Subgroups — #36460075

Order 22784.

642 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	C₂×#19151553	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151370	C₂	1	2	yes	maximal
3	#26019969	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151355	C₂	1	2	yes	maximal
5	C₂×#341×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	1	yes	characteristic
7	C₂×#15667971	C₂²	1	2	yes	—
8	C₂×#15667971	C₂²	1	1	yes	characteristic
9	#19151553	—	2	4	no	—
10	C₂×#15667922	C₄	1	2	yes	complemented
11	C₂×C₈×Dic₈₉	C₄	1	2	yes	complemented
12	#19151370	C₄	1	8	yes	complemented
13	#19151361	—	2	2	no	—
14	#19151355	—	2	4	no	—
15	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#341×C₈₉	—	2	2	no	—
18	C₂×#13083558	—	2	2	no	—
19	C₂×#13083558	—	2	1	no	—
20	C₂×#13083558	—	2	1	no	—
21	C₂×#13083558	C₂×C₄	1	4	yes	—
22	C₂²×#10623494	C₂×C₄	1	2	yes	—
23	#15667971	—	2	2	no	—
24	#15667971	—	2	2	no	—
25	#15667971	—	2	2	no	—
26	#15667971	—	2	2	no	—
27	#15667971	C₂×C₄	1	4	yes	—
28	#15667971	D₄	1	4	yes	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄⋊Dic₈₉	—	2	1	no	—
31	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
32	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
33	C₂×C₄×Dic₈₉	—	2	2	no	—
34	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
35	#15667922	—	2	4	no	—
36	C₈×Dic₈₉	—	2	4	no	—
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
43	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₈₉	—	2	2	no	—
45	C₂²×C₈×C₈₉	—	2	2	no	—
46	C₂²×C₈×C₈₉	—	2	2	no	—
47	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	D₄	1	2	yes	—
50	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
51	C₂²×Dic₈₉	—	2	2	no	—
52	C₂²×Dic₈₉	—	4	2	no	—
53	#13083558	—	2	4	no	—
54	#13083558	—	2	4	no	—
55	C₂×#10623494	—	2	2	no	—
56	C₂×#10623494	—	2	2	no	—
57	C₂×#10623494	C₄²	1	4	yes	—
58	C₄⋊Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	2	4	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	C₂²⋊Dic₈₉	—	2	4	no	—
62	C₂²⋊Dic₈₉	—	2	4	no	—
63	C₂²⋊Dic₈₉	—	4	2	no	—
64	C₄×Dic₈₉	—	2	2	no	—
65	C₄×Dic₈₉	—	2	4	no	—
66	C₄×Dic₈₉	—	4	2	no	—
67	C₄×Dic₈₉	C₂²⋊C₄	1	4	yes	—
68	C₂⁴×C₈₉	#41	1	1	yes	characteristic
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	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	#41	1	2	yes	—
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	1	no	—
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	—	2	1	no	—
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	—	2	4	no	—
89	C₂×C₈×C₈₉	—	2	4	no	—
90	C₂×C₈×C₈₉	—	4	2	no	—
91	C₂×C₈×C₈₉	—	4	2	no	—
92	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
94	C₂³×C₈₉	—	4	2	no	—
95	C₂³×C₈₉	#117	1	1	yes	characteristic
96	C₂³×C₈₉	#117	1	2	yes	—
97	C₂³×C₈₉	#123	1	2	yes	—
98	C₂³×C₈₉	#124	1	1	yes	characteristic
99	C₂³×C₈₉	#125	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	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	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₈₉	#121	1	2	yes	—
112	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₈₉	#123	1	2	yes	—
114	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#126	1	2	yes	—
117	C₂×C₄×C₈₉	#126	1	2	yes	—
118	C₂×Dic₈₉	—	2	4	no	—
119	C₂×Dic₈₉	—	4	4	no	—
120	C₂×Dic₈₉	—	4	4	no	—
121	C₂×Dic₈₉	—	8	4	no	—
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	—	2	2	no	—
124	C₈×C₈₉	—	4	4	no	—
125	#10623494	—	2	4	no	—
126	C₂²×C₈₉	—	2	2	no	—
127	C₂²×C₈₉	—	2	2	no	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	—	4	2	no	—
130	C₂²×C₈₉	—	4	2	no	—
131	C₂²×C₈₉	#376	1	1	yes	characteristic
132	C₂²×C₈₉	#381	1	1	yes	characteristic
133	C₂²×C₈₉	#382	1	1	yes	characteristic
134	C₂²×C₈₉	#387	1	2	yes	—
135	C₂²×C₈₉	#388	1	2	yes	—
136	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈₉	#380	1	1	yes	characteristic
139	C₄×C₈₉	#383	1	1	yes	characteristic
140	C₄×C₈₉	#387	1	2	yes	—
141	C₄×C₈₉	#388	1	2	yes	—
142	Dic₈₉	—	4	4	no	—
143	Dic₈₉	—	8	4	no	—
144	#10804	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	—	2	2	no	—
147	C₂×C₈₉	—	2	1	no	—
148	C₂×C₈₉	—	2	2	no	—
149	C₂×C₈₉	#1434	1	1	yes	characteristic
150	C₂×C₈₉	#1833	1	2	yes	—
151	C₂×#405	—	89	1	no	—
152	C₂×#344	—	89	2	no	—
153	C₂×#341	D₈₉	1	1	yes	characteristic
154	#1735	—	89	1	no	—
155	C₂×H(3,ℤ/4)	—	89	2	no	—
156	C₈₉	#10804	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	89	1	no	—
