Subgroups — #36460167

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667923	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151362	C₂	1	2	yes	maximal
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×#15667923	—	2	4	no	—
10	C₂×#15667923	C₄	1	4	yes	—
11	#19151362	—	2	4	no	—
12	#19151362	C₄	1	8	yes	complemented
13	#19151353	—	2	2	no	—
14	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	—	2	2	no	—
17	C₂×#13083558	—	2	2	no	—
18	C₂×#13083558	—	2	2	no	—
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	C₂×C₄	1	4	yes	—
23	C₂²×#10623494	—	2	2	no	—
24	C₂²×#10623494	—	2	2	no	—
25	C₂²×#10623494	—	2	1	no	—
26	C₂²×#10623494	—	2	1	no	—
27	C₂²×#10623494	—	2	2	no	—
28	C₂²×#10623494	—	2	1	no	—
29	C₂²×#10623494	—	2	1	no	—
30	C₂²×#10623494	C₂×C₄	1	4	yes	—
31	C₂²×#10623494	C₂×C₄	1	2	yes	—
32	C₂²×#10623494	C₂×C₄	1	2	yes	—
33	C₂²×#10623494	D₄	1	4	yes	—
34	#15667923	—	2	8	no	—
35	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₈₉	—	2	2	no	—
38	C₂×M₁₆×C₈₉	—	2	2	no	—
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	C₂²×C₈×C₈₉	—	2	2	no	—
44	C₂×C₄²×C₈₉	—	2	2	no	—
45	C₂×C₄²×C₈₉	—	2	2	no	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₈₉	D₄	1	2	yes	—
48	#13083558	—	2	4	no	—
49	#13083558	—	2	4	no	—
50	#13083558	—	4	2	no	—
51	#13083558	—	4	2	no	—
52	C₂×#10623494	—	2	2	no	—
53	C₂×#10623494	—	2	2	no	—
54	C₂×#10623494	—	2	2	no	—
55	C₂×#10623494	—	2	2	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	—	2	4	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₂×#10623494	—	4	2	no	—
63	C₂×#10623494	C₄²	1	4	yes	—
64	C₂×#10623494	C₂²⋊C₄	1	4	yes	—
65	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	—	2	4	no	—
67	C₂²×C₄×C₈₉	—	4	4	no	—
68	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	#41	1	2	yes	—
74	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	2	yes	—
79	M₁₆×C₈₉	—	2	4	no	—
80	M₁₆×C₈₉	—	2	4	no	—
81	M₁₆×C₈₉	—	4	2	no	—
82	M₁₆×C₈₉	—	4	2	no	—
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₂×C₈×C₈₉	—	4	2	no	—
85	C₂×C₈×C₈₉	—	4	2	no	—
86	C₄²×C₈₉	—	2	2	no	—
87	C₄²×C₈₉	—	2	2	no	—
88	C₄²×C₈₉	—	2	4	no	—
89	C₂³×C₈₉	—	2	4	no	—
90	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
92	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
93	C₂³×C₈₉	#125	1	1	yes	characteristic
94	C₂³×C₈₉	#125	1	1	yes	characteristic
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	—	2	8	no	—
97	C₂×C₄×C₈₉	—	4	8	no	—
98	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#117	1	2	yes	—
102	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#117	1	2	yes	—
105	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₈₉	#121	1	2	yes	—
110	C₂×C₄×C₈₉	#123	1	2	yes	—
111	C₂×C₄×C₈₉	#123	1	2	yes	—
112	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#126	1	2	yes	—
115	C₂×C₄×C₈₉	#126	1	2	yes	—
116	C₈×C₈₉	—	4	4	no	—
117	#10623494	—	2	4	no	—
118	#10623494	—	2	4	no	—
119	#10623494	—	4	4	no	—
120	C₂²×C₈₉	—	2	4	no	—
121	C₂²×C₈₉	—	2	2	no	—
122	C₂²×C₈₉	—	2	2	no	—
123	C₂²×C₈₉	—	2	4	no	—
124	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₈₉	#380	1	1	yes	characteristic
126	C₂²×C₈₉	#382	1	1	yes	characteristic
127	C₂²×C₈₉	#387	1	2	yes	—
128	C₂²×C₈₉	#387	1	2	yes	—
129	C₂²×C₈₉	#430	1	4	yes	—
130	C₄×C₈₉	—	4	4	no	—
131	C₄×C₈₉	#376	1	1	yes	characteristic
132	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₈₉	#381	1	1	yes	characteristic
134	C₄×C₈₉	#383	1	1	yes	characteristic
135	C₄×C₈₉	#388	1	2	yes	—
136	C₄×C₈₉	#388	1	2	yes	—
137	#10823	—	89	1	no	maximal, Sylow
138	C₂×C₈₉	—	2	2	no	—
139	C₂×C₈₉	—	2	2	no	—
140	C₂×C₈₉	#1434	1	1	yes	characteristic
141	C₂×C₈₉	#1778	1	2	yes	—
142	C₂×C₈₉	#1811	1	2	yes	—
143	C₂×C₈₉	#1838	1	2	yes	—
144	C₂²×#97	—	89	1	no	—
145	C₂×#341	—	89	2	no	—
146	C₂×#341	—	89	2	no	—
147	#1728	—	89	1	no	—
148	#1728	D₈₉	1	1	yes	characteristic
149	C₈₉	#10823	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	—	89	2	no	—
