Subgroups — #36460336

Order 22784.

806 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667947	C₂	1	2	yes	maximal
2	C₂²×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151362	C₂	1	2	yes	maximal
4	#26019946	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂³×#10623494	C₂²	1	1	yes	characteristic
8	C₂³×#10623494	C₂²	1	1	yes	characteristic
9	C₂×#15667947	—	2	4	no	—
10	C₂×#15667947	—	2	4	no	—
11	C₂×#15667947	C₄	1	8	yes	complemented
12	C₂×C₄×#10623494	—	2	4	no	—
13	C₂×C₄×#10623494	C₄	1	4	yes	—
14	#19151362	—	2	4	no	—
15	#19151353	—	2	2	no	—
16	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#336×C₈₉	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
25	C₂²×#10623494	—	2	2	no	—
26	C₂²×#10623494	—	2	2	no	—
27	C₂²×#10623494	—	2	2	no	—
28	C₂²×#10623494	C₂×C₄	1	2	yes	—
29	C₂²×#10623494	C₂×C₄	1	4	yes	—
30	C₂²×#10623494	C₂×C₄	1	2	yes	—
31	C₂²×#10623494	D₄	1	2	yes	—
32	C₂²×#10623494	D₄	1	2	yes	—
33	C₂²×#10623494	D₄	1	4	yes	—
34	C₂²×#10623494	Q₈	1	2	yes	—
35	C₂²×#10623494	Q₈	1	2	yes	—
36	#15667947	—	2	16	no	—
37	C₄×#10623494	—	2	8	no	—
38	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
39	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₈₉	—	2	2	no	—
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	—	2	2	no	—
44	C₂×M₁₆×C₈₉	—	2	2	no	—
45	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₈₉	—	2	2	no	—
47	C₂×C₄²×C₈₉	—	2	2	no	—
48	C₂×C₄²×C₈₉	—	2	2	no	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₈₉	D₄	1	2	yes	—
51	#13083558	—	2	4	no	—
52	#13083558	—	2	4	no	—
53	#13083558	—	4	2	no	—
54	#13083558	—	4	2	no	—
55	C₂×#10623494	—	2	2	no	—
56	C₂×#10623494	—	2	2	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	—	2	4	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₂×#10623494	C₄²	1	4	yes	—
63	C₂×#10623494	C₂²⋊C₄	1	4	yes	—
64	C₂×#10623494	C₂²⋊C₄	1	8	yes	—
65	C₂×#10623494	C₄⋊C₄	1	4	yes	—
66	C₂×#10623494	C₄⋊C₄	1	4	yes	—
67	C₂×#10623494	C₄⋊C₄	1	8	yes	—
68	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	—	2	8	no	—
71	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	2	yes	—
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	M₁₆×C₈₉	—	2	4	no	—
83	M₁₆×C₈₉	—	2	4	no	—
84	M₁₆×C₈₉	—	4	2	no	—
85	M₁₆×C₈₉	—	4	2	no	—
86	C₂×C₈×C₈₉	—	2	4	no	—
87	C₂×C₈×C₈₉	—	4	2	no	—
88	C₂×C₈×C₈₉	—	4	2	no	—
89	C₄²×C₈₉	—	2	2	no	—
90	C₄²×C₈₉	—	2	2	no	—
91	C₄²×C₈₉	—	2	4	no	—
92	C₂³×C₈₉	—	2	4	no	—
93	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
95	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
96	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
97	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₈₉	—	2	16	no	—
99	C₂×C₄×C₈₉	—	2	4	no	—
100	C₂×C₄×C₈₉	—	2	4	no	—
101	C₂×C₄×C₈₉	#108	1	8	yes	complemented
102	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₈₉	#121	1	2	yes	—
114	C₂×C₄×C₈₉	#121	1	2	yes	—
115	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₈₉	#124	1	2	yes	—
117	C₂×C₄×C₈₉	#125	1	2	yes	—
118	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
120	C₈×C₈₉	—	4	4	no	—
121	#10623494	—	2	4	no	—
122	#10623494	—	4	4	no	—
123	#10623494	#95	1	8	yes	—
124	C₂²×C₈₉	—	2	4	no	—
125	C₂²×C₈₉	—	2	2	no	—
126	C₂²×C₈₉	—	2	2	no	—
127	C₂²×C₈₉	—	2	4	no	—
128	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₈₉	#380	1	1	yes	characteristic
130	C₂²×C₈₉	#384	1	1	yes	characteristic
131	C₂²×C₈₉	#389	1	2	yes	—
132	C₂²×C₈₉	#391	1	2	yes	—
133	C₂²×C₈₉	C₂×#108	1	4	yes	—
134	C₄×C₈₉	—	2	8	no	—
135	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
136	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈₉	#380	1	1	yes	characteristic
138	C₄×C₈₉	#382	1	1	yes	characteristic
139	C₄×C₈₉	#389	1	2	yes	—
140	C₄×C₈₉	#390	1	2	yes	—
141	#10864	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	—	2	2	no	—
143	C₂×C₈₉	—	2	2	no	—
144	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
145	C₂×C₈₉	C₄×#108	1	2	yes	—
146	C₂×C₈₉	#1810	1	2	yes	—
147	C₂×C₈₉	#1848	1	2	yes	—
148	C₂²×#108	—	89	2	no	—
149	C₂²×C₄×C₈	—	89	1	no	—
150	C₂×#341	—	89	2	no	—
151	#1728	—	89	1	no	—
152	#1728	D₈₉	1	1	yes	characteristic
153	C₈₉	#10864	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	89	2	no	—
