Subgroups — #36462712

Order 22784.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667948	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151370	C₂	1	2	yes	maximal
3	#26019979	C₂	1	2	yes	maximal
4	#26019948	C₂	1	1	yes	maximal, characteristic
5	C₂×#341×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂²×#13083558	C₂²	1	1	yes	characteristic
8	C₂×#15667971	C₂²	1	1	yes	characteristic
9	C₂×#15667948	—	2	2	no	—
10	C₂×#15667948	—	2	4	no	—
11	#19151370	—	2	4	no	—
12	#19151363	—	2	4	no	—
13	#19151352	—	2	2	no	—
14	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#341×C₈₉	C₄	1	4	yes	complemented
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	1	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	D₄	1	2	yes	—
26	C₂×#13083558	D₄	1	4	yes	—
27	C₂²×#10623494	—	2	1	no	—
28	C₂²×#10623494	—	2	2	no	—
29	#15667971	—	2	2	no	—
30	#15667971	—	2	2	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×C₂²⋊Dic₈₉	Q₈	1	2	yes	—
33	C₂×C₄×Dic₈₉	—	2	1	no	—
34	#15667948	—	2	8	no	—
35	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₈₉	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₈₉	—	2	2	no	—
43	C₂²×C₈×C₈₉	—	2	2	no	—
44	C₂²×C₈×C₈₉	—	2	2	no	—
45	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₈₉	D₄	1	2	yes	—
47	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
48	C₂²×Dic₈₉	—	2	2	no	—
49	#13083558	—	2	4	no	—
50	#13083558	—	2	4	no	—
51	#13083558	—	2	4	no	—
52	#13083558	—	4	2	no	—
53	#13083558	—	4	2	no	—
54	#13083558	—	4	2	no	—
55	C₂×#10623494	—	2	2	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₂×#10623494	—	4	1	no	—
60	C₂×#10623494	—	4	1	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₂²⋊Dic₈₉	—	2	4	no	—
63	C₄×Dic₈₉	—	4	1	no	—
64	C₄×Dic₈₉	—	4	1	no	—
65	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	#41	1	2	yes	—
72	C₂²×C₄×C₈₉	#41	1	2	yes	—
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	M₁₆×C₈₉	—	2	4	no	—
77	M₁₆×C₈₉	—	2	4	no	—
78	M₁₆×C₈₉	—	4	2	no	—
79	M₁₆×C₈₉	—	4	2	no	—
80	C₂×C₈×C₈₉	—	2	4	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	2	4	no	—
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	—	4	2	no	—
86	C₂×C₈×C₈₉	—	4	2	no	—
87	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
88	C₂³×C₈₉	—	4	2	no	—
89	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
90	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
91	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂³×C₈₉	#125	1	2	yes	—
93	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#117	1	2	yes	—
96	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	2	yes	—
100	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₈₉	#123	1	2	yes	—
103	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#123	1	2	yes	—
105	C₂×C₄×C₈₉	#124	1	2	yes	—
106	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#125	1	2	yes	—
110	C₂×C₄×C₈₉	#126	1	2	yes	—
111	C₂×C₄×C₈₉	#126	1	2	yes	—
112	C₂×Dic₈₉	—	4	4	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₈×C₈₉	—	2	4	no	—
115	C₈×C₈₉	—	4	4	no	—
116	#10623494	—	4	4	no	—
117	#10623494	—	4	2	no	—
118	C₂²×C₈₉	—	2	2	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	—	4	2	no	—
122	C₂²×C₈₉	—	4	2	no	—
123	C₂²×C₈₉	#387	1	2	yes	—
124	C₂²×C₈₉	#389	1	1	yes	characteristic
125	C₂²×C₈₉	#393	1	1	yes	characteristic
126	C₂²×C₈₉	#398	1	2	yes	—
127	C₂²×C₈₉	#400	1	1	yes	characteristic
128	C₄×C₈₉	#382	1	1	yes	characteristic
129	C₄×C₈₉	#385	1	1	yes	characteristic
130	C₄×C₈₉	#388	1	2	yes	—
131	C₄×C₈₉	#394	1	1	yes	characteristic
132	C₄×C₈₉	#397	1	1	yes	characteristic
133	C₄×C₈₉	#400	1	2	yes	—
134	Dic₈₉	—	8	2	no	—
135	#11272	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	—	2	1	no	—
137	C₂×C₈₉	—	2	2	no	—
138	C₂×C₈₉	—	2	2	no	—
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	#1442	1	1	yes	characteristic
141	C₂×C₈₉	#1953	1	2	yes	—
142	C₂²×#108	—	89	1	no	—
143	C₂×#344	—	89	2	no	—
144	#1738	—	89	2	no	—
145	C₂×#341	D₈₉	1	1	yes	characteristic
146	#1729	—	89	1	no	—