151	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
152	C₂³×C₈	—	89	1	no	—
153	C₂³×C₈	—	89	1	no	—
154	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
155	C₂×#97	—	89	4	no	—
156	C₂×#97	—	178	4	no	—
157	#341	—	89	8	no	—
158	#341	—	178	4	no	—
159	#336	—	2	2	no	—
160	#336	—	178	2	no	—
161	C₂³×C₄	Dic₁₇₈	1	2	yes	—
162	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
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	—	178	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	C₄×D₈₉	1	4	yes	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	89	4	no	—
177	C₂²×C₈	—	89	2	no	—
178	C₂²×C₈	—	89	2	no	—
179	C₂²×C₈	—	89	4	no	—
180	C₂²×C₈	—	178	2	no	—
181	C₂²×C₈	—	178	2	no	—
182	C₂²×C₈	—	178	1	no	—
183	C₂²×C₈	—	178	1	no	—
184	C₂²×C₈	—	178	2	no	—
185	C₂²×C₈	—	178	1	no	—
186	C₂²×C₈	—	178	1	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₄×D₈₉	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
191	#97	—	178	8	no	—
192	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	#13083553	1	4	yes	—
196	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
198	C₂²×C₄	D₄×D₈₉	1	2	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
200	C₂²×C₄	#13083573	1	2	yes	—
201	C₂²×C₄	#13083573	1	1	yes	characteristic
202	C₂²×C₄	#13083573	1	1	yes	characteristic
203	C₂²×C₄	#13083574	1	2	yes	—
204	C₂²×C₄	#13083575	1	1	yes	characteristic
205	C₂²×C₄	#13083575	1	1	yes	characteristic
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	356	2	no	—
211	M₁₆	—	356	2	no	—
212	M₁₆	—	178	4	no	—
213	M₁₆	—	178	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	89	4	no	—
218	C₂×C₈	—	89	4	no	—
219	C₂×C₈	—	356	2	no	—
220	C₂×C₈	—	356	2	no	—
221	C₂×C₈	—	178	4	no	—
222	C₂×C₈	—	178	2	no	—
223	C₂×C₈	—	178	2	no	—
224	C₂×C₈	—	178	2	no	—
225	C₂×C₈	—	178	4	no	—
226	C₂×C₈	—	178	4	no	—
227	C₂×C₈	—	178	4	no	—
228	C₂×C₈	—	178	4	no	—
229	C₂×C₈	—	178	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₂³	—	2	4	no	—
234	C₂³	C₂×#13083553	1	1	yes	characteristic
235	C₂³	C₄×Dic₁₇₈	1	2	yes	—
236	C₂³	#15667989	1	2	yes	—
237	C₂³	#15668003	1	1	yes	characteristic
238	C₂³	#15668003	1	1	yes	characteristic
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
244	C₂×C₄	#15667968	1	1	yes	characteristic
245	C₂×C₄	#15667968	1	2	yes	—
246	C₂×C₄	#15667969	1	1	yes	characteristic
247	C₂×C₄	#15667970	1	1	yes	characteristic
248	C₂×C₄	#15667970	1	2	yes	—
249	C₂×C₄	#15667970	1	1	yes	characteristic
250	C₂×C₄	#15667972	1	2	yes	—
251	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
252	C₂×C₄	#15667974	1	2	yes	—
253	C₂×C₄	#15667984	1	2	yes	—
254	C₂×C₄	#15667994	1	2	yes	—
255	C₂×C₄	#15667997	1	2	yes	—
256	C₂×C₄	#15667999	1	1	yes	characteristic
257	C₂×C₄	#15668001	1	1	yes	characteristic
258	C₂×C₄	#15668005	1	2	yes	—
259	C₂×C₄	#15668007	1	2	yes	—
260	C₈	—	4	4	no	—
261	C₈	—	356	4	no	—
262	C₈	—	178	4	no	—
263	C₈	—	178	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	C₄×#13083553	1	1	yes	characteristic
269	C₂²	#19151456	1	1	yes	characteristic
270	C₂²	#19151463	1	1	yes	characteristic
271	C₂²	#19151481	1	2	yes	—
272	C₂²	#19151485	1	2	yes	—
273	C₂²	#19151646	1	4	yes	—
274	C₄	—	4	4	no	—
275	C₄	#19151443	1	1	yes	characteristic
276	C₄	#19151445	1	1	yes	characteristic
277	C₄	#19151460	1	1	yes	characteristic
278	C₄	#19151465	1	1	yes	characteristic
279	C₄	#19151488	1	2	yes	—
280	C₄	#19151492	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	2	no	—
283	C₂	#26018478	1	1	yes	characteristic
284	C₂	#26020159	1	2	yes	—
285	C₂	#26020292	1	2	yes	—
286	C₂	#26020392	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	#10823
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
12. #19151362	C₄	semidirect
149. C₈₉	#10823	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667923	2
C₂×#19151362	2
#26019946	2
C₈₉×#1728	2
#10823	89