Subgroups — #36444957

Order 22784.

506 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151552	C₂	1	1	yes	maximal, characteristic
2	#26018564	C₂	1	2	yes	maximal
3	#26018535	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018096	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13083557	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₈₉	C₄	1	1	yes	characteristic
8	C₂×#15667969	C₂²	1	1	yes	characteristic
9	#19151651	C₂²	1	2	yes	—
10	#19151552	—	2	1	no	—
11	#19151552	—	2	1	no	—
12	#19151552	—	2	1	no	—
13	#19151552	—	2	1	no	—
14	C₂×#15667923	C₂²	1	1	yes	characteristic
15	C₂×#15667922	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
17	#19151323	C₄	1	4	yes	—
18	#19151321	C₄	1	4	yes	—
19	#19151319	—	2	1	no	—
20	C₈×#10623494	—	2	1	no	—
21	#431×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#322×C₈₉	—	2	1	no	—
24	C₈²×C₈₉	—	2	1	no	—
25	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×#13083557	—	2	1	no	—
27	C₂×#13083557	—	2	1	no	—
28	C₂×#13083557	—	2	2	no	—
29	C₂×#13083557	C₈	1	4	yes	complemented
30	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
31	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₈₉	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	1	no	—
34	C₂×C₈×D₈₉	—	2	2	no	—
35	C₂×C₈×D₈₉	C₈	1	4	yes	complemented
36	#15667969	—	2	1	no	—
37	#15667969	—	2	1	no	—
38	#15667969	—	2	2	no	—
39	C₂×#13083553	C₂×C₄	1	2	yes	—
40	C₂×#13083551	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄⋊#10623494	—	2	2	no	—
43	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
44	#15667923	—	2	1	no	—
45	#15667923	—	2	1	no	—
46	#15667922	—	2	1	no	—
47	#15667922	—	2	1	no	—
48	C₄×#10623494	—	2	2	no	—
49	C₈×Dic₈₉	—	2	1	no	—
50	C₈×Dic₈₉	—	2	1	no	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	—	2	2	no	—
55	C₂²⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₈₉	—	2	2	no	—
57	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₂³×D₈₉	C₄²	1	1	yes	characteristic
62	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₄²	1	2	yes	—
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
70	#13083557	—	2	4	no	—
71	#13083557	—	2	4	no	—
72	C₂×#10623494	—	2	1	no	—
73	C₂×#10623494	—	2	1	no	—
74	C₂×#10623494	—	2	1	no	—
75	C₂×#10623494	—	2	1	no	—
76	C₂×#10623494	—	2	4	no	—
77	C₂×#10623494	C₂×C₈	1	2	yes	—
78	C₂×#10623494	C₂×C₈	1	2	yes	—
79	C₈×D₈₉	—	2	4	no	—
80	C₈×D₈₉	—	2	4	no	—
81	#13083553	—	2	2	no	—
82	#13083553	—	2	2	no	—
83	#13083551	—	2	2	no	—
84	#13083551	—	2	2	no	—
85	C₄×Dic₈₉	—	2	1	no	—
86	C₄×Dic₈₉	—	2	1	no	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	—	2	4	no	—
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
95	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
96	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
97	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
98	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₈₉	#41	1	2	yes	—
101	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
102	C₂²×D₈₉	—	2	1	no	—
103	C₂²×D₈₉	—	2	1	no	—
104	C₂²×D₈₉	—	4	2	no	—