147	C₈₉	#11272	1	1	yes	characteristic, complemented, Sylow
148	C₂²×M₁₆	—	89	2	no	—
149	C₂²×M₁₆	—	89	1	no	—
150	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
151	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
152	C₂×#117	—	89	1	no	—
153	C₂×#108	—	178	4	no	—
154	C₂×#108	—	178	2	no	—
155	#344	—	178	4	no	—
156	#341	—	178	4	no	—
157	#341	Dic₈₉	1	4	yes	complemented
158	#336	—	178	2	no	—
159	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	89	2	no	—
166	C₂×M₁₆	—	89	4	no	—
167	C₂×M₁₆	—	178	2	no	—
168	C₂×M₁₆	—	178	2	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	—	178	1	no	—
175	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	178	1	no	—
180	C₂²×C₈	—	178	2	no	—
181	C₂²×C₈	Dic₁₇₈	1	2	yes	—
182	C₂²×C₈	D₃₅₆	1	2	yes	—
183	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
184	#117	—	178	2	no	—
185	#117	—	178	2	no	—
186	C₂×C₄⋊C₄	—	178	1	no	—
187	C₂×C₂²⋊C₄	—	89	2	no	—
188	C₂×C₄²	—	178	1	no	—
189	#108	—	178	8	no	—
190	C₂⁴	Q₈×D₈₉	1	1	yes	characteristic
191	C₂²×C₄	—	178	2	no	—
192	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
193	C₂²×C₄	D₄×D₈₉	1	2	yes	—
194	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
197	C₂²×C₄	#13083573	1	2	yes	—
198	C₂²×C₄	#13083573	1	1	yes	characteristic
199	C₂²×C₄	#13083574	1	2	yes	—
200	C₂²×C₄	#13083574	1	2	yes	—
201	C₂²×C₄	#13083575	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	356	2	no	—
207	M₁₆	—	356	2	no	—
208	M₁₆	—	356	2	no	—
209	M₁₆	—	178	4	no	—
210	M₁₆	—	178	4	no	—
211	M₁₆	—	178	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	356	2	no	—
220	C₂×C₈	—	356	1	no	—
221	C₂×C₈	—	356	2	no	—
222	C₂×C₈	—	356	1	no	—
223	C₂×C₈	—	178	2	no	—
224	C₂×C₈	—	178	4	no	—
225	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
226	C₄⋊C₄	—	356	2	no	—
227	C₂²⋊C₄	—	178	4	no	—
228	C₄²	—	356	1	no	—
229	C₄²	—	356	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	Q₈×Dic₈₉	1	1	yes	characteristic
232	C₂³	#15667990	1	2	yes	—
233	C₂³	#15667992	1	1	yes	characteristic
234	C₂³	#15668002	1	2	yes	—
235	C₂³	#15668011	1	1	yes	characteristic
236	C₂×C₄	—	356	4	no	—
237	C₂×C₄	—	356	2	no	—
238	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
239	C₂×C₄	#15667971	1	2	yes	—
240	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
241	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
242	C₂×C₄	#15667982	1	1	yes	characteristic
243	C₂×C₄	#15667985	1	2	yes	—
244	C₂×C₄	#15667987	1	1	yes	characteristic
245	C₂×C₄	#15667989	1	2	yes	—
246	C₂×C₄	#15667994	1	2	yes	—
247	C₂×C₄	#15667996	1	1	yes	characteristic
248	C₂×C₄	#15667997	1	2	yes	—
249	C₂×C₄	#15667998	1	2	yes	—
250	C₂×C₄	#15668000	1	1	yes	characteristic
251	C₂×C₄	#15668000	1	1	yes	characteristic
252	C₂×C₄	#15668003	1	1	yes	characteristic
253	C₂×C₄	#15668004	1	2	yes	—
254	C₂×C₄	#15668005	1	2	yes	—
255	C₂×C₄	#15668007	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	4	4	no	—
258	C₈	—	356	4	no	—
259	C₈	—	356	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	#19151484	1	2	yes	—
266	C₂²	#19151497	1	1	yes	characteristic
267	C₂²	#19151511	1	1	yes	characteristic
268	C₂²	#19151532	1	2	yes	—
269	C₂²	#19151538	1	1	yes	characteristic
270	C₄	—	712	2	no	—
271	C₄	#19151464	1	1	yes	characteristic
272	C₄	#19151473	1	1	yes	characteristic
273	C₄	#19151491	1	2	yes	—
274	C₄	#19151516	1	1	yes	characteristic
275	C₄	#19151525	1	1	yes	characteristic
276	C₄	#19151540	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	1	no	—
281	C₂	#26018509	1	1	yes	characteristic
282	C₂	#26021043	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667987
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#341×C₈₉	C₂
socle	C₂²×C₈₉	#389
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#341	D₈₉
hypocenter	C₈₉	#11272
LCS	G ▷ C₂²×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
16. #341×C₈₉	C₄	semidirect
147. C₈₉	#11272	semidirect
157. #341	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667948	2
C₂×#19151370	2
#26019979	2
#26019948	2
C₂×#341×C₈₉	2
#11272	89