158	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
159	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
160	C₂×#117	—	89	1	no	—
161	C₂×#117	—	89	2	no	—
162	#405	—	178	4	no	—
163	C₂×#97	—	89	2	no	—
164	C₂×C₄×C₈	—	89	2	no	—
165	#344	—	89	8	no	—
166	#341	—	2	2	no	—
167	#340	—	178	2	no	—
168	H(3,ℤ/4)	—	178	4	no	—
169	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
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	—	2	2	no	—
175	C₂×M₁₆	—	89	4	no	—
176	C₂×M₁₆	—	178	2	no	—
177	C₂×M₁₆	—	178	1	no	—
178	C₂×M₁₆	—	178	1	no	—
179	C₂×M₁₆	C₄×D₈₉	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	89	2	no	—
184	C₂²×C₈	Dic₁₇₈	1	2	yes	—
185	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
188	#117	—	89	4	no	—
189	#117	—	89	4	no	—
190	#117	—	178	2	no	—
191	#117	—	178	2	no	—
192	#117	—	178	2	no	—
193	#117	—	178	2	no	—
194	C₂×C₄⋊C₄	—	178	2	no	—
195	C₂×C₄⋊C₄	—	178	1	no	—
196	C₂×C₂²⋊C₄	—	178	2	no	—
197	C₂×C₂²⋊C₄	—	178	1	no	—
198	C₂×C₄²	—	89	2	no	—
199	C₂×C₄²	—	178	2	no	—
200	#97	—	178	4	no	—
201	C₄×C₈	—	178	4	no	—
202	C₂⁴	#13083575	1	1	yes	characteristic
203	C₂²×C₄	—	356	2	no	—
204	C₂²×C₄	—	178	2	no	—
205	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
207	C₂²×C₄	D₄×D₈₉	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
210	C₂²×C₄	#13083573	1	2	yes	—
211	C₂²×C₄	#13083573	1	1	yes	characteristic
212	C₂²×C₄	#13083573	1	1	yes	characteristic
213	C₂²×C₄	#13083574	1	2	yes	—
214	C₂²×C₄	#13083575	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	178	4	no	—
220	M₁₆	—	178	4	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	89	4	no	—
231	C₂×C₈	—	178	2	no	—
232	C₂×C₈	—	178	2	no	—
233	C₂×C₈	#13083553	1	2	yes	—
234	C₂×C₈	#13083553	1	2	yes	—
235	C₄⋊C₄	—	356	2	no	—
236	C₄⋊C₄	—	178	4	no	—
237	C₄⋊C₄	—	178	4	no	—
238	C₂²⋊C₄	—	356	2	no	—
239	C₂²⋊C₄	—	178	4	no	—
240	C₂²⋊C₄	—	178	4	no	—
241	C₄²	—	89	4	no	—
242	C₄²	—	356	2	no	—
243	C₄²	—	178	4	no	—
244	C₄²	—	178	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	#15667970	1	1	yes	characteristic
247	C₂³	#15667970	1	2	yes	—
248	C₂³	#15667997	1	2	yes	—
249	C₂³	#15668001	1	1	yes	characteristic
250	C₂³	#15668003	1	1	yes	characteristic
251	C₂×C₄	—	356	4	no	—
252	C₂×C₄	—	356	4	no	—
253	C₂×C₄	—	178	4	no	—
254	C₂×C₄	—	712	4	no	—
255	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
256	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
257	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
258	C₂×C₄	#15667968	1	1	yes	characteristic
259	C₂×C₄	#15667968	1	2	yes	—
260	C₂×C₄	#15667969	1	1	yes	characteristic
261	C₂×C₄	#15667970	1	1	yes	characteristic
262	C₂×C₄	#15667972	1	2	yes	—
263	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
264	C₂×C₄	#15667974	1	2	yes	—
265	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
266	C₂×C₄	#15667984	1	2	yes	—
267	C₂×C₄	#15667989	1	2	yes	—
268	C₂×C₄	#15667994	1	2	yes	—
269	C₂×C₄	#15667999	1	1	yes	characteristic
270	C₂×C₄	#15668003	1	1	yes	characteristic
271	C₂×C₄	#15668005	1	2	yes	—
272	C₂×C₄	#15668007	1	2	yes	—
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	4	4	no	—
276	C₈	—	178	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#19151443	1	1	yes	characteristic
283	C₂²	#19151460	1	1	yes	characteristic
284	C₂²	#19151463	1	1	yes	characteristic
285	C₂²	#19151485	1	2	yes	—
286	C₂²	#19151488	1	2	yes	—
287	C₄	—	356	4	no	—
288	C₄	—	712	4	no	—
289	C₄	#19151445	1	1	yes	characteristic
290	C₄	C₄×#13083553	1	1	yes	characteristic
291	C₄	#19151456	1	1	yes	characteristic
292	C₄	#19151465	1	1	yes	characteristic
293	C₄	#19151481	1	2	yes	—
294	C₄	#19151492	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#26018478	1	1	yes	characteristic
300	C₂	#26020377	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667970
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#341×C₈₉	C₂
socle	C₂²×C₈₉	#382
prime radical	C₂⁴×C₈₉	#41
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#341	D₈₉
hypocenter	C₈₉	#10804
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
10. C₂×#15667922	C₄	semidirect
11. C₂×C₈×Dic₈₉	C₄	semidirect
12. #19151370	C₄	semidirect
156. C₈₉	#10804	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151553	2
C₂×#19151370	2
#26019969	2
C₂×#19151355	2
C₂×#341×C₈₉	2
#10804	89