105	C₂²×D₈₉	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×Dic₈₉	—	2	1	no	—
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	2	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	C₂×Dic₈₉	C₄×C₈	1	2	yes	—
120	C₄×D₈₉	—	2	2	no	—
121	C₄×D₈₉	—	2	2	no	—
122	C₄×D₈₉	—	2	2	no	—
123	C₄×D₈₉	—	2	2	no	—
124	C₈×C₈₉	—	4	4	no	—
125	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
126	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
127	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
128	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
129	#10623494	—	2	2	no	—
130	#10623494	—	2	2	no	—
131	#10623494	—	4	4	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
135	C₂²×C₈₉	C₄×M₁₆	1	1	yes	characteristic
136	C₂×D₈₉	—	2	2	no	—
137	C₂×D₈₉	—	4	2	no	—
138	C₂×D₈₉	—	4	2	no	—
139	C₂×D₈₉	—	4	2	no	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₄×C₈₉	#431	1	1	yes	characteristic
145	C₄×C₈₉	#431	1	1	yes	characteristic
146	Dic₈₉	—	2	2	no	—
147	Dic₈₉	—	4	2	no	—
148	#8156	—	89	1	no	maximal, Sylow
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	—	2	1	no	—
151	C₂×C₈₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
152	C₂×C₈₉	#1757	1	2	yes	—
153	D₈₉	—	4	2	no	—
154	C₂×#405	—	89	1	no	—
155	#1458	—	89	2	no	—
156	#1450	—	89	1	no	—
157	#1450	D₈₉	1	1	yes	characteristic, complemented
158	#1281	—	89	2	no	—
159	C₈₉	#8156	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	89	1	no	—
161	C₂³×C₈	—	89	1	no	—
162	C₂×#117	—	89	1	no	—
163	#431	—	89	2	no	—
164	#431	C₂×D₈₉	1	2	yes	—
165	#405	—	178	1	no	—
166	#405	—	178	1	no	—
167	#405	—	178	1	no	—
168	#405	—	178	1	no	—
169	C₂×#97	—	89	1	no	—
170	C₂×#97	—	89	1	no	—
171	C₂×C₄×C₈	—	89	1	no	—
172	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
173	#324	—	89	4	no	—
174	#323	—	89	4	no	—
175	#322	—	2	1	no	—
176	#322	—	178	1	no	—
177	C₈²	—	2	1	no	—
178	C₈²	—	178	1	no	—
179	C₂³×C₄	—	89	1	no	—
180	C₂×M₁₆	—	89	4	no	—
181	C₂×M₁₆	—	178	1	no	—
182	C₂×M₁₆	—	178	1	no	—
183	C₂×M₁₆	—	178	2	no	—
184	C₂²×C₈	—	89	1	no	—
185	C₂²×C₈	—	89	1	no	—
186	C₂²×C₈	—	89	4	no	—
187	C₂²×C₈	—	178	1	no	—
188	C₂²×C₈	—	178	1	no	—
189	C₂²×C₈	—	178	2	no	—
190	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
192	#117	—	178	1	no	—
193	#117	—	178	1	no	—
194	#117	—	178	2	no	—
195	C₂×C₄⋊C₄	—	89	2	no	—
196	C₂×C₂²⋊C₄	—	89	2	no	—
197	C₂×C₄²	—	89	1	no	—
198	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	178	2	no	—
201	C₂²⋊C₈	—	89	4	no	—
202	C₂²⋊C₈	C₄×D₈₉	1	4	yes	—
203	#97	—	178	1	no	—
204	#97	—	178	1	no	—
205	#97	—	178	1	no	—
206	#97	—	178	1	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	—	178	1	no	—
209	C₄×C₈	—	178	1	no	—
210	C₄×C₈	—	178	2	no	—
211	C₄×C₈	D₃₅₆	1	1	yes	characteristic
212	C₄×C₈	D₃₅₆	1	1	yes	characteristic
213	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
214	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
215	C₂⁴	—	89	1	no	—
216	C₂²×C₄	—	89	1	no	—
217	C₂²×C₄	—	89	2	no	—
218	C₂²×C₄	—	89	2	no	—
219	C₂²×C₄	—	89	2	no	—
220	C₂²×C₄	—	178	1	no	—
221	C₂²×C₄	—	178	1	no	—
222	C₂²×C₄	—	178	1	no	—
223	C₂²×C₄	—	178	1	no	—
