂×C₈	#13083574	1	2	yes	—
242	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
243	C₄⋊C₄	—	178	2	no	—
244	C₄⋊C₄	—	178	1	no	—
245	C₂²⋊C₄	—	178	2	no	—
246	C₂²⋊C₄	—	178	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	89	2	no	—
249	C₄²	—	178	2	no	—
250	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
251	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
252	C₁₆	—	2	4	no	—
253	C₁₆	—	178	4	no	—
254	C₂³	—	89	2	no	—
255	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
256	Q₈	—	178	1	no	—
257	Q₈	—	178	1	no	—
258	D₄	—	178	2	no	—
259	D₄	—	178	2	no	—
260	D₄	—	178	1	no	—
261	D₄	—	178	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	8	1	no	—
265	C₂×C₄	—	89	2	no	—
266	C₂×C₄	—	178	2	no	—
267	C₂×C₄	—	178	2	no	—
268	C₂×C₄	—	178	2	no	—
269	C₂×C₄	—	712	2	no	—
270	C₂×C₄	#15667969	1	1	yes	characteristic
271	C₂×C₄	#15667969	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
273	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
274	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	89	4	no	—
279	C₈	—	178	2	no	—
280	C₈	#15667969	1	2	yes	—
281	C₈	C₂×#13083574	1	2	yes	—
282	C₂²	—	4	1	no	—
283	C₂²	—	356	2	no	—
284	C₂²	—	178	2	no	—
285	C₂²	#19151653	1	1	yes	characteristic
286	C₂²	#19151653	1	1	yes	characteristic
287	C₂²	C₂×M₁₆×D₈₉	1	1	yes	characteristic
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	356	2	no	—
291	C₄	—	178	2	no	—
292	C₄	#19151653	1	1	yes	characteristic
293	C₄	#19151653	1	1	yes	characteristic
294	C₄	C₂×#15667969	1	1	yes	characteristic
295	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	356	2	no	—
300	C₂	C₂×#19151653	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#15667969
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#347×C₈₉	C₂
socle	C₂×C₈₉	C₂×#432
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#347	D₈₉
hypocenter	C₈₉	#33426
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
2. #26022066	C₂	semidirect
3. #26022062	C₂	semidirect
4. #26022017	C₂	semidirect
6. C₂×#19151376	C₂	semidirect
7. C₂×#347×C₈₉	C₂	semidirect
157. C₂×#347	D₈₉	semidirect
158. C₈₉	#33426	semidirect

Maximal subgroups

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

subgroup	index
#26028677	2
#26022066	2
#26022062	2
#26022017	2
#26021872	2
C₂×#19151376	2
C₂×#347×C₈₉	2
#33426	89