Subgroups — #36463144

Order 22784.

666 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151647	C₂	1	2	yes	maximal
2	#26028816	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151362	C₂	1	2	yes	maximal
4	#26019949	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂²×#13083558	C₂²	1	1	yes	characteristic
8	C₂³×#10623494	C₂²	1	1	yes	characteristic
9	#19151647	—	2	4	no	—
10	#19151647	—	2	4	no	—
11	C₂×#15667948	—	2	4	no	—
12	#19151625	—	2	4	no	—
13	#19151622	—	2	2	no	—
14	#19151362	—	2	4	no	—
15	#19151353	—	2	2	no	—
16	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
18	#343×C₈₉	C₄	1	4	yes	—
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	1	no	—
26	C₂×#13083558	—	2	2	no	—
27	C₂×#13083558	D₄	1	4	yes	—
28	C₂×#13083558	D₄	1	2	yes	—
29	C₂²×#10623494	—	2	2	no	—
30	C₂²×#10623494	—	2	2	no	—
31	C₂²×#10623494	—	2	1	no	—
32	C₂²×#10623494	—	2	1	no	—
33	C₂²×#10623494	—	2	1	no	—
34	C₂²×#10623494	—	2	2	no	—
35	C₂²×#10623494	Q₈	1	2	yes	—
36	#15667948	—	2	16	no	—
37	C₄⋊#10623494	—	2	8	no	—
38	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₈₉	—	2	2	no	—
40	C₂×M₁₆×C₈₉	—	2	2	no	—
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₈₉	D₄	1	4	yes	—
44	C₂×M₁₆×C₈₉	Q₈	1	4	yes	—
45	C₂²×C₈×C₈₉	—	2	2	no	—
46	#117×C₈₉	—	2	2	no	—
47	#117×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₈₉	Q₈	1	2	yes	—
50	C₂×C₄²×C₈₉	D₄	1	2	yes	—
51	#13083558	—	2	4	no	—
52	#13083558	—	2	4	no	—
53	#13083558	—	2	4	no	—
54	#13083558	—	4	2	no	—
55	#13083558	—	4	2	no	—
56	#13083558	—	4	2	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	—	2	2	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₂×#10623494	—	4	1	no	—
63	C₂×#10623494	—	4	2	no	—
64	C₂×#10623494	—	4	2	no	—
65	C₂×#10623494	—	4	1	no	—
66	C₂×#10623494	—	4	1	no	—
67	C₂×#10623494	—	4	1	no	—
68	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	2	yes	—
79	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
80	M₁₆×C₈₉	—	2	4	no	—
81	M₁₆×C₈₉	—	2	4	no	—
82	M₁₆×C₈₉	—	2	4	no	—
83	M₁₆×C₈₉	C₄⋊C₄	1	8	yes	—
84	C₂×C₈×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	—	2	4	no	—
86	C₂×C₈×C₈₉	—	2	4	no	—
87	C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₂²⋊C₄×C₈₉	—	2	4	no	—
89	C₄²×C₈₉	—	2	2	no	—
90	C₄²×C₈₉	—	2	2	no	—
91	C₂³×C₈₉	—	4	2	no	—
92	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
94	C₂³×C₈₉	#125	1	1	yes	characteristic
95	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
96	C₂³×C₈₉	C₄⋊Q₈	1	2	yes	—
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	4	4	no	—
99	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#120	1	2	yes	—
104	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#121	1	2	yes	—
106	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#123	1	2	yes	—
112	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#125	1	2	yes	—
114	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
117	C₈×C₈₉	—	2	8	no	—
118	#10623494	—	4	4	no	—
119	#10623494	—	4	2	no	—
120	#10623494	—	4	2	no	—
121	C₂²×C₈₉	—	2	2	no	—
122	C₂²×C₈₉	—	2	4	no	—
123	C₂²×C₈₉	—	4	1	no	—
124	C₂²×C₈₉	—	4	2	no	—
125	C₂²×C₈₉	—	4	1	no	—
126	C₂²×C₈₉	#384	1	1	yes	characteristic
127	C₂²×C₈₉	#385	1	1	yes	characteristic
128	C₂²×C₈₉	#389	1	1	yes	characteristic
129	C₂²×C₈₉	#395	1	2	yes	—
130	C₂²×C₈₉	#398	1	2	yes	—
131	C₄×C₈₉	—	4	4	no	—
132	C₄×C₈₉	#385	1	2	yes	—
133	C₄×C₈₉	#389	1	2	yes	—
134	C₄×C₈₉	#392	1	1	yes	characteristic
135	C₄×C₈₉	#395	1	1	yes	characteristic
136	C₄×C₈₉	#398	1	1	yes	characteristic
137	C₄×C₈₉	#399	1	1	yes	characteristic
138	#11343	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	—	2	2	no	—
142	C₂×C₈₉	—	2	2	no	—
143	C₂×C₈₉	#1444	1	1	yes	characteristic
144	C₂×C₈₉	#1858	1	2	yes	—
145	C₂×#430	—	89	2	no	—
146	#2905	—	89	1	no	—
147	#1742	D₈₉	1	1	yes	characteristic
148	C₂×#341	—	89	2	no	—
149	#1729	—	89	1	no	—
150	C₈₉	#11343	1	1	yes	characteristic, complemented, Sylow
151	C₂²×M₁₆	—	89	2	no	—
152	C₂²×M₁₆	—	89	1	no	—