224	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
226	M₁₆	—	178	4	no	—
227	M₁₆	—	178	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	89	2	no	—
230	C₂×C₈	—	89	2	no	—
231	C₂×C₈	—	178	4	no	—
232	C₂×C₈	—	178	1	no	—
233	C₂×C₈	—	178	1	no	—
234	C₂×C₈	—	178	1	no	—
235	C₂×C₈	—	178	1	no	—
236	C₂×C₈	—	178	4	no	—
237	C₂×C₈	—	178	4	no	—
238	C₂×C₈	#13083551	1	1	yes	characteristic
239	C₂×C₈	#13083551	1	1	yes	characteristic
240	C₂×C₈	#13083551	1	1	yes	characteristic
241	C₂×C₈	#13083551	1	1	yes	characteristic
242	C₂×C₈	C₈×D₈₉	1	2	yes	—
243	C₂×C₈	C₈×D₈₉	1	2	yes	—
244	C₂×C₈	#13083557	1	2	yes	—
245	C₂×C₈	#13083557	1	2	yes	—
246	C₄⋊C₄	—	178	2	no	—
247	C₄⋊C₄	—	178	2	no	—
248	C₂²⋊C₄	—	178	2	no	—
249	C₂²⋊C₄	—	178	2	no	—
250	C₄²	—	178	1	no	—
251	C₄²	—	178	1	no	—
252	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
253	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
254	C₄²	#13083574	1	2	yes	—
255	C₂³	—	89	2	no	—
256	C₂³	—	356	2	no	—
257	C₂³	—	178	1	no	—
258	C₂³	—	178	1	no	—
259	C₂³	C₄²×D₈₉	1	1	yes	characteristic
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	89	2	no	—
262	C₂×C₄	—	356	2	no	—
263	C₂×C₄	—	178	2	no	—
264	C₂×C₄	—	178	1	no	—
265	C₂×C₄	—	178	1	no	—
266	C₂×C₄	—	178	2	no	—
267	C₂×C₄	—	178	2	no	—
268	C₂×C₄	—	178	2	no	—
269	C₂×C₄	—	178	2	no	—
270	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
271	C₂×C₄	#15667969	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
273	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
274	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
276	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
277	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
278	C₈	—	4	4	no	—
279	C₈	—	356	4	no	—
280	C₈	—	178	2	no	—
281	C₈	—	178	2	no	—
282	C₈	#15667924	1	2	yes	—
283	C₈	#15667924	1	2	yes	—
284	C₈	#15667924	1	2	yes	—
285	C₈	#15667924	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	356	2	no	—
288	C₂²	—	356	2	no	—
289	C₂²	—	356	2	no	—
290	C₂²	—	178	2	no	—
291	C₂²	C₄×#13083551	1	1	yes	characteristic
292	C₂²	C₄×C₈×D₈₉	1	1	yes	characteristic
293	C₂²	C₄×#13083557	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	356	2	no	—
297	C₄	—	178	2	no	—
298	C₄	C₂×#15667924	1	1	yes	characteristic
299	C₄	C₂×#15667924	1	1	yes	characteristic
300	C₄	#19151649	1	1	yes	characteristic
301	C₄	#19151649	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	356	2	no	—
305	C₂	C₄×#15667924	1	1	yes	characteristic
306	C₂	#26020053	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#13083551
commutator	C₂²×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1450	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂³×D₈₉	C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1450	D₈₉
hypocenter	C₈₉	#8156
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #26018535	C₂	semidirect
4. #26018096	C₂	semidirect
5. C₈₉×#1450	C₂	semidirect
29. C₂×#13083557	C₈	semidirect
35. C₂×C₈×D₈₉	C₈	semidirect
157. #1450	D₈₉	semidirect
159. C₈₉	#8156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151552	2
#26018564	2
#26018535	2
#26018096	2
C₈₉×#1450	2
#8156	89