155	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
156	C₂³×C₈	—	89	1	no	—
157	C₂³×C₈	—	89	1	no	—
158	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
159	C₂×#108	—	89	8	no	—
160	C₂×#108	—	178	4	no	—
161	C₂×#108	—	178	4	no	—
162	C₂×C₄×C₈	—	89	4	no	—
163	C₂×C₄×C₈	—	178	4	no	—
164	#341	—	178	4	no	—
165	#336	—	2	2	no	—
166	#336	—	178	2	no	—
167	C₂³×C₄	Dic₁₇₈	1	2	yes	—
168	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	89	4	no	—
175	C₂×M₁₆	—	178	2	no	—
176	C₂×M₁₆	—	178	2	no	—
177	C₂×M₁₆	—	178	2	no	—
178	C₂×M₁₆	—	178	2	no	—
179	C₂×M₁₆	—	178	2	no	—
180	C₂×M₁₆	C₄×D₈₉	1	4	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	89	2	no	—
183	C₂²×C₈	—	89	4	no	—
184	C₂²×C₈	—	89	4	no	—
185	C₂²×C₈	—	89	2	no	—
186	C₂²×C₈	—	89	2	no	—
187	C₂²×C₈	—	89	2	no	—
188	C₂²×C₈	—	89	2	no	—
189	C₂²×C₈	—	89	2	no	—
190	C₂²×C₈	—	178	2	no	—
191	C₂²×C₈	—	178	2	no	—
192	C₂²×C₈	—	178	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₄×D₈₉	1	2	yes	—
196	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
197	#108	—	178	16	no	—
198	C₄×C₈	—	178	8	no	—
199	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	#13083553	1	4	yes	—
203	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
205	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₈₉	1	2	yes	—
207	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
210	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
211	C₂²×C₄	#13083573	1	2	yes	—
212	C₂²×C₄	#13083574	1	2	yes	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	356	2	no	—
218	M₁₆	—	356	2	no	—
219	M₁₆	—	178	4	no	—
220	M₁₆	—	178	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	89	4	no	—
225	C₂×C₈	—	89	4	no	—
226	C₂×C₈	—	89	8	no	—
227	C₂×C₈	—	89	4	no	—
228	C₂×C₈	—	89	8	no	—
229	C₂×C₈	—	89	4	no	—
230	C₂×C₈	—	356	2	no	—
231	C₂×C₈	—	356	2	no	—
232	C₂×C₈	—	178	4	no	—
233	C₂×C₈	—	178	2	no	—
234	C₂×C₈	—	178	4	no	—
235	C₂×C₈	—	178	4	no	—
236	C₂×C₈	—	178	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	2	no	—
240	C₂³	—	2	4	no	—
241	C₂³	C₂×#13083553	1	1	yes	characteristic
242	C₂³	C₄×Dic₁₇₈	1	2	yes	—
243	C₂³	#15667989	1	2	yes	—
244	C₂³	#15668011	1	1	yes	characteristic
245	C₂³	#15668011	1	1	yes	characteristic
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	#15667947	1	8	yes	complemented
250	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊C₄×D₈₉	1	2	yes	—
253	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
254	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
255	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
256	C₂×C₄	#15667968	1	1	yes	characteristic
257	C₂×C₄	#15667969	1	1	yes	characteristic
258	C₂×C₄	#15667972	1	2	yes	—
259	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
260	C₂×C₄	#15667977	1	2	yes	—
261	C₂×C₄	#15667984	1	2	yes	—
262	C₂×C₄	#15667985	1	2	yes	—
263	C₂×C₄	#15667990	1	2	yes	—
264	C₂×C₄	#15667998	1	2	yes	—
265	C₂×C₄	#15668002	1	2	yes	—
266	C₂×C₄	#15668008	1	1	yes	characteristic
267	C₂×C₄	#15668010	1	1	yes	characteristic
268	C₈	—	4	4	no	—
269	C₈	—	89	8	no	—
270	C₈	—	356	4	no	—
271	C₈	—	178	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	4	no	—
276	C₂²	C₄×#13083553	1	1	yes	characteristic
277	C₂²	#19151456	1	1	yes	characteristic
278	C₂²	#19151467	1	1	yes	characteristic
279	C₂²	#19151495	1	2	yes	—
280	C₂²	#19151502	1	2	yes	—
281	C₂²	C₂×#15667947	1	4	yes	—
282	C₄	—	2	8	no	—
283	C₄	#95×D₈₉	1	1	yes	characteristic
284	C₄	#19151449	1	1	yes	characteristic
285	C₄	#19151457	1	1	yes	characteristic
286	C₄	#19151462	1	1	yes	characteristic
287	C₄	#19151493	1	2	yes	—
288	C₄	#19151498	1	2	yes	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	#26018480	1	1	yes	characteristic
292	C₂	C₄×#15667947	1	2	yes	—
293	C₂	#26020288	1	2	yes	—
294	C₂	#26020423	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#1728	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#1728	D₈₉
hypocenter	C₈₉	#10864
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
11. C₂×#15667947	C₄	semidirect
101. C₂×C₄×C₈₉	#108	semidirect
153. C₈₉	#10864	semidirect
249. C₂×C₄	#15667947	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667947	2
C₂²×C₄×#10623494	2
C₂×#19151362	2
#26019946	2
C₈₉×#1728	2
#10864	89