153	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
154	C₂³×C₈	—	89	1	no	—
155	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
156	#430	—	178	4	no	—
157	#430	—	178	4	no	—
158	C₂×#108	—	178	4	no	—
159	#424	—	178	4	no	—
160	#423	—	178	2	no	—
161	#343	Dic₈₉	1	4	yes	—
162	#341	—	178	4	no	—
163	#336	—	178	2	no	—
164	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	89	4	no	—
169	C₂×M₁₆	—	89	2	no	—
170	C₂×M₁₆	—	178	2	no	—
171	C₂×M₁₆	—	178	2	no	—
172	C₂×M₁₆	—	178	2	no	—
173	C₂×M₁₆	—	178	2	no	—
174	C₂×M₁₆	—	178	2	no	—
175	C₂×M₁₆	—	178	2	no	—
176	C₂×M₁₆	—	178	2	no	—
177	C₂×M₁₆	—	178	1	no	—
178	C₂×M₁₆	Dic₁₇₈	1	4	yes	—
179	C₂×M₁₆	D₃₅₆	1	4	yes	—
180	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	89	2	no	—
183	C₂²×C₈	—	178	2	no	—
184	C₂²×C₈	—	178	1	no	—
185	C₂²×C₈	—	178	2	no	—
186	C₂²×C₈	—	178	1	no	—
187	C₂²×C₈	—	178	1	no	—
188	C₂²×C₈	—	178	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
192	C₂×C₂²⋊C₄	Dic₁₇₈	1	2	yes	—
193	C₂×C₄²	D₃₅₆	1	2	yes	—
194	#108	—	178	16	no	—
195	C₄⋊C₈	—	178	8	no	—
196	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
199	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
200	C₂²×C₄	D₄×D₈₉	1	2	yes	—
201	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
202	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
203	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
205	C₂²×C₄	#13083573	1	1	yes	characteristic
206	C₂²×C₄	#13083573	1	2	yes	—
207	C₂²×C₄	#13083575	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	356	2	no	—
212	M₁₆	—	356	2	no	—
213	M₁₆	—	356	2	no	—
214	M₁₆	—	178	4	no	—
215	M₁₆	—	178	4	no	—
216	M₁₆	—	178	4	no	—
217	M₁₆	C₄⋊Dic₈₉	1	8	yes	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	356	2	no	—
222	C₂×C₈	—	356	1	no	—
223	C₂×C₈	—	356	2	no	—
224	C₂×C₈	—	356	2	no	—
225	C₂×C₈	—	356	1	no	—
226	C₂×C₈	—	356	1	no	—
227	C₂×C₈	—	356	1	no	—
228	C₂×C₈	—	178	4	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	—	178	4	no	—
231	C₂×C₈	—	178	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
238	C₂³	#15667989	1	2	yes	—
239	C₂³	#15668003	1	1	yes	characteristic
240	C₂³	#15668011	1	1	yes	characteristic
241	C₂³	#15668012	1	2	yes	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
245	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
246	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
247	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
248	C₂×C₄	#15667982	1	2	yes	—
249	C₂×C₄	#15667982	1	1	yes	characteristic
250	C₂×C₄	#15667987	1	2	yes	—
251	C₂×C₄	#15667987	1	1	yes	characteristic
252	C₂×C₄	#15667989	1	2	yes	—
253	C₂×C₄	#15667989	1	2	yes	—
254	C₂×C₄	#15667992	1	2	yes	—
255	C₂×C₄	#15667992	1	1	yes	characteristic
256	C₂×C₄	#15667996	1	2	yes	—
257	C₂×C₄	#15667996	1	1	yes	characteristic
258	C₂×C₄	#15668003	1	2	yes	—
259	C₂×C₄	#15668009	1	1	yes	characteristic
260	C₂×C₄	#15668011	1	2	yes	—
261	C₂×C₄	#15668012	1	1	yes	characteristic
262	C₈	—	2	8	no	—
263	C₈	—	356	4	no	—
264	C₈	—	356	2	no	—
265	C₈	—	356	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	#19151468	1	1	yes	characteristic
272	C₂²	#19151473	1	1	yes	characteristic
273	C₂²	#19151497	1	1	yes	characteristic
274	C₂²	#19151519	1	2	yes	—
275	C₂²	#19151530	1	2	yes	—
276	C₄	—	4	4	no	—
277	C₄	#19151473	1	2	yes	—
278	C₄	#19151497	1	2	yes	—
279	C₄	#19151508	1	1	yes	characteristic
280	C₄	#19151519	1	1	yes	characteristic
281	C₄	#19151530	1	1	yes	characteristic
282	C₄	#19151535	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	#26018517	1	1	yes	characteristic
288	C₂	#26020474	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667982
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#1742	C₂
socle	C₂²×C₈₉	#385
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#1742	D₈₉
hypocenter	C₈₉	#11343
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
150. C₈₉	#11343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151647	2
#26028816	2
C₂×#19151362	2
#26019949	2
C₈₉×#1742	2
#